Decentralized model-reference adaptive control for a class of uncertain large-scale time-varying delayed systems with series nonlinearities

Chaos, Solitons and Fractals 33 (2007) 1558–1568 www.elsevier.com/locate/chaos

Decentralized model-reference adaptive control for a class of uncertain large-scale time-varying delayed systems with series nonlinearities Meei-Ling Hung a

a,*

, Jun-Juh Yan

b

Department of Electrical Engineering, Far-East College, No. 49 Jung-Haw Road, Hsin-Shih Town, Tainan 744, Taiwan, ROC b Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC Accepted 6 March 2006

Abstract In this paper, the problem of model-reference adaptive control for large-scale time-varying delayed systems with series nonlinearities is investigated. By applying the theory of variable structure control, we propose an adaptive controller, which is both memoryless and decentralized, to derive the error subsystem between the local model state and plant state to zero. The proposed variable structure control is able to ensure the stability of a sliding manifold of the composite system even though the control input is nonlinear. The main difficulty for handling the effects of interconnected terms is well solved by a new proposed adaptation mechanism. Finally, a numerical example is illustrated to demonstrate the validity of the derived controller.  2006 Elsevier Ltd. All rights reserved.

1. Introduction Time delays, due to the information transmission between subsystems, naturally exist in large-scale systems and hence the control problem becomes more important than that in systems without time delays. So far, there are two well-known approaches to design robust adaptive control for uncertain systems. They are Lyapunov or Riccati-equation approach [1,4,7,9,13,16] and the variable structure control [2,5,6,10–12,15,18,19]. The Riccati-equation approach uses continuous control laws to achieve the control goal. However, the transient performance is not considered in their control design. Another approach is based on variable structure control. It is a modification of the conventional feedback control method and possesses many attractive advantages such as easy realization, fast response, good transient performance and insensitive to variation in plant parameters or external disturbances. So far, many works [2,5,6,10,12,18,19] have successfully applied the concept of variable structure control to deal with the robust stabilization problem for uncertain large-scale systems. However, the controllers in [2,10,12] were not decentralized since they all included coupling terms on the sliding surface, which implied that the control is coupled and is not local control. Thus they are adequate only when the global information can be used. To overcome this drawback, Xu et al. [18] utilized the *

Corresponding author. E-mail addresses: [email protected] (M.-L. Hung), [email protected] (J.-J. Yan).

0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.004

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1559

weighted time-varying sum to deal with the interconnected terms. However, Xu et al. only coped with no delayed largescale systems and in his approach, the M-matrix might not guarantee the global hitting condition as one of the switching modes is attained [5]. Moreover, in the above works except [5,6] the control schemes are all under the assumption of linear input. However, because of the physical limitation in control systems, there are many nonlinearities in control input and their effects cannot be neglected in analysis of control design and realization. Hsu [5,6] proposed robust variable structure controllers for uncertain systems with nonlinear inputs. However, they were only able to deal with no delay systems. For the above reasons, it is necessary to propose a new method to deal with the decentralized control problem for uncertain large-scale time-varying delayed systems with input nonlinearities. To design the memoryless decentralized model-reference adaptive control (DMRAC) for large-scale time-varying delayed systems with series nonlinearities, there exist two main difficulties. One is how to handle the effects of interconnected delayed terms without using the information of global states or time-varying delays. The other is how to overcome the input nonlinearities such that the occurrence of sliding manifold can be guaranteed. In this paper, we propose the solutions of above two problems. A new scheme of memoryless DMRAC with an adaptation mechanism is proposed. Appling this scheme, due to the adaptation mechanism, the interconnected effects can be handled without using the global states or time-varying delay information, and the existence of the composite sliding mode can be also guaranteed even under the effect of input nonlinearity. In particular, it shows that the uncertain nonlinear large-scale system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. Finally, a numerical example is illustrated to demonstrate the validity of the proposed memoryless DMRAC. For simplicity, in the following section, WT denotes the transpose of W, and kWk represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. In 2 Rn·n represents the identity matrix. k(W) denote the eigenvalues of W and kmax(W) represents the eigenvalue of W with the maximum real part.

