on d®-type iterative learning control

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On D-type Iterative Learning Control 1

YangQuan Chen and Kevin L. MooreCenter for Self-Organizing and Intelligent Systems (CSOIS)

Dept. of Electrical and Computer EngineeringUMC 4160, College of Engineering, 4160 Old Main Hill

Utah State University, Logan, UT 84322-4160, USA

Abstract

The classical Arimoto D-type iterative learning control(ILC) updating law uses the first order derivative (withtransfer function s) of tracking error. By using thefractional order derivative, it is proposed in this paperto use a fractional order ILC updating law where thefractional order derivative (with transfer function s, (0, 1]) of tracking error is employed. The basic ideais explained in some detail together with a frequency

domain design method. For implementation, a directdiscretization technique is introduced. An example isgiven to show the benefits of the D-type ILC. Severalremarks on future research topics are briefly discussedin the finally section.Keywords: Tracking control; iterative learning con-trol; D-type ILC; P-type ILC; fractional-order differen-tiator; Tustin operator; recursive expansion; continuedfraction expansion.

1 Introduction

The term iterative learning control (ILC) was coinedby Arimoto and his associates [1] for a better control of

systems performing repetitive tasks. In general, thereare two types of ILC updating laws: D-type and P-type. The original D-type ILC updating law [1] is

uk+1(t) = uk(t) + d

dtek(t) (1)

where is the learning gain to be designed based onprior knowledge about the system under investigation.uk(t) is the control signal at the k-the iteration while

ek(t)= yd(t)yk(t) is the tracking error signal between

the actual output trajectory yk(t) and the desired oneyd(t) at the k-the iteration. Here, time t [0, T], whereT is known and finite. The input (u) and output (y)

mapping is not fully known.The above ILC scheme actually only works for rel-ative degree one systems [2]. Later, motivated by thetracking control problem of a robot manipulator, whichis a relative degree two system, without accelerationmeasurement, a P-type ILC scheme is preferred

uk+1(t) = uk(t) + ek(t). (2)1This work is supported in part by U.S. Army Automo-

tive and Armaments Command (TACOM) Intelligent Mobil-ity Program (agreement no. DAAE07-95-3-0023. Correspond-ing author: Dr YangQuan Chen. E-mail: [email protected];Tel. 01-435-7970148; Fax: 01-435-7972003. URL:http://www.crosswinds.net/~yqchen .

We can unify the above updating laws in the frequencydomain in the form of

Uk+1(s) = Uk(s) + sEk(s) (3)

where, when = 0, (3) is a P-type ILC and, when = 1, it is a D-type ILC. Detailed literature reviewson ILC research can be found in [3, 4]. Most of theexisting work has focused on the analysis issue of ILCschemes. However, the convergence conditions found inthe literature are typically not sufficient for actual ILC

applications. Therefore, in recent years increasing ef-forts have been made on the designissue of ILC. Thesecan be observed from the latest books [5, 4] and thededicated ILC web server [6]. A recent survey on theILC designissue [7] has documented various practicallytested design schemes, mainly for robotic manipulators.

The focus of this paper is to go in-between P-typeand D-type ILC schemes, i.e., we consider in this pa-per that in (3) is a real number between 0 and1. We name this kind of generalized scheme D-type( (0, 1] ) ILC. For example, we can define a D0.5-type ILC updating law according to

uk+1(t) = uk(t) + d0.5

dt0.5ek(t). (4)

This kind of fractional order ILC scheme is actuallyan application of fractional calculus which is a 300-years-old topic. The theory of fractional-order deriva-tive was developed mainly in the 19-th century. Recentbooks [8, 9, 10, 11] provide a good source of referenceson fractional calculus. However, applying fractional-order calculus to dynamic systems control is a recenttopic of interest [12, 13, 14, 15, 16]. For example, PIDcontrollers, which dominate industrial control practice,have been modified using the notion of fractional-orderintegrators and differentiators. It has been shown thatthe extra degree of freedom from the use of fractional-order integrators and differentiators make it possibleto further improve the performance of traditional PIDcontrollers. For fractional-order systems, the fractionalcontroller CRONE [17] has been developed while in [18]and [19, 20] PD controller and the PID controllerwere proposed respectively. For pioneering work onthis regard, we cite [21, 22].

