sliding sr

7/30/2019 Sliding SR

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European Journal of Scientific ResearchISSN 1450-216X Vol.50 No.3 (2011), pp.363-380 EuroJournals Publishing, Inc. 2011http://www.eurojournals.com/ejsr.htm

Performance Comparison of PI and Sliding Mode for SpeedControl Applications of SR Motor

Muhammad Rafiq Muhammad Ali Jinnah University, Islamabad, Pakistan

E-mail: [email protected]

Saeed-ur-RehmanCentre for Advanced Studies in Engineering (CASE), Islamabad, Pakistan

Fazal-ur-Rehman Muhammad Ali Jinnah University, Islamabad, Pakistan

Qarab Raza ButtCentre for Advanced Studies in Engineering (CASE), Islamabad, Pakistan

Abstract

In this paper, a performance comparison of Proportional Integral (PI) Controllerwith Sliding Mode Controller is presented for speed control of Switched Reluctance Motor(SR Motor). A robust controller is also suggested for high performance speed regulationand tracking problem of SR motor. The suggested scheme is based on higher order slidingmode (HOSM) technique. The proposed controller also guarantees that the motor speedconverges to the desired speed significantly faster than other conventional techniques. Theeffectiveness of the proposed controller is confirmed by simulation results. The robustnessof the proposed controller to parametric variations is also validated through simulationstudies.

Keywords: SR motor, sliding mode controller (SMC), higher order sliding modecontroller (HOSMC), speed regulation, tracking control, super twistingalgorithm.

I. IntroductionSwitched reluctance motor is a doubly salient machine in which torque is produced by the tendency of the rotor poles to align themselves with the poles of the excited motor phases. The simple geometricalconstruction is one of its attributes because there are no windings on the rotor and hence itsmanufacturing cost is low as compared to other motor drives. SR motor has the advantages of robustness, high efficiency and high torque in low speed (Miller, 2001). However along with theseadvantages, SR motor has some drawbacks as it has highly nonlinear magnetic structure and motorparameters are time varying, so modern control techniques are required to control SR motors. Severalnonlinear control techniques such as back stepping, sliding mode, artificial neural network, fuzzy logic,feedback linearization, etc. have been developed for the control of SR motors. Alrifai et al ., used back stepping approach and developed speed controller for SR motor. The proposed controller takes inputs


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Performance Comparison of PI and Sliding Mode forSpeed Control Applications of SR Motor 364

in the form of rotor position, rotor speed, phase currents and reference speed and finds out the requiredphase currents to keep the motor speed near to the reference speed. The simulation results showed thatthe proposed controller is better than PI controller in providing faster dynamic response. A self tuningfuzzy PI controller with artificial neural network was suggested for speed regulation problem of SRmotor (Karakas and Soner, 2007). The simulation results presented in their work showed improvedperformance as compared to fuzzy PI controllers. Haiqing et al ., 1996 gave the idea of feedback linearization and proposed controller for speed regulation purpose of SR motor. The proposedcontroller was then compared with conventional PI controller and shown to be robust in presence of uncertainties and unknown disturbances.

Figure 1: A cross-sectional view of 3-Phase SR motor with 6-Stator and 8-Rotor poles.

Sliding mode control has achieved much importance in the last two decades. Due to itssimplicity, high accuracy, fast dynamic response and robustness, it has received researchers attentionand a variety of new algorithms have been developed (Levant, 1993). A comparison of sliding modewith PI and fuzzy controller was investigated in Inanc and Ozbulur, 2003 for speed regulation problemof SR motor. The proposed controller was capable of removing low frequency oscillations and shownto be more effective and robust than PI and fuzzy controllers. The performance comparison of slidingmode control with PI control for SR motor was also reported in Tahour et al., 2008. The simulationresults reflected that the proposed controller was superior to PI controller. Forrai et al., 1998 proposedsliding-mode controller for SR motor to control speed but their work did not include magneticsaturation. It is important to note that SR motor is usually operated in magnetic saturation in order toincrease its output torque. A comprehensive study of sliding mode, PID and fuzzy logic controllers wasreported in Singh et al., 1998 for speed regulation problem of SR motor. The simulation resultsindicated that the performance of sliding mode controller was better than PID and fuzzy logiccontrollers. The conventional sliding-mode control suffers from the inherent problem of chatteringwhich leads to high wear and tear of mechanical parts and as a result high heat losses are caused inelectrical power circuits (Perruquetti and Barbot, 2002). This problem can be overcome by usinghigher order sliding mode (HOSM) technique (Levant, 1987). HOSM has been used for a number of engineering applications (Butt and Bhatti, 2008; Butt et al., 2009; Qaiser et al., 2009; Orani et al.,


