Research ArticleA MRAC Principle for a Single-Link Electrically DrivenRobot with Parameter Uncertainties

Carlos Aguilar-Avelar and Javier Moreno-Valenzuela

Instituto Politecnico Nacional-CITEDI Ave Instituto Politecnico Nacional 1310 Nueva Tijuana 22435 Tijuana BC Mexico

Correspondence should be addressed to Javier Moreno-Valenzuela morenocitedimx

Received 2 July 2016 Accepted 14 November 2016 Published 15 January 2017

Academic Editor Sigurdur F Hafstein

Copyright copy 2017 C Aguilar-Avelar and J Moreno-Valenzuela This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this paper a model reference adaptive control (MRAC) principle for a one-degree-of-freedom rigid-link electrically driven robotis presented The proposed control methodology addresses the problem of trajectory tracking with parameter uncertainties in thedynamic model of the system and proposes adaptation laws for the electrical and mechanical parameters Closed-loop stability isrigorously discussed proving that the tracking error trajectories converge to the origin exponentially With the aim of performingexperimental comparisons two control schemes are also revisited theoretically and experimentally one is an algorithm previouslyreported in the literature and the other is an adaptive controller derived under the assumption that the electrical dynamics of theactuator are negligible All the discussed controllers have been implemented in an experimental setup consisting in a rigid-linkrobot actuated with brushed DC motor The comparison indicates that better results are obtained with the new MRAC scheme

1 Introduction

Adaptive control is a design technique focused on high-performance applications in control of dynamical systems inorder to deal with parametric uncertainties characterized by aset of unknown constant parameters On the other hand thedesign of adaptive controllers requires knowing the structureof the dynamic model of the system [1]

An advantage of adaptive control considering that theparameters of the plant changes during operation is that thecontroller compensates the changes in the model parametersachieving better performance In a nonadaptive systemwhere the parameters of the plant are considered constantsthese changes can compromise the good performance of theoverall system Another advantage of the adaptive controlsystems is that the knowledge of the plant parameters todesign the controller is not necessary [2]

Direct current (DC) motors are widely used as actuatorsin different kind of robots Due to its dynamics DC motorscan be described appropriately by a linear dynamic modelwhich is sensitive to parametric variations noise interferenceand parametric uncertainties Therefore conventional con-trol schemes can not yield a convenient performance in the

presence of the aforementioned disturbances [3] Taking intoaccount these facts this document presents a new adaptivecontrol scheme in addition to a theoretical and experimentalcomparative study of adaptive algorithms for the trajectory-tracking control of a one-degree-of-freedom (DOF) rigid-link electrically driven (RLED) robot

In [4] an adaptive tracking controller for a DC motorwas presented Simulation results are given where the DCmotor shaft follows a desired sinusoidal trajectory with anacceptable tracking performance In [5] a summary of pre-vious works on trajectory-tracking control for manipulatorrobots was presented in which the electrical dynamics of theactuators are neglected in the design procedure In [6] theidea of considering the electrical dynamics of the actuators asa part of the system dynamics was introduced and a robustcontroller for the trajectory tracking was presented Thiscontroller guarantees that the tracking error is uniformly ulti-mately bounded In addition an adaptive controller for a rigidrobot was presented in [5] where the electrical dynamics ofthe actuators are considered in the controller design In [7]the electrical dynamics were added to the controller designfor an adaptive algorithm previously reported in literatureThis is for the position regulation problem on rigid robots

HindawiComplexityVolume 2017 Article ID 9296012 13 pageshttpsdoiorg10115520179296012

2 Complexity

actuated with a brushedDCmotor However only simulationresults were provided In [8] a formal stability analysiswas introduced for a voltage based controller designed formanipulator robots although the problem of parametricuncertainties was not addressed

Let us notice that much of the recent literature in thecontrol of DC motors does not take into account the loaddynamics but pays attention to other important problemswith practical relevance In [9 10] the velocity control ofa DC motor was addressed by taking into account thedynamics of a DC-DC power converter Those approacheswere supported by real-time experiments The integrationof barrier Lyapunov function to avoid the violation of stateconstraints was used in [11] for the development of DCmotor controller In [12] an output-feedback control wasproposed for the positioning problem of a lineal actuatordriven by a DC motor where the saturation of the controlinput and the presence of nonlinear friction are consideredin the development of the new methodology

The application of model reference adaptive control(MRAC) for a RLED robot is introduced in this document Ingeneral the MRAC-based adaptive controllers are designedusing a reference model which describes the desired charac-teristics of the plant to be controlled The use of a referencemodel facilitates the analysis of the adaptive system andprovides a stability framework since the controller designand stability analysis was performed in two steps Regardingthis the MRAC scheme was analyzed in [13] in the presenceof unmodelled dynamics where the analysis showed theexistence of robustness and this fact was confirmed by meansof simulations In [14] the application of a MRAC schemefor an induction motor was presented shown to be robustto parametric variations In [15] a MRAC-based controllerwas presented in order to solve the speed control problemof a DC motor In this case the classic adaptation scheme isreplaced with fuzzy linear adaptation to improve the systemperformance at low speed and variable load conditionsNevertheless results are only validated with simulations In[16] an adaptive output-feedback controller that belongs tothe class of MRAC schemes was presented This algorithmis applied to manipulators neglecting the actuator dynamicsand uses a linear observer to estimate the states for amanipulator In [17] a MRAC scheme was presented inorder to control robot manipulators However the electricaldynamics of the actuators are not considered in the controllerdesign More recently the velocity control of DC motor wasexplored in [18] by using a MRAC scheme together with avariable structure extension However electrical dynamicswere also neglected Other interesting applications of theMRAC philosophy for nonlinear systems can be found in forexample [19ndash22]

More recently intelligent control techniques have beenapplied to control this kind of systems For instance fuzzyand neural networks based controllers were introduced in[23ndash32] However in most cases only simulation results areprovided In [33ndash35] some sliding-mode based controllerswere presented for robot manipulators where the electri-cal dynamics of the actuator are neglected However in[36] a neural-network-based terminal sliding-mode control

scheme was proposed for robotic manipulators includingactuator dynamics In [37] a robust optimal voltage controlof electrically driven robot manipulators was presentedParticle swarm optimization is used to optimize the controldesign parameters Also a comparative study between voltageand torque based controllers is presented which confirmsthe superiority of the voltage control strategy In [38] arobust discrete repetitive control of electrically driven robotmanipulators was introduced where the control problem isthe tracking of a periodic trajectory However in all of theaforementioned work only simulation results were givenmaking comparing with the controllers presented in thisdocument difficult

The main contributions of this document are the inclu-sion of the electrical dynamics of the actuator in the controllaw and the problem of parametric uncertainties whichare considered in a united form in addition to the designand analysis of a new adaptive control algorithm for thetrajectory-tracking control of a 1-DOF RLED robot Specif-ically the novel adaptive algorithm presented is based on aMRAC principle It is noteworthy that the application of theMRAC scheme for the control of RLED robots deserves adeeper study since only a few studies have been reported

Additionally an experimental performance comparisonof our solution with respect to some other adaptive algo-rithms is presented For the comparative study this paperrevisits two adaptive controllers an algorithm previouslyreported in literature [39] and an adaptive controller derivedunder the assumption that the electrical dynamics of theactuator is negligibleThe newMRAC scheme and the othersare tested in real time in an experimental platform consistingin a 1-DOF RLED robot which is affected by gravitationalforce

This paper is organized as follows Section 2 is devoted tosummarizing the RLED robot dynamic model In additiona brief discussion about the control goal is presented Theproposed MRAC scheme is described in Section 3 An adap-tive controller previously reported in literature is given inSection 4 In Section 5 an adaptive controller obtained fromthe assumption that the electrical dynamics of the actuatorare negligible is presented A brief description of the exper-imental platform results of the real-time implementationof the new controller and the performance comparison arepresented in Section 6 Finally in Section 7 some concludingremarks are given

2 RLED Robot Dynamic Modeland Control Goal

The dynamic model of a 1-DOF RLED robot actuated by abrushed DC motor is presented as in [39] and is written as

119869 + 119861 + 119873 sin (119902) = 119894 (1)

119871 119889119889119905 119894 + 119877119894 + 119870119861 = V (2)

where 119869 is the grouped rotor inertia 119861 is the groupedviscous friction 119873 is the constant of grouped gravitational

Complexity 3

load 119871 is the inductance 119877 is the electric resistance and119870119861 is the back electromotive force constant all these beingpositive constants Furthermore 119902(119905) (119905) and (119905) representposition velocity and acceleration respectively Finally 119894(119905)and V(119905) represent current and the input voltage respectively

It is possible to represent (1)-(2) as a linear regressionmodel which separates the constant parameters from thestates This representation is especially useful in parametricidentification and in the design of adaptive controllersThenthe mechanical part of the system model described in (1) isrewritten in a linear regression form as

119884119898 (119902 ) 120579119898 = 119894 (3)

where

119884119898 = [ sin (119902)] (4)

120579119898 = [119869 119861 119873]119879 (5)

with 119884119898 in (4) being the regression matrix of the mechanicalpart and 120579119898 in (5) being the vector of constant parameters ofthe mechanical part

Similarly the electrical part of the system modeldescribed in (2) can be represented as

119884119890 ( 119889119889119905 119894 119894 ) 120579119890 = V (6)

where

119884119890 = [ 119889119889119905 119894 119894 ] (7)

120579119890 = [119871 119877 119870119861]119879 (8)

with119884119890 in (7) being the regressionmatrix of the electrical partand 120579119890 in (8) being the vector of constant parameters of theelectrical part

Let us consider a time-varying twice differentiable signal119902119889(119905) which denotes the desired trajectory of the jointposition Then the control goal consists in ensuring that thetrajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) satisfies

lim119905rarrinfin

(119905) = 0 (9)

This paper introduces three controllers that ensure limit (9)

3 Proposed MRAC Principle

A new MRAC-based algorithm is presented in this sectionwhere in the first stage of design a control action is proposedfor the tracking between the real model and the referencemodel After this a complementary control action is definedfor the tracking between the reference model and a desiredtrajectory Hence the control objective defined in (9) issatisfied

31 Reference Model The proposed reference model forsystems (1)-(2) is defined as

119903 + 119903 + sin (119902) = 119894119903 (10)

119889119889119905 119894119903 + 119894119903 + 119861119903 = V + 119906 (11)

where (119905) is the estimated grouped rotor inertia (119905) isthe estimated grouped viscous friction (119905) is the estimatedconstant of grouped gravitational load (119905) is the estimatedinductance (119905) is the estimated electric resistance and119861(119905) is the estimated back electromotive force constant allthese being positive constants Furthermore 119903(119905) and 119903(119905)represent velocity and acceleration of the reference modelrespectively Also 119894119903(119905) and V(119905) represent current and theinput voltage of the referencemodel respectively Finally 119906(119905)is an auxiliary control signal to be defined Note that V(119905)represents also the input of systems (1)-(2)

It is possible to express the reference dynamic modeldescribed in (10)-(11) as a linear regression model as wellThen the mechanical part of the reference model in (10) isrewritten in a linear regression form as

119884119898119903 (119902 119903 119903) 119898119903 = 119894119903 (12)

where

119884119898119903 = [119903 119903 sin (119902)] (13)

119898119903 = [ ]119879 (14)

with 119884119898119903(119905) being the regression matrix of the mechanicalpart of the reference model and 119898119903(119905) being the vector ofestimated parameters of the mechanical part of the referencemodel

Similarly the electrical part of the reference modeldescribed in (11) can be expressed as