In the existing literature, to our best knowledge,there is no effort in developing fractional order deriv-ative based ILC schemes. In this paper, we extendedthe classical Arimoto D-type iterative learning control(ILC) updating law to a fractional order ILC updat-ing law D-type ILC where the fractional order

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derivative (with transfer function s, (0, 1]) of thetracking error is employed. The basic idea is explainedin some detail together with a frequency domain con-vergence analysis. For implementation, a recursive di-rect discretization technique is introduced. An exam-ple is given to show that by using the D-type ILCscheme the learning convergence can be tuned to bemore monotonic in the presence of strong time-varyingnonlinearities.

2 Problem Formulation

2.1 Fractional Order Differential Equation(FODE)

Note: This introductory subsection is similar to thatfound in a second paper by the authors in these pro-ceedings1. The material is duplicated here to preservecontinuity in the development of the paper.

The fractional calculus is a generalization of integra-tion and differentiation to non-integer order operators[8, 9, 10, 11]. The idea of fractional calculus has beenknown since the development of the normal calculus,

with the first reference probably being associated withLeibniz and LHospital in 1695. A fundamental oper-ator aD

t , a generalization of differential and integral

operators, is introduced as follows.

aDt =

d

dt() > 0,

1 () = 0,ta(d)

() < 0.

There are two commonly-used definitions for gen-eral fractional differentiation and integral, i.e., theGrunwald definition and the Riemann-Liouville defi-nition [8, 10, 11]. The Grunwald definition is that

aDt f(t) = lim

h0

1

h

[ tah

]j=0

(1)j

j

f(t jh), (5)

where [] is a flooring-operator, while the Riemann-Liouville definition is given by

aDt f(t) =

1

(n )dn

dtn

ta

f()

(t )n+1 d, (6)

for (n 1 < < n) and where (x) is the well knownEulers gamma function.

The formula for the Laplace transform of the

Riemann-Liouville fractional derivative (6) has theform [23]:

0

ept 0Dt f(t) dt =

= pF(p) n1k=0

pk 0Dk1t f(t)

t=0

, (7)

1Analytical stability bound for a class of delayed fractional-order dynamic systems, YangQuan Chen and Kevin L. Moore,in Proceedings of 2001 IEEE Conference on Decision and Con-trol, Orlando, Florida, December 2001.

for (n 1 < n).For numerical calculation of fractional-order deriva-

tion we can use the relation (8) derived from theGrunwald definition (5). This relation has the follow-ing form:

(tL)Dt f(t) h

N(t)j=0

bjf(t jh), (8)

where L is the memory length, h is the step size ofthe calculation,

N(t) = min

t

h

,

L

h

, (9)

where bj is the binomial coefficient given by the follow-ing recursive formula:

b0 = 1, bj =

1 1 +

j

bj1. (10)

To solve the fractional-order differential equations

(FODE), the Laplacian transformation of the Mittag-Leffler function in two parameters was proposed as aneffective means [23]. A two-parameter function of theMittag-Leffler type is defined by the series expansion:

E,(z) =k=0

zk

(k + ), (, > 0). (11)

In fact, it is shown [23] that the Mittag-Leffler functionis a generalization of exponential function ez, i.e.,

E1,1(z) =k=0

zk

(k + 1)=

k=0

zk

k!= ez.

In general, a LTI fractional-order controlled systemcan be described by

an Dnt y(t) + + a1 D1t y(t) + a0 D0t y(t) = (12)

bm Dmt u(t) + + b1 D1t u(t) + b0 D0t u(t),

where u(t) and y(t) are control and controlled signalsrespectively; k, k (k = 0, 1, 2, ) are generally realnumbers, n > > 1 > 0, m > > 1 > 0and ak, bk (k = 0, 1, ) are arbitrary constants. Notethat here (0D

t Dt ).