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365 Muhammad Rafiq, Saeed-ur-Rehman, Fazal-ur-Rehman and Qarab Raza Butt

2009). Defoort et al., 2006, 2009; Nollet et al., 2008 worked on stepper motors and designed thirdorder sliding-mode controllers for position tracking problems. The proposed controllers were based ongeometric homogeneity and integral term was augmented in the controller to cater for uncertainties.The robustness of the developed controller against parametric variations and load disturbances wasalso reported. Laghrouchi et al., 2004 investigated MIMO nonlinear systems and developed HOSMcontroller for position control of permanent magnet synchronous motor. Finite time convergence andgood robustness were reported. Huangfu et al., 2008 extended this work for fault detection and

isolation. A double differentiator based on HOSM was reported in Bartolini et al., 2003 for robustspeed and torque estimation of induction motor. The simulation results showed that the proposedcontroller was a good estimator and robust against measurement errors. Floquet et al., 2000 alsoworked on induction motor and developed second order sliding-mode controller for speed trackingproblems. To estimate rotor flux, first order sliding mode observer was designed and their techniquedid not require any torque estimation. Traore et al., 2008 extended this work and proposed HOSMCbased observer. The observer estimated the motor parameters and the estimated values were given tothe controller for speed tracking application. Damiano et al., 2004 applied HOSM on DC motors forspeed control applications. Their scheme did not require the exact knowledge about system parametersand the simulation results showed the performance to be far better than that achievable from a PIcontroller.

This paper compares the performance of PI controller with sliding mode controller (SMC) anda robust controller based on HOSM control technique is then proposed for better performance. The restof the paper is organized as follows: Section II describes the mathematical model of the SR motor;Section III introduces the Sliding Mode, PI and Higher-Order Sliding-Mode techniques. Controllerdesign for both regulation and tracking problems are also developed in this Section. Simulation resultsare discussed in Section IV and Section V concludes the paper by summarizing the main contributions.

II. Mathematical Model of the SystemFor any controller design, the important step is to develop the reliable mathematical model thatrepresents the system dynamics under various operating conditions. The system under consideration is3-Phase 6/8 SR motor whose parameters are given in Table1. The control oriented mathematical modelof SR motor consists of electrical and mechanical dynamic subsystems (Rafiq et al., 2009), which aredescribed below.

Figure 2: HOSMC based derived system for speed control of SR motor

HOSMC

-

+

6/8 SRMotor

PowerInverter

DCSupply

re f

A. Electrical Subsystem

Because of the concentrated nature of phase windings, the mutual inductance between various phasesof SR motor is negligibly small. Thus the voltage applied to any one phase of the SR motor can beaccurately described as


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( )

,(t) (t) 1, 2,3

j j

j j j

d iu R i j

dt

= + = (1)

Where , , and j j ju R i represent the input, resistance, and current in the jth phase. ( ), j ji is

the flux linkage in the j th phase which is a the nonlinear function of rotor position and phase current.

For simplicity of notation, the explicit dependence of ju

and ji

on time t will be omitted in theremaining part of the paper. The decoupling between motor phases leads to the following expressionfor the time derivative of phase flux linkage:

( ) ( ) ( ) , , , j j j j j j j j

d i i i did dt dt i dt

= +

(2)

Substituting (2) in (1) we have:

( ) ( ) , , j j j j j j j j j

i i diu R i

i dt

= + +

(3)

which can be re-written in the following form

( ) ( )1 , ,( ) ( ) j j j j j j j j j

i idi u R idt i

=

(4)

Where( )

, j j

j

i

i

represents the self-inductance of the phase and

( ),

j ji

is the back

EMF produced in the j th phase.

