sliding-mode control for nonlinear state-delayed systems using neural-network approximation

Sliding-mode control for nonlinear state-delayed systems using neural-network approximation

Y. Niu, J. Lam, X. Wang and D.W.C. Ho

Abstract: The sliding-mode control problem is studied for a class of state-delayed systems with mismatched parameter uncertainties, unknown nonlinearities and external disturbances. By integrating neural-network approximation and the Lyapunov theory into the sliding-mode technique, a neural-network-based sliding-mode control scheme is proposed. The major advantage of the present work over traditional sliding-mode designs is the relaxation of the requirement that the unknown nonlinearities are to be bounded. By means of linear matrix inequalities, a sufficient condition for ensuring the asymptotic stability of the sliding-mode dynamics restricted to the defined sliding surface is given. Further, by utilising a neural-network model to approximate the unknown nonlinearity, a sliding-mode control scheme is proposed to guarantee that the system state trajectory is attracted to the designed sliding surface.

1 Introduction

Over thc past decades, the analysis and synthesis of time- delay systems have been one of the most active research areas in system sciences. This is because time delays are commonly encountered in practical applications and may lead to poor system perfomiance or even instability. Although a number of significant rcsults on the analysis and synthesis of linear time-delay systems have been obtained, sec [I-61 and the references therein, there are relatively few reports on the control design problem involving nonlinear time-delay systems with unmodelled dynamics or unknown nonlinearities. Moreover, in the above papers thc admissible nonlinear uncertainties are usually assumed to satisfy certain conditions, such as the norm- boundedness condition in [7, 81. Very often, this leads to conservative sufficient conditions.

Sliding-mode control (SMC) has proven to be an effec- tive robust control strategy for incompletely modelled or uncertain systems through applications [9]. An SMC system can be considered as a variable-structure one with the control structure intentionally switched using some high- speed-switching (often discontinuous) feedback control. The basic idea of SMC is to drive the state trajectory of the system onto some specified smooth manifold (sliding surface) passing through the zero state, and maintain the trajectory on it for all subsequent time. On this sliding surface, the system acquires certain desired properties on

Q IEE. 2003 IEE Proceediri~s online no. 2003032 I UOI: 10.10491ip-cta:2nn3n321 Papcr first received 15th May and i n revised form 18th October 2002 Y. Niu and X. Wang arc with the School of Information Science and Engineering, East China University of Scicncr & Technology, Shanghai 200237, PR China 1. Lam is with the Department of Mechanical Engineering, University of Hung KO"& Pokfulam Road, Hong Kong D.W.C. Ho is with the Department of Mathematics, City University of Hung Kong, Tat Chee Avenue, Hong Kong

I€€ Prr,c.-Coniml Tltiiru~,, Appl.. h l . 150. No. 3. M q 2003

prescribed decay speed, good transient performance, and robustness in the face of uncertainties or pemrbations.

Since its early appearance in the 19SOs, SMC has received continual attention and some recent developments on SMC involving time-delay systems are given in [10-17]. In these SMC schemes, the construction of the fecdback control action utilises the past information of the state. Although a memoryless controller has the implementation advantage, improved performance can be obtained when such a delayed-feedback control action is employed [ l l ] . Yang et al. [ I O ] gave an output-feedback sliding-mode control law for uncertain systems with unmeasurable states. Hu et al. [I21 and Li ef al. [13] preseuted sliding-mode control approaches for uncertain time-delay systems with mismatched uncertainties, respectively. A linear-transformation approach was used in [I41 to produce a delay-free system. Hu ef al. [IS] used a transformation for uncertain input-delay systems and designed a sliding-mode controller for the transformed systems. Most of these SMC designs for uncertain time- delay systems involved norm-bounded nonlinearities which are simplistically treated as external disturbances.

For the control of complex nonlinear systems, on the other hand, adaptive-neural control schemes have shown great promises in delivering satisfactory control performance which is otherwise impossible using linear control techniques. I t is now a well established fact that any well behaved nonlinear function can be approximated with arbitrary accuracy by a multilayer neural network [18, 191. Based on this powerhl approximation capability, a wealth of literature focusing on neural-network-based control schemes aiming to solve highly nonlinear control problems have appeared see [20-231 and the references therein'. However, despite the many applications of neural controllers for nonlinear systems which have been reported, the use of the neural-network technique to solve control problems for nonlinear time-delay systems is relatively limited.