119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 = V + 119906 (15)

where

119884119890119903 = [ 119889119889119905 119894119903 119894119903 119903] 119890119903 = [ 119861]119879

(16)

with119884119890119903(119905) being the regressionmatrix of the electrical part ofthe reference model and 119890119903(119905) being the vector of estimatedparameters of the electrical part of the reference model

32 Error Model Nowwe proceed to compute the open-looperror dynamics of the electrical and mechanical part Theerror model of the mechanical part is given by the differencebetween the mechanical part of the reference model in (12)and themechanical part of the realmodel in (3) Analogouslythe errormodel of the electrical part is given by the difference

4 Complexity

between the electrical part of the reference model in (15) andthe electrical part of the real model in (6)

The error model is expressed as

119884119898119903 (119902 119903 119903) 119898119903 minus 119884119898 (119902 ) 120579119898 = 119894 (17)

119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 minus 119884119890 (119889119894119889119905 119894 ) 120579119890 = 119906 (18)

where 119894 = 119894119903 minus 119894The adaptation errors are defined as follows

119898119903 = 120579119898 minus 119898119903 isin R3 (19)

119890119903 = 120579119890 minus 119890119903 isin R3 (20)

In order to define the error model in terms of theparameter adaptation error it is possible to substitute theexpression of 119898119903(119905) from (19) into the error model in (17)such that

119884119898119903 (119902 119903 119903) 120579119898 minus 119884119898 (119902 ) 120579119898minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (21)

Writing explicitly the first two terms in (21) doing suitablecancellations and defining the position error between thereference model and the real model as

119902 = 119902119903 minus 119902 (22)

the error model of the mechanical part can be written as

119869 + 119861 minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (23)

It is worth remarking that if the parameter estimation errorof the mechanical part 119898119903(119905) is equal to zero in (23) then theremaining elements describe the error model for a referencemodel with known parameters

Similarly proceeding the error model of the electricalpart in (18) can be rewritten as

119871 119889119889119905 119894 + 119877119894 + 119870119861 minus 119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 = 119906 (24)

As well as for the mechanical part of the error model noticethat if the parameter estimation error of the electrical part119890119903(119905) is equal to zero in (24) then the remaining elementsin (24) describe the error model for a reference model withknown parameters Systems (23)-(24) describe the trackingerror dynamics between the referencemodel (10)-(11) and theactual model (1)-(2)

33 Controller for the Tracking between Reference Modeland Real Model As can be seen from the error dynamics(23)-(24) the input 119906(119905) must be designed such that theconvergence of the electric current error 119894(119905) and the positionerror 119902(119905) is ensured In other words the control signal 119906(119905)is introduced in order to ensure the tracking between thereference model (10)-(11) and the actual model (1)-(2) With

this aim 119906(119905) is proposed as feed-forward compensation interms of the electrical dynamics plus a proportional-integralpart in terms of a change of variable 120588(119905) that is

119906 = + 119910 minus 1198960120588 minus 119896119894120585 (25)

where 1198960 and 119896119894 are positive constants and the change ofvariable

120588 = 119894 minus 119910 (26)

is defined in order to ensure that the current error 119894(119905)introduces the compensation 119910(119905) to the mechanical errordynamics (23) In other words the proportional part 1198960120588 andthe integral part 119896119894120585 with

= 120588 (27)

are introduced to ensure that 119894 = 119910 On the other hand thecompensation 119910(119905) given as

119910 = minus119896119889119911 minus 119896119901119902 (28)

introduces a proportional term in 119902(119905) and a filtered derivativecompensation 119911(119905) defined as

= minus119896119891 (119911 minus ) (29)

with 119896119901 119896119889 and 119896119891 being positive constantsTaking into account the definition of the electrical adap-

tation error 119890119903(119905) in (20) in particular the definitions of (119905)and (119905) the control signal 119906(119905) in (25) can be expressed as

119906 = 119871 + 119877119910 minus 1198960120588 minus 119896119894120585 minus minus 119910 (30)

Substituting the definitions of 119894(119905) in (26) and 119910(119905) in (28) into(23) the expression of themechanical part of the error modelis rewritten as

119869 + 119861 + 119896119889119911 + 119896119901119902 minus 119884119898119903119898119903 = 120588 (31)

Analogously by substituting the control signal 119906(119905) in (30)into the the electrical part of the error model in (24) thisyields

119871 + [119877 + 1198960] 120588 + 119870119861 + 119896119894120585 minus 119884119890119906119890119903 = 0 (32)

where

119884119890119906 = [ 119889119889119905 (119894119903 minus 119910) (119894119903 minus 119910) 119903] (33)

The closed-loop systems (31)-(32) result from applying thecontroller 119906(119905) in (25) to the open-loop system in (23)-(24)

34 Adaptation Law We propose the following adaptationlaws

119889119889119905 119898119903 = minusΓ119898119903119884119879119898119903 ( + 120572119902) (34)

119889119889119905 119890119903 = minusΓ119890119903119884119879119890119906120588 (35)

Complexity 5

where 120572 is a positive constant sufficiently small 119884119898119903(119905) isdefined in (13) 119884119890119906(119905) is defined in (33) and

Γ119898119903 = diag (1205741198981199031 1205741198981199032 1205741198981199033) gt 0Γ119890119903 = diag (1205741198901199031 1205741198901199032 1205741198901199033) gt 0 (36)

are defined as the adaptation gains of the mechanical andelectrical part respectively

35 Analysis for Tracking between the Reference Model andReal Model We propose the following energy function

V = 12057621198711205882 + 12057621198961198941205852 + 121198692 + 121198961199011199022 + 12119896minus1119891 1198961198891199112+ 120572119869119902 + 12057221198611199022 + 12 119879119898119903Γminus1119898119903119898119903 + 1205762 119879119890119903Γminus1119890119903 119890119903

(37)

with 119902(119905) defined in (22) 120588(119905) in (26) 120585(119905) in (27) 119911(119905) in (29)119898119903(119905) in (19) and 119890119903(119905) in (20)

It is possible to prove that for120572 small enoughV in (37) isa positive definite and radially unbounded function in termsof the states of the reference model error described by (27)(29) (31) and (32) as well as the adaptation laws in (34)-(35)

To ensure that the tracking error between the referencemodel and the actual model which is expressed by 119902(119905)is a bounded signal it is required that V in (37) be adecreasing function that is the time derivative V to benegative semidefinite

By performing all the suitable substitutions and conduct-ing the algebra the time derivative of the function (37) isgiven by

V = minus120576 [119877 + 1198960] 1205882 minus 120576119870119861120588 minus 1198612 + 120588 minus 1198961198891199112+ 1205721198692 minus 120572119896119889119902119911 minus 1205721198961199011199022 + 120572119902120588 (38)

which can be expressed in a matrix form as

V = minusx119879Px (39)

where

x = [119902 119911 120588]119879 (40)

P =[[[[[[[[[[[

120572119896119901 0 1205721198961198892 minus12057220 119861 minus 120572119869 0 120576119870119861 minus 121205721198961198892 0 119896119889 0minus1205722 120576119870119861 minus 12 0 120576 [119877 + 1198960]

]]]]]]]]]]]

(41)

Since (39) is a quadratic form by proving that P = P119879in (41) is a positive definite matrix we ensure that V is anegative semidefinite function We can prove this fact byusing Sylvesterrsquos criterion to determine conditions to ensurethat P is positive definite Then it can be shown that by

selecting sufficiently large values for 119896119901 and 1198960 as well assufficiently small values for 120572 and 120576 there exist conditionsfor P to be a positive definite matrix Finally it is possible toinvoke Barbalatrsquos lemma [40] to prove that

lim119905rarrinfin

x (119905) = 0 (42)

where x(119905) is defined in (40) Hence the tracking error 119902(119905)between the referencemodel and the real model tends to zeroas time 119905 increases36 Controller for Desired Trajectory Tracking Once thetracking error between the reference model and the actualmodel 119902(119905) is proven to be convergent we propose the controlvoltage V(119905) in order to ensure that the reference modelposition 119902119903(119905) tracks a desired time-varying trajectory 119902119889(119905)With this aim the tracking error signals are defined as

120575 = 119894119889 minus 119894119903 (43)

119890 = 119902119889 minus 119902119903 (44)

and the control voltage V(119905) is proposed as

V = minus119906 + 120593 (45)

where 119906(119905) is defined in (25) and

120593 = 119861119889 + 119889119889119905 119894119889 + 119894119889 + 119896119900V120575 (46)

119894119889 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 (47)

Proceeding to obtain the closed-loop equations in termsof 120575(119905) in (43) and 119890(119905) in (44) we substitute (47) into (43)where solving for 119894119903(119905) we obtain

119894119903 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 minus 120575 (48)

where 119896119901V and 119896119889V are positive constants Solving (10) for119903(119905) and substituting into the second time derivative of thetracking error in (44) one obtains

119890 = 119889minusminus1 [119894119903 minus 119903 minus sin (119902)] (49)

Substituting (48) into (49) and after some simplifications weobtain the tracking error dynamics of the mechanical part asfollows

119890 + [ + 119896119889V] 119890 + 119896119901V119890 = 120575 (50)

Similarly to obtain the closed-loop equation for theelectrical part in terms of the tracking error 120575(119905) we calculatethe time derivative of (43) andmultiplying by (119905) this yields

119889119889119905120575 = [ 119889119889119905 119894119889 minus 119889119889119905 119894119903] (51)

Solving (11) for (119889119889119905)119894119903(119905) substituting into (51) and per-forming the algebra we obtain the error dynamics of theelectrical part as follows

+ [ + 119896119900V] 120575 = minus119861 119890 (52)

6 Complexity

The system formed by (50) and (52) is linear and time-varying It can be demonstrated that if (119905) (119905) (119905) (119905)and 119861(119905) are strictly positives for all 119905 ge 0 we have that

lim119905rarrinfin

[119890 (119905)120575 (119905)] = 0 (53)

with exponential rate of convergence It is noteworthy thatthe limits (42) and (53) imply that the control goal in (9) issatisfied

Finally it is worth remarking that the total control voltageV(119905) to be applied to the open-loop system dynamics in (1)-(2)is given in (45) with the auxiliary control signals 119906(119905) in (25)which ensures the tracking between reference model and realmodel 120593(119905) in (46) which introduces the desired dynamicsto the system the parameter adaptation laws

120579119898119903(119905) in (34)120579119890119903(119905) in (35) and definitions therein

4 Adaptive Controller Proposed byDawson Hu and Burg (1998)

In this section an adaptive controller proposed in [39] ispresented and the stability analysis is briefly reviewed Thepurpose of presenting this algorithm is tomake a comparisonwith respect to the MRAC scheme introduced in this paperThus the trajectory-tracking adaptive controller is written as

V = 119882119890119890 + 119870119890119894 + 119903 (54)

where 119870119890 is a positive constant 119903(119905) is the filtered trackingerror given by

119903 = 119902 + 120572 (55)

with120572 being a positive constant (119905) stands for the trajectory-tracking error defined as

= 119902119889 minus 119902 (56)

where 119902119889(119905) is the desired trajectory (119905) is the current errordefined by

= 119894119889 minus 119894 (57)

where 119894119889(119905) is the desired current which is expressed as

119894119889 = 119882120591120591 + 119870119904119903 (58)

with the positive constant119870119904119882120591 = [119889 + 120572 119902 sin (119902)] (59)

defined as the regressor of the mechanical part and

120591 = [ ]119879 isin R3 (60)

is an estimation of the parameters of the mechanical part 120579120591Similarly

119890 = [119869 119861119869 119861 119873119869 ]119879 isin R6 (61)

represents the estimation of 120579119890 which is a vector of groupedparameters containing parameters from both mechanicaland electrical part Likewise