B. Mechanical Subsystem

Mechanical subsystem can be expressed by the following relation.

( )( )

2

2

1,

e j L

d T i B T

dt J

= (5)

where is the rotor angle and is the rotor speed. J and B are the moment of inertia and coefficientof friction, respectively. ( ),e jT i is the total electromagnetic torque which is equal to the sum of individual torques produced by all motor phases. For simplicity, the explicit dependence of eT on and i is being omitted in the remaining part of the paper.

3

1( , )e j jT i == (6)

where ( ), j jT i is the torque of the j th phase.

( )( ),

,c j

j j

W i

T i

= (7)where cW is co-energy.

( ) ( ) ji

c j j j j0

, i , i diW = (8)Now (7) takes the form as

( ) ( )0

, , ji

j j j j jT i i di

= (9)


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Now, the complete dynamic model of the SR motor can be expressed in the following state-space form

d dt

= (10)

( )( )1 ,e j Ld T i B T dt J

= (11)

( ) ( ) ( ) ( )1

, ,( ) ( ) j j j j j

j j j j

i idi t u t R i t dt i

=

(12)

For speed control application, (11) can be rewritten as

( )1 e Ld

T B T dt J

= = (13)

By differentiating above equation we get1

e LdT dT d d

Bdt J dt dt dt =

(14)

( )31

,1

j j L

j

dT i dT d B

dt J dt dt

=

= (15)

( ) ( )3 31 1

, ,1

j j j j j L j j

j

T i T idi dT d B

dt J i dt dt

= = = + (16)

Substituting (12) into (16) and leads to the following equation:

( ) ( ) ( ) ( )1

, , , ,1 3 31 1

T i i i T i dT d j j j j j j j j Lu R i B j j j j jdt J i i dt j j

= + = =

(17)

( ) ( ) ( )1

3

1

, , ,1

j j j j j j

j j j j j

T i i i R i J i i

=

=

( ) ( ) ( )1

3 3

1 1

, , ,1

j j j j j j L j j j

j j

T i T i idT B u

dt J i i

= =

+ + (18)

This can be written in a compact form as:

( ) ( ), , , , , Ld

i B T i udt

= + (19)

Where u is the input vector consisting of 3-phase voltages. The scalar function and vectorfunction are defined as:

( )( ) ( ) ( )

1

3

1

, , ,1, , , ,

j j j j j j

L j j j j j

T i i ii B T R i

J i i

=

=

( )31

, j j L j

T i d T B

d t

=

+

(20)

( )( ) ( )

1

3

1

, ,1,

j j j j

j j j

T i ii

J i i

=

=

(21)

For simplicity of notation, the explicit dependence of u on time t and & vectors on, , , , Li B T will be omitted in the remaining part of the paper.


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III. Controllers StructuresTo optimize the power consumption of SR motor, a commutation scheme has been incorporated whiledesigning the controllers. This commutation scheme minimizes input power by energizing at the mosttwo out of three phases of same polarity at any instant. Conversely, at least one, and at the most twophases of opposite polarity are kept off to avoid the negative contribution which causes power loss atthe own expenses. Thus the motor is driven in a power efficient manner by this commutation scheme.The details of commutation scheme can be found in previous work of author(s) in [25]. The controllercould be incorporated in motor control as shown in Figure 2.

Figure 3: Graphical representation of SMC minimizing the error to the origin

Sliding mode

Switching surface

S = 0

Reachingsurface

Reaching mode

Desired Final value

Design of Sliding Mode Controller (SMC)

The objective of the sliding mode controller is to design switching surface and control law which isresponsible to force the system trajectories towards the switching surface. On switching surface, the

system is insensitive to certain parameter variations and unknown disturbances (Slotine and Weiping,1991; Utkin et al., 2002).The basic idea behind the design of any speed controller is to minimize the speed error. i.e. to

minimize the error to the origin of a plane formed by ( ) e t and ( )e t & . Graphically it can be seen inFigure 3.