In this paper, a neural-network-based sliding- mode control approach is presented to solve the problem of robust control for state-delayed systems in the presence

233

of mismatched parametric uncertainties, unknown nonlinearities and external disturbance. It will be seen that the neural-network-based control can overcome certain difficulties encountered in traditional control approaches for nonlinear time-delay systems, such as the assumption that the nonlinearities are of the norm-bounded type. The sliding surface used is a linear function of the current system states. An LMI-based sufficient condition is given to guarantee the asymptotic stability of the sliding-mode dynamics. Furthermore, a sliding-mode controller based on a neural-network approximation of the nonlinearity is presented.

2 Problem formulation and preliminaries

In this paper, 1 1 . 1 1 denotes the Euclidean norm of a vector or the spectral norm of a matrix, l l . l l F denotes the Frobenius norm of a matrix. For a real symmetric matrix, M > 0 means that M is positive definite. I is used to represent an identity matrix of appropriate dimensions. Matrices, if their dimensions are not stated, will be assumed to have compa- tible dimensions.

Consider the uncertain nonlinear stated-delay system described by

X(I) = ( A + M(t) )x( t ) + ( A d + M d ( t ) ) ~ ( f - d)

+ B[u(t) + f (x ( t ) , ~ ( t - d) ) + r(t)l (1)

X ( t ) = r#J(f). f E [-d, 01 (2)

where x(t) E R" is the state, u(i) E R" is the control input, A , Ad , and B are known real constant matrices of appropriate dimensions, AA(t) and Mdt) are unknown time- varying-system parameter uncertainties, d is the delay time, 4(t) is a continuous vector-valued initial function, f {x ( t ) , x ( f - d ) } ER"' is an unknown nonlinear function which represents state-dependent uncertainty, ~ ( t ) is a time- varying bounded external disturbance satisfying Ilr(i)ll 5 p with known constant p? 0.

Remark I : System description (1)-(2) incorporates both nonlinear uncertainty f (x(t), w(t ~ d ) ) (which may result from unmodelled dynamics) and unknown time-varying disturbance ~ ( t ) entering the system through the input channels. Moreover, unlike earlier developments [7, 81, nonlinear function f (x( t ) , X( I - d ) ) is not required to be

Make the following assumptions for the system of ( l t ( 2 ) .

Assumption I : Matrix B is of full rank.

Assumption 2: The parameter uncertainties are of the form

norm-bounded in this paper. a

[M(t) M d ( f ) l HF(r)[E Ed1 (3)

where H , E and Ed are known constant matrices, F(r) is an unknown time-varying matrix and satisfying

FT(t)F(t) 5 I (4)

It was mentioned in Section 1 that multilayer neural networks have good capability in function approximation. Hence, in this work, the following three-layer feedfonvard neural network will be used for functional approximation:

g(2) = W'o( V'Z) (5)

234

where Z=(sr- I)' is the input vector (the - 1 term denotes the input bias) with

= [ ] X(t - d )

Y E R'2"+l)"/ and W E RIXm are the weights of input- to-hidden layer and the hidden-to-output layer, respectively, V includes the threshold

4 4 = I.,(xJ & 2 ) " ' ~ l ( x l ) 1 7

where

i = I , . . . , I 1

1 + exp(-rjxi) .,(.,) =

and ri > 0, i = I , . . . , I , are constants. According to Funahashi [18], it can be shown that, for

an arbitrary constant p > 0, there exists an integer I (the number of hidden neurons) and real constant optimal weight matrices WI and V* such that the unknown nonlinear function f(.) can be approximated as follows:

f ( z ) = W*Tu(V*TZ) + E ( ? ) Vx(f) E R C R" (6)

where ~ ( d ) is the approximation error vector satisfying Ila(Z)II 5 11, and R is a compact set. Notice that the exis- tence of the time-delay term ~ ( t - d) in the nonlinear functionf(x(t), x(t ~ d ) ) does not affect the approximation ability of neural network (see [24, 251 and references therein).

