Sliding mode control: a new contribution usingan integral action in the boundary layer

Sofiane Mahieddine-Mahmoud, Rafiou Ramanou, Laid Kefsi & Larbi Chrifi-Alaoui.Laboratoire des Technologies Innovantes (L.T.I),

IUT de l'Aisne, Departement G.E.I.I,13 Av F. Mitterrand. 02880 Cuffies, France,

Sofiane.Mahieddine@ u-picardiefrlarbi. alaoui@ u-picardie.fr

Abstract -In this paper, the nonlinear sliding mode control on the sliding phase [17]. The control structure is changed(SMC) combined with non linear Integral corrector is according to a reselected switching logic. The switchingproposed. Initially, NSMC is applied only when the states of the changes are based on the instantaneous values of systemsystem are far away from the setpoint; if the state system is states along the trajectory [6].close to the reference an integral component was added. The .nstability and the robustness of the closed-loop system are High frequency control switchig leads to the so-calledproven analytically using the Lyapunov synthesis approach. chattering effect which is exhibited by high frequencyThe proposed method attenuates the effect of both vibration of the controlled plant and can be dangerous inuncertainties and external disturbances, moreover eliminates applications, a number of methods were proposed tothe chattering phenomenon introduced by classical sliding- overcome these difficulties; for example [16] proposed anmode control. The simulation of this method to Induction i omotor gives encouraging results which are presented andcommented in this paper. to the expression of u the term sign (S) bay sat(S). Our

strategy consists in adding to the sliding mode structure anIndex Terms: Sliding Mode Control, Field Orientation integral corrector when the state trajectory approach to theControl, Robust control Induction Motors. reference.

The aim of this paper is to develop a nonlinear sliding modeI. INTRODUCTION based on a non linear integral corrector. The controller

In the last years, the sliding mode (SMC) technique has design is presented in section two. The section 3 deals withbeen widely studied and developed for the control and state the modelling of the induction motor. In section 4, weestimation problems since the studies of Utkin [17]. This discuss the controller design and its implementation on thecontrol technique gives a good steady state performances induction machine by simulation.and good dynamic behavior in the presence of systemparameters variation and disturbances [4]. Several methods II. SLIDING MODE CONTROLof applying the sliding mode control to induction motordrives have been presented [1, 9, and 10]. All of these A. General conceptmethods have common features; here the analysis and designof the sliding mode controller are based on the mathematical We consider a nonlinear system given bymodel of the induction motor.

Most of the early works in the area are in the Russian kc f1 (x) + B(x)u (I)literature (see Utkin [17] and the references within). X2 f2 (X) +w(t) (II) (1)Subsequently, various SMC algorithms (Slotine [16]) have y = h(x2)been successfully used for trajectory tracking problems.Successful results have been reported in terms of eliminating where, xe s m . n are the state vector,disturbances, addressing nonlinearities, and achieving I 2acceptable control in the presence of modeling errors. A u e U a- 9jm is the control vector u =[u1, U2.. Urn]Tpopular SMC approach for trajectory tracking problems is bounded as u <U0 for iI to m, with U0 > 0, the vectorsbased on Lypunov's method. This approach yieldsmultivariable designs that produce sliding mode on the field f1(x), f1 fn-Jm andf2(x), f2 :n -m areintersection of several switching surfaces. continuous nd smooth, rank(B) = m y e Y a P k ithThe control laws are designed so that the system'

trajectory could always reaches the sliding surface. This output vector, we Wa q represent the disturbances.known as the reaching phase. Once on the sliding surface, The aim is to give the system state into the set S(x) = 0the control structure is changed discontinuously to maintainthe system on the sliding surface. At this stage, the system is S(x) =[S1 '2 Sm 1T S(x)ERe (2)

1-4244-0726-5/06/$20.OO '2006 IEEE 681

as .. asS, is the commutation surface v =a X2ref + a X2refThis involves computing the control input as following:

then S(x2, x32) reached in the finite time to and confined to

+ f S, > manifold Ds such that:u =l- if S < O (3 s = t(X2 X2 QI I|S('2, x2 )|| < 40}

where S7 is the boundary layer thickness

ui designed to guarantee convergence of the motionprojection on the state space.

We can check the stability of the sliding surface by usingA sliding surface should be designed such that: the Lyapunov theorem. Let us choose the following positive

function V(S) > 0 such ass=s(2,X )0 (4) 1

V(S) - STS (7)2

with x2 = - X2refThe derivative of this Lyapunov function is defined as

Notations. negative, thus guaranteeing motion of the state trajectory to7 Cjjm is a vector the manifold.