119882119890 = [1199081198901 1199081198902 1199081198903 1199081198904 1199081198905 1199081198906] (62)

with

1199081198901 = 119894 minus 119870119904119894 minus 1205721198941199081198902 = 119870119904 minus + 1205721199081198903 = 1198941199081198904 = 1199081198905 = 119870119904 sin (119902) minus sin (119902) + 120572 sin (119902) 1199081198906 = 119902119889 + 120572119889 +119882120591Γ120591119882119879120591 119903 + 119870119904119889 + 119870119904120572 119902

+ cos (119902)

(63)

being the regressor of the vector of grouped parameters

41 Adaptation Law By defining the adaptation errors as

120591 = 120579120591 minus 120591 isin R3

119890 = 120579119890 minus 119890 isin R6 (64)

the adaptation laws for the update of the mechanical param-eters 120591(119905) and grouped electrical parameters 119890(119905) are givenby

119889119889119905 120591 = minusΓ120591119882119879120591 119903119889119889119905 119890 = minusΓ119890119882119879119890

(65)

where

Γ120591 = diag (1205741205911 1205741205912 1205741205913) gt 0Γ119890 = diag (1205741198901 1205741198902 1205741198906) gt 0 (66)

are the adaptation gains of the mechanical and electricalparts respectively

The closed-loop system is obtained by substituting thecontrol action V(119905) defined in (54) into the dynamic model(1)-(2) By expressing in terms of the filtered tracking error119903(119905) the current error (119905) and the adaptation errors 120591(119905) and119890(119905) the closed-loop system is given by

119889119889119905[[[[[[[

119903120591

119890

]]]]]]]=[[[[[[[

119869minus1 [119882120591120591 minus 119870119904119903 + ]119871minus1 [119882119890119890 minus 119870119890119894 minus 119903]

minusΓ120591119882119879120591 119903minusΓ119890119882119879119890

]]]]]]] (67)

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 3

load 119871 is the inductance 119877 is the electric resistance and119870119861 is the back electromotive force constant all these beingpositive constants Furthermore 119902(119905) (119905) and (119905) representposition velocity and acceleration respectively Finally 119894(119905)and V(119905) represent current and the input voltage respectively

It is possible to represent (1)-(2) as a linear regressionmodel which separates the constant parameters from thestates This representation is especially useful in parametricidentification and in the design of adaptive controllersThenthe mechanical part of the system model described in (1) isrewritten in a linear regression form as

119884119898 (119902 ) 120579119898 = 119894 (3)

where

119884119898 = [ sin (119902)] (4)

120579119898 = [119869 119861 119873]119879 (5)

with 119884119898 in (4) being the regression matrix of the mechanicalpart and 120579119898 in (5) being the vector of constant parameters ofthe mechanical part

Similarly the electrical part of the system modeldescribed in (2) can be represented as

119884119890 ( 119889119889119905 119894 119894 ) 120579119890 = V (6)

where

119884119890 = [ 119889119889119905 119894 119894 ] (7)

120579119890 = [119871 119877 119870119861]119879 (8)

with119884119890 in (7) being the regressionmatrix of the electrical partand 120579119890 in (8) being the vector of constant parameters of theelectrical part

Let us consider a time-varying twice differentiable signal119902119889(119905) which denotes the desired trajectory of the jointposition Then the control goal consists in ensuring that thetrajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) satisfies

lim119905rarrinfin

(119905) = 0 (9)

This paper introduces three controllers that ensure limit (9)

3 Proposed MRAC Principle

A new MRAC-based algorithm is presented in this sectionwhere in the first stage of design a control action is proposedfor the tracking between the real model and the referencemodel After this a complementary control action is definedfor the tracking between the reference model and a desiredtrajectory Hence the control objective defined in (9) issatisfied

31 Reference Model The proposed reference model forsystems (1)-(2) is defined as

119903 + 119903 + sin (119902) = 119894119903 (10)

119889119889119905 119894119903 + 119894119903 + 119861119903 = V + 119906 (11)

where (119905) is the estimated grouped rotor inertia (119905) isthe estimated grouped viscous friction (119905) is the estimatedconstant of grouped gravitational load (119905) is the estimatedinductance (119905) is the estimated electric resistance and119861(119905) is the estimated back electromotive force constant allthese being positive constants Furthermore 119903(119905) and 119903(119905)represent velocity and acceleration of the reference modelrespectively Also 119894119903(119905) and V(119905) represent current and theinput voltage of the referencemodel respectively Finally 119906(119905)is an auxiliary control signal to be defined Note that V(119905)represents also the input of systems (1)-(2)

It is possible to express the reference dynamic modeldescribed in (10)-(11) as a linear regression model as wellThen the mechanical part of the reference model in (10) isrewritten in a linear regression form as

119884119898119903 (119902 119903 119903) 119898119903 = 119894119903 (12)

where

119884119898119903 = [119903 119903 sin (119902)] (13)

119898119903 = [ ]119879 (14)

with 119884119898119903(119905) being the regression matrix of the mechanicalpart of the reference model and 119898119903(119905) being the vector ofestimated parameters of the mechanical part of the referencemodel

Similarly the electrical part of the reference modeldescribed in (11) can be expressed as

119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 = V + 119906 (15)

where

119884119890119903 = [ 119889119889119905 119894119903 119894119903 119903] 119890119903 = [ 119861]119879

(16)

with119884119890119903(119905) being the regressionmatrix of the electrical part ofthe reference model and 119890119903(119905) being the vector of estimatedparameters of the electrical part of the reference model

32 Error Model Nowwe proceed to compute the open-looperror dynamics of the electrical and mechanical part Theerror model of the mechanical part is given by the differencebetween the mechanical part of the reference model in (12)and themechanical part of the realmodel in (3) Analogouslythe errormodel of the electrical part is given by the difference

4 Complexity

between the electrical part of the reference model in (15) andthe electrical part of the real model in (6)

The error model is expressed as

119884119898119903 (119902 119903 119903) 119898119903 minus 119884119898 (119902 ) 120579119898 = 119894 (17)

119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 minus 119884119890 (119889119894119889119905 119894 ) 120579119890 = 119906 (18)

where 119894 = 119894119903 minus 119894The adaptation errors are defined as follows

119898119903 = 120579119898 minus 119898119903 isin R3 (19)

119890119903 = 120579119890 minus 119890119903 isin R3 (20)

In order to define the error model in terms of theparameter adaptation error it is possible to substitute theexpression of 119898119903(119905) from (19) into the error model in (17)such that

119884119898119903 (119902 119903 119903) 120579119898 minus 119884119898 (119902 ) 120579119898minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (21)

Writing explicitly the first two terms in (21) doing suitablecancellations and defining the position error between thereference model and the real model as

119902 = 119902119903 minus 119902 (22)

the error model of the mechanical part can be written as

119869 + 119861 minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (23)

It is worth remarking that if the parameter estimation errorof the mechanical part 119898119903(119905) is equal to zero in (23) then theremaining elements describe the error model for a referencemodel with known parameters

Similarly proceeding the error model of the electricalpart in (18) can be rewritten as

119871 119889119889119905 119894 + 119877119894 + 119870119861 minus 119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 = 119906 (24)

As well as for the mechanical part of the error model noticethat if the parameter estimation error of the electrical part119890119903(119905) is equal to zero in (24) then the remaining elementsin (24) describe the error model for a reference model withknown parameters Systems (23)-(24) describe the trackingerror dynamics between the referencemodel (10)-(11) and theactual model (1)-(2)

33 Controller for the Tracking between Reference Modeland Real Model As can be seen from the error dynamics(23)-(24) the input 119906(119905) must be designed such that theconvergence of the electric current error 119894(119905) and the positionerror 119902(119905) is ensured In other words the control signal 119906(119905)is introduced in order to ensure the tracking between thereference model (10)-(11) and the actual model (1)-(2) With

this aim 119906(119905) is proposed as feed-forward compensation interms of the electrical dynamics plus a proportional-integralpart in terms of a change of variable 120588(119905) that is

119906 = + 119910 minus 1198960120588 minus 119896119894120585 (25)

where 1198960 and 119896119894 are positive constants and the change ofvariable

120588 = 119894 minus 119910 (26)

is defined in order to ensure that the current error 119894(119905)introduces the compensation 119910(119905) to the mechanical errordynamics (23) In other words the proportional part 1198960120588 andthe integral part 119896119894120585 with

= 120588 (27)

are introduced to ensure that 119894 = 119910 On the other hand thecompensation 119910(119905) given as

119910 = minus119896119889119911 minus 119896119901119902 (28)

introduces a proportional term in 119902(119905) and a filtered derivativecompensation 119911(119905) defined as

= minus119896119891 (119911 minus ) (29)

with 119896119901 119896119889 and 119896119891 being positive constantsTaking into account the definition of the electrical adap-

tation error 119890119903(119905) in (20) in particular the definitions of (119905)and (119905) the control signal 119906(119905) in (25) can be expressed as

119906 = 119871 + 119877119910 minus 1198960120588 minus 119896119894120585 minus minus 119910 (30)

Substituting the definitions of 119894(119905) in (26) and 119910(119905) in (28) into(23) the expression of themechanical part of the error modelis rewritten as

119869 + 119861 + 119896119889119911 + 119896119901119902 minus 119884119898119903119898119903 = 120588 (31)

Analogously by substituting the control signal 119906(119905) in (30)into the the electrical part of the error model in (24) thisyields

119871 + [119877 + 1198960] 120588 + 119870119861 + 119896119894120585 minus 119884119890119906119890119903 = 0 (32)

where

119884119890119906 = [ 119889119889119905 (119894119903 minus 119910) (119894119903 minus 119910) 119903] (33)

The closed-loop systems (31)-(32) result from applying thecontroller 119906(119905) in (25) to the open-loop system in (23)-(24)

34 Adaptation Law We propose the following adaptationlaws

119889119889119905 119898119903 = minusΓ119898119903119884119879119898119903 ( + 120572119902) (34)

119889119889119905 119890119903 = minusΓ119890119903119884119879119890119906120588 (35)

Complexity 5

where 120572 is a positive constant sufficiently small 119884119898119903(119905) isdefined in (13) 119884119890119906(119905) is defined in (33) and

Γ119898119903 = diag (1205741198981199031 1205741198981199032 1205741198981199033) gt 0Γ119890119903 = diag (1205741198901199031 1205741198901199032 1205741198901199033) gt 0 (36)

are defined as the adaptation gains of the mechanical andelectrical part respectively

35 Analysis for Tracking between the Reference Model andReal Model We propose the following energy function

V = 12057621198711205882 + 12057621198961198941205852 + 121198692 + 121198961199011199022 + 12119896minus1119891 1198961198891199112+ 120572119869119902 + 12057221198611199022 + 12 119879119898119903Γminus1119898119903119898119903 + 1205762 119879119890119903Γminus1119890119903 119890119903

(37)

with 119902(119905) defined in (22) 120588(119905) in (26) 120585(119905) in (27) 119911(119905) in (29)119898119903(119905) in (19) and 119890119903(119905) in (20)

It is possible to prove that for120572 small enoughV in (37) isa positive definite and radially unbounded function in termsof the states of the reference model error described by (27)(29) (31) and (32) as well as the adaptation laws in (34)-(35)

To ensure that the tracking error between the referencemodel and the actual model which is expressed by 119902(119905)is a bounded signal it is required that V in (37) be adecreasing function that is the time derivative V to benegative semidefinite

By performing all the suitable substitutions and conduct-ing the algebra the time derivative of the function (37) isgiven by