In general, the ideal weights w* and v* are unknown and need to be estimated in the controller design. Let W and V be the estimates of W* and v*,_respectively, and t>e cqrresponding estimation errors be W = W - w* and V = V - V*. For the neural-estimation modcl (6), we have the following property.

Lemma I : [ 2 3 ] For the nonlin_ear function f ( z ) , the function approximation error f ( z ) = f ( z ) - f ( z ) with j ( z ) = k'u( P'Z), can be written as

j ( Z ) = k'(U - &Vi) + W r U F ' i + 5 (7)

(8)

where the residual term 5 can bc expressed as

5 = k'U"'i + W*'O"''i) - .(i)

with U= u(krZ), U'= ~'("'7,) is Lhe corresponding Jaco- bian matrix, and o( P'Z) --f 0 as V'i + 0.

Lemma 2: [26] The residual term 5 can be bounded by a linear-in-parameter function, i.e.

11511 < .'Y ( 9 )

function vectory is equal to [ I , I IZI~, I I Z ~ ~ ~ I W I I ~ , I I Z I I I I V I I ~ ] ?

where the unknown parameter vector 01 E R4 is composed of optimal weight matrices and constants,-and the known

Let B denote an estimate of a and the estimation error be .i =& - a. In the following Sections, we *will develop the adaptive laws for the parameters W V and &. We conclude this Section by introducing a matrix inequality that will be used in the proof of our result.

Lemma 3: [27] Given matrices Q, H and E of appropriate dimensions and Q = QT then

Q + HF(t)E + ETFT(t)HT < 0 (10) for all F(t) satisfying F(t)'F(t) 5 I , if and only if there exists some constant S < 0 such that

(1 1 ) Q + SHH' + 6 - l ~ ' ~ < o IEE Pm.~Con~ol T h e q Appl.. Yo/. 150, A'". 3. ,&f<z? 2003

3 Sliding-surface design and stability analysis

Consider the sliding surface as a linear function of the current system states given by

S(t) = rB'P-'x(/) (12)

where PER""" is a positive-definite matrix to be designed, and r E R"'""' is some nonsingular matrix. For simplicity, r is chosen as the identity matrix in this work. As in the methods in [ 9 ] , define a transformation matrix and the associated vector v as

where v , ( t ) E W-" and y 2 ( / ) E R", B is any basis of the null space of B: that is, B is an orthpgonal complement of B. Note that, for a given matrix B, B is nonunique but any choice satisfyin the Londition is acceptable. It is easily shown that T = [PB B], and v2(t) = (B7P-'B)-ls(t). Using the transformation T, we obtain the ( n - m) reduccd-order sliding-mode dynamics restricted to the sliding surface s(t) = 0 as follows:

9

Vl(t) = (BTPB)-'b'(A + M(t ) )PBv, ( t )

+ (BrPB)-'B'{Ad + M,,(f))PLh,(t - d ) (14)

The following theorem gives a sufficient condition for ensuring the stability ofthe sliding-mode dynamics in (14).

Theorem I : The (n - M) reduced-order sliding mode dynamics (14) is asymptotically stable, if there exist matriccs P > 0, Q > 0 and scalar 6 > 0 such that the following linear matrix inequality holds:

[ EPB

Z(AP + PA' + GHH')B + Q BTPA:B

B'PET < 0 (15) 1 BTA,PB B'PE'

-Q E,PB -61

Pro($ Choose a Lyapunov functional candidate as

V { v l ( f ) , t ) = v:( t )bTPBv,(f) + uT(r)Qvl(T)dT (16) 6, Taking the time derivative along the trajectory of the sliding-mode dynamics in (14), one obtains

P(VI(/), t) = 2v:(t)B'{A + M(f )JPB" , ( / )

+ 2vT(r)Br(.4, + ~ ~ ~ ~ ( t ) ) ~ i f v , ( r - d)

+ v:(t)ev,(t) - vT(t ~ d)ev,( t - 4

Obviously, one has &vl(t), t) < 0 if 0 < 0 for v l ( r ) # 0. In the following, it is established that 0 < 0 if linear matrix inequality (15) holds.