17 = [71 ¢ 72 . 7hlm] V(S) = s S < 0 (8)K E Stnmxm is a diagonal matrix and defined such that:

using equation (4) we have:

K(ql) = diag{q1,q2,qrl2¢ 173 ,.*.*.*,q17} S = F(x) +Au (9)

Denoting where

1/| = 71/172 ....,i7mY|] F(x) = Alff(x) +A2 + A3 w(t) +v

Proposition 1 The function F(x) and the matrixA(x), are not exactlyknown, but estimated by F(x) and A(x) where

The dynamics f1 (x) and f2 (x) are not exactly known, AA A ~~~~~~~~F(x)= F(x) + AF(x) and A(x) = A(x) + lA4(x).

but estimated byj,(x) andj2(x) ,the following control Fx)=Fx (xanAx =Ax+ Ax.

but estimatedbosystem1 is vnd by(:thefolowing And replacing u(x, t) of (9) by relation (5) we havelaw applied to system (1) is given by:

u = - Al (F(x) + K(q) sign(S)) (5) S = IF - Au - K(q) sign(S) (10)F(x)= 1(x) +A2x2+v (6) FAalAaIA ai

The estimated error on AA(x)K(q) = diag{ql, q2Y q. I}and qi > niN = /n,, n2, .....,nrn T7is a parameters vector related to the L/\a.11 ..... al1a.,

disturbance and to the modeling uncertainties. AF1The expression of the matrix A ,A/, A2 and A3 are given and F(x) assumed to be bounded by some

by LAFm]as af2^A = . B(x) has full rank. known matrix a and vector r suchthat:aiX2 ax

A-2 axs n2 AX aS + , A3= asnd A\ajj<.aj,i,I=1,...,m and iZf|<fl,

N=Fr+au|= /n1,.......,aT (11)

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The gain K is chosen positive to satisfy attractivity andstability conditions, K(q), qi > n1 .., then, the trajectory of F iAl (F+K(q) sat(S))if X2r-2>1

state reached the surface S(i2, x2 ) = 0 in the finite time to u = A (F+K(q)st(S))+u if<2( . )and confined to manifold D -.

Remark I where

We know that sliding-mode techniques generate Ur °KJ(X2re(T)-X2(r)dr) (18)undesirable chattering, and then a smooth function replacesthe discontinuous part of the control action [15]. Thus, the Ki is the integral constant matrix, Ki E 9imx(n-m).sign(S) becomes Then, forgoing consideration can be summarised by the

sign(Si )if ISi > (0i following algorithmsat(S)= Sii=1,2, ...,m (12)

f if S1 i) Tracking error by sliding mode if||S|| > 0.-o= l 02..... 0mT ii) Elimination of chattering phenomenon,

h Yic2.kin ] n = y| iS the boundary layer interpolation of sat(S) function in the boundarythickness. layer if||S| . 0In this case

iii) Correction of the error by integrator correctoru =-A (F+ K(q) sat(S)) (13) f X2ref-X2 <..

We note that the algorithm is characterised by threeFor ||S(x, t)JJ > 0 the system trajectories are guaranteed to stages.converge to the boundary layer by the law (8), but when Stepi: the system trajectory reaches the manifold in finite||S(x, t)JJ < 0 the structure of control is changed and the time to, this phase is known as the reaching phase;

and then the control structure is changed

sign(S,) becomes , we have so: discontinuously until ||S|| < 0 . At this stage the system(Oi is in the sliding phase.

S = (F-AAF) -A A K(q)K(y7)' S Step2: In this stage we have a smooth control S(x, t) locallythen stable in the manifold when the disturbances assumed

to be constant or varying slowly.K (y)K(q7A'SA + S (14) Step3: Eliminate steady-state of the error by the new

corrector.

AA-1 exists and positive, so not (chattering problem). III. INDUCTION MOTOR SYSTEM

= K(o)K(q) AA' (F-A A F) = £ ]T (15) Under the assumptions of linearity of the magnetic circuit,and neglecting the iron losses in a tree-phase squirrel cage

If £ assumed to be constant (or negligible dynamics). induction motor, the 4th order non-linear model (d-q frame)of the induction motor is

lim S(t) = (16)to ds -77 Id.s + Os Iqs + Vfdr + P cOr Vfqr + a utds

Therefore we have an error created by - (disturbance and dt Trparameter uncertainty) dIqs ds pWsIds 1lIqs-PCOrVJdr+ l//qr +azUqsB. Integrator corrector in the boundary layer dt Tr (19)

dV/fdr Lm 119

In this section, our objective consists in reconstituting a dt Ids fr qcontrol law to eliminate steady-state error created by d, L 1disturbance; we added an integrator when the trajectories of qr= 1 Iq-9s YVdr - T YWqrstate approach to its reference. dt r, qs