Using the Laplacian transformation method [23, 24,25], the transfer function of (12) can be written as

Gs(j) =Y(j)

U(j)=

mk=0 bk(j)

knk=0 ak(j)

k. (13)

2.2 D-type ILCAs discussed in the introduction section, in-between P-type and D-type ILC schemes, by using the fractionalorder derivative notion mentioned in Sec. 2.1, the fol-lowing learning updating law can be used:

uk+1(t) = uk(t) + d

dtek(t) (14)

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with its frequency domain form given in (3).Here we can regard d

dt as a special type of filter.Unlike any conventional FIR or IIR filter, which hasfinite memory, a fractional order derivative is a filterwhich requires all past information by its definition.

It should be noted that here we have only two de-sign parameters: and . The design guidelines forchoosing these parameters will be given in Sec. 3.

Note that s is an infinite dimensional linear op-erator. In the continuous time domain it can be ap-proximated by a nice recursive approximation schemeproposed in [26]. Given a frequency range of practicalinterest, [L, H], we can immediately get a ra-tional transfer function of finite order which is a fit tothe given fractional order differentiator s. In practice,we need to set the transitional frequency range muchlarger than the [L, H], for example, [0.1L, 10H].Using this transitional frequency range and a desiredorder of approximation, 2N+ 1, the following formulaeare the so-called Oustaloup-Recursive-Approximation(ORA):

limN

DN(s) = D(s) = s, (15)

where

DN(s) = (uH

)N

k=N

1 + s/k1 + s/k

, (16)

and

u =

HL, (17)

k = L(HL

)(k+N+0.50.5)/(2N+1), (18)

k = L(HL

)(k+N+0.5+0.5)/(2N+1). (19)

In the implementation of the fractional order deriva-tive used in our simulations below, we use these ORAformulae to approximate s.

3 Convergence Analysis

In this section, we shall perform a convergence analysisof the proposed D-type ILC in a general setting. Itis a common practice to apply ILC as a feedforwardcontroller in addition to the feedback controller. Thefeedforward-feedback configuration of ILC algorithmshas already been a standard consideration in eitheranalysis or design [7] work on ILC. ILC, originallyproposed as an open-loop control [1], has been con-sidered as a feedforward control in addition to an ex-isting feedback controller. As suggested in [7], whenILC starts to have a substantial impact on how con-trol is actually done in industry, it will be the linearbased ILC that leads the way. In engineering practice,to design a control system, it is very fundamental tohave a linear proximal model for frequencies below afrequency of interest, say, c. For a well designed feed-back controlled system, it is almost sure that its fre-quency response can be well approximated by a linearsystems, i.e., Gc(s), the closed-loop transfer function.

Therefore, at this point, it is understood that, the plantconsidered in the following is a linearly major part ofthe feedback controlled plantwhich may be nonlinear ingeneral. Therefore, the plant transfer function is Gc(s)which should be taken as the closed loop transfer func-tion of the plant under feedback control. Here, thefeedback controller is not necessarily a linear feedbackcontroller. It could be nonlinear feedback control, e.g.,feedback linearization scheme etc.

For Gc(s), the input is u(t) and output y(t). Wecan assume that Gc(s) is BIBO and internally stable.Given the desired output trajectory yd(t), t [0, T], weassume we can find a unique desired input ud(t) suchthat

yd(t) = Gcud(t). (20)

From the updating law in frequency domain (3), wehave

Uk+1(s) = Uk(s) + s(Yd(s) Yk(s))

= Uk(s) + sGc(s)(Ud(s) Uk(s))

= (1 sGc(s))Uk(s) + sGc(s)Ud(s).(21)

Let

= 1 sGc(s). (22)It is easy to write (21) as follows:

Uk+1(s) = kUk(s) +

1 k1 s

Gc(s)Ud(s).(23)