2. Problem formulation The decentralized variable structure model-reference adaptive control system consists of a local reference model, a controlled plant subjected to time-varying delays and input nonlinearity, a decentralized controller and an adaptation mechanism, as shown in Fig. 1. In it, the decentralized variable structure control is established only by using available local information and an adaptation parameter. Thus, the controller is easy to implement. Now, consider the local plant Xi shown in Fig. 1 expressed as Xi : x_ i ðtÞ ¼ Ai xi ðtÞ þ

N X

Aij ðtÞxj ðt  sij ðtÞÞ þ Bi /i ðui ðtÞÞ þ Bi fi ðxi ; ui ; pi ; tÞ;

i ¼ 1; 2; . . . ; N;

ð1Þ

j6¼i j¼1

where xi ðtÞ 2 Rni , ui ðtÞ 2 Rmi , pi ðtÞ 2 Rqi and fi ðxi ; ui ; pi ; tÞ 2 Rmi , respectively, represent the state variable, control input, uncertain parameter, and external perturbation of subsystem Xi with

local reference model Ri

yi

x j (t − τ ij (t ))

vi fi vi ei

decentralized variable structure control

u i (t )

ui

coupling states

local plant

Σ

ei

xi

Ωi

βˆ i Si xi adaptation mechanism βˆ ( t )

S i (t )

i

Fig. 1. The decentralized variable structure model-reference adaptive control system.

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ni ¼ n;

i¼1

N X

mi ¼ m;

ð2Þ

i¼1

sij(t) > 0 denote the time-varying delays in interconnections and are bounded by smax, i.e., sij(t) 6 smax, "i, j. Ai and Bi are with appropriate dimensions. Aij(t) are the interconnection terms including the uncertainties. /i : Rmi ! Rmi is a continuous function satisfying /i(0) = 0. The state vector of composite system X is defined as  T ð3Þ X ðtÞ ¼ xT1 xT2 . . . xTN . For the lumped uncertainty f(xi, ui, pi; t), there exist three positive constants k1,i, k2,i and k3,i such that the norm of the lumped uncertainties satisfies: kfi ðxi ; ui ; pi ; tÞk 6 k 1;i kxi ðtÞk þ k 2;i kui ðtÞk þ k 3;i ;

8ðxi ; ui ; pi ; tÞ 2 Rni  Rmi  Rqi  R.

ð4Þ

In (1), the nonlinear input /i(ui(t)) applied to the subsystem Xi satisfies uTi ðtÞ/i ðui ðtÞÞ P hi uTi ðtÞui ðtÞ;

ð5Þ

where hi is a positive non-zero constant. The parameter hi is often called as the gain reduction tolerance. An example of a scalar nonlinear function is shown in Fig. 2. Let the local reference model Ri for the ith subsystem in Fig. 1 be given as Ri : y_ i ðtÞ ¼ Ami y i ðtÞ þ Bmi vi ðtÞ;

ð6Þ

where y i ðtÞ 2 Rni is the state variable of the local model. vi ðtÞ 2 Rmi is the local bounded reference input. Ami, Bmi are the known constant matrices with appropriate dimensions. Ami is stable. We define the local error vector: ei ðtÞ ¼ y i ðtÞ  xi ðtÞ.

ð7Þ

Our objective is to design a memoryless DMRAC for each subsystem so that the state of every local subsystem can follow that of a local reference model, i.e. lim ei ðtÞ ¼ 0;

t!1

i ¼ 1; 2; . . . ; N .

Fig. 2. A scale nonlinear function /(u(t)) with the gain reduction tolerance h.

ð8Þ

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1561

The error dynamic equation of the subsystem is e_ i ðtÞ ¼ y_ i ðtÞ  x_ i ðtÞ ¼ Ami ei ðtÞ þ ðAmi  Ai Þxi ðtÞ þ Bmi vi ðtÞ 

N X

Aij ðtÞxj ðt  sij ðtÞÞ  Bi fi ðxi ; ui ; pi ; tÞ  Bi /i ðui ðtÞÞ.

