Sliding mode control of systems with distributed delay

IFAC

Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001

~

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SLIDING MODE CONTROL OF SYSTEMS WITH DISTRIBUTED DELAY F. Gouaisbaut, M. Dambrine, J.P. Richard

LAIL cnrs upresa 8021, Ecole Centrale de Litle, BP 48, 59651 Villeneuve d'Ascq CEDEX - FRANCE. e-mail: !rederic.gouaisbaut@ec-hHe.!r FAX: (+33) 3 20 33 54 18

Abstract: The present article is devoted to the construction of sliding mode controllers for a class of distributed, possibly time-varying, delay systems. By the use of Lyapunov-Krasovskii functionnals or Lyapunov-Razumikhin functions, the reduced system is proven to be asymptotically stable for all delays less than an upperbound, which is optimized via LMIs. Finally, an example illustrates our approach. Copyright © 2001 IFAC Keywords: Distributed Delay system, Robust Control, Sliding Mode Control, LMI

1. INTRODUCTION

knowledge, the only result concerned with SMC and distributed delays (with constant value) was provided in (Zheng et al., 1995) : the controller was based on the predictor approach proposed in (Fiagbedzi and Pearson, 1987; Pearson and Pandiscio Jr, 1997). The resulting control law needs to solve a transcendental matrix equation. The present paper considers uncertain systems with distributed possibly time-varying but unknown state-delay and additive perturbations which may be nonvanishing. It aims at designing the sliding surface in such a way that it maximizes the calculable set of admissible delays, where "admissible" means here those that don't destabilize the closed-loop, relay-delay system. The outline of the paper is as follows : after some notations, Section 3 is devoted to the transformation of the original system into a suitable form for sliding mode design called regular form. Then in Section 4, a sliding mode controller is developed by means of an LMI approach for distributed and constant time delays. This approach is extended to the case of a time-varying delay in Section 5. In both sections, the reduced system is proven to be asymptotically stable for any value of the delay less than a bound obtained by solving a convex optimization problem.

Sliding mode control (SMC), based on the use of switching control laws, belongs to the variable structure approaches. It is known to efficient in many problems of robust stabilization, as non-holonomic systems for instance or, more generally, systems which do not fit the Brocket's conditions (see (Perruquetti and Barbot, 2001 (to appear); Misawa and Utkin, 2001)). However, the presence of delays in combination with switching devices makes the design more complex (see for instance (Fridman et al., 1996)). Some general frameworks were proposed (infinite dimensional systems (Orlov, 2000), differential inclusions for delay systems (Kolmanovskii and Myshkis, 1999)), but the concrete control results were mainly concerning systems with discrete delays (systems with point-wise delays). In (Choi, 1999; Shyu and Yan, 1993; Koshkouei and Zinober, 1996; Shyu and Yan, 1993) as here, results are considering delayed state variables, but the delays are pointwise and the control laws are independent of the delay value (and, consequently, may be more conservative than delaydependent results (Gouaisbaut et al., September 1999; Gouaisbaut et al., 2001)). To our best

225

Lastly, in Section 6, an illustrative example shows the effectiveness of the method. In our opinion, this work has two original features:

where (5)

(1) It considers distributed delays with weak hypotheses (except linearity). (2) It can be implemented via classical LMI algorithms.

Lemma 1. (Choi, 1997) The original system (1) is equivalent to i(t)

A = (~ll ~12)

In this paper, the following system, with possibly time-varying delay T = T(t) ~ 0, will be considered:

{

x(t)

+ Ad.!

= 1J(t),

x(w )dw + Bu(t)

.1 21 .1 22

+ f, (t, x,),

for t E [-T,O] , T = T(t)

_(iF .1X13(iJT X 13)-1 13T .1B(SB)-1 ) , S.1X13(13 T X13)-1 SAB(SB)-1 Ad (1A dlld21 ~dI2) A d22 =

_ (iJ T AdX13(13T Xi3)-1 13 T A dB(SB)-I) SAdX13(iJ T X13)-1 SA dB(SB)-1 ,

(1)

The following notations are used: x(t) E lR n; A and Ad are constant n x n matrices; B is a n x m matrix, u E lR m is the input vector; h is a term representing the neglected dynamics and external disturbances,T is a uniformly bounded delay (T :::; Tsup ) .. Xt is the function associated with x and defined on [-T,O] by xt(B) = x(t+B). 1J is the initial function defined on [-Tsup , 0] and can be eventually discontinuous.