( ) ( ) ( )ref e t t t = (22)

Where ( )ref t is the desired speed. In this section, speed controller based on sliding-modetechnique is derived. The first step is to design a sliding surface, which is taken to be as below.

S e e = +& (23)

A candidate Lyapunov function is taken as 21

2V S = that leads to

dV SS

dt = &where

S e e = +& && &(24)

( )( ( ))ref ref S t t = + +& & &&& (25)

( )( ( ( ))ref ref dV

S u t t dt

= + + +& && (26)

Speed regulation and tracking problem are now considered one by one.


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Case-1: Regulation ProblemThe purpose of regulation problem is to stabilize the motor speed at a desired constant value. i.e.

( )ref ref t = and ( ) ( )0, 0.ref ref t t = =&&& For proving that the proposed control law guarantees theconstant speed requirement, first consider the Proposition 1.

Proposition 1 : The following SMC will stabilize the speed to its desired value as t

( )( )1 S u sign S = + + (27)

Proof: Substituting (27) in (26), we get

( )( ) S dV

S sign S dt

= + (28)

And simplifying we get

( ) 0S dV

S sign S dt

= < (29)

It is evident from (29), 0dV dt

= only when 0.S = This ensures that the control law defined in

(27) would guarantee that ( ) ref t when t

Case-2: Tracking ProblemThe objective of tracking problem is to follow the time varying reference signal keeping the trackingerror to a minimum. To prove that the SMC will follow the reference signal, we consider theProposition 2

Proposition 2 : The following SMC ensures that the motor speed follows a time varyingreference signal when t

( ) ( )1 { ( ( )}S ref ref u sign S t t = + + +&& & (30)

Proof: By combining (26) and (30), one can obtain:

( ) ( )( ) ( )( )( )( ) ( )S ref ref ref ref dV S sign S t t t t dt

= + + + +&& & && & (31)

Simplifying we get

( ) 0S dV

S sign S dt

= < (32)

From (32), it is clear that 0dV dt

= only when 0.S = This ensures that the control law defined

in (30) will guarantee that the motor speed follows the time-varying reference signal in the limit. Inboth the above cases, it is shown that V is positive definite and V & is negative definite, therefore thecontrol law u would guarantee that ( ) ( )ref t t as t .

Design of PI ControllerPI controllers are widely used in industries due to their simplicity of design, and low cost. Moreover,their implementation in analogue or digital hardware is simple and easy. Under limited operatingconditions, they perform well and also their steady state performance is good. The PI controller for theabove system can be expressed as

( ) ( ) p iu K e t K e t dt = + (33)Where pK and iK are the proportional and integral gain constants. Since the motor is highly

nonlinear and state variables are coupled, so it is not easy to predict the system dynamics with the helpof mathematical solution. Therefore classical pole placement method is not applicable for designing PIcontroller. For a certain operating point, the gains pK and iK are tuned using trial and error method.


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The system model of SR motor is highly nonlinear as stated earlier and tuning of PI controller gains forthe entire input/output space would be highly tedious job. Online tuning would require a lot of computational resources and real time response could still be in question. The numerical values of pK

and iK for optimum performance are given in Appendix 1.

Table 1: Parameters Of Switched RELUCTANCE MOTOR

Parameter ValueNo. of phases 3No. of stator poles 6No. of rotor poles 8Inertia (J) 20.1 . N ms Coefficient of friction(B) 0.1 . N ms Phase rsistance 4.7 DC voltage supply 250 v

Figure 4: Twisting algorithm phase trajectory

O

Design of Higher Order Sliding Mode (HOSM)The higher order sliding mode (HOSM) technique generalizes the basic sliding mode idea by acting onhigher order derivatives of sliding variable instead of first order derivative. This technique gives thesame robustness and performance as conventional sliding mode with an extra benefit of chatteringremoval (Levant, 1987). The sliding order is a number of continuous total time derivatives of S in thevicinity of sliding mode. The r th order sliding mode ( r -sliding) is determined by the equations