V = minus120576 [119877 + 1198960] 1205882 minus 120576119870119861120588 minus 1198612 + 120588 minus 1198961198891199112+ 1205721198692 minus 120572119896119889119902119911 minus 1205721198961199011199022 + 120572119902120588 (38)

which can be expressed in a matrix form as

V = minusx119879Px (39)

where

x = [119902 119911 120588]119879 (40)

P =[[[[[[[[[[[

120572119896119901 0 1205721198961198892 minus12057220 119861 minus 120572119869 0 120576119870119861 minus 121205721198961198892 0 119896119889 0minus1205722 120576119870119861 minus 12 0 120576 [119877 + 1198960]

]]]]]]]]]]]

(41)

Since (39) is a quadratic form by proving that P = P119879in (41) is a positive definite matrix we ensure that V is anegative semidefinite function We can prove this fact byusing Sylvesterrsquos criterion to determine conditions to ensurethat P is positive definite Then it can be shown that by

selecting sufficiently large values for 119896119901 and 1198960 as well assufficiently small values for 120572 and 120576 there exist conditionsfor P to be a positive definite matrix Finally it is possible toinvoke Barbalatrsquos lemma [40] to prove that

lim119905rarrinfin

x (119905) = 0 (42)

where x(119905) is defined in (40) Hence the tracking error 119902(119905)between the referencemodel and the real model tends to zeroas time 119905 increases36 Controller for Desired Trajectory Tracking Once thetracking error between the reference model and the actualmodel 119902(119905) is proven to be convergent we propose the controlvoltage V(119905) in order to ensure that the reference modelposition 119902119903(119905) tracks a desired time-varying trajectory 119902119889(119905)With this aim the tracking error signals are defined as

120575 = 119894119889 minus 119894119903 (43)

119890 = 119902119889 minus 119902119903 (44)

and the control voltage V(119905) is proposed as

V = minus119906 + 120593 (45)

where 119906(119905) is defined in (25) and

120593 = 119861119889 + 119889119889119905 119894119889 + 119894119889 + 119896119900V120575 (46)

119894119889 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 (47)

Proceeding to obtain the closed-loop equations in termsof 120575(119905) in (43) and 119890(119905) in (44) we substitute (47) into (43)where solving for 119894119903(119905) we obtain

119894119903 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 minus 120575 (48)

where 119896119901V and 119896119889V are positive constants Solving (10) for119903(119905) and substituting into the second time derivative of thetracking error in (44) one obtains

119890 = 119889minusminus1 [119894119903 minus 119903 minus sin (119902)] (49)

Substituting (48) into (49) and after some simplifications weobtain the tracking error dynamics of the mechanical part asfollows

119890 + [ + 119896119889V] 119890 + 119896119901V119890 = 120575 (50)

Similarly to obtain the closed-loop equation for theelectrical part in terms of the tracking error 120575(119905) we calculatethe time derivative of (43) andmultiplying by (119905) this yields

119889119889119905120575 = [ 119889119889119905 119894119889 minus 119889119889119905 119894119903] (51)

Solving (11) for (119889119889119905)119894119903(119905) substituting into (51) and per-forming the algebra we obtain the error dynamics of theelectrical part as follows

+ [ + 119896119900V] 120575 = minus119861 119890 (52)

6 Complexity

The system formed by (50) and (52) is linear and time-varying It can be demonstrated that if (119905) (119905) (119905) (119905)and 119861(119905) are strictly positives for all 119905 ge 0 we have that

lim119905rarrinfin

[119890 (119905)120575 (119905)] = 0 (53)

with exponential rate of convergence It is noteworthy thatthe limits (42) and (53) imply that the control goal in (9) issatisfied

Finally it is worth remarking that the total control voltageV(119905) to be applied to the open-loop system dynamics in (1)-(2)is given in (45) with the auxiliary control signals 119906(119905) in (25)which ensures the tracking between reference model and realmodel 120593(119905) in (46) which introduces the desired dynamicsto the system the parameter adaptation laws

120579119898119903(119905) in (34)120579119890119903(119905) in (35) and definitions therein

4 Adaptive Controller Proposed byDawson Hu and Burg (1998)

In this section an adaptive controller proposed in [39] ispresented and the stability analysis is briefly reviewed Thepurpose of presenting this algorithm is tomake a comparisonwith respect to the MRAC scheme introduced in this paperThus the trajectory-tracking adaptive controller is written as

V = 119882119890119890 + 119870119890119894 + 119903 (54)

where 119870119890 is a positive constant 119903(119905) is the filtered trackingerror given by

119903 = 119902 + 120572 (55)

with120572 being a positive constant (119905) stands for the trajectory-tracking error defined as

= 119902119889 minus 119902 (56)

where 119902119889(119905) is the desired trajectory (119905) is the current errordefined by

= 119894119889 minus 119894 (57)

where 119894119889(119905) is the desired current which is expressed as

119894119889 = 119882120591120591 + 119870119904119903 (58)

with the positive constant119870119904119882120591 = [119889 + 120572 119902 sin (119902)] (59)

defined as the regressor of the mechanical part and

120591 = [ ]119879 isin R3 (60)

is an estimation of the parameters of the mechanical part 120579120591Similarly

119890 = [119869 119861119869 119861 119873119869 ]119879 isin R6 (61)

represents the estimation of 120579119890 which is a vector of groupedparameters containing parameters from both mechanicaland electrical part Likewise

119882119890 = [1199081198901 1199081198902 1199081198903 1199081198904 1199081198905 1199081198906] (62)

with

1199081198901 = 119894 minus 119870119904119894 minus 1205721198941199081198902 = 119870119904 minus + 1205721199081198903 = 1198941199081198904 = 1199081198905 = 119870119904 sin (119902) minus sin (119902) + 120572 sin (119902) 1199081198906 = 119902119889 + 120572119889 +119882120591Γ120591119882119879120591 119903 + 119870119904119889 + 119870119904120572 119902

+ cos (119902)

(63)

being the regressor of the vector of grouped parameters

41 Adaptation Law By defining the adaptation errors as

120591 = 120579120591 minus 120591 isin R3

119890 = 120579119890 minus 119890 isin R6 (64)

the adaptation laws for the update of the mechanical param-eters 120591(119905) and grouped electrical parameters 119890(119905) are givenby

119889119889119905 120591 = minusΓ120591119882119879120591 119903119889119889119905 119890 = minusΓ119890119882119879119890

(65)

where

Γ120591 = diag (1205741205911 1205741205912 1205741205913) gt 0Γ119890 = diag (1205741198901 1205741198902 1205741198906) gt 0 (66)

are the adaptation gains of the mechanical and electricalparts respectively

The closed-loop system is obtained by substituting thecontrol action V(119905) defined in (54) into the dynamic model(1)-(2) By expressing in terms of the filtered tracking error119903(119905) the current error (119905) and the adaptation errors 120591(119905) and119890(119905) the closed-loop system is given by

119889119889119905[[[[[[[

119903120591

119890

]]]]]]]=[[[[[[[

119869minus1 [119882120591120591 minus 119870119904119903 + ]119871minus1 [119882119890119890 minus 119870119890119894 minus 119903]

minusΓ120591119882119879120591 119903minusΓ119890119882119879119890

]]]]]]] (67)

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Complexity

between the electrical part of the reference model in (15) andthe electrical part of the real model in (6)

The error model is expressed as

119884119898119903 (119902 119903 119903) 119898119903 minus 119884119898 (119902 ) 120579119898 = 119894 (17)

119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 minus 119884119890 (119889119894119889119905 119894 ) 120579119890 = 119906 (18)

where 119894 = 119894119903 minus 119894The adaptation errors are defined as follows

119898119903 = 120579119898 minus 119898119903 isin R3 (19)

119890119903 = 120579119890 minus 119890119903 isin R3 (20)

In order to define the error model in terms of theparameter adaptation error it is possible to substitute theexpression of 119898119903(119905) from (19) into the error model in (17)such that

119884119898119903 (119902 119903 119903) 120579119898 minus 119884119898 (119902 ) 120579119898minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (21)

Writing explicitly the first two terms in (21) doing suitablecancellations and defining the position error between thereference model and the real model as

119902 = 119902119903 minus 119902 (22)

the error model of the mechanical part can be written as

119869 + 119861 minus 119884119898119903 (119902 119903 119903) 119898119903 = 119894 (23)

It is worth remarking that if the parameter estimation errorof the mechanical part 119898119903(119905) is equal to zero in (23) then theremaining elements describe the error model for a referencemodel with known parameters

Similarly proceeding the error model of the electricalpart in (18) can be rewritten as

119871 119889119889119905 119894 + 119877119894 + 119870119861 minus 119884119890119903 ( 119889119889119905 119894119903 119894119903 119903) 119890119903 = 119906 (24)

As well as for the mechanical part of the error model noticethat if the parameter estimation error of the electrical part119890119903(119905) is equal to zero in (24) then the remaining elementsin (24) describe the error model for a reference model withknown parameters Systems (23)-(24) describe the trackingerror dynamics between the referencemodel (10)-(11) and theactual model (1)-(2)

33 Controller for the Tracking between Reference Modeland Real Model As can be seen from the error dynamics(23)-(24) the input 119906(119905) must be designed such that theconvergence of the electric current error 119894(119905) and the positionerror 119902(119905) is ensured In other words the control signal 119906(119905)is introduced in order to ensure the tracking between thereference model (10)-(11) and the actual model (1)-(2) With

this aim 119906(119905) is proposed as feed-forward compensation interms of the electrical dynamics plus a proportional-integralpart in terms of a change of variable 120588(119905) that is

119906 = + 119910 minus 1198960120588 minus 119896119894120585 (25)

where 1198960 and 119896119894 are positive constants and the change ofvariable

120588 = 119894 minus 119910 (26)

is defined in order to ensure that the current error 119894(119905)introduces the compensation 119910(119905) to the mechanical errordynamics (23) In other words the proportional part 1198960120588 andthe integral part 119896119894120585 with

= 120588 (27)

are introduced to ensure that 119894 = 119910 On the other hand thecompensation 119910(119905) given as

119910 = minus119896119889119911 minus 119896119901119902 (28)

introduces a proportional term in 119902(119905) and a filtered derivativecompensation 119911(119905) defined as

= minus119896119891 (119911 minus ) (29)

with 119896119901 119896119889 and 119896119891 being positive constantsTaking into account the definition of the electrical adap-

tation error 119890119903(119905) in (20) in particular the definitions of (119905)and (119905) the control signal 119906(119905) in (25) can be expressed as

119906 = 119871 + 119877119910 minus 1198960120588 minus 119896119894120585 minus minus 119910 (30)

Substituting the definitions of 119894(119905) in (26) and 119910(119905) in (28) into(23) the expression of themechanical part of the error modelis rewritten as

119869 + 119861 + 119896119889119911 + 119896119901119902 minus 119884119898119903119898119903 = 120588 (31)

Analogously by substituting the control signal 119906(119905) in (30)into the the electrical part of the error model in (24) thisyields

119871 + [119877 + 1198960] 120588 + 119870119861 + 119896119894120585 minus 119884119890119906119890119903 = 0 (32)

where

119884119890119906 = [ 119889119889119905 (119894119903 minus 119910) (119894119903 minus 119910) 119903] (33)

The closed-loop systems (31)-(32) result from applying thecontroller 119906(119905) in (25) to the open-loop system in (23)-(24)

34 Adaptation Law We propose the following adaptationlaws

119889119889119905 119898119903 = minusΓ119898119903119884119879119898119903 ( + 120572119902) (34)

119889119889119905 119890119903 = minusΓ119890119903119884119879119890119906120588 (35)