Where UdS,Uqsarth control inputs, I sI sare the

sttruret adltrtVrare the roo fluq.In this case the control law is defined by sttrcret,ad' d Yqr r h oo lx

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IL2 R RL2 LmLIsLr 11= 7LL2 TLLLm X12 +,L,5 a + a L2r srm Ff//(X) = -77 xii + Tm - + T X21 + PXI2X221 Tr X21 Tr

QC= COS O=s-) rI r =)p2r L XiL's f1(x)= f12(x)=- xx1 -77x12 -ppx22 x12a is the leakage coefficient, Rr andR are respectively Tr X21

the rotor and stator resistances, Lr and L, are respectively -PX IX22

the rotor and stator inductances, Lm is the mutual

inductance, T7 and T, are the rotor and stator constant f21(x) m - __

times, p is the number of pairs of poles, Ws, O)r and W)sl are f2(x) Tr r

respectively the electrical rotor, stator and slip speed. Lf22(X) = u x21x12 - X22

The mechanical modelling part of the system is given by: The Global control drive the trajectory state to the surface

dtr ___qS Wdr-[d.s Wqr )-f Q r--defined by S(2,x2)=0dt /P(IqsVdr -ds k'qr jQ r (20)

PL TrF 01where,u= JL; A Is the friction coefficient, TL is a S(X2x2)=[o ]KL2]+K kj0L j (24)load torque and J is the inertia.

Using the system equations (19), with the d-q axisattached to the module of the rotor flux space vector, we Where x2 = X2 - X2r the tracking error in the variablehave: x2, K is a constant matrix, kl k2 > 0

Ydr = Y'r and 'Aqr 0 The expression ofthe matrix A A, A2 andA3 aregivenConsequently, the induction motor model established in by

the rotor flux field co-ordinate is thenb

0A as af2 B(x)=1 m which has full rank

dId,,, _ Z +i

+ P )x2 Ix1 L 0 a21jdt Tdssqs T Ydr ds when x21 .0.

-qs s Idsl-77 Iqs-PpvrV/dr + a Uqs as af2 Lm 01,dt (21) a~2ax L 0 X21 j

d/Vfdr L11 0

69=p mIds (22) Tr °1 T0Z

dt Tre Tr ar i 2 a

xl = fi (x) + B(x)u (23) Remark 2~~+ ~3

d qsV/drj r il

with___k2]and

(OS=P r+L Iq (2

The model of the IM is presented in the state-space form: V #l2ref - 01x2ek1 =(x)+B(x)u(2) Rmr2

X2 =f2 (X) +w(t)Assume that the only measured variable of the state of the

FX11FJd l FX21 FMrl IM are the stator current and the rotor speed. The rotor fluxwhere x1 =L0 2j=LI j' X2 LX221L0r j Wr and position O, can be estimated by means of stator

current and speed measurements (direct method) and are

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Vfdr L ds (25) °200Trs +1II

LmIqs )t~0Os=f(PQr+ dt0CTr /dr 0 0.5 1 1.5 tim ) 35 4 4.5

IV. SIMULATION RESULTS

To verify the performance of the proposed method, the 0o 15 time(-) 3 3:5 4 4.5sliding-mode-controller was simulated in EMATLAB/SIMULINK. Z

We present different tests to justify our approach: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5The first one represents a nominal test with T1 = Tnominal ,thesecond tests is a robustness test with the variation of the Fig 1 speed QDr , rotor flux (r and load torqueparameter indicated in the table, we present the simulation profile(nominal system).with 50°0 variation of Rr and Rs, and 20% variation of Ls, Lr 20and L.1

The parameters of the induction motor are defined in the ¢ 0following table.

-10

TABLE -20INDUCTION MOTOR CHARACTERISTICS 0 time(s)

2

P Rated Power 1,5 kw 300us ~~voltage 220 V 200

Rated current 6. JA . [ -

r Nominal speed 1430 rpm >-ioo-200

Rs stator resistance 1.47±50% 0 -3000 05 1 1.5 2 2.5 3 3.5 4 4.5

time(s)Rr rotor resistance O. 79 ±50%QmsL | stator inductance 0.105±200%H Fig 2 Stator current and voltage (nominal system)Lr rotor inductance 0.094 ±20% H Robustness TestLm Mutual inductance 0.094 ±20% H

number of pairs of |2 1c o00poles P0-

Inertia 7103 kg.m2s - -1000|

the friction coefficient 2,910-3Nm/rad/s 0 0 5 1 .5 2 2 5 .5

Nominal Test 0.40.2 0.