( 1) 0r S S S S = = = = =& &&

The main problem of HOSM is the increment of the demanded information. For example any r -sliding controller needs the information about ( 1), , r S S S S &&& in order to keep 0.S = Super twistingalgorithm is one of the popular algorithms among the second order sliding mode algorithms that doesnot require this extra piece of information. This algorithm has been developed and analysed forsystems which has relative degree one in order to avoid chattering. The trajectories of the algorithmtwist around the origin in the phase portrait of sliding variable as shown in Figure 4. The super twistingalgorithm has the advantage over other algorithms in that it does not require the time derivative of sliding variable. The control law used in this algorithm is composed of two components. The firstcomponent 1v is defined in term of discontinuous time derivative while the other component 2v is acontinuous function of sliding variable (Fridman and Levant, 2002; Khan et al. , 2003).


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1 2u v v= + (34)

( )11

1S

u when uv

sign S when u

>=

& (35)

( )( )

0 02

0

s

s

S sign S when S S v

S sign S when S S

>=

(36)

The super twisting algorithm converges in finite time and the corresponding sufficientconditions are:

22

4 ( ) , , 0 0.5

( ) M S

S sm m m S

+

> < +

(37)

Where 0 , , , , S mS are some positive constants. When the system is linearly dependenton ,u then control law can be expressed as:

( )0 1 su S sign S v

= + (38)

( )1 S v sign S = & (39)

The super twisting algorithm converges exponentially to the origin when 1 s =

. The designedcontrol law using super twisting algorithm for speed regulation and speed tracking problem finallytakes the form as given below

Speed RegulationThe control signal u computed for speed regulation can be written as:

( )( )0.5 11 S u S sign S v = + + + (40)( )1 S v sign S = & (41)

Speed TrackingThe control signal u for tracking a reference speed is:

( ) ( )( )0.51 ( ( ))S ref ref u S sign S t t = + + + &&& (42)( )1 S v sign S = & (43)

Figure 5: Speed responses of PI, SMC & HOSMC for a step command.

0 0.1 0 .2 0.3 0. 4 0 .5 0 .6 0. 7 0 .8 0 .9 10

1

2

3

4

5

6

7

8

9

10

PISM CH O S M C


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Figure 6: A close up view of responses of PI, SMC & HOSMC for a step command. The high magnitudes of chattering of SMC and speed ripples of PI are clearly noticeable.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

9.97

9.98

9.99

10

10.01

10.02

10.03

10.04

10.05

Time ( sec )

S p e e d

( r a d

/ s e c )

PISMCHOSMC

Figure 7: Error plots of speed response of PI, SMC & HOSMC for a step command.

0 0 .1 0 .2 0. 3 0. 4 0 . 5 0 . 6 0 . 7 0 .8 0 .9 1

0

2

4

6

8

10PISM CH O S M C

Figure 8: A close up view of error plots of speed response of PI, SMC & HOSMC to a step command. Thereduced amount of error magnitude is clearly visible.

0 0 .1 0 .2 0. 3 0. 4 0 . 5 0 . 6 0 . 7 0 .8 0 .9 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

PISM CH O S M C


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The following section gives the results of simulation testing of control algorithms designedabove.

IV. Simulation ResultsThe mathematical model presented in the previous section is simulated using MATLAB/SIMULINKsoftware. The dynamic equations are programmed in S-function; which is a dynamic equationprogramming routine. A template of this routine is available along with MATLAB. The GUI for thissimulation is created using SIMULINK blocks. The physical and functional parameters of SR motorused for simulation testing are given in Table1.