B=

In this section, we consider that the delay constant but unknown.

We will use the following assumptions:

AI) The pair of matrix (A, B) is controllable. A2) The perturbation term h satisfies the classical matching conditions, i.e.

ATp+PA = -I,

(2)

and ml

(3)

u(t) where

The aim is to design a sliding mode controller for (1). We first transform the original system into a special form, appropriate for sliding mode control (classically called regular form (Lukyanov and Utkin, 1981)). Let us choose the following sliding surface s(x) = Sx = B T X-IX = 0,

IISBIIlJ1(xt(B)) m

)

= ml +OE[-Trnax,O] sup [0 x(t + w)dw 11 ( + 11 SAd 10

°

, the surface s(x) = is globally attractive in finite time, and the solution x = of the system (6) is asymptotically stable for T E [0, Tmax ]' where T max is the solution of the following optimization problem: -1 _ . ( -1) (9) Tmax-mznT ,

where S E jR(n-m)xn, and define a nonsingular t~ansformation detM:j; 0,

Ps

= _(SB)-I( -As(x) + SAx(t) + m IIPsll)' (8)

3. REGULAR FORM

= Mx(t),

°

> a real number.

Assume that conditions (AI) - (A2) - (A3) hold. Then, with the control law

where lJ1 is a known functional of Xt. A3) B is full-rank: rank(B) = m.

z(t)

is

T

Theorem 2. Let A E jRnxn be an Hurwitz matrix, P is the solution of the Lyapunov equation

and is bounded as follows

IIfll < lJ1(xd,

(7)

4. SLIDING MODE CONTROL SYNTHESIS: CASE OF A CONSTANT DELAY

Ilell=1 norm of the n x n matrix M. Finally M will denote an orthogonal complement of M.

= Bf(x(t), t),

(S~ ).

In the following, we first present a sliding mode control for a system with one constant delay, then the result is extended to the case of systems with a time-varying delay.

We will denote Ilell the Euclidean norm of the n-vector e, and IIMII = sup IIMell the spectral

h(x(t), t)

(6)

where

2. NOTATIONS AND ASSUMPTIONS

x( t) = Ax(t)

= Az(t)+Ad l~T z(w)dw+B(u(t)+f),

under the constraints (

(4)

226

(1,

°

et

E lR) :

[

1 T- 0+ YA dX ]_ XAI -Y

0] < 0,

The derivative of V along the solutions of the reduced system (16) gives us :

[BBT 0 BB T

(7

X>O Y> 0 Y -aBB T > 0

V(Zlt)

+2z[(t) PA dll

(10)

Remark 3. In practice, the stable matrix A allows to choose the dynamics of s when the solution of the system is far from the manifold sex) = o.

By using the following inequality:

2Z1(t)PAdlll~Tzl(w)dw ~ TZl(t)PAdllR-1AIllPZl(t)

The proof is decomposed into two subproblems: firstly, prove the attractivity of the surface in finite time; secondly, prove the asymptotic stability of the reduced system (on the surface).

+

.

V(Zlt) .

With N(T)

Proof. Let us consider the function

= sT(x(t))Ps(x(t)),

I

Ps

+ SB! - mIIPsll)'

'T

SA ll

(13)

or (14)

The vector Z appearing in the regular form (6) is partitioned into Z = [Zl zzV, where Zl E lR n-m, Zz E lR m.

+ A.dlll

t-T

Zl(W).

= sex) =

Theorem 4. Let A E jRnxn be an Hurwitz matrix, P is the solution of the Lyapunov equation

It

ATp+PA=-I,

(16)

and

Proof. Let us choose the following LyapunovKrasoYskii's functional t V = z[(t)Pz 1(t) + Zl (cf RZ 1(c) dcdw , t-T Jw

r

(17)

where P, RE R(n-m)x(n-m) are positive definite matrices.