ð9Þ

j6¼i j¼1

Now we shall use the variable structure control for the controlled systems. Therefore, we first define the associated sliding submanifold as S i ðtÞ ¼ ri ei ðtÞ 

Z

t

ðri Ami þ K i Þei ðkÞ dk;

i ¼ 1; 2; . . . ; N;

ð10Þ

0

where S i ðtÞ 2 Rmi , ri 2 Rmi ni and K i 2 Rmi ni , matrix ri ¼ Bgi results in ri Bi ¼ I mi . Bgi is the generalized inverse of Bi. The matrix Ki is chosen to satisfy kmax ðAmi þ Bi K i Þ < 0.

ð11Þ

The composite sliding manifold can be described as  T SðtÞ ¼ S T1 S T2 . . . S TN .

ð12Þ

To ensure the achievement of model reference’s objective, we make the following assumptions. Assumption 1. (Ami, Bi), i = 1, 2, . . . , N are controllable pairs. Assumption 2. There exist matrix functions Hij, Hmi, Gmi of appropriate dimensions such that [6,18,20–23]: ðaÞ Aij ¼ Bi H ij ;

kH ij k 6 cij ; i 6¼ j; i; j ¼ 1; 2; . . . ; N.

ðbÞ ðAmi  Ai Þ ¼ Bi H mi ; ðcÞ Bmi ¼ Bi Gmi ;

kH mi k 6 ai ; i ¼ 1; 2; . . . ; N .

kGmi k 6 gi ; i ¼ 1; 2; . . . ; N .

ð13aÞ ð13bÞ ð13cÞ

where cij, ai, gi are unknown positive constants. Remark 1. These conditions in (13) are the so-called matching conditions [3]. The assumptions of matching conditions restrict the structure of the systems that we considered. However, many mechanical systems do belong to this class of systems [23] and many works [6,18,20–23] had also been proposed based on the matching conditions. In this paper, although it is still under the limitation of matching condition, a novel DMRAC will be proposed. Our purpose is to design a memoryless DMRAC such that the error state trajectories of controlled composite system with input nonlinearity can reach the composite sliding manifold and slide to the equilibrium point of the error state space along the sliding surfaces. In consequence, the major problem is how to determine a memoryless adaptive decentralized control such that the existence of the sliding mode can be guaranteed even with the input nonlinearity. The solution is proposed in the following section.

3. Memoryless DMRAC design To design a DMRAC to achieve limt!1 SðtÞ ¼ 0, we first introduce the following Barbalat lemma. Lemma 1 (Barbalat lemma [14]). If w : R ! R is a uniformly continuous function for t P 0 and if the limit of the integral Z t jwðkÞj dk; ð14Þ lim t!1

0

exists and is finite, then lim wðtÞ ¼ 0.

t!1

ð15Þ

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To ensure the occurrence of the sliding motion, a memoryless DMRAC is proposed as S i ðtÞ ; kS i ðtÞk 8 > > ni < ^ wi ðxi ; vi ; ei ; bi ; tÞ ¼ ½ai þ k 1;i kxi ðtÞk þ kK i kkei ðtÞk hi  k 2;i > > : ^i ; tÞ ui ðtÞ ¼ wi ðxi ; vi ; ei ; b

0

1

N BX

þ gi kvi ðtÞk þ k 3;i þ B @

j6¼i j¼1

ð16Þ

9 > > =

C ^ cij C Abi ðtÞ>; > ;

ð17Þ

where hi > k2,i and n > 1 are positive constants specified by the designer. The adaptation mechanism is given as 0 1 N BX C ^_ i ðtÞ ¼ B b cij C @ AkS i ðtÞk;

^ i ð0Þ ¼ b ^i0 ; b

ð18Þ

j6¼i j¼1

^i ðtÞ. It is obvious b ^ i ðtÞ > 0 for all t > 0. Obviously; the adaptive ^i0 is the positive and bounded initial value of b where b law can be also rewritten in the integral form as 0 1 Z N BX C t ^i ðtÞ ¼ b ^i0 þ B C b c kS i ðkÞk dk; i ¼ 1; 2; . . . ; N . ð19Þ ij A @ j6¼i j¼1

0

Theorem 1. Consider the error dynamic equation of the subsystem (9) with input nonlinearity, if the control law of memoryless DMRAC is given by (16)–(18), then the system trajectory converges to the sliding manifold S(t) = 0. Proof. Consider the following Lyapunov function candidate: V ðtÞ ¼

