Simulation Modelling Practice and Theory 25 (2012) 106–117

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Simulation Modelling Practice and Theory

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A second order sliding mode control design of a switched reluctancemotor using super twisting algorithm

Muhammad Rafiq a,⇑, Saeed-ur Rehman b, Fazal-ur Rehman a, Qarab Raza Butt b, Irfan Awan c

a Muhammad Ali Jinnah University, Islamabad, Pakistanb Centre for Advanced Studies in Engineering (CASE), Islamabad, Pakistanc School of Computing, Informatics and Media, University of Bradford, UK

a r t i c l e i n f o

Article history:Received 20 December 2011Received in revised form 1 March 2012Accepted 2 March 2012Available online 2 April 2012

Keywords:SR motorSliding mode controlRegulationTrackingSuper twisting algorithm

1569-190X/$ - see front matter � 2012 Elsevier B.Vhttp://dx.doi.org/10.1016/j.simpat.2012.03.001

⇑ Corresponding author. Tel.: +447438577950.E-mail addresses: rafiq_mufti@yahoo.com (M. Ra

a b s t r a c t

A novel robust technique for speed control application of variable reluctance motor is pro-posed. The suggested scheme is model based and uses a mathematical model of an SRmotor, and Second Order Sliding Mode Control (SOSMC) with Super-Twisting algorithm.Sliding mode controllers for SR motor were reported before but super twisting SOSMC havean added advantage of reduced chattering which is one of the main focuses of this work.The proposed controller gives fast dynamic response with no overshoot and nearly zerosteady state error. The effectiveness of the proposed controller and its robustness toparameter variations is also confirmed by simulation results.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Switched Reluctance Motors (SRMs) have gained considerable attention due to its robust, and simple mechanical con-struction, high efficiency, and ease to maintain a high torque at low speed. Due to its mechanical design structure an SR mo-tor is very suitable for operations at high speed [1]. SRMs are typically operated in magnetic saturation and phase torque is ahighly nonlinear function of phase current and rotor position. Many nonlinear control techniques have been proposed for thecontrol of SRMs. Sliding mode is one of these techniques and has gained much popularity in such applications due to its sim-ple structure, intrinsic robustness and potential to control nonlinear systems [2]. This technique has been applied on variousengineering problems (for example [3]). In [4,5], sliding mode control was reported for SR motor to regulate its speed buttheir research did not cater for magnetic saturation.

SR motor has inherently a problem of torque ripples particularly in variable speed application. Sliding mode techniquehas provided ripple free torque for SR motor (see for example [6–8]). Ref. [9] designed flux linkage controller for SR motorto reduce torque ripple. The proposed controller was based on integral sliding mode technique. Both the properties of PI andsliding mode controls were incorporated in the controller and better results were reported. A similar attempt was also madein [10] to minimize torque ripples in SR motor. The controller was designed to remove the low frequency oscillations andfurther it was applied for speed regulation problems and was shown to be more efficient and robust than conventional con-trollers. Ref. [11] used the idea of dynamic sliding mode technique for speed regulation of SR motor. The performance of thedesigned controller was shown to have chattering reduction in output.

. All rights reserved.

fiq), I.U.Awan@Bradford.ac.uk (I. Awan).

M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117 107

The conventional sliding-mode control experiences inherent problem of chattering. A number of techniques have beenintroduced in the literature for chattering reduction, i.e. one of the techniques is the use of Fuzzy Sliding Mode Control(FSMC) [12]. Another popular technique for the chattering reduction is the use of Higher Order Sliding Mode (HOSM) control[13]. HOSM has been used for a number of engineering problems [14–18]. Ref. [19] investigated HOSM technique and em-ployed it on DC motors having less data about system parameters showing its better performance. Ref. [20] proposed HOSMcontroller for a class of MIMO nonlinear uncertain systems. The effectiveness of the proposed controller was shown throughexperimental results. Ref. [21] used the same system and designed HOSM observer to suppress chattering inherent in con-ventional sliding mode technique. Application of HOSM control on induction motor was also reported in [22–24] for variouspurposes. In [23,24], HOSM based observer was used for speed tracking problems. The system parameters like magnetic flux,angular velocity, stator phase current and load torque were estimated by the observer by eliminating the need of mechanicalsensors.