Complexity 5

where 120572 is a positive constant sufficiently small 119884119898119903(119905) isdefined in (13) 119884119890119906(119905) is defined in (33) and

Γ119898119903 = diag (1205741198981199031 1205741198981199032 1205741198981199033) gt 0Γ119890119903 = diag (1205741198901199031 1205741198901199032 1205741198901199033) gt 0 (36)

are defined as the adaptation gains of the mechanical andelectrical part respectively

35 Analysis for Tracking between the Reference Model andReal Model We propose the following energy function

V = 12057621198711205882 + 12057621198961198941205852 + 121198692 + 121198961199011199022 + 12119896minus1119891 1198961198891199112+ 120572119869119902 + 12057221198611199022 + 12 119879119898119903Γminus1119898119903119898119903 + 1205762 119879119890119903Γminus1119890119903 119890119903

(37)

with 119902(119905) defined in (22) 120588(119905) in (26) 120585(119905) in (27) 119911(119905) in (29)119898119903(119905) in (19) and 119890119903(119905) in (20)

It is possible to prove that for120572 small enoughV in (37) isa positive definite and radially unbounded function in termsof the states of the reference model error described by (27)(29) (31) and (32) as well as the adaptation laws in (34)-(35)

To ensure that the tracking error between the referencemodel and the actual model which is expressed by 119902(119905)is a bounded signal it is required that V in (37) be adecreasing function that is the time derivative V to benegative semidefinite

By performing all the suitable substitutions and conduct-ing the algebra the time derivative of the function (37) isgiven by

V = minus120576 [119877 + 1198960] 1205882 minus 120576119870119861120588 minus 1198612 + 120588 minus 1198961198891199112+ 1205721198692 minus 120572119896119889119902119911 minus 1205721198961199011199022 + 120572119902120588 (38)

which can be expressed in a matrix form as

V = minusx119879Px (39)

where

x = [119902 119911 120588]119879 (40)

P =[[[[[[[[[[[

120572119896119901 0 1205721198961198892 minus12057220 119861 minus 120572119869 0 120576119870119861 minus 121205721198961198892 0 119896119889 0minus1205722 120576119870119861 minus 12 0 120576 [119877 + 1198960]

]]]]]]]]]]]

(41)

Since (39) is a quadratic form by proving that P = P119879in (41) is a positive definite matrix we ensure that V is anegative semidefinite function We can prove this fact byusing Sylvesterrsquos criterion to determine conditions to ensurethat P is positive definite Then it can be shown that by

selecting sufficiently large values for 119896119901 and 1198960 as well assufficiently small values for 120572 and 120576 there exist conditionsfor P to be a positive definite matrix Finally it is possible toinvoke Barbalatrsquos lemma [40] to prove that

lim119905rarrinfin

x (119905) = 0 (42)

where x(119905) is defined in (40) Hence the tracking error 119902(119905)between the referencemodel and the real model tends to zeroas time 119905 increases36 Controller for Desired Trajectory Tracking Once thetracking error between the reference model and the actualmodel 119902(119905) is proven to be convergent we propose the controlvoltage V(119905) in order to ensure that the reference modelposition 119902119903(119905) tracks a desired time-varying trajectory 119902119889(119905)With this aim the tracking error signals are defined as

120575 = 119894119889 minus 119894119903 (43)

119890 = 119902119889 minus 119902119903 (44)

and the control voltage V(119905) is proposed as

V = minus119906 + 120593 (45)

where 119906(119905) is defined in (25) and

120593 = 119861119889 + 119889119889119905 119894119889 + 119894119889 + 119896119900V120575 (46)

119894119889 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 (47)

Proceeding to obtain the closed-loop equations in termsof 120575(119905) in (43) and 119890(119905) in (44) we substitute (47) into (43)where solving for 119894119903(119905) we obtain

119894119903 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 minus 120575 (48)

where 119896119901V and 119896119889V are positive constants Solving (10) for119903(119905) and substituting into the second time derivative of thetracking error in (44) one obtains

119890 = 119889minusminus1 [119894119903 minus 119903 minus sin (119902)] (49)

Substituting (48) into (49) and after some simplifications weobtain the tracking error dynamics of the mechanical part asfollows

119890 + [ + 119896119889V] 119890 + 119896119901V119890 = 120575 (50)

Similarly to obtain the closed-loop equation for theelectrical part in terms of the tracking error 120575(119905) we calculatethe time derivative of (43) andmultiplying by (119905) this yields

119889119889119905120575 = [ 119889119889119905 119894119889 minus 119889119889119905 119894119903] (51)

Solving (11) for (119889119889119905)119894119903(119905) substituting into (51) and per-forming the algebra we obtain the error dynamics of theelectrical part as follows

+ [ + 119896119900V] 120575 = minus119861 119890 (52)

6 Complexity

The system formed by (50) and (52) is linear and time-varying It can be demonstrated that if (119905) (119905) (119905) (119905)and 119861(119905) are strictly positives for all 119905 ge 0 we have that

lim119905rarrinfin

[119890 (119905)120575 (119905)] = 0 (53)

with exponential rate of convergence It is noteworthy thatthe limits (42) and (53) imply that the control goal in (9) issatisfied

Finally it is worth remarking that the total control voltageV(119905) to be applied to the open-loop system dynamics in (1)-(2)is given in (45) with the auxiliary control signals 119906(119905) in (25)which ensures the tracking between reference model and realmodel 120593(119905) in (46) which introduces the desired dynamicsto the system the parameter adaptation laws

120579119898119903(119905) in (34)120579119890119903(119905) in (35) and definitions therein

4 Adaptive Controller Proposed byDawson Hu and Burg (1998)

In this section an adaptive controller proposed in [39] ispresented and the stability analysis is briefly reviewed Thepurpose of presenting this algorithm is tomake a comparisonwith respect to the MRAC scheme introduced in this paperThus the trajectory-tracking adaptive controller is written as

V = 119882119890119890 + 119870119890119894 + 119903 (54)

where 119870119890 is a positive constant 119903(119905) is the filtered trackingerror given by

119903 = 119902 + 120572 (55)

with120572 being a positive constant (119905) stands for the trajectory-tracking error defined as

= 119902119889 minus 119902 (56)

where 119902119889(119905) is the desired trajectory (119905) is the current errordefined by

= 119894119889 minus 119894 (57)

where 119894119889(119905) is the desired current which is expressed as

119894119889 = 119882120591120591 + 119870119904119903 (58)

with the positive constant119870119904119882120591 = [119889 + 120572 119902 sin (119902)] (59)

defined as the regressor of the mechanical part and

120591 = [ ]119879 isin R3 (60)

is an estimation of the parameters of the mechanical part 120579120591Similarly

119890 = [119869 119861119869 119861 119873119869 ]119879 isin R6 (61)

represents the estimation of 120579119890 which is a vector of groupedparameters containing parameters from both mechanicaland electrical part Likewise

119882119890 = [1199081198901 1199081198902 1199081198903 1199081198904 1199081198905 1199081198906] (62)

with

1199081198901 = 119894 minus 119870119904119894 minus 1205721198941199081198902 = 119870119904 minus + 1205721199081198903 = 1198941199081198904 = 1199081198905 = 119870119904 sin (119902) minus sin (119902) + 120572 sin (119902) 1199081198906 = 119902119889 + 120572119889 +119882120591Γ120591119882119879120591 119903 + 119870119904119889 + 119870119904120572 119902

+ cos (119902)

(63)

being the regressor of the vector of grouped parameters

41 Adaptation Law By defining the adaptation errors as

120591 = 120579120591 minus 120591 isin R3

119890 = 120579119890 minus 119890 isin R6 (64)

the adaptation laws for the update of the mechanical param-eters 120591(119905) and grouped electrical parameters 119890(119905) are givenby

119889119889119905 120591 = minusΓ120591119882119879120591 119903119889119889119905 119890 = minusΓ119890119882119879119890

(65)

where

Γ120591 = diag (1205741205911 1205741205912 1205741205913) gt 0Γ119890 = diag (1205741198901 1205741198902 1205741198906) gt 0 (66)

are the adaptation gains of the mechanical and electricalparts respectively

The closed-loop system is obtained by substituting thecontrol action V(119905) defined in (54) into the dynamic model(1)-(2) By expressing in terms of the filtered tracking error119903(119905) the current error (119905) and the adaptation errors 120591(119905) and119890(119905) the closed-loop system is given by

119889119889119905[[[[[[[

119903120591

119890

]]]]]]]=[[[[[[[

119869minus1 [119882120591120591 minus 119870119904119903 + ]119871minus1 [119882119890119890 minus 119870119890119894 minus 119903]

minusΓ120591119882119879120591 119903minusΓ119890119882119879119890

]]]]]]] (67)

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 5

where 120572 is a positive constant sufficiently small 119884119898119903(119905) isdefined in (13) 119884119890119906(119905) is defined in (33) and

Γ119898119903 = diag (1205741198981199031 1205741198981199032 1205741198981199033) gt 0Γ119890119903 = diag (1205741198901199031 1205741198901199032 1205741198901199033) gt 0 (36)

are defined as the adaptation gains of the mechanical andelectrical part respectively

35 Analysis for Tracking between the Reference Model andReal Model We propose the following energy function

V = 12057621198711205882 + 12057621198961198941205852 + 121198692 + 121198961199011199022 + 12119896minus1119891 1198961198891199112+ 120572119869119902 + 12057221198611199022 + 12 119879119898119903Γminus1119898119903119898119903 + 1205762 119879119890119903Γminus1119890119903 119890119903

(37)

with 119902(119905) defined in (22) 120588(119905) in (26) 120585(119905) in (27) 119911(119905) in (29)119898119903(119905) in (19) and 119890119903(119905) in (20)

It is possible to prove that for120572 small enoughV in (37) isa positive definite and radially unbounded function in termsof the states of the reference model error described by (27)(29) (31) and (32) as well as the adaptation laws in (34)-(35)

To ensure that the tracking error between the referencemodel and the actual model which is expressed by 119902(119905)is a bounded signal it is required that V in (37) be adecreasing function that is the time derivative V to benegative semidefinite

By performing all the suitable substitutions and conduct-ing the algebra the time derivative of the function (37) isgiven by

V = minus120576 [119877 + 1198960] 1205882 minus 120576119870119861120588 minus 1198612 + 120588 minus 1198961198891199112+ 1205721198692 minus 120572119896119889119902119911 minus 1205721198961199011199022 + 120572119902120588 (38)

which can be expressed in a matrix form as

V = minusx119879Px (39)

where

x = [119902 119911 120588]119879 (40)

P =[[[[[[[[[[[

120572119896119901 0 1205721198961198892 minus12057220 119861 minus 120572119869 0 120576119870119861 minus 121205721198961198892 0 119896119889 0minus1205722 120576119870119861 minus 12 0 120576 [119877 + 1198960]

]]]]]]]]]]]

(41)

Since (39) is a quadratic form by proving that P = P119879in (41) is a positive definite matrix we ensure that V is anegative semidefinite function We can prove this fact byusing Sylvesterrsquos criterion to determine conditions to ensurethat P is positive definite Then it can be shown that by

selecting sufficiently large values for 119896119901 and 1198960 as well assufficiently small values for 120572 and 120576 there exist conditionsfor P to be a positive definite matrix Finally it is possible toinvoke Barbalatrsquos lemma [40] to prove that

lim119905rarrinfin

x (119905) = 0 (42)

where x(119905) is defined in (40) Hence the tracking error 119902(119905)between the referencemodel and the real model tends to zeroas time 119905 increases36 Controller for Desired Trajectory Tracking Once thetracking error between the reference model and the actualmodel 119902(119905) is proven to be convergent we propose the controlvoltage V(119905) in order to ensure that the reference modelposition 119902119903(119905) tracks a desired time-varying trajectory 119902119889(119905)With this aim the tracking error signals are defined as