First, it is shown from ( 3 ) that 0 < 0 is equivalent to

x F'(t)(H'B 0 ) < 0 (18)

where

1 B'(AP + PAT)B + Q B'AdPB

B'PA:B -Q 0, =

By applying lemma 3 , matrix inequality ( I 8) holds for F(t) satisfying F(/)'F(t) 51 if there cxists a constant 6 > 0 such that

x [EPB EdPBl < 0 (19)

That is,

B'(AP+ PA' + SHH' + G-'PE'EP)B + Q Z(PA; + S-'PE:EP)B

B'(A,P + S-'PE~E,P)B < ] (20) -Q + s-'B~PE;E,PB

[ which, by Schur's complement, is equivalent to (15). Hence, i'(v,(r), t ) < O which implies that the system states will be asymptotically stable on the sliding surface

Remark 2: Consider the special case that parameter uncertainties M(f) and MAC) are matched, i.e. M(t) and M A C ) are represented as

s(/) = 0. 0

M(t) = BD(f ) M,,(f) = BD,(t) (21)

where D(t) and Ddt) are unknown matrix functions. Then, system ( I ) is rewritten as

i ( t ) = Ax(t) + A,x(t - d ) + B[lr(t) + f a ( x ( / ) , . ~ ( t - d ) ) + T(f)I (22)

with fa(x(t), x(t - d)) =f(x(t), x(r - d ) ) + D l ( t M t ) + D2(f)X(t - d ) . Similarly to the arguments in Section 2, one obtains the neural estimate of fo(.) as follows:

7 f o ( ~ o ) W;'~(Vz' io) + E & ) V Z ~ = [x'(t) ~ ' ( t ~ d) ]

(23)

where the arguments, W8, V;, and co(.) are analogous to those in (6). Furthermore, LMI (15) in theorem I is reduced to

For the case of nonlinear time-varying time-delay systems, where the time-varying delay d(t) is a positive Fontinuous differentiable function satisfying 0 < d(t) 5 d, d(t) 5 h < 1 with h > 0 is a scalar, one can obtain the following delay- dependent stability condition.

Proposition I : The (n-m) reduced-order sliding-mode dynamics (14) is asymptotically stable, if there exist

235

matrices P> 0, Ql > 0, Q2 > 0 and scalar y > 0 such that the following lincar matrix inequallty:

B'(AP + PA7 + yHH')B + Q, + jQz B'PAiB

EPB

1 B'A,PB BrPET

EdPB ' -?I

-(I -h )Ql B'PE: (25)

[ Proof: Choosing the Lyapunov functional as follows

thus, onc obtains, for v , ( t ) # 0,

with

= - [ E'[M + M ( t ) ) P +PiA + M(t) l ' lB + Q, + dQ2

B7{Ad + M,(t)]PB

- - B'P(A, + Md(t)}'b

1 -(I - h)Qi

As in the proof procedure in theorem I , it can be shown that, if there exist matrices P > 0, Q , > 0 , QZ > 0 and scalar y > 0 such that the linear matrix inequality (25) is satisfied one obtains S < 0. Thus, it follows from (28) that i'o(ul(/), d(t) , t) c 0. 0

4 SMC design with neural-network approximation

This Section describes the design of a neural-network- based SMC law and weight update rules such that the system state trajectories are attracted to the abovc defined sliding surface whenever x(t) E R.