The paper is organized as follows: In next section, control oriented mathematical model of the SR motor is describedalong with commutation scheme. Higher order sliding mode is briefly explained in Section 3, with brief details of supertwisting controller design algorithm. Second order super twisting controller design for both regulation and tracking controlsis proposed in Section 4. Simulation results are discussed in Section 5, and Section 6 concludes the work presented in thispaper.

2. Mathematical model of SR motor

In order to study the dynamics of SR motor and to synthesize its controllers, a mathematical model of the system isrequired. A lot of work has been done on modeling and design of SR motor. Several numbers of techniques have been foundin literature for estimating the motor parameters [25–28]. The mathematical model used in our work is a specific 3-phasecommercial SR motor whose parameters are listed in Section 5. This model has been taken from [29]. The model consists ofelectrical and mechanical dynamical subsystems which is given in state space form as:

hJBijuj

Te

xTj

TL

kj

Rj

dhdt¼ x ð1Þ

dxdt¼ 1

JðTe � Bx� TLÞ ð2Þ

dij

dt¼ @kjðh; ijÞ

@ij

� ��1

uj � Rij �x@kjðh; ijÞ@h

� �ð3Þ

where j = 1, 2, 3 represent the phase number and

Rotor positionMoment of inertia (rotor)Coefficient of frictionCurrent in the jth phaseVoltages of jth phase = [u1 u2 u3]T

Total electromagnetic torque =P3

j¼1Tjðh; ijÞAngular velocity of rotorElectromagnetic torque of the jth phaseLoad torqueFlux linkages in jth phaseResistance to the jth phase = R

2.1. Commutation scheme

The mathematical model of a three phase 6-Stator and 8-Rotor poles Switched Reluctance (SR) motor is used. In this con-figuration, an electrical angle of 2p radians is equivalent to p/4 radians rotation of mechanical angle. This mechanical angleof p/4 radians is further divided into 12 regions ranging from R1 to R12 as explained above. In region R1, positive torque isproduced via phase B, while phase C and phase A provide negative torques. Similar phases are energized for region R2. In R3,phases B and C provide positive torque while negative torque is produced by phase A. So if the positive torque is required,only the phases B and C should be energized that may give higher net torque instead of energizing all phases which will col-lectively give less net torque due to the counter effect of negative torque produced by phase A in this region. On the otherhand, if only negative torque is required, only phase A should be energized. There is no need to energize phases B and C dueto their positive torques in the region, which will ultimately increase the net torque. A similar pattern is adapted in remain-ing regions and only one or at the most two phases of desired polarity are energized. From the above discussion it is clearthat torque depends upon the rotor position. Ultimately the proposed scheme will save net power, which will increase thesystem efficiency.

108 M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117

3. Second Order Sliding Mode (SOSM)

Sliding mode design is accomplished into two steps. First, define the switching surface/sliding variable from the givensystem states and then design the control law in such a way that the system trajectories are forced towards the switchingsurface. The Higher-Order Sliding-Mode (HOSM) technique generalizes the basic sliding-mode idea by incorporating higher-order derivatives of the sliding variable. The inclusion of higher-order derivatives, while maintaining the same robustnessand performance as that of the conventional sliding-mode, leads to a reduction in the undesirable chattering effect inherentin the sliding-mode technique [30].

HOSM technique achieves this performance enhancement by incorporating the knowledge about the higher-order deriv-atives of the sliding variable. For example, an nth-order sliding-mode controller requires the information abouts; _s;€s; . . . ; sðn�1Þ in order to keep s = 0. The knowledge about the values of higher-order derivatives of the sliding variableseems to be a constraining requirement. However, this apparent restriction can be avoided by the use of Super-TwistingAlgorithm. super-twisting algorithm was employed for the systems having relative degree one for the purpose of reductionin chattering [31–35]. It has the advantage over other algorithms in that it does not demand the time derivatives of slidingvariable. The control law used in this algorithm is composed of two terms. The first term va is a discontinuous time derivativefunction whereas the second term vb is a continuous function of sliding variable [36].

u ¼ va þ vb ð4Þ

_va ¼�u when juj > 1�vsignðsÞ when juj 6 1

�ð5Þ

vb ¼�ljs0jgsignðsÞ when jsj > s0

�ljsjgsignðsÞ when jsj 6 s0

(ð6Þ

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

v >UCm

;l2 P4U

C2m

CMðvþUÞCmðv�UÞ ; 0 < g 6 0:5 ð7Þ

where v, l, s0, U, Cm are some positive constants. When controlled system is linearly dependent on u, the control law can besimplified as:

u ¼ �ljsjgsignðsÞ þ va ð8Þ_va ¼ �v signðsÞ ð9Þ

when g = 1 the algorithm converges to the origin exponentially.