120575 = 119894119889 minus 119894119903 (43)

119890 = 119902119889 minus 119902119903 (44)

and the control voltage V(119905) is proposed as

V = minus119906 + 120593 (45)

where 119906(119905) is defined in (25) and

120593 = 119861119889 + 119889119889119905 119894119889 + 119894119889 + 119896119900V120575 (46)

119894119889 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 (47)

Proceeding to obtain the closed-loop equations in termsof 120575(119905) in (43) and 119890(119905) in (44) we substitute (47) into (43)where solving for 119894119903(119905) we obtain

119894119903 = 119889 + 119889 + sin (119902) + 119896119889V 119890 + 119896119901V119890 minus 120575 (48)

where 119896119901V and 119896119889V are positive constants Solving (10) for119903(119905) and substituting into the second time derivative of thetracking error in (44) one obtains

119890 = 119889minusminus1 [119894119903 minus 119903 minus sin (119902)] (49)

Substituting (48) into (49) and after some simplifications weobtain the tracking error dynamics of the mechanical part asfollows

119890 + [ + 119896119889V] 119890 + 119896119901V119890 = 120575 (50)

Similarly to obtain the closed-loop equation for theelectrical part in terms of the tracking error 120575(119905) we calculatethe time derivative of (43) andmultiplying by (119905) this yields

119889119889119905120575 = [ 119889119889119905 119894119889 minus 119889119889119905 119894119903] (51)

Solving (11) for (119889119889119905)119894119903(119905) substituting into (51) and per-forming the algebra we obtain the error dynamics of theelectrical part as follows

+ [ + 119896119900V] 120575 = minus119861 119890 (52)

6 Complexity

The system formed by (50) and (52) is linear and time-varying It can be demonstrated that if (119905) (119905) (119905) (119905)and 119861(119905) are strictly positives for all 119905 ge 0 we have that

lim119905rarrinfin

[119890 (119905)120575 (119905)] = 0 (53)

with exponential rate of convergence It is noteworthy thatthe limits (42) and (53) imply that the control goal in (9) issatisfied

Finally it is worth remarking that the total control voltageV(119905) to be applied to the open-loop system dynamics in (1)-(2)is given in (45) with the auxiliary control signals 119906(119905) in (25)which ensures the tracking between reference model and realmodel 120593(119905) in (46) which introduces the desired dynamicsto the system the parameter adaptation laws

120579119898119903(119905) in (34)120579119890119903(119905) in (35) and definitions therein

4 Adaptive Controller Proposed byDawson Hu and Burg (1998)

In this section an adaptive controller proposed in [39] ispresented and the stability analysis is briefly reviewed Thepurpose of presenting this algorithm is tomake a comparisonwith respect to the MRAC scheme introduced in this paperThus the trajectory-tracking adaptive controller is written as

V = 119882119890119890 + 119870119890119894 + 119903 (54)

where 119870119890 is a positive constant 119903(119905) is the filtered trackingerror given by

119903 = 119902 + 120572 (55)

with120572 being a positive constant (119905) stands for the trajectory-tracking error defined as

= 119902119889 minus 119902 (56)

where 119902119889(119905) is the desired trajectory (119905) is the current errordefined by

= 119894119889 minus 119894 (57)

where 119894119889(119905) is the desired current which is expressed as

119894119889 = 119882120591120591 + 119870119904119903 (58)

with the positive constant119870119904119882120591 = [119889 + 120572 119902 sin (119902)] (59)

defined as the regressor of the mechanical part and

120591 = [ ]119879 isin R3 (60)

is an estimation of the parameters of the mechanical part 120579120591Similarly

119890 = [119869 119861119869 119861 119873119869 ]119879 isin R6 (61)

represents the estimation of 120579119890 which is a vector of groupedparameters containing parameters from both mechanicaland electrical part Likewise

119882119890 = [1199081198901 1199081198902 1199081198903 1199081198904 1199081198905 1199081198906] (62)

with

1199081198901 = 119894 minus 119870119904119894 minus 1205721198941199081198902 = 119870119904 minus + 1205721199081198903 = 1198941199081198904 = 1199081198905 = 119870119904 sin (119902) minus sin (119902) + 120572 sin (119902) 1199081198906 = 119902119889 + 120572119889 +119882120591Γ120591119882119879120591 119903 + 119870119904119889 + 119870119904120572 119902

+ cos (119902)

(63)

being the regressor of the vector of grouped parameters

41 Adaptation Law By defining the adaptation errors as

120591 = 120579120591 minus 120591 isin R3

119890 = 120579119890 minus 119890 isin R6 (64)

the adaptation laws for the update of the mechanical param-eters 120591(119905) and grouped electrical parameters 119890(119905) are givenby

119889119889119905 120591 = minusΓ120591119882119879120591 119903119889119889119905 119890 = minusΓ119890119882119879119890

(65)

where

Γ120591 = diag (1205741205911 1205741205912 1205741205913) gt 0Γ119890 = diag (1205741198901 1205741198902 1205741198906) gt 0 (66)

are the adaptation gains of the mechanical and electricalparts respectively

The closed-loop system is obtained by substituting thecontrol action V(119905) defined in (54) into the dynamic model(1)-(2) By expressing in terms of the filtered tracking error119903(119905) the current error (119905) and the adaptation errors 120591(119905) and119890(119905) the closed-loop system is given by

119889119889119905[[[[[[[

119903120591

119890

]]]]]]]=[[[[[[[

119869minus1 [119882120591120591 minus 119870119904119903 + ]119871minus1 [119882119890119890 minus 119870119890119894 minus 119903]

minusΓ120591119882119879120591 119903minusΓ119890119882119879119890

]]]]]]] (67)

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Complexity

The system formed by (50) and (52) is linear and time-varying It can be demonstrated that if (119905) (119905) (119905) (119905)and 119861(119905) are strictly positives for all 119905 ge 0 we have that

lim119905rarrinfin

[119890 (119905)120575 (119905)] = 0 (53)

with exponential rate of convergence It is noteworthy thatthe limits (42) and (53) imply that the control goal in (9) issatisfied

Finally it is worth remarking that the total control voltageV(119905) to be applied to the open-loop system dynamics in (1)-(2)is given in (45) with the auxiliary control signals 119906(119905) in (25)which ensures the tracking between reference model and realmodel 120593(119905) in (46) which introduces the desired dynamicsto the system the parameter adaptation laws

120579119898119903(119905) in (34)120579119890119903(119905) in (35) and definitions therein

4 Adaptive Controller Proposed byDawson Hu and Burg (1998)

In this section an adaptive controller proposed in [39] ispresented and the stability analysis is briefly reviewed Thepurpose of presenting this algorithm is tomake a comparisonwith respect to the MRAC scheme introduced in this paperThus the trajectory-tracking adaptive controller is written as

V = 119882119890119890 + 119870119890119894 + 119903 (54)

where 119870119890 is a positive constant 119903(119905) is the filtered trackingerror given by

119903 = 119902 + 120572 (55)

with120572 being a positive constant (119905) stands for the trajectory-tracking error defined as

= 119902119889 minus 119902 (56)

where 119902119889(119905) is the desired trajectory (119905) is the current errordefined by

= 119894119889 minus 119894 (57)

where 119894119889(119905) is the desired current which is expressed as

119894119889 = 119882120591120591 + 119870119904119903 (58)

with the positive constant119870119904119882120591 = [119889 + 120572 119902 sin (119902)] (59)

defined as the regressor of the mechanical part and

120591 = [ ]119879 isin R3 (60)

is an estimation of the parameters of the mechanical part 120579120591Similarly

119890 = [119869 119861119869 119861 119873119869 ]119879 isin R6 (61)

represents the estimation of 120579119890 which is a vector of groupedparameters containing parameters from both mechanicaland electrical part Likewise

119882119890 = [1199081198901 1199081198902 1199081198903 1199081198904 1199081198905 1199081198906] (62)

with

1199081198901 = 119894 minus 119870119904119894 minus 1205721198941199081198902 = 119870119904 minus + 1205721199081198903 = 1198941199081198904 = 1199081198905 = 119870119904 sin (119902) minus sin (119902) + 120572 sin (119902) 1199081198906 = 119902119889 + 120572119889 +119882120591Γ120591119882119879120591 119903 + 119870119904119889 + 119870119904120572 119902

+ cos (119902)

(63)

being the regressor of the vector of grouped parameters

41 Adaptation Law By defining the adaptation errors as

120591 = 120579120591 minus 120591 isin R3

119890 = 120579119890 minus 119890 isin R6 (64)

the adaptation laws for the update of the mechanical param-eters 120591(119905) and grouped electrical parameters 119890(119905) are givenby

119889119889119905 120591 = minusΓ120591119882119879120591 119903119889119889119905 119890 = minusΓ119890119882119879119890

(65)

where

Γ120591 = diag (1205741205911 1205741205912 1205741205913) gt 0Γ119890 = diag (1205741198901 1205741198902 1205741198906) gt 0 (66)

are the adaptation gains of the mechanical and electricalparts respectively

The closed-loop system is obtained by substituting thecontrol action V(119905) defined in (54) into the dynamic model(1)-(2) By expressing in terms of the filtered tracking error119903(119905) the current error (119905) and the adaptation errors 120591(119905) and119890(119905) the closed-loop system is given by

119889119889119905[[[[[[[

119903120591

119890

]]]]]]]=[[[[[[[

119869minus1 [119882120591120591 minus 119870119904119903 + ]119871minus1 [119882119890119890 minus 119870119890119894 minus 119903]

minusΓ120591119882119879120591 119903minusΓ119890119882119879119890

]]]]]]] (67)

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 7

42 Stability Analysis To investigate stability of the equilib-rium point at the origin of system (67) the following Lya-punov candidate function is proposed

V = 121198691199032 + 121198711198942 + 12 119879120591 Γminus1120591 120591 + 12 119879119890 Γminus1119890 119890 (68)

where the time derivative of V along the trajectories of theclosed-loop system (67) is given by

V = minus1198701199041199032 minus 1198701198901198942 (69)

Hence V in (69) is negative or zero Then the origin of thestate space is Lyapunov stable Invoking Barbalatrsquos lemma ispossible to prove that

lim119905rarrinfin

[119903 (119905) (119905)] = [00] (70)

is satisfied Also it can be shown that the tracking error (119905)approaches zero as time 119905 increases that is

lim119905rarrinfin

(119905) = 0 (71)

See [39] for further details about the stability analysis

5 Adaptive PD Controller NeglectingElectrical Dynamics

In this section an adaptive PD controller is presented Thisalgorithm is derived under the assumption that the electricaldynamics of the actuator are neglected In literature this isbased on the assumption that the inductance 119871 is sufficientlysmall such that the current 119894(119905) can be approached by

119894 = V119877 minus 119870119861119877 (72)

Substituting (72) into the mechanical part of model (1) theexpression

119877119869 + [119877119861 + 119870119861] + 119877119873 sin (119902) = V (73)

is obtained Note that system (73) can be controlled by usinga PD-type scheme plus adaptive compensation to ensureconvergence of the error (119905)

By assuming that the current error (119905) converges quicklyto zero the resistance119877 is compensated by the controller andredefining the parameters of model (73) as

119869119877 = 119877119869119861119877 = 119877119861 + 119870119861119873119877 = 119877119873