When

| |< 1, (24)we can conclude that

limkUk(s) = Ud(s). (25)

The convergence condition is that

| 1 (j)Gc(j) |< 1, . (26)

Remark 3.1 The convergence condition (26) tells uswhy a D-type ILC is normally required. The reason isthat the Bode plot of Gc(j) normally has a low passcharacteristic. Of course, the idealGc(j) of a feedbackcontrolled system is a constant 1. But this is not phys-ically possible in practice. Also ideally, it is desirableto replace the term

(j) in (26) with the inverse

of Gc(j). But this is also practically not possible be-cause we may not have full knowledge about Gc(j).However, it is reasonable to expect to use a filter withhigh-pass characteristics to compromise or dilute thelow-pass characteristics of Gc(j) with phase advance-ment. The classical Arimotos D-type ILC [1] scheme,i.e., when = 1, just fulfills the above argument. InD-type ILC, with the role of s diluting Gc(j), it isrelatively easier to find a learning gain with satis-

factory rate of convergence. The above discussion alsoexplains why a P-type ILC may work, but will convergevery slowly.

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Remark 3.2 The D-type ILC provides an effectivetuning knob in the Bode plot or in the Nyquist plane

for fitting or averaging such that for a wider range offrequency (j)Gc(i) is better shaped, which in turnfacilitates the proper choice of .

Remark 3.3 The zero-phase low-pass filter plus apure phase lead is a key implementation skill in ILC

applications [7]. Following the above remarks, we cansee that the major objective is to dilute Gc(j). Thisidea is not to actively dilute Gc(j) by using a high-pass filter but simply cutoff the high frequency part. Thephase lead element can play a role in compensating thelow pass characteristic of Gc(j) but its effect is lim-ited. Therefore, introducing a generals to replace thezero-phase low pass filter plus a pure phase lead makessense here.

4 Implementation Issues

In general, there are two discretization methods for s:direct discretization and indirect discretization. In in-direct discretization methods [26, 27], as also simplyintroduced in Sec. 2.2, two steps are required, i.e., fre-quency domain fitting in the continuous time domainfirst and then discretizing the fit s-transfer function.Other frequency-domain fitting methods can also beused but without guaranteeing the stable minimum-phase discretization.

In this Section, we focus on the direct discretizationmethod using the well known Tustin operator. It isa straightforward scheme to discretize the fractional-order derivative. The discrete transfer function is sta-ble and minimum phase. The recursive formula fordiscretization with different orders of approximation

simplifies the programming efforts. In general, the dis-cretization of the fractional-order differentiator s (is a real number) can be expressed by the so-calledgenerating function s = (z1). This generating func-tion and its expansion determine both the form ofthe approximation and the coefficients [28]. For ex-ample, when a backward difference rule is used, i.e.,(z1) = (1 z1)/Ts, where Ts is the sampling pe-riod, performing the power series expansion (PSE) of(1 z1) gives the discretization formula (8).

The trapezoidal (Tustin) rule as a generating func-

tion is considered here: ((z1)) =

2Ts

1z1

1+z1

To get a recursive formula similar to (10), here we intro-

duce the so-called Muir-recursion originally used in geo-physical data processing with applications to petroleumprospecting [29]. The Muir-recursion (motivated incomputing the vertical plane wave reflection responsevia the impedance of a stack of n-layered earth) canbe used in recursive discretization of fractional-orderdifferentiator of Tustin generating function:

((z1)) = (2

Ts)

1 z11 + z1

= (2

Ts) lim

n

An(z1, )

An(z1,) (27)

where A0(z1, ) = 1, and

An(z1, ) = An1(z

1, ) + cnznAn1(z, ), (28)

and

cn =

/n ; n is odd;

0 ; n is even.(29)

Therefore,

s ( 2Ts

)An(z1, )

An(z1,) .