4. Sliding mode controller design

A speed controller minimizes the speed error, i.e.

eðtÞ ¼ xðtÞ �xref ðtÞ ð10Þ

where xref(t) is the desired speed. The sliding variable in this case is taken to be 0s0 with n > 0 as an arbitrary positive definiteconstant. Sliding variable is defined by the equation

s ¼ _eþ ne ð11Þ

Convergence of sliding variable is analyzed by considering a nonempty manifold S = {e = e = 0} and the Lyapunov based en-ergy function of the sliding variable V ¼ 1

2 s2 which gives _V ¼ s_s where

_s ¼ €eþ n _e ð12Þ_s ¼ €xðtÞ þ n _xðtÞ � ð €xref ðtÞ þ n _xref ðtÞÞ ð13Þ_V ¼ sð €xðtÞ þ n _xðtÞ � ð €xref ðtÞ þ n _xref ðtÞÞÞ ð14Þ

To ensure the convergence of designed controller, the derivative of Lyapunov function of s must be a decaying function,i.e. _V ¼ s_s < 0. To show this, the Model is first brought into a compact form to avoid unnecessary detailed equations as below.Eq. (2) is differentiated and (3) is put into the resulting equation to obtain the following equation.

€x ¼ 1JP3j¼1

@Tjðh; ijÞ@ij

@kjðh; ijÞ@ij

� ��1

uj � Rjij �x@kjðh; ijÞ@h

� �þx

P3j¼1

@Tjðh; ijÞ@h

� B _x� dTL

dt

( )ð15Þ

This can simply be written in a compact form as:

M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117 109

€x ¼ f h;x; i; _x;ddt

i;B;R; TL; _TL

� �þ gjðh; ij; Þuj ð16Þ

where f is a scalar and g is a row vector function which can be defined as:

f h;x; i; _x;ddt

i;B;R; TL; _TL

� �¼ 1

JP3j¼1

@Tjðh; ijÞ@ij

@kjðh; ijÞ@ij

� ��1

�Rjij �x@kjðh; ijÞ@h

� �þx

P3j¼1

@Tjðh; ijÞ@h

� B _x� dTL

dt

( )ð17Þ

g ¼ gjðh; ij; Þ ¼ g1ðh; i1; Þ g2ðh; i2; Þ g3ðh; i3; Þ½ � ¼ 1J

@T1ðh;i1Þ@i1

@k1ðh;i1Þ@i1

� ��1� �@T2ðh;i2Þ

@i2@k2ðh;i2Þ

@i2

� ��1� �@T3ðh;i3Þ

@i3@k3ðh;i3Þ

@i3

� ��1� �

266666664

377777775

T

j ¼ 1;2;3 ð18Þ

More detail about the derivation of these equations can be found in [37,38]. Regulation and tracking problems are dealtwith separately here although regulator is a special case of tracking.

4.1. Regulation problem

The regulator maintains constant motor speed for a given range of load variations, i.e. x(t) = xref for all times and_xref ðtÞ ¼ 0. The regulator has the following shape:

u ¼ u1 u2 u3½ �T ¼ �g�ðf þ n _xþ v:signðsÞÞ ð19Þ

where v is the gain of FOSMC, and a positive constant number and g⁄ is a Moore–Penrose pseudo inverse of row vector g. Thevector g has already been defined in (18). The Pseudo-Inverse of g can be written as g� ¼ g�1 g�2 g�3½ �T and is a column vec-tor. It is assumed that following conditions are satisfied.

1. g⁄g is Hermitian,2. gg⁄ is Hermitian,3. g⁄gg⁄ = g⁄, and4. gg⁄g = g

For further details about generalized Moore–Penrose pseudo inverse of non-square vectors; see [39,40].Convergence: Substituting (16) into (14), the following equation is obtained.