We present the result of the first test, figure (1) is: the 0 0.5 1 1.5 2time(s) 2.5 3 3.5 4 4.5

system response with nominal load torque; the desired 10

speed, varied from 0 to nominal speed, the speed attraction z 01-5proves the interest of our algorithm. 0405 2 3 4 45

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5time(s)

Fig 3. Speed D,r rotor flux (or and load torque profile( Rr, and

R]s +50% and Ls Lr and Lm and -20%.).The dynamic performance is affected by assuming that the

plant parameters have deviated from their nominal values.For -20%0 increase in each Ls Lr and Lm and ±50°/O increase inRr, andRsThe most powerful merit of the sliding mode controlcombined with integral corrector in the boundary layer isthat the system is robust with the respect to parameter

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uncertainty (identification problem, modelling problem [10] M.O. Mahmoudi, et al, "Cascade sliding mode control of a fieldfig(3,4)) and disturbances. The simulation results show a oriented induction machine drive", Eur. Phys. J. AP 7, pp 217 225,lgood trackingina spitne ofthepresenes of thesults siabe 1999.good tracking in spite of the presence of the variable [11] R.J. Mantz, P.F. Puleston and H. De Battista, "Output overshoots indisturbances. systems with integral action operating in sliding mode", Automatica,

1999, pp. 1141-1147.[12] R. Marino, P. Tomei, "Nonlinear control design", Printice-Hall, Inc.

20 1995.[13] W. Perruquetti and J.P. Barbot, "Sliding mode control in engineering",

10 Marcel Dekker, 2002.[14] H. Sira-Ramirez and S5K. Spurgeon "Robust Sliding Mode Control

Using Measured Outputs" Journal of Mathematical Systems, Vol. 6,

[15] J.E. Slotine and W Li "Applied nonlinear contror' Printice-Hall-20 International, Inc. 1991.

0 0.5 1 1.5 tl(2 2 3 4 [16] J.J.E Slotine et al "Sliding controller design for non-linear systems"International Journal ofControl, 40, 1984, pp. 421-434.

300 w w w w w w [17] V. Utkin, "Sliding mode control in electromechanical system". Taylor200 & Francis 1999.100 [18] V. Utkin, "Variable Structure Systems with Sliding Mode," IEEE

0 Transactions on Automatic Control, vol. 22, 1977, pp. 2 12-222.-100

-200

-3000 0.5 1 1.5 2 2.5 3 3.5 4 4.5

time(s)Fig 4: Stator current and voltage( Rr, and Rs +50% and Ls

Lr and Lm and -20%.)

V. CONCLUSIONS

In this paper we have presented a new control strategy basedon a combination between integral corrector and nonlinearsliding mode controller. We have noticed the presence ofsteady-state errors, in the case of uncertainties systems, whenthe sat(S) function is interpolated to the classical slidingmode control. Those inconvenient are overcame by theproposed control strategy. This method was validated bysimulation on a nonlinear system (Induction motor). Theobtained results prove the viability of this control method,and presented good performances in tern of robustness to thesystem disturbances and uncertainties.

VI. REFERENCES[1] A. Benchaib et al, "Sliding mode Input-Output Linearization and

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[3] P.Caon et J.-P.Hautier, "Modelisation et commande de la machineasynchrone", EDITION TECHNIP, 1995.

[4] M.O. Efe et al "Variable structure control of a class of uncertainsystems" Automatica 40 (2004) 59 - 64.

[5] K. Erbatur, "A study, on robustness property of sliding modecontrollers: A novel design and experimental investigations" IEEETransaction on Industrial Electronics, vol, 46, no, 5, 1999.

[6] A.F. Fillipov. "Differential equations with discontinuous right handsides" Mathematicheskii Sbornik, 51, 1960, pp. 99-128.

[7] W. Gao, "Variable Structure Control of Nonlinear Systems, A newApproach", IEEE Transaction on Industrial electronics, vol, 40, no. 1,1993.

[8] R. Ghosh-Roy, N. Olgac, "Robust Nonlinear Control via Movingsliding surface n-th order case ", IEEE Conference on Decision &Control, USA 1997.

[9] 5. M. Mahieddine et al "Sliding mode control based on fieldorientation for an induction motor" IEEE Industrial ElectronicsSociety, JECON, North Carolina November 6 - 10, 2005 pp. 18 1-186.

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