<0,

In this section, we consider that the delay T = T(t) is varying, but remains uniformly bounded:

t

Zl(t) = A ll z 1(t)

' ' 1 'T + A.llS + TSRS + TPAdllRA dll

5. SLIDING MODE CONTROL SYNTHESIS: CASE OF A TIME-VARYING DELAY.

4.2 Asymptotic stability of the reduced system

Once in sliding mode, the equations sex)

'T' ' - 1 'T = AllP+PA ll +TR+TPAdllR AdllP.

with Q = SRS. By choosing Q = (ETy E), with Y a positive definite matrix, and by using schur's complement, the inequality is known to be equivalent to (10). •

(15)

This last inequality is known to prove the finite time convergence of the system (6) towards the surface (Utkin, 1991). •

olead to the reduced system:

T

Zl (t)Nz1(t),

ET AXE + ETXATE + TQ +TET Ad X EQ-l ETX AIE < 0,

Vet) < -s(xfsex) - 2ml VAmin(P)VV(t)

< -2ml VAmin(Ph/V(t).

~

If there exists P and R two positive definite matrices and a real Tmax such that N(Tmax ) is a negative definite matrix, then the reduced system is asymptotically stable for all delays T E [0, T max ]. To optimize the upper bound T max , let post and pre-multiply N by S = p- 1 and take P = eB T X B)-I. The inequality N < 0 becomes

(11)

Its derivative along the trajectories of (6) with (8) is Vet) = -sT(x)s(x) + 2s T (x)P(SAx(t) (12) t

l~T zi(w)Rz1(w)dw,

where R is a positive definite matrix, we obtain an estimate of V :

4.1 Attractivity of the surface

+ SAd t-T x(w)dw

l~T Zl (w)dw

-l~T zi(w)Rz1(w)dw + Tzi(t)Rz1(t)dw.

where 0 = AX + X AT and X, Y E lR n

F(t)

= zi (t) (Ail P + PAll) Zl (t)

ml

> 0 a real number.

Assume that conditions (AI) - (A2) - (A3) hold. Then, with the control law (8) the surface sex) = is globally attractive in finite time, and the system is asymptotically stable for T E [0, T max], where T max is the solution of the following optimization problem: -1 . ( -1) , (18) Tmax=mznT

°

227

under the constraints ( a E lR) :

e [ XAI

AdX] [BB -X -a 0

T

0 ] BB T

< 0,

(19)

X >0, where

e = T- 1 (AX + XA T ) + X,

X E lR nxn.

As previously, the proof is decomposed into two subproblems: firstly, prove the attractivity of the surface in finite time; secondly, prove the asymptotic stability of the reduced system (on the surface). As the proof of the attractivity of the surface remains the same, we do not repeat it. The proof of the second part is the following:

'~

(20) is a positive definite

= z[(t)

T

A

+2Z 1 (t)P.4.dll

.

I

.

.-

I

__

, !

To

'1"'iI!

T-1(A.X+XAT)+qXAdX]

-x

XAJ

[

[BBT

0

-a

0 ] BBT < O. (23)

For q = 1,the inequality (23) is exactly the constraint (19). By continuity, if we assume that there exists X, T, a, Q, 13 such that (19) is satisfied, then there exists q1 > 1 such that (23) is satisfied too. •

(21)

with q > 1 is satisfied for 0 E [-T(t),O]. The derivative of V along the solutions of the reduced system (16) give us :

~r(Zlt)

~.

By using the Schur's complement, we derive that (22) is equivalent to

= z[(t)Pzdt),

+ B)) < qV(x(t»,

;

Fig. 1. Response of (24) with control (8) and T = 1.99.

where P E lR(n-m)x(n-m) matrix. following the approach proposed by Razumikhin, let us assume that the following inequality: V(x(t

if

i '. "

Proof. Once in sliding mode, the equations sex) = sex) = 0 lead to the reduced system (16). Let us choose the following function of LyapunovRazumikhin : "(t)

.'

(.AL P + P.4. ll ) Zl (t)

i

t.

6. EXAMPLE

t.-T

zl(w)dw.

Consider system (1) with

By using the following inequality: 2z 1(t) PA dll

i~T zl(w)dw ~ TZ1PAdllP-1.4.IllPZ1 +

l~T z[(w)Pz1(w)dw,

A

=

we derive that: B, = .