N X  1 T S i ðtÞS i ðtÞ þ q2i ðtÞ ; 2 i¼1

ð20Þ

where qi(t) 2 R denotes the adaptation error which will be defined later. Taking the derivative of V(t) with respect to time t and substituting (9) into the differentiation of Si(t), one has V_ ðtÞ ¼

N X i¼1

¼

ðS Ti ðtÞS_ i ðtÞ þ qi ðtÞq_ i ðtÞÞ

8 > N > < X i¼1

9 3 > > N = X 7 6 7 þ qi ðtÞq_ i ðtÞ . S Ti ðtÞ6 r ðA  A Þx ðtÞ þ r B v ðtÞ  ðr A ðtÞx ðt  s ðtÞÞÞ  r B / ðu ðtÞÞ  r B f ðx ; u ; p ; tÞ  K e ðtÞ i mi i i i mi i i ij j ij i i i i i i i i i i i i 5 4 > > > > j6¼i : ; 2

j¼1

ð21Þ

By using (4), (13) and the inequality kABk 6 kAkkBk and ri Bi ¼ I mi , the above equation can be rewritten as 8 > N > < X

9 3 > > N = X 7 6 7 _ þ q V_ ðtÞ ¼ S Ti ðtÞ6 H x ðtÞ þ G v ðtÞ  ðH ðtÞx ðt  s ðtÞÞÞ  / ðu ðtÞÞ  f ðx ; u ;p ;tÞ  K e ðtÞ ðtÞ q ðtÞ mi i mi i ij j ij i i i i i i i i i i 5 4 > > > i¼1 > j6¼i : ; j¼1 8 9 3 2 > > > N > N < = X X 7 6 7  S T ðtÞ/i ðui ðtÞÞ þ qi ðtÞq_ i ðtÞ . kS i ðtÞk6 ða þ k Þkx ðtÞk þ g kv ðtÞk þ kK kke ðtÞk þ k þ ðc kx ðt  s ðtÞÞkÞ þ k ku ðtÞk 6 i 1;i i i i i 3;i j ij 2;i i i ij i 5 4 > > > i¼1 > j6¼i : ; 2

j¼1

ð22Þ

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From (16) and (17), it is obvious that ^i ; tÞ. k 2;i kui ðtÞk ¼ k 2;i wi ðxi ; vi ; ei ; b

ð23Þ

Furthermore, from (5) and (16), we have T ^ i ; tÞ S i ðtÞ / ðui ðtÞÞ P hi uT ðtÞui ðtÞ ¼ hi w2 ðxi ; vi ; ei ; b ^ i ; tÞ. uTi ðtÞ/i ðui ðtÞÞ ¼ wi ðxi ; vi ; ei ; b i i kS i ðtÞk i

ð24Þ

Therefore, we have the following inequality: ^i ; tÞkS i ðtÞk. S Ti ðtÞ/i ðui ðtÞÞ 6 hi wi ðxi ; vi ; ei ; b

ð25Þ

Now assume there exist some constants large enough (i.e. bi ! 1) to satisfy 1 > bi P kxj ðkÞk;

j 6¼ i; i; j ¼ 1; 2; . . . ; N ;

for all k 2 ½ smax

1 .

ð26Þ

^i ðtÞ  b denote the adaptation error. Because bi is assumed to be constant, then the following expression Let qi ðtÞ ¼ b i keeps valid ^_ i ðtÞ. q_ i ðtÞ ¼ b