_V ¼ sðf þ g:uþ n _xÞ ð20Þ

Plugging (19) into (20), we get

_V ¼ sðf � f � n _x� vsignðsÞ þ n _xÞ ð21Þ) _V ¼ �v :s:signðsÞ < 0 ð22Þ

Eq. (22) implies that _V ¼ 0 if s = 0. Hence convergence is guaranteed, i.e. x(t) ? xref as t ?1.

4.2. Tracking problem

In tracking, the controller is required to follow the time varying reference signal while keeping the tracking error mini-mum. In this case the controller has the following appearance:

u ¼ u1 u2 u3½ �T ¼ �g�f þ n _xðtÞ þ v :signðsÞ�ð €xref ðtÞ þ n _xref ðtÞÞ

� �ð23Þ

Convergence: Combining (14) and (16), following is obtained:

_V ¼ sðf þ g:uþ n _xðtÞ � ð _xref ðtÞ þ n _xref ðtÞÞÞ ð24Þ

Substituting (23) in (24) gives

_V ¼ �s:v :signðsÞ < 0 ð25Þ

From (25), it is obvious that _V < 0, and would become zero only when s = 0. In both cases, it is shown that V is positive def-inite and _V is negative definite, therefore the control law u will guarantee that x(t) ? xref(t) as t ?1. SOSMC is designed as

110 M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117

discussed earlier in sliding mode controller design section. The control law designed on the super-twisting algorithm takesthe following form for speed regulation:

u ¼ u1 u2 u3½ �T ¼ �g�ðf þ n _xðtÞ þ ljsj0:5signðsÞÞ þ va ð26Þ_va ¼ �v :signðsÞ ð27Þ

For tracking, it takes the following shape

u ¼ u1 u2 u3½ �T ¼ �g�ðf þ n _xðtÞ þ ljsj0:5signðsÞ � ð _xref ðtÞ þ l _xref ðtÞÞÞ þ va ð28Þ_va ¼ �v :signðsÞ ð29Þ

Note: The value obtained from integration of _va is a single scalar and will be added to each element of vector u

4.3. Dynamic Sliding Mode Control (DSMC)

Dynamic sliding mode control is used to reduce the chattering issue encountered in conventional sliding mode techniqueby incorporating dynamic sliding surface. The surface is either control dependant or control independent. In our case slidingsurface is designed as:

s ¼ uþ g�ðf þ k1 _xþ k2ðx�xref ÞÞ ðk1; k2 are some postive constants:Þ ð30Þ

The control law is defined as

u ¼ �g�ðf þ k1 _xþ k2ðx�xref ÞÞ þ v ð31Þ_v ¼ �WsignðsÞ ðW is nonzero positive constant:Þ ð32Þ

5. Simulation results and discussions

To simulate the mathematical model derived in the previous section, MATLAB/SIMULINK software was used. The dynamicequations are programmed in S-function; which is a template function given with the Matlab package, used to program thedifferential equations. SIMULINK interface is used to create the GUI for these simulations. Parameters and constants of SRmotor used for simulation testing are given in below.

No of phases = 3,

0 1 2 3 4 5 60

100

200

Time (sec)

FOSM

C

Controller efforts

0 1 2 3 4 5 60

50

100

150

200

250

Time (sec)

SOSM

C

Fig. 1. Controller efforts.

No. of stator poles = 6,

No. of rotor poles = 8, Rotor inertia (J) = 0.1 N ms2

Coefficient of friction (B) = 0.1 N ms

Phase resistance = 4.7X DC voltage supply = 250 V

The input voltages of SR motor phases as controlled by FOSMC and SOSMC are plotted in Figs. 1–3. From Fig. 1, the per-

formance of both controllers is not clearly distinguishable, therefore two close ups are plotted in Figs. 2 and 3 which showthe effectiveness of SOSMC in power saving as the phase voltage values in SOSMC are low being less in area under the curveindicating saving in power consumption.

The speed responses of conventional First Order Sliding Mode Controller (FOSMC) and proposed design (SOSMC) areshown in Fig. 4 when motor is directed to move to reference speed of 30 rad/s starting from rest. It is clear that the motorspeed reaches the desired value within 1 s. Both FOSMC and SOSMC show nearly the same performance in reaching the de-sired value in terms of time required with FOSMC leading the other. When the desired speed is reached, the behavior of bothcontrollers varies with higher magnitude of chattering observed in trajectory produced by FOSMC. To clearly observe thechattering phenomenon, a close up view of response of both controllers is also plotted. The higher magnitude of chatteringproduced by FOSMC is clearly visible. To remove this, an SOSMC is proposed. Proposed design proves to be a good choice for

M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117 111

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

100

200

Time (sec)

FOSM

C

Controller efforts in Transient stage

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

50

100

150

200

250

Time (sec)

SOSM

C

Fig. 2. Controller efforts in transient state.