T

V(Zl/.) ~ Zl (t)Nz 1(t), 1\T() _ (A.i1 P A+ PAll1 +ATTqp) • h were 1Y T +TPAdll P - AdllP If there exists a positive definite matrix P and a real T max such that N(Tmax ) is negative definite, then the reduced system is asymptotically stable for T E [0, Tmax ]' To optimize the upper bound Tmax , let us post and pre-multiply N by S = p- 1 and let us assume that P = (iJTxiJ)-l. The inequality N < 0 becomes AT

SA ll

and z(t) =

A

AT

(~) 1

U]

,p= (

~

=

(-1 0 0.2 0) , -0.1 0.25 1 -2-1 (24) )

sin(zl(t-T)) foe t E [-T, 01-

(25)

Varying delay: Now, if we don't restrict to constant delays, then according to Theorem 4, the system is asymptotically stabilized for T(t) < 0.83 which is of course more constraining. The simulation leads to Fig.2-Fig.3.

or, equivalently iJT AXiJ + iJT "'C4. TiJ + TqS -T - -T T+TB AdXBPB XAdB < O.

2 0.25 0 0.8 1 ) ,Ad 1.75 o 2 3

Constant delay: By using semi-definite programming and theorem 2, we find that the system (24) with control (8) is asymptotically stable for all constant delays T < 1.99. The simulation provided in Fig.1.was obtained with a first-order integration scheme of step 0.001.

+ AllS + TqS + TPAdllSPSA dll < 0, A

(

(22)

228

"~'l " "I

the relay control systems with time delay. In: 35th IEEE CDC'96. Kobe, Japan. pp. 46014606. Gouaisbaut, F., M. Dambrine and J.P. Richard (2001). Sliding mode control of perturbed systems with time varying delays. In: Proc. 1th IFA C Symposium on System Structure and Control, SSSC'01. Prague. Gouaisbaut, F., W. Perruquetti, Y. Orlov and J.P. Richard (September 1999). A sliding mode controller for linear time delay systems. In: Proc. ECC'99 European Control Conference. Karlsruhe. Kolmanovskii, V.B. and A. Myshkis (1999). Introduction to the theory and applications of functional differential equations. Kluwer Acad.. Dordrecht. Koshkouei, A.J. and A.S.I. Zinober (1996). Sliding mode time-delay systems.. In: Proc. International Workshop on VSS. Toskyo. pp. 97-10l. Lukyanov, A.G. and V.I. Utkin (1981). Methods of reducing equations of dynamics systems to regular form. Automat. Remote Control 42, 413-420. Misawa, E., A. and V. Utkin (2001). Special issue on sliding mode control. Journal of Dynamic Systems, Measurement and Control. Orlov, Y.V. (2000). Discontinuous unit feedback control of uncertain infinite-dimensional systems. IEEE Tmns. Aut. Control 45(5), 834843. Pearson, A.E. and A. A. Pandiscio Jr (1997). Control of time lag systems via reducing transformations. In: Imacs '97 Worid Congress. Berlin. pp. 9-14. Perruquetti, W. and J.P. Barbot (2001 (to appear)). Sliding Mode Control in Engineering. Marcel Dekker. Shyu, K.K. and J.J. Yan (1993). Robust stability of uncertain time-delay systems and its stabilization by variable structure control. Int. J. of Control 57, 237-246. Utkin, V.I. (1991). Sliding Modes in Control Optimization. CCES Springer-Verlag. Zheng, F., M. Cheng and W-B. Gao (1995). Variable structure control of time-delay systems with a simulation study on stabilizing combustion in liquid propellant rockect motors. Automatica 31(7), 1031-1037.

::\tJH~H1I

':,t·::I,~·:1, :

.

,

:

,-

'!

,:/.{. . . . . . . . . . . •

X

U

i:

.,

,--,--,,--,....-,

)~

:! ':"-1' '~

I

,

I

'!

I'

Fig. 2. Response of (24) with control (8) and a time varying delay Tmax(t) = 0.83.

Fig. 3. Implemented delay (saw) 7. CONCLUSION In this paper, we have developped a sliding mode control appoach to distributed time delay systems. Two cases were considered: the case of a constant distributed delay and the case of a time varying delay. Both approaches lead to LMI optimization problems, which can be solved very efficiently.

8. REFERENCES Choi, H.H. (1997). A new method for variable structure control system design : A linear matrix inequality approach. Automatica 33, 2089-2092. Choi, H.H. (1999). An LMI approach to sliding mode control design for a class of uncertain time delay systems. In: Proc. ECC'99. Karlsruhe, Germany. Fiagbedzi, Y.A. and A.E. Pearson (1987). A multistage reduction technique for feedback stabilizing distributed time-lag systems. A utomatica 23(3), 311-326. Fridman, L.M., E. Fridman and E.I. Shustin (1996). Steady modes and sliding modes in

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