(74)

our adaptive controller for system (73) is written as follows

V = 119884119898119889119898 + 119870119901119898 + 119870119889119898 119902 (75)

with119884119898119889 = [119889 119889 sin (119902)] 119898 = [119877 119877 119877]119879 (76)

and (119905) is obtained from (56) and 119870119901119898 and 119870119889119898 are definedas positive constants Note that the controller (75) has a PD-type structure in the position error (119905)51 Adaptation Law The adaptation law for the update of themechanical parameters 119898(119905) is given by

119889119889119905 119898 = Γ119898119884119879119898119889 [120598 + 119902] (77)

where

Γ119898 = diag (1205741198981 1205741198982 1205741198983) gt 0 (78)

is the adaptation gains of themechanical part and 120598 is a strictlypositive constant

52 Stability Analysis By defining the adaptation error of themechanical part as

119898 = 120579119898 minus 119898 isin R3 (79)

the closed-loop system formed by the substitution of thecontroller in (75) into the open-loop system in (73) can bewritten as 119889119889119905 = 119902

119869119877 119889119889119905 119902 = minus [119861119877 + 119870119889119898] 119902 minus 119870119901119898 + 119884119898119889119898119889119889119905 119898 = minusΓ119898119884119879119898119889 [120598 + 119902]

(80)

It is possible to show that the origin of state space is anequilibrium point

Let us consider that the constant 120598 involved in theadaptation law (77) satisfies

120598 lt minradic119870119901119898119869119877 119870119901119898 [119861119877 + 119870119889119898]

119870119901119898119869119877 + (14) [119861119877 + 119870119889119898]2 (81)

Furthermore under (81)

V = 121198701199011198982 + 12119869119877 1199022 + 120598119869119877 119902 + 12 119879119898Γminus1119898 119898 (82)

is a candidate Lyapunov function for the state space origin ofsystem (80) The time derivative of V along the closed-loopsystem trajectories is given in a matrix form by

V = minus[119902]119879

sdot [[

120598119870119901119898 minus1205982 [119861119877 + 119870119889119898]minus 1205982 [119861119877 + 119870119889119898] [119861119877 + 119870119889119898] minus 120598119869119877]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119876

[119902] (83)

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Complexity

It is possible to show that (81) is a sufficient condition thatguarantees the positive definiteness of the matrix 119876 in (83)and invoking Barbalatrsquos lemma it is possible to prove that

lim119905rarrinfin

[1003816100381610038161003816 (119905)1003816100381610038161003816 10038161003816100381610038161003816 119902 (119905)10038161003816100381610038161003816]119879 = 0 (84)

is satisfied

6 Experimental Evaluation

In this section a brief description of the experimental setupand the results obtained from the implementation of thedescribed adaptive controllers are presented

61 Experimental Platform For the experimental validationof the adaptive controllers presented in this paper thetestbed was constructed using a brushed DC motor fromAdvanced Motion Controls model MBR2303NI (Nema 34brushed direct currentmotor)mounted on an aluminumplateand a pendular load attached to the shaft See Figure 1 for apicture of the experimental platformThis experimental setupwas instrumented with the following components

(i) A PC with119882119894119899119889119900119908119904 119883119875(ii) A servoamplifier Advanced Motion Controls model

30A20ACV(iii) An optical encoder US Digital model HB6M-2500-

250-IE-D-H with a resolution of 2500 pulses perrevolution

(iv) An inductive filter Advanced Motion Controls modelBFC10010

(v) A current sensor FW Bellmodel NT-5(vi) A DAQ Sensoray 626

The control algorithms were implemented in 119878119894119898119906119897119894119899119896 andthe PC interacts with the DAQ through the Real-TimeWindows Target libraries at 1 millisecond of sample timeFinally Figure 1 shows a diagram of the overall experimentalplatform

The desired trajectory was selected as

119902119889 (119905) = 2 sin (3119905) + 15 [rad] (85)

Thephysicalmeaning of this trajectory is a displacement fromthe downward position to the upward position of the jointpassing through the point where the load is more affected bythe gravitational force

62 Control Gains The selected control gains and adaptationgains are shown in Tables 1ndash3 Specifically Table 1 shows thegains for the proposed MRAC algorithm in (45) Table 2shows the gains for the adaptive controller in (54) previouslyintroduced by [39] and Table 3 contains the gains for theadaptive PD controller in (75)

The tuning process was carried out in an experimentalform starting with sufficiently small values Then each gainwas increased looking for an improvement of the trajectory-tracking performance Related to the control gains small

Table 1 Control and adaptation gains for theMRACalgorithm (45)

Gain Value119896119901 30119896119894 10119896119889 10119896119891 1001198960 10119896119901V 30119896119900V 10119896119889V 03120572 011205741198981199031 0011205741198981199032 0011205741198981199033 501205741198901199031 0011205741198901199032 0751205741198901199033 005

Table 2 Control and adaptation gains for the DHB algorithm (54)

Gain Value119870119904 10119870119890 10120572 101205741205911 0011205741205912 0011205741205913 501205741198901 0031205741198902 0031205741198903 031205741198904 0031205741198905 0031205741198906 003

Table 3 Control and adaptation gains for the APD algorithm (75)

Gain Value119870119901119898 30119870119889119898 10120598 101205741198981 0011205741198982 0011205741198983 10

valueswere preferred this is in order to exhibit the adaptationperformance of each algorithm In general it is possible to saythat all the controllers were tuned in a similar manner

63 Experimental Results The following shows the obtainedexperimental results Figures 2-3 depict the comparisonbetween the desired the reference and the actual signals forthe position and electric current respectively obtained fromthe application of the proposed model reference adaptivecontroller in (45) As can be seen the obtained experimental

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 9

PCDAQ

Sensoray 626Servoamplifier

30A20AC

NT-5

DC motor

Optical encoder

Current sensor

Figure 1 Block diagram of the experimental platform of a 1-DOF RLED robot actuated by a brushed DC motor

0 2 4 6 8 10minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion

(rad

)

Time (s)

qd(t)qr(t)

q(t)

Figure 2 Time evolution of the joint position 119902(119905) and the modelreference position 119902119903(119905) during the tracking of the desired trajectory119902119889(119905) for the MRAC scheme

results validate the theory since the tracking between thereference and the desired signals is achieved while thetracking between the actual and the reference signals is alsoaccomplished

Additionally Figures 4ndash6 show the performance com-parison of the proposed model reference adaptive controller(MRAC) in (45) the adaptive controller proposed by Daw-son Hu and Burg (DHB) in (54) and the adaptive PD(APD) controller neglecting the electrical dynamics in (75)Specifically Figure 4 depicts the comparison between theactual position 119902(119905) and the desired position 119902119889(119905) in (85)for the aforementioned adaptive controllers In Figure 5the comparison of the electric current signals 119894(119905) can beseen Finally Figure 6 depicts the applied control voltage V(119905)generated for each control algorithm

It can be observed from Figure 4 that the MRAC algo-rithm converges much faster than the others This can be

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent (

A)

id(t)

ir(t)

i(t)

Figure 3 Time evolution of the electric current 119894(119905) the modelreference current 119894119903(119905) and the desired current 119894119889(119905) for the MRACscheme

possibly attributed to the overparameterization of the DHBalgorithm and to the simplicity of the APD controller sincethe DHB adapts nine parameters against only six parametersadapted by the MRAC making the MRAC adaptation lawseasier to tune On the other hand the fact that the APDneglects the dynamics of the electric actuator produces highererror values It was also observed in Figure 4 that the APDalgorithm presents better tracking performance than theDHB controller during the first seconds of experiment whichmay be due to the tuning of the DHB controller and the factthat the APD only responds to the magnitude of the positionerror without the delay introduced by the electric dynamicsof the actuatorHowever after the first seconds of experimentthe tracking performance of the DHB becomes much betterthan the APD

Furthermore in Figure 7 the time evolution of theestimated parameters (119905) (119905) and (119905) of the mechanicalpart is shown for the MRAC and the DHB algorithms

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Complexity

0 2 4 6 8 10minus15

minus1

minus05

0

05

1

15

2

25

3

35

4

Posit

ion q(t) (

rad)

Time (s)

APDDHBMRAC

qd(t)

Figure 4 Time evolution of the joint position 119902(119905) during the track-ing of the desired trajectory 119902119889(119905) for all the compared algorithms

0 2 4 6 8 10minus3

minus2

minus1

0

1

2

3

4

5

6

Time (s)

Curr

ent i(t) (

A)

APDDHBMRAC

Figure 5 Time evolution of the electric current 119894(119905) for all thecompared algorithms

Similarly in Figure 8 the time evolution of the estimatedparameters (119905) (119905) and 119861(119905) of the electrical part isshown The mechanical parameters of the APD controller in(75) are not shown in Figure 7 because their values are outof scale and do not provide any meaningful information tothe comparison Likewise the estimated values of the lumpedparameters 119890(119905) in (61) from the DHB controller are alsoomitted

It is worth remembering that matrix P in (41) should bepositive definite in order to ensure that the trajectories x(119905)

0 2 4 6 8 10minus2

minus1

0

1

2

3

4

5

6

7

Volta

ge (

t) (v

)

Time (s)

APDDHBMRAC

Figure 6 Time evolution of the applied voltage V(119905) for all thecompared algorithms

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

01

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

002

004

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time (s)

Time (s)

Time (s)

N(A

)B

(As

rad)

J(A

s2r

ad)

Figure 7 Time evolution of the estimated parameters of themechanical part for the DHB and MRAC algorithms

in (40) converge to zero as time 119905 approaches infinity Withthe aim of demonstrating that this condition is satisfied it ispossible to use the final values of the estimated parameters inFigures 7 and 8 for 119905 = 10 [s] that is (10) = 00275 (10) =00478 (10) = 08604 and 119861(10) = 00364 in addition tothe control gains shown in Table 1 for the MRAC algorithmHence by selecting an appropriate value for 120576 in (41) for

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 11

0 1 2 3 4 5 6 7 8 9 10minus005

0

005

DHBMRAC

0 1 2 3 4 5 6 7 8 9 100

05

1

0 1 2 3 4 5 6 7 8 9 100

002

004

006

Time (s)

Time (s)

Time (s)

KB

(Nm

A)

R(Ω

)L

(H)

Figure 8 Time evolution of the estimated parameters of theelectrical part for the DHB and MRAC algorithms

example 120576 = 5 the matrix P is positive definite with theeigenvalues 1205821 = 93203 1205822 = 10036 1205823 = 02962 and 1205824 =00270Therefore if rough estimates of the systemparametersare available then the provided stability analysis specificallythe conditions to ensure the positive definiteness of matrix Pin (41) can be used as tuning rules for the controller gains

With the aim of complementing the results of theexperimental comparison shown in Figures 4ndash6 Figure 9depicts the comparison between the error signals for thethree controllers where the exponential convergence rateof the trajectory-tracking error (119905) = 119902119889(119905) minus 119902(119905) can beseen Furthermore Table 4 presents a comparative of themaximum error obtained during the experiments for 15 le119905 le 20 [s] Specifically we compute max|(119905)| for eachcontroller during the mentioned time interval Table 4 alsoshows the maximum percentage of variation for the trackingerror with respect to the total displacement of the desiredtrajectory 119902119889(119905) that is

100 times max 1003816100381610038161003816 (119905)1003816100381610038161003816max 119902119889 (119905) minusmin 119902119889 (119905) (86)

Table 4 also shows the RMS value of the applied controlvoltages for each algorithm for a time interval of 15 le 119905 le20 [s]

The results shown in Table 4 suggest that the controllerMRAC performs better than the rest of the controllers of thecomparison yielding the smallest maximum deviation withrespect to the desired trajectory The controller DHB showsthe second best tracking performance on this studyTheAPDcontroller presents the worst tracking performance