Note from sliding-mode control theory that, under ideal sliding mode, one has

s( t ) = 0 S(t) = 0 (29)

When M(t) = 0, M d ( t ) = 0 and the nonlinear functionf(-) is replaced by its neural estimatc, the virtual equivalent control is then given by

U,(() = -(B'P-'B)-IB7P-' (Ax( / ) + A d X ( t - 4 1

k'o( P'Z) (30)

236

Since thcrc exist parameter uncertainties and unknown nonlinearities in practical system ( I ) , to stabilise the time-delay system (I), au&ment ueq(l) with a robustifying control term u,-(t) which results in the following SMC law:

u(t) = U& + U,.@) (3 1)

with

U,(t) = -(B'P-'B)-' jlB'P-'Hll,(ljEx(t)/I

+ IIE,x(t - d)jl)sgn(s) - (B 'P- 'B) - 's (~)

- (B7P-'B)-' llB7P-'BjlF(&'y + p)sgn(s) (32)

Substituting control law (31) into the equation for s(t) and utilizing lemma I, one obtains

S( t ) = B'P-I[{A + M(t) }x( t ) + {A,/ + M,(t)]x(t - d ) ]

+ B7P-'B[u,(t) + 1dr ( t ) +f(x(t). ~ ( t - d)) + r(t)l = B'P-'{AA(t)x(t) + M~l(/)x(t - d ) } + B'P-IB

x [fw X ( t - d) ) + 7(t) - krcr(Prz)1 + B'P-'Bu,(~)

- ~ ~ p - l ~ [ k r ( e - e , P r t ) + ki.Typj,

+ 5 - r(t)l (33)

Theorem 2: Consider system (1)-(2) under assumptions I and 2 with switching surface (12). If the network weights are adaptively updated by the following rules

= B'P-'[AA(t)x(t) + Md(t )x( t - d) ] + B'P-'Bu,(t)

k = r l [ (u - u ' ~ 7 i ) s r ( r ) B 7 P ~ ' B ) (34)

( 3 5 )

& = ~ ~ I I B ' P ~ ' B l l ~ / l ~ ( ~ ) l / ~ (36)

where rl > 0, r, > 0, and r3 > 0, then the neural-network based SMC law (31)-(32) guarantees that every solution trajectory of the closed-loop system is attracted to the sliding surface ~ ( t ) = 0. Proof( Consider the following Lyapunov function candidate:

^ . V = r , { i ( ~ ' ( t ) B ' P ~ ~ B ) k ' u ]

Utilising the weight-updating rules (34) and (39 , after simplification one obtains

L = sr(t)BrP-'(HF(t)Ex(t) + HF(t)E,+(r - d ) ]

- sT(t)(BrP-'B)(C - r( t ) ] + sr( t ) (BrP-'B)u,( / ) - T -'L

+ a r, 01 5 I I ~ r P - ' H I I F { I I ~ ~ ( / ) I I + llE(py(f - d)ll)lls(t)ll

+ Il~ 'P~'~I I , I ls(~) l l ( l ICl l + P )

+~"( I ) (B 'P- 'B)u , . (~ ) + 2r; ld (39) Utilising lemma 2, one obtains

L 5 I I ~ r ~ ~ ' ~ l I ~ ~ / I ~ . ~ ( ~ ) l l + I IEA- d)llllls(~)ll + IIB'P-'BII,I/s(t)/l(ary + P )

+ S ~ ( ~ ) ( B ' P - ' B ) U , . ( ~ ) +err;'&

+ l l B ~ P - ~ B l l ~ l l ~ ~ f ~ l l ~ a ~ y + / I )

+ s ~ ( ~ ) ( B ~ P - ~ B ) u , ( ~ ) + (&'r;I& - IIB~P-~BII, I I~(~)II~'~) (40)

= IIBrP-'HIIFIIIEx(t)l l + IIEd-r(t - d)l l l l ls(t) l l

substituting the updating law (36) and the 'robustifying' control term (32) into (40), one obtains

L 5 -IIs(t)I12 + I/B'P-'HII,(IIE~(/)/I

+ IIEAt -d)lIllls(~)ll + I I ~ ' P ~ ' B I I ~ I I ~ ( ~ ) I / ( ~ ~ Y + P I

- s '~~ ) l IB 'P~ '~ / I .~ I I~~ (~ ) I I + /IEdx(f - d)llIsgn(s)

- J(~ ) I IB~P-~BI I , (&~Y + p)sgn(s)

5 -lls(t)ll2 (41)

Note that when s(t) # 0, we have L < 0. This implies that all signals s(t) , Vand Fi are bound, and .s(f) asymptotically converges to zero. 0 Remurk 3: Once the state trajectory hits the sliding surface, it remains on the surface for all subsequent time. Since theorem 1 shows that the system is asymptotically stable on the sliding surface, it is concluded that the closed-loop system resulting from (Ip(2) and the adaptive

n Remark 4: When the uncertainties M(f) and M,,(t) are represented as in (21), and then the system ( I ) is rewritten as (22), i.e.

i ( t ) = Ax(t) +Adx(f - d) + B[u(t)

rules (34)-(36) is asymptotically stable.