5 An Illustrative Example

To demonstrate the benefits from using the proposedD-type ILC, a single link direct joint driven manip-ulator model is used for the simulation study. Thedynamics of the system is described by

(t) =1

J(u(t) F(t)) + 1

J(

1

2m + M)gl sin (t) (30)

where (t) is the angular position of the manipulator;u(t) is the applied joint torque; F(t) is the frictiontorque; m, l are the mass and length of the manip-ulator respectively, M is the mass of the tip load, g isthe gravitational acceleration, and J is the moment ofinertia w.r.t the joint and is given by J = Ml2 +ml2/3.The friction torque F(t) is

F(t) =

f+ + B+, 0,f + B, < 0,

where the Coulomb friction f+ = 8.43Nm and f =

8.26Nm; the viscous friction coefficients B+ =

4.94Nm/rad/s and B

= 3.45Nm/rad/s [30, page 257].For simulation, the parameters are assumed as fol-

lows: m = 2 kg, M = 4 kg, l = 0.5 m. g = 9.8 m./s2.It is also assumed that the desired tracking trajectoriesare specified as

d(t) = b + (b f)(154 65 103) (31)d(t) = (b f)(603 304 302) (32)

where = t/(tf t0). In the simulation,b = 0

, f = 90, t0 = 0, and tf = 1. sec.

The RK-4 method is used to numerically inte-grate the state equation with constant time stepTs = 0.01 second. The initial states at each ILC

repetition are all set to 0. The ILC iterationsend when eb1 1 and at the same time wheneb2 1/sec. where eb1 = supt[0,1] | d(t) (t) |and eb2

= supt[0,1] | d(t) (t) |. In the ILC

updating law (14), the tracking error is defined by

ek(t) = d(t) k(t). Here we only assume that theangular velocity is available. We also assume thatthat accurate system dynamics are not known andthat the estimated J is 50% of the true value in thesimulations. Then, for = 1, the best choice of is J, which makes = 0 [4]. In the following, we fix

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the learning gain = J/2 which implies that we haveassumed 50% uncertainties in parameter J aroundits nominal value actually used in the simulations.

We consider five cases, with =0, 0.25, 0.5, 0.75, 1.0. For = 0, only the statevariables are used for ILC updating and this is thuscalled P-type ILC. For = 1, the derivative of statevariable is to be used which involves the angularacceleration and thus this is called D-type. In our

simulation, the acceleration is obtained by digitaldifferentiation, which is a 5-point FIR filter given by

(k) ((k2)8(k1)+8(k+1)(k+2))/(12Ts).Using the Muir-recursive expansion of Tustin operatorin Sec. 4, for Ts = 0.01 sec., given N=5, we have

s0.25 3.761z5 0.9402z4 + 0.09402z3 0.3173z2 + 0.04701z 0.188

z5 + 0.25z4 + 0.025z3 + 0.08437z2 + 0.0125z + 0.05

s0.5 14.14z5 7.071z4 + 1.414z3 2.475z2 + 0.7071z 1.414

z5 + 0.5z4 + 0.1z3 + 0.175z2 + 0.05z + 0.1

s0.75

53.18z5 39.89z4 + 11.97z3 14.79z2 + 5.983z 7.977

z5 + 0.75z4 + 0.225z3 + 0.2781z2 + 0.1125z + 0.15

10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

eb

1

vs. ILC iter. no.

iteration number

eb

1

(deg)

=0

=0.25

=0.5

=0.75

=1

Figure 1: Comparison of D-type ILC with different :maximum absolute angular tracking errors.

Figs. 1-2 summarize the simulation results for com-parisons of the maximum absolute tracking errors overiteration number for different values of . We canobserve from Fig. 1 and Fig. 2 that, when = 0,the pure P-type ILC scheme gives a quite slow con-vergence. Due to the nonlinear frictional effect, thepure D-type ( = 1) gives a quite fast convergence butwith some significant oscillations in its convergence be-havior. When starting from = 0 and increasing by 0.25, the convergence behavior gets more and moreimproved. The most notable improvement is that thelearning convergence gets more and more monotonicwhen fractional order derivative is used. Especially,for the given exit conditions, learning control does notconverge within 60 iterations for conventional P-typeand D-type schemes. However, for = 0.75, the learn-ing converges at the 25-th iteration.