0 2 4 6 8 10minus02

0

02

04

06

08

1

12

14

16

Time (s)

RefAPD

DHBMRAC

Erro

rq(t)

(rad

)Figure 9 Time evolution of the trajectory-tracking error (119905) for allthe compared algorithms

Table 4 Quantification of the trajectory-tracking error (119905) and theapplied control voltaje V(119905) for 15 le 119905 le 20Indexalgorithm APD DHB MRACmax|(119905)| [rad] 01949 00406 00402max|(119905)| []lowast 48726 10141 10051RMS(119905) [rad] 01094 00266 00217RMSV(119905) [119881rms] 11215 11247 11267lowastWith respect to the total displacement of the desired trajectory 119902119889(119905) see(86)

Note that the RMS value of the applied voltage V(119905) givenin Table 4 is similar for the three controllers Therefore itis possible to say that the proposed MRAC algorithm yieldsbetter performance with practically the same applied energy

It was also observed that the MRAC controller and theDHB scheme (controllers with armature current feedback)performed much better than the APD algorithm giving animprovement of 80 and 75 respectively However fromthe engineering point of view the APD adaptive controllermay be preferred in some cases since is a simpler controllerwith respect to those that use armature current feedback Apossible advantage of the APD controller is that the appliedcontrol voltage V(119905) is a noise free signal (see Figure 6) as aresult of not including the dirty current measurement 119894(119905)7 Concluding Remarks

This paper has dealt with the trajectory-tracking control ofa RLED robot with 1-DOF for which the parameters of thedynamic model are unknown A new controller has beenproposed based on aMRACprinciple Asymptotic tracking ofthe desired trajectory has been proved theoretically and prac-ticallyThe new controller has been compared with respect to

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Complexity

an adaptive controller previously reported in literature anda controller designed by neglecting the electrical part of thesystem

The new scheme showed better performance even withthe adaptation of less parameters with respect to the DHBscheme avoiding with this the overparameterization of themodel In other words the new MRAC only requires adap-tation of the six parameters present in the system while theDHB scheme is based on adaption of nine parameters

The results for the three controllers given in Figure 9 showthat the position error signal (119905) is kept in a certain boundafter some finite time 119879 gt 0 (practical stability) The reasonsare the quantization disturbance of position measurementnoisy current measurement and discrete implementationamong others

The experimental comparison indicated that in generalbetter results in the tracking performance are achieved ifcurrent feedback is added to the controller This is clearlyobserved from Figure 9 The reason is that the assumptionof neglecting the electrical dynamics is not valid for someoperating conditions This study suggests that good perfor-mance can be obtained with feedback of position velocityand a ldquosmall amountrdquo of current feedback

Finally as future work the noise reduction in the controlsignal V(119905) can be investigated which would improve thetracking performance This can be done by using an esti-mation of the current signal 119894(119905) instead of using a noisymeasurement of such signal avoiding the requirement ofa current sensor In this sense some preliminary work ispresented in [41 42] where the adaptive control of robotmanipulators is considered including the actuator dynamics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by CONACyT Project no176587 and SIP-IPN Mexico

References

[1] S Zak Systems and Control OxfordUniversity Press NewYorkNY USA 2003

[2] M Koksal F Yenici and A N Asya ldquoPosition control of apermanent magnet DC motor by model reference adaptivecontrolrdquo in Proceedings of the IEEE International Symposiumon Industrial Electronics (ISIE rsquo07) pp 112ndash117 Istanbul TurkeyJune 2007

[3] B Bhushan andM Singh ldquoAdaptive control of DCmotor usingbacterial foraging algorithmrdquo Applied Soft Computing vol 11no 8 pp 4913ndash4920 2011

[4] S Hwang and R Carmichael ldquoAdaptive tracking control for aDCmotorrdquo inProceedings of the 1st IEEERegional Conference onAerospace Control Systems pp 231ndash235 Westlake Village CalifUSA May 1993

[5] R Guenther and L Hsu ldquoVariable structure adaptive cascadecontrol of rigid-link electrically-driven robot manipulatorsrdquo

in Proceedings of the 32nd IEEE Conference on Decision andControl Part 3 (of 4) pp 2137ndash2142 San Antonio Tex USADecember 1993

[6] D M Dawson Z Qu and J J Carroll ldquoTracking control ofrigid-link electrically-driven robot manipulatorsrdquo InternationalJournal of Control vol 56 no 5 pp 991ndash1006 1992

[7] V M Hernandez-Guzman V Santibanez and G HerreraldquoControl of rigid robots equipped with brushed DC-motorsas actuatorsrdquo International Journal of Control Automation andSystems vol 5 no 6 pp 718ndash724 2007

[8] J Orrante-Sakanassi V Santibanez and J Moreno-ValenzuelaldquoStability analysis of a voltage-based controller for robotmanip-ulatorsrdquo International Journal of Advanced Robotic Systems vol10 article 22 2013

[9] R Silva-Ortigoza J R Garcıa-Sanchez J M Alba-Martınez etal ldquoTwo-stage control design of a buck converterDC motorsystem without velocity measurements via a ΣminusΔ- modulatorrdquoMathematical Problems in Engineering vol 2013 Article ID929316 11 pages 2013

[10] R Silva-Ortigoza V M Hernandez-Guzman M Antonio-Cruz and D Munoz-Carrillo ldquoDCDC buck power converteras a smooth starter for a DC motor based on a hierarchicalcontrolrdquo IEEE Transactions on Power Electronics vol 30 no 2pp 1076ndash1084 2015

[11] R Bai ldquoNeural network control-based adaptive design for aclass of DC motor systems with the full state constraintsrdquoNeurocomputing vol 168 pp 65ndash69 2015

[12] N Wang J Yu and W Lin ldquoPositioning control for a linearactuator with nonlinear friction and input saturation usingoutput-feedback controlrdquoComplexity vol 21 no 2 pp 191ndash2002016

[13] Z J Chen and P A Cook ldquoRobustness of model-referenceadaptive control systems with unmodelled dynamicsrdquo Interna-tional Journal of Control vol 39 no 1 pp 201ndash214 1984

[14] E Unkauf and D Torrey ldquoDirect model reference control ofan induction motorrdquo in Proceedings of the IEEE 10th AnnualApplied Power Electronics Conference (APEC rsquo95) pp 192ndash196IEEE Dallas Tex USA March 1995

[15] A S Kumar M S Rao and Y S K Babu ldquoModel referencelinear adaptive control of DC motor using fuzzy controllerrdquo inProceedings of the IEEERegion 10 Conference (TENCON rsquo08) pp1ndash5 Hyderabad India November 2008

[16] H M Schwartz ldquoModel reference adaptive control for roboticmanipulators without velocity measurementsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 8 no 3pp 279ndash285 1994

[17] H Schwartz ldquoAn MRAC output feedback controller for robotmanipulatorsrdquo in Proceedings of the 7th Mediterranean Confer-ence on Control and Automation pp 2292ndash2301 Haifa IsraelJune 1999

[18] B Rashidi M Esmaeilpour andM R Homaeinezhad ldquoPreciseangular speed control of permanentmagnet DCmotors in pres-ence of high modeling uncertainties via sliding mode observer-based model reference adaptive algorithmrdquo Mechatronics vol28 pp 79ndash95 2015

[19] C Aguilar-Avelar and J Moreno-Valenzuela ldquoModel referenceadaptive control for the trajectory tracking of a DC motor withpendular loadrdquo in Proceedings of the 14th IASTED InternationalConference Intelligent Systems and Control pp 266ndash273 Marinadel Rey Calif USA November 2013

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 13

[20] L HsuM C Teixeira R R Costa and E Assuncao ldquoLyapunovdesign of multivariable MRAC via generalized passivationrdquoAsian Journal of Control vol 17 no 5 pp 1484ndash1497 2015

[21] M Hosseinzadeh and M J Yazdanpanah ldquoPerformance en-hanced model reference adaptive control through switchingnon-quadratic Lyapunov functionsrdquo Systems amp Control Lettersvol 76 pp 47ndash55 2015

[22] A P Nair N Selvaganesan and V R Lalithambika ldquoLyapunovbased PDPID in model reference adaptive control for satellitelaunch vehicle systemsrdquo Aerospace Science and Technology vol51 pp 70ndash77 2016

[23] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014

[24] R-J Wai and R Muthusamy ldquoFuzzy-neural-network inheritedsliding-mode control for robot manipulator including actuatordynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 274ndash287 2013

[25] S Sefriti J Boumhidi R Naoual and Y Boumhidi ldquoAdaptiveneural network sliding mode control for electrically-drivenrobot manipulatorsrdquoControl Engineering and Applied Informat-ics vol 14 no 4 pp 27ndash32 2012

[26] MM Fateh and S Khorashadizadeh ldquoRobust control of electri-cally driven robots by adaptive fuzzy estimation of uncertaintyrdquoNonlinear Dynamics An International Journal of NonlinearDynamics and Chaos in Engineering Systems vol 69 no 3 pp1465ndash1477 2012

[27] R-J Lian ldquoIntelligent controller for robotic motion controlrdquoIEEE Transactions on Industrial Electronics vol 58 no 11 pp5220ndash5230 2011

[28] G G Rigatos ldquoAdaptive fuzzy control of DCmotors using stateand output feedbackrdquo Electric Power Systems Research vol 79no 11 pp 1579ndash1592 2009

[29] S N Huang K K Tan and T H Lee ldquoAdaptive neural networkalgorithm for control design of rigid-link electrically drivenrobotsrdquo Neurocomputing vol 71 no 4-6 pp 885ndash894 2008

[30] R-JWai andP-CChen ldquoRobust neural-fuzzy-network controlfor robot manipulator including actuator dynamicsrdquo IEEETransactions on Industrial Electronics vol 53 no 4 pp 1328ndash1349 2006

[31] J P Hwang and E Kim ldquoRobust tracking control of anelectrically driven robot adaptive fuzzy logic approachrdquo IEEETransactions on Fuzzy Systems vol 14 no 2 pp 232ndash247 2006

[32] C Kwan F L Lewis and D M Dawson ldquoRobust neural-network control of rigid-link electrically driven robotsrdquo IEEETransactions onNeural Networks vol 9 no 4 pp 581ndash588 1998

[33] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011

[34] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[35] H Hu and P-Y Woo ldquoFuzzy supervisory sliding-mode andneural-network control for robotic manipulatorsrdquo IEEE Trans-actions on Industrial Electronics vol 53 no 3 pp 929ndash9402006

[36] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009

[37] M M Fateh and S Khorashadizadeh ldquoOptimal robust voltagecontrol of electrically driven robot manipulatorsrdquo NonlinearDynamics vol 70 no 2 pp 1445ndash1458 2012

[38] M M Fateh H A Tehrani and S M Karbassi ldquoRepetitivecontrol of electrically driven robot manipulatorsrdquo InternationalJournal of Systems Science Principles andApplications of Systemsand Integration vol 44 no 4 pp 775ndash785 2013

[39] D M Dawson J Hu and T C Burg Nonlinear Control ofElectric Machinery Marcel Dekker New York NY USA 1998

[40] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[41] Y S Khaligh and M Namvar ldquoAdaptive control of robotmanipulators including actuator dynamics and without jointtorque measurementrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo10) pp 4639ndash4644 Anchorage Alaska USA May 2010

[42] H Kazemi andMNamvar ldquoAdaptive compensation of actuatordynamics in manipulators without joint torque measurementrdquoinProceedings of the 52nd IEEEConference onDecision andCon-trol (CDC rsquo13) pp 2294ndash2299 IEEE Firenze Italy December2013

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of