3 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10

100

200

Time (sec)

FOSM

C

Controller efforts in steady stage

3 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10

50

100

150

200

250

Time (sec)

SOSM

C

Fig. 3. Controller efforts in steady state.

0 1 2 3 4 5 60

10

20

30

Time (sec)

Spee

d (r

ad/s

)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

29.4

29.6

29.8

30

Time (sec)

Spee

d cl

oseu

p (r

ad/s

)

SOSMC

FOSMC

SOSMCFOSMC

Fig. 4. Speed response when motor is commanded to attain the reference speed 30 rad/s.

112 M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117

chattering removal which is evident from the same plot. The error plots of desired speed and trajectory produced by bothcontrollers are given in Fig. 5. The close up view clearly shows the higher magnitude of chattering caused by FOSMC, whilethe proposed design; SOSMC, causes chattering of very low magnitude.

Better performance of SOSMC can also be observed from Fig. 6 when SR motor experiences a sudden change in torqueload from 0 to 2 N-m at t = 2.0 s until t = 2.1 s and again from 0 to 4 N-m at t = 3.0 s until t = 3.1 s. Both controllers areput to test against this change. It is visible from this Figure that despite the sudden change in load torque, the SOSMCwas able to maintain the desired speed without a significant dip in speed of motor where FOSMC was unable to give thatperformance and gave bigger oscillation/chattering in speed during the period while trying to compensate the disturbancecaused by sudden change in load. The average value of trajectory is farther from the desired one in case of FOSMC.

The proposed controller behaves well for speed tracking problems also. Fig. 7 demonstrate the speed response when mo-tor is directed to follow a sine wave. The controller is able to maintain its tracking performance, and gives some oscillationswhen the speed changes its sign while crossing the zero axes. From this figure it is also evident that this error may be due tothe phase difference of both signals. It is shown that the proposed controller was able to force the motor speed close to thedesired trajectory in all the above cases.

0 1 2 3 4 5 6

0

10

20

30

Time (sec)

Spee

d er

ror

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-0.1

0

0.1

0.2

0.3

0.4

Time (sec)

Spee

d er

ror

clos

eup

SOSMCFOSMC

SOSMCFOSMC

Fig. 5. Error response when motor is commanded to attain the reference speed 30 rad/s.

0 1 2 3 4 5 60

10

20

30

Time (sec)Sp

eed

(rad

/s)

Sudden change in Torque load

0 1 2 3 4 5 6

26

27

28

29

30

Time (sec)

Spee

d (r

ad/s

)

Sudden change in Torque load (close up )

SOSMCFOSMC

SOSMCFOSMC

2 N-m4 N-m

2 N-m

4 N-m

Fig. 6. Speed response during sudden change in torque load.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-20

-10

0

10

20

Time (sec)

Spee

d (r

ad/s

)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-6-4-20246

Time (sec)

Err

or

SOSMCFOSMCRef.signal

SOSMC

FOSMC

Fig. 7. Tracking performance & respective errors.

M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117 113

5.1. Robustness of the proposed control

High performance applications demand that the proposed design should be robust against parameter variations andexternal disturbances; the most common of which is load torque in case of an SR motor. In Fig. 6, a comparison of two con-trollers was shown when SR motor is suddenly subjected to a disturbance in the form of a sudden external load of 2 N-m and4 N-m. A comparison of the robustness of both FOSMC and super twisting SOSMC controllers is immediately evident fromthe plots. It can be seen that the higher order sliding mode controller (SOSMC) is more robust than the FOSMC and outper-forms it in maintaining the motor speed against the suddenly applied external load.

Next, the robustness of controllers is compared on parametric variations. The responses of synthesized controllers areinvestigated against changes in moment of inertia J, which could be due to engagement of load; stator phase resistance R,

0 1 2 3 4 5 60

10

20

30

Time (sec)Sp

eed

(rad

/s)

Variation in J

0 1 2 3 4 5 626

27

28

29

30

Time (sec)

Spee

d (r

ad/s

)

Variation in J closeup

JJ / 2J * 2

J

J / 2

J * 2

Fig. 8. Variations in J.