It should also be noted that this example is a non-linear example with no feedback controller.

10 20 30 40 50 600

20

40

60

80

100

120

140

160

180

eb

2

vs. ILC iter. no.

iteration number

eb

2

(deg.

/sec.)

=0

=0.25

=0.5

=0.75

=1

Figure 2: Comparison of D-type ILC with different :maximum absolute angular velocity trackingerrors.

6 Concluding Remarks

In this paper, it is proposed to use the fractional orderderivative notion to extend the classical Arimoto D-type ILC to the D-type. The basic idea is explainedin some detail together with a frequency domain con-vergence analysis. The rationale behind the D-typeILC is explained in the form of three remarks. Forimplementation, a recursive direct discretization tech-nique is introduced. An example is given to show thebenefits from the D-type ILC. It is observed that withD-type ILC scheme, the learning process convergencesmore monotonically.

Several remarks on the future research topics arebriefly discussed in the following: It is illustrated in Sec. 5 that D-type ILC also

works for nonlinear systems. A time-domain analysisof this effect will be interesting. It remains to define ways to design a set of ex-

periments such that and can be well-tuned basedon the experimental data. We may consider relay au-totuning technique used in PID controller parametersettings. We can consider a general fractional order trans-

fer function (13) to replace s in (3). A more practicalway is to consider PID scheme [19, 20]. A fractionallead-lag scheme is also possible [16].

It will be interesting to compare the proposed D-

type ILC design with those using the H loop shapingor other robust control techniques. Along the iteration axis, we can consider the

fractional-order difference. This may become a sys-tematic way of designing high-order ILC. Using the QFT (qualitative feedback control) tech-

niques [31, 32], we may propose frequency templatesfor ILC design in frequency domain. We can also makesimultaneous design for feedback (re)design and feed-forward (ILC) design. We can also consider the case when is a complex

number.

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References

[1] S. Arimoto, S. Kawamura, and F. Miyazaki,Bettering operation of robots by learning, J. of Ro-botic Systems, vol. 1, no. 2, pp. 123140, 1984.

[2] T. Sugie and T. Ono, An iterative learning con-trol law for dynamical systems, Automatica, vol. 27,no. 4, pp. 729732, 1991.

[3] K. L. Moore, Iterative learning control - an

expository overview, Applied & Computational Con-trols, Signal Processing, and Circuits, vol. 1, no. 1.

[4] YangQuan Chen and Changyun Wen, IterativeLearning Control: Convergence, Robustness and Appli-cations, vol. LNCIS-248 ofLecture Notes series on Con-trol and Information Science, Springer-Verlag, London,1999.

[5] Zeungnam Bien and Jian-Xin Xu, IterativeLearning Control - Analysis, Design, Integration andApplications, Kluwer Academic Publishers, 1998.

[6] YangQuan Chen, Dedicated webserver for Iterative Learning Control research,http://cicserver.ee.nus.edu.sg/~ilc, Oct. 1998.

[7] Richard W. Longman, Designing IterativeLearning and Repetitive Controllers. In Z. Bien andXu J.-X. eds, Iterative Learning Control - Analysis,Design, Integration and Application, Kluwer Acad-emic Publishers, pp. 107145, 1998.

[8] K. B. Oldham and J. Spanier, The FractionalCalculus, Academic Press, New York, 1974.

[9] S. G. Samko, A. A. Kilbas, and O. I. Marichev,Fractional integrals and derivatives and some of theirapplications, Nauka i technika, Minsk, 1987.

[10] K. S. Miller and B. Ross, An Introduction to theFractional Calculus and Fractional Differential Equa-tions, Wiley, New York, 1993.

[11] I. Podlubny, Fractional Differential Equations,Academic Press, San Diego, 1999.

[12] Boris J. Lurie, Three-parameter tunable tilt-integral-derivative (TID) controller, US PatentUS5371670, 1994.