Figures 5-8 show the speed responses of PI, SMC and proposed design (HOSMC) when SRmotor is operating at a reference speed of 10 rad/s. It can be observed that the motor speed convergesto the desired speed within 0.07 seconds. PI controller exhibits large speed ripples in steady state.Chattering is observed in SMC which is revealed by Figure 6. The proposed controller shows an initialovershoot which is reasonable and then tracks the reference speed closely. Figure 7 illustrates therespective error plots and Figure 8 gives its close up view. It is important to note that as far aschattering and large speed ripples are concerned, the proposed controller has outperformed the othertwo controllers. Figure 9 gives the speed responses when SR motor is commanded to follow thereference with sudden change in torque load. Initially the external torque load is zero and thensuddenly a torque load of 4 N-m is applied at t = 0.4 sec. which results in a slightly larger ripples inmotor speed. Finally the external torque load is withdrawn at t = 0.5 sec. and speed ripples decreaseslightly. It is clear from Figure 10 that the maximum speed ripple using PI control is 7%. Using SMC,it is 6% and using proposed HOSMC design it is just 2.5% that makes a better choice. It clearlyoutperforms the other two. The same performance of the proposed controller can be observed fromFigures 11-12 when SR motor is commanded to operate at 10 rad/s from its standstill position withtorque load of 4 N-m. It can be seen that the proposed controller produces exceptional performancewhere as the dynamic responses of SMC and PI controller are not up to the mark. Chattering and 4%steady state error can be observed in SMC; and PI controller exhibits a slight drop in speed response intransient state.

Figure 13-15 show a comparison of tracking performance of proposed controller with PI andSMC while following a Sinusoidal trajectory. Figure 14 gives a clear picture of the controller output atdifferent points of one sinusoidal cycle. It is visible that SMC is showing higher magnitude of chattering, while proposed controller is producing smaller spikes whenever sinusoid crosses the datum(zero) line. It is a well documented fact that high frequency or high magnitude of chattering of a slidingcontrol is dangerous when an implementation is done and an actuator has to obey a sliding/switchingcontrol command. Therefore, it is established that the proposed controller (HOSMC) is producingbetter overall results for SR motor control.

Figure 9: Speed response of controllers against sudden change in torque load.

0 0 .1 0 .2 0. 3 0. 4 0 . 5 0 . 6 0 . 7 0 .8 0 .9 19

9. 1

9. 2

9. 3

9. 4

9. 5

9. 6

9. 7

9. 8

9. 9

10

PI

SM C

H O S M C


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Figure 10: A close up view of speed response of controllers against sudden change in torque load.

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

1

2

3

4

5

6

7

8

9

10

P I

S MC

H OSM C

Figure 11: Speed response with torque load.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9

10

Time ( sec )

S p e e

d ( r a

d / s e c

)

PISMCHOSMC

Figure 12: A close up view of speed response of controllers with torque load.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

Time ( sec )

S p e e

d ( r a

d / s e c

)

PISMCHOSMC

Figure 13: Speed response of controllers against the reference signal defined by ( ) 10 (2= ref t sin t )

0 0 .5 1 1. 5 2 2 . 5 3 3 . 5 4 4 .5 5

-1 0

-8

-6

-4

-2

0

2

4

6

8

10

PISM CH O S M CRef. Signal


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Figure 14: A close up view of Speed response against a reference signal ( ) 10 (2= ref t sin t ).

3 3 .1 3 .2 3. 3 3. 4 3 . 5 3 . 6 3 . 7 3 .8 3 .9 4

-10

-8

-6

-4

-2

0

2

4

6

8

10PISM CH O S M CR e f . S i g n a l

Figure 15: Error between speed response of PI, SMC and HOSMC ( ) 10 (2= ref t sin t )

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

23

Time ( sec )

S p e e d E r r o r

( r a d / s )

PI

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10123

Time ( sec )

S p e e d E r r o r

( r a d / s )

SMC

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2

3

Time ( sec )

S p e e d E r r o r

( r a d / s )

HOSMC

A. Robustness of the Proposed Design

For high performance applications the proposed scheme should be robust to parameter variations.