0 1 2 3 4 5 60

10

20

30

Time (sec)

Spee

d (r

ad/s

)

Variation in R

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 628

28.5

29

29.5

30

Time (sec)

Spee

d (r

ad/s

)

Variation in R closeup

R

R / 2

R * 2

R

R / 2

R * 2

Fig. 9. Variations in R.

114 M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117

which could vary due to temperature variations in winding during operation [41] and coefficient of viscous friction B as amodel uncertainty, since this parameter is usually provided by the manufacturer of mechanical components of motor,and may not be accurate when the motor parts are assembled into a complete and operational SR motor assembly. Therefore,it is difficult to find its exact value, and is usually estimated [42]. Changes in moment of inertia J, stator resistance R andcoefficient of friction B are analyzed through simulations. In these simulation experiments, one parameter is allowed tochange at one time during one experiment; the other parameters are kept constant. The motor is directed to move from0 to desired speed of 30 rad/s under no torque load. Changes are made during the time intervals t = 0 to t = 1 s and t = 3to t = 3.1 s.

Fig. 8 elaborates the controller response when changes in moment of inertia are carried out. Changes are analyzed bydecreasing the nominal value 50% and then increasing the nominal value 100%. It can be observed that the performancehas been improved due to increment in moment of inertia but in case of decrement, it slows down the dynamic response,

0 1 2 3 4 5 60

10

20

30

Spee

d (r

ad/s

)

Variation in B

0 1 2 3 4 5 622

24

26

28

30

Time (sec)

Spee

d (r

ad/s

)

Variation in B closeup

BB / 2B * 2

BB / 2B * 2

Time (sec)

Fig. 10. Variations in B.

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

11

Time (sec)

Spee

d (r

ad/s

ec)

SOSMCDSMC

Fig. 11. Speed response of DSMC and SOSMC to a step command.

M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117 115

since the proposed controller SOSMC reaches the desired speed a little bit late. Fig. 9 shows the speed response when thesame changes in phase resistance are made. It is clear from this figure that the performance remains the same but onlythe dynamic response is affected. The decrement in phase resistance gives good dynamic response but the increment inphase resistance causes the dynamic response slower.

Fig. 10 illustrates the speed response during variation in coefficient of friction. It is noted that while decreasing the coef-ficient of friction up to 50% of its nominal value, the dynamic response has been improved. On the other hand, by increasingthe coefficient of friction up to 100% of its nominal value, poor dynamic response is observed. It is clear that the proposedSOSMC controller maintains desired speed even in the presence of parameter variations. Therefore it can be concluded that itis also robust against parametric variations and unknown disturbances.

A comparison of SOSMC was also carried out with Dynamic Sliding Mode Controller (DSMC) for its effectiveness in chat-tering removal. Fig. 11 illustrates the results of both controllers; from which it is clear that the rise time of DSMC is less thanSOSMC but its overshoot is a bit higher than SOSMC. The close up view of responses of both the controllers in Fig. 12 showsthat SOSMC outperforms the DSMC not only in tracking the desired speed but also in chattering reduction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.4

9.5

9.6

9.7

9.8

9.9

10

10.1

Time (sec)

Spee

d (r

ad/s

ec)

SOSMCDSMC

Fig. 12. A close-up view of speed response of DSMC and SOSMC to a step command.

116 M. Rafiq et al. / Simulation Modelling Practice and Theory 25 (2012) 106–117

6. Conclusion

Regulation and Tracking control and chattering elimination from output speed of SR motor were addressed. The inherentchattering in FOSMC may be harmful to SR motor; therefore SOSMC based on super twisting algorithm is proposed as a solu-tion to the problem of chattering. It guarantees the asymptotic convergence of motor speed to a desired value. The simula-tion experiments verify that the performance of SOSMC is better than conventional FOSMC. The behavior of controllers isaddressed under ideal conditions and a number of simulation results are presented for comparison. The performance testingof SOSMC against parametric variations is also carried out and simulation results show that it is robust against parametervariations and disturbances also.

Acknowledgement

This research was financially supported by Higher Education Commission (HEC) Pakistan. The authors are grateful to theresearch fellows at CASPR at Mohammad Ali Jinnah University (MAJU) Islamabad and at CASE for their useful technicalsupport.

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