+fo(x(t), X(f - d ) ) + ml (42)

Utilising the neural estimate (23) off&), lemmas 1 and 2, we arrive at

A , . " ( - p

fn(zo) = *;(U - UP,TZ,) + w,o v,zo + 50 (43)

and 11&ll <miyo , where the unknown parameter vector an E R4 is composed of optimal weights and constants, and the known function vector yo is equal to [I; Iltoll, l\iollIIWallh lliolIl~VoI~r]? Let a,, denote the estimate of an, and the estimation error be & =ao - mo. As for the procedure in theorem 2, it is easily established that, if the parameter update rules are given as follows,

kn = rIo{(e - CrrP,'i,)sr(t)BrP-lB)

ko = r3o IIB'P-'BII, lb(t)llYn

(44)

(46)

= T20[&,{.~r(t)BTP-'B) kiu'] (45)

/E€ Pmc.-Conirol Theorv Appl., Fbl. /SO No. 3, Mqy Zll/l3

for some Tlo > 0, rzn > 0, and r30 > 0, then the following control law (47)-(49),

U ( t ) = Ii& + I i r ( / ) (47)

(48)

Iieq(t) = -(BTP-'B)-'BrP-' [Ax( / ) + A,x(t - d ) ] k r o pr-

- o ( nzu)

x (GY, + /i)ssn(s)l (49)

I,(t) = - ( B r F ' B ) - ' [ s ( t ) + I/B'P-'BII,

can guarantee that every solution trajectory of the closed- loop system is attracted to the sliding surface s ( / )=O. As explained in Remark 4, the closed-loop system is

Remark 5: To implement the neural network for real-time applications, the adaptation rules may he integrated on-chip along with the realised network, where they are performed in parallel for optimal efficiency. In particular, the continuous-time updating algorithm of network weights described by a set of differential equations is suitable for hardware implementation using analogue VLSl techniques. Note out that some learning rules may be performed outside the chip for greater flexibility. For detailed hardware solution, refer to [28-301 and references therein. In this paper, the network weights may he randomly initia-

n Remark 6: The design of the proposed neural-network control scheme is independent of the form and the complexities of system nonlinearities. Hence. it can be applied to a large class of nonlinear state-delayed systems in the form (1)-(2). The results obtained in this work are semiglobal in the sense that they are valid as long as x ( f ) E R, where the set R can he arbitrarily large. In the case when the neural network approximation (6) holds on R", that is, R = W, then the stability results become

Remark 7: It is well known that, due to the sign function sgn(s) involving in the control law (32), chattering phenomenon may occur in the control signals, which is undesirable as i t may incite high-frequency unmodelled dynamics and even lead to the instability of controlled system. To alleviate chattering, there are a number of approaches proposed, for instance, the use of a boundary layer in [3l], the prefiltering of the control signal in [32], and the introduction of a differentiable approximation given by s/(llsll +a) for some small scalar 6 > 0 [33] (we have chosen this approach for illustration in the numerical

asymptotically stable. n

lised, or simply set to zero.

global. n

simulation in Section 5 ) . A

5 Numerical simulation

Consider the following nonlinear uncertain state-delayed system ,

where

2 1 0.1 0.1

0 0 -I

231

0.3 0.4 0.2 sin(t) 0

H = 1 . 3 0.5 0.41, F(t) = [ ; si:

0.2 0.3 0.2 cos@) ] 0.2 0 0.2 0.1 0.2 0

'I

1

I I-0.01 +x: ( t )

11 +4(t - d))l-rj(f - d)l

=f(x( t ) .x (r - d))