Changes in moment of inertia J, stator phase resistance R and coefficient of friction B are investigatedthrough simulations. The simulation studies are undertaken by changing one parameter at a time whilekeeping other parameters unchanged. The motor is commanded to accelerate from rest to referencespeed of 10 rad/sec under no torque load. Figures 16-17 show the motor responses of PI, SMC andHOSMC when moment of inertia is decreased by 50% and then increased by 100% of its originalvalue. It can be seen that PI control exhibits poor dynamic response. Overshoot and speed ripples arequite significant. The speed response of SMC is however excellent but it suffers from the chatteringeffect; which is an undesirable phenomena. An increase or decrease of moment of inertia J does nothave any effect on the performance of the proposed Higher Order Sliding Mode technique. Thedecrease of moment of inertia gives good performance but it slows down the dynamic response.

Figures 18-19 show the response when there are changes in phase resistance. It can be noted

that the decrease in phase resistance causes large overshoot in SMC and HOSMC than PI controlwhich is affected by speed ripples. The increase in phase resistance slows down the dynamicresponses. The response of SMC is slower than PI and Proposed Controller.

Figures 20-21 show the speed response when coefficient of friction is decreased by 50%, andthen increased by 100% of its original value. It can be seen that the increment or decrement of coefficient of friction does not impose any effect on the performance of the proposed technique butonly affects the dynamic response. The decrease/increase of coefficient of friction in PI control resultsin high overshoot and large speed ripples. The performance of SMC is affected at transient stage andcauses more chattering. It is very much clear from Figures 16-21 that the proposed controller(HOSMC) is insensitive to parametric variations and a robust tracking performance is achieved inpresence of the uncertain parameters.


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Figure 16: Speed response with variations in moment of inertia J.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d

( r a

d /

s e c

)

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d

( r a

d

/ s e c

)

SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d

( r a

d /

s e c

)

HOSMC

JJ / 2J * 2

JJ / 2J * 2

JJ /2J * 2

Figure 17: A close up view of speed response of controllers with changes in J.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

Time (sec )

S p e e

d (

r a

d /

s e c

)

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

Time (sec)

S p e e

d

( r a

d /

s e c

)

SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

Time (sec )

S p e e

d (

r a

d /

s e c

)

HOSMC

JJ / 2J * 2

JJ / 2J * 2

JJ / 2J * 2

Figure 18: Speed response of controllers with variations in R.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)

SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)

HOSMC

RR / 2R * 2

RR / 2R * 2

RR / 2R * 2


15/17


Figure 19: A close up view of speed response of controllers with changes in R.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.9

10

10.1

Time ( sec )

S p e e

d (

r a

d /

s e c

)

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.9

10

10.1

Time ( sec )

S p e e

d (

r a

d /

s e c

)

SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.9

10

10.1

Time ( sec )

S p e e

d (

r a

d /

s e c

)

HOSMC

RR / 2R * 2

RR / 2R * 2

RR / 2R * 2

Figure 20: Speed response of controllers with changes in B.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

24

6

8

10

Time ( sec )

S p e e

d (

r a

d /

s e c

)

HOSMC

BB / 2B * 2

B

B / 2B * 2

BB / 2B * 2

Figure 21: A close up view of speed response of controllers with changes in B.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

10.1

Time ( sec )

S p e e d ( r a d / s e c )

PI

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

10.1

Time ( sec )


SMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.8

9.9

10

10.1

Time ( sec )


HOSMC

BB / 2B * 2

BB / 2B * 2

BB / 2B * 2


16/17


ConclusionRegulation and tracking control problems of SR motor has been addressed. PI controller exhibits largespeed ripple and inherent chattering in SMC may be harmful to SR motor, therefore HOSMC isintroduced to handle these problems. The performance of the controllers is validated throughsimulations. The proposed controller also guarantees that the motor speed converges to the desiredspeed considerably faster than other conventional techniques. The simulation results show that theperformance of proposed controller is better than conventional SMC and PI controller. A number of simulation results are presented for comparison. The results prove the robustness of the proposedcontroller against parameter variations and load disturbances.

AcknowledgmentThe authors are grateful to Higher Education Commission (HEC) of Pakistan for financial support.

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Appendix -1 PI controller gains: 10 0.05 p iK K = =

SMC Parameters: 9680 1.0S = = HOSMC Parameters: 5200 1.0S = =

sliding sr

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