[13] Igor Podlubny, Fractional-order systems andPID-controllers, IEEE Trans. Automatic Control,vol. 44, no. 1, pp. 208214, 1999.

[14] A. Oustaloup, B. Mathieu, and P. Lanusse, TheCRONE control of resonant plants: application to aflexible transmission, European Journal of Control,vol. 1, no. 2, 1995.

[15] A. Oustaloup, X. Moreau, and M. Nouillant,The CRONE suspension, Control Engineering Prac-tice, vol. 4, no. 8, pp. 11011108, 1996.

[16] H.-F. Raynaud and A. ZergamKnoh, State-space representation for fractional order controllers,Automatica, vol. 36, pp. 10171021, 2000.

[17] A. Oustaloup, La Derivation non Entiere, HER-MES, Paris, 1995.

[18] L. Dorcak, Numerical models for simulation thefractional - order control systems, in UEF SAV, TheAcademy of Sciences Institute of Experimental Physics,Kosice, Slovak Republic, 1994, pp. 6268.

[19] I. Petras, L. Dorcak, and I. Kostial, Controlquality enhancement by fractional order controllers,Acta Montanistica Slovaca, vol. 2, pp. 143148, 1998.

[20] I. Podlubny, Fractional-order systems andfractional-order controllers, in UEF-03-94, The Acad-emy of Sciences Institute of Experimental Physics,Kosice, Slovak Republic, 1994.

[21] A. Oustaloup, Linear feedback control systems

of fractional order between 1 and 2, in Proc. of theIEEE Symposium on Circuit and Systems, Chicago,USA, 4 1981.

[22] M. Axtell and E. M. Bise, Fractional calculusapplications in control systems, in Proc. of the IEEE1990 Nat. Aerospace and Electronics Conf., New York,USA, 1990, pp. 563566.

[23] I. Podlubny, The Laplace transform method forlinear differential equations of the fractional order,in Proc. of the 9th International BERG Conference,Kosice, Slovak Republic (in Slovak), 9 1997, pp. 119119.

[24] L. Dorcak I. Petras, The frequency method for

stability investigation of fractional control systems,SACTA journal, vol. 2, no. 1-2, pp. 7585, 1999.

[25] I. Petras, L. Dorcak, P. OLeary, Vinagre B.M., and I. Podlubny, The modelling and analysis offractional-order control systems in frequency domain,in Proceedings of ICCC2000, High Tatras, Slovak Re-public, May 23-26 2000, pp. 261 264.

[26] A. Oustaloup, F. Levron, F. Nanot, and B. Math-ieu, Frequency band complex non integer differentia-tor : characterization and synthesis, IEEE Transac-tions on Circuits and Systems I: Fundamental Theoryand Applications, vol. 47, no. 1, pp. 2540, 1 2000.

[27] A. Chareff, H. H. Sun, Y. Y. Tsao, and B. Onaral,

Fractal system as represented by singularity func-tion, IEEE Trans. on Automatic Control, vol. 37, pp.14651470, 1992.

[28] C. H. Lubich, Discretized fractional calculus,SIAM J. Math. Anal., vol. 17, no. 3, pp. 704719, 1986.

[29] Jon F. Claerbout, Fundamentals of Geophys-ical Data Processing with applications to petroleumprospecting, Blackwell Scientific Publications, 1976.

[30] P. I. Corke and B. Armstrong-Helouvry, Ameta-study of PUMA 560 dynamics: a critical ap-praisal of literature data, Robotica, vol. 13, pp. 253258, 1995.

[31] C. H. Houpis and S. J. Rasmussen, Classical feed-back control with MATLAB, vol. 6 ofControl Engineer-ing Series of Reference Books and Textbooks, MarcelDekker, Inc., New York, 2000.

[32] Boris J. Lurie and Paul J. Enright, Quantitativefeedback theory: fundamental and applications, vol. 3of Control Engineering Series of Reference Books andTextbooks, Marcel Dekker, Inc., New York, 1999.

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