1

.(f) = [ 0.1 + 'd - .i;]

i and time-delay d= 1, initial function +(f)=(2 2 2); t ~ [ - - l , 01, x(O)=[O, -1, I ]? Note that the open-loop nonlinear state-delayed system is unstable. By solving LMI (15), we obtain P > 0 and sliding-surface variable s(r) in (12) as follows:

0.6467 0.4224 0.3283

P = 0.4224 0.7609 -0.2712

0.3283 -0.2712 0.9490

1.7081 0.1731 -0.5415

3.0590 -1.8928 -0.5457 ]x(r) [ S ( f ) =

The nonlinear termf(x(t), x(t - d)) does not belong to the usual norm-hounded type, and 7(f) is nonsmooth and exhibits a significant disturbance at f =2 with Ilr(f)lI,5 10.

The three-layer neural network (5) was selected with 20 hidden-layer neurons, i.e., I = 20, and r, = 2 , i = 1, . . . , 20. Let the design parameters he r, = diag(2, 2,. . . , 2) t

, r ,=diag( l , I , . . . , and r 3 = d i a g R20"20

( I , 1, 1, 1). The initial values k(O), V(O), and & ( O ) are all chosen to be 0. To prevent the control signals from chattering, replace sgn{s(r)} with .s(t)/{ Ils(t)ll + O . O l } in (32)

o.2p-. 0 : - _ - _ _ - - * . , , . .. . .. , : ....-

.. ..... ... .... -0.2; .......

4 . 4 : x3 ''.. ,..'.\

-1.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

t is

Fig. 1

238

Trajectories ofstate variable x(Q

0.4 r

-1.01 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

V S

Fig. 2 Trajectories of sliding variable S(Q

in the simulation as in practice. The sliding-mode control law, u(t) = ueq(t) + u,(t), has

0.4306 1.0002 0.2151

2.2341 -0.0002 0.1 173 0.0215 0.1000 0.0430 - [ 0.1 117 -0.0000 0.0235

x(r - I ) 1 u,,(t) = -

- F P U ( P i )

]SW s(r) - [ 0.7464 -0.3464

Ils(f)ll + 0.01 -0.3464 0.5586

-4.3575 7.0269 1 lls(t)ll + 0.01

x

- [ 9.3893 -4.3575 (&%+ lO)s(t)

Figs. 1-3 present the simulation results of the closed-loop system. I t is shown that the sliding-mode control law (3 1 t ( 32 ) using neural-network approximation of the nonlinear term can effectively eliminate the effects of parameter uncertainties and nonlinearities. The Figures demonstrate that the state trajectories are attracted towards the designed sliding surface and the closed-loop system asymptotically stable. Moreover, it can be seen from Figs. 1 and 2 that there are notable variations in the curves of state variables and sliding-mode variables around r = 1 and f =2. The former is caused by the effect

-25 I 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

tis

Fig. 3 Control signal U ( / )

IEE Pm.-Cunlrul Theon. Appl.. yid I S 0 I\". 3, M q 20113

of the time delay d = 1 in the statc-dclayed differential equation, while the latter is due to the nonlinear function ~ ( t ) which has a significant disturbance at f = 2 . Never- theless, these undesirable effects are effectively attenuated and the system-state trajectories arc quickly driven to the sliding surface. It can also be observed that the states of closed-loop system eventually die down to zero.

6 Conclusions

We have considered the sliding-mode control problem of a class of state-delayed systems in the presence of mismatched parametric uncertainties and unknown nonlinearities. By utilising a multilayer neural network to approximate the unknown nonlinear function, we have proposed a neural-network-based sliding-mode control approach that can guarantee that the system-state trajectory is attracted to the defined sliding surface. Further an LMI-based sufficient condition for the asymptotic stability of the sliding-mode dynamics is derived by means of a Lyapunov approach. The key features of the present work are that the commonly used norm-bounding condition on the unknown nonlinearities is weakened, and that there is no need to have a priuri knowledge of the hounds of ideal neural-network weights and approximation errors.

7 Acknowledgment

The research was partly supported by RGC grant HKU 7 I03/0l P.

8 References

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sliding-mode control for nonlinear state-delayed systems using neural-network approximation

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