ð27Þ

Inserting (17), (23), (25) and (27) into the right hand side of inequality (22), this yields 8 9 2 0 1 3 > > > N > N < = X 6 BX C 7 _ ^ ^ B C 7 b  ðh kS i ðtÞk6 ða þ k Þkx ðtÞk þ g kv ðtÞk þ kK kke ðtÞk þ k þ c  k Þw ðx ;v ;e ; b ;tÞkS ðtÞk þ q ðtÞ b ðtÞ V_ ðtÞ 6 1;i i i i i 3;i i 2;i i i i i ij A i 5 i i i i i 4 i @ > > > i¼1 > j6¼i : ; j¼1 9 8 0 0 1 1 qi ðtÞ > > > zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ = < N > N N X BX BX C C _ ^i ðtÞÞ ðni  1ÞkS i ðtÞkB ^i ðtÞ . ^i ðtÞ þ q ðtÞb C C 6 ð1  ni Þ½kS i ðtÞkððai þ k 1;i Þkxi ðtÞk þ gi kvi ðtÞk þ kK i kkei ðtÞk þ k 3;i Þ þ B c ðtÞk ðb  b c kS b i ij A i ij A i @ @ > > > i¼1 > j6¼i j6¼i ; : j¼1

j¼1

ð28Þ By Eq. (18) and ni > 1, one can obtain 8 9 0 1 > > > N > N < = X X B C ^ B C _V ðtÞ 6  ðni  1ÞkS i ðtÞk@ cij Abi ðtÞ ¼ wðtÞ 6 0, > > > i¼1 > j6¼i : ;

ð29Þ

j¼1

where wðtÞ ¼

PN

i¼1

 ðni  1Þ

V ð0Þ P V ðtÞ þ

Z

 PN



 ^ b c ðtÞkS ðtÞk . Integrating the above equation from zero to t, it yields j6¼i ij i i

j¼1 t

Z

t

wðkÞ dk P 0

wðkÞ dk.

ð30Þ

0

As t goes infinite, the above integral is always less than or equal to V(0). However, V(0) is positive and finite, thus according to Barbalat lemma (see Lemma 1), we obtain 8 9 0 1 > > > > N N < = X BX C ^ B C ðni  1Þ@ cij Abi ðtÞkS i ðtÞk ¼ 0. lim wðtÞ ¼ lim ð31Þ t!1 t!1 > > > i¼1 > j6¼i : ; j¼1

^i ðtÞ > 0 for all t > 0 and ni > 1 is chosen. Thus Eq. (31) implies Si(t) ! 0 as t ! 1. Hence the proof is Furthermore cij, b achieved completely. h Remark 2. It is noteworthy that the unknown but existing constants bi satisfying 1 > bi P kxj(t)k, j 5 i, i, j = 1, 2, . . . , N, for all 1 > t P smax, is only introduced in the proof process and it is not necessary to know the precise value of bi for the memoryless DMRAC design. Thus we can suppose that bi is a constant large enough, i.e. bi ! 1, such that the assumption of (26) will always hold. Remark 3. From the proof of Theorem 1 and inequality (30), it is observed that V(t) in (20) is bounded since V(0) is finite. This also implies that kSi(t)k and qi(t) are bounded for all t > 0. And the system trajectories converge to Si(t) = 0

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^i ðtÞ is bounded from (19) and kei(t)k = kyi(t)  xi(t)k will also decay to zero only by selecting in a finite time. Thus b appropriate matrices Ki to satisfy Eq. (11). Furthermore, since the state yi(t) of interested reference model is always bounded, kxi(t)k is bounded. After assuring the occurrence of the sliding manifold, the problem left is how to ensure the stability of the error dynamic equation of (9). We will discuss in the following remark. Remark 4. It is well known that when the system operates in the sliding manifold S(t) = 0, it satisfies the equations [8,17] S_ i ðtÞ ¼ 0 and

S i ðtÞ ¼ 0; i ¼ 1; 2; . . . ; N.

ð32Þ

Therefore, by differentiating (10) with respect to time and substituting from (9), the local equivalent control /eq,i(ui(t)) in the sliding manifold satisfies S_ i ðtÞ ¼ K i ei ðtÞ þ ri ðAmi  Ai Þxi ðtÞ þ ri Bmi vi ðtÞ 

N X

ri Aij ðtÞxj ðt  sij ðtÞÞ  fi ðxi ; ui ; pi ; tÞ  /eq;i ðui ðtÞÞ ¼ 0;

ð33Þ

j6¼i j¼1

where ri Bi ¼ Bgi Bi ¼ I mi has been introduced. From (33), the equivalent control /eq,i(ui(t)) in the sliding mode is given by /eq;i ðui ðtÞÞ ¼ K i ei ðtÞ þ ri ðAmi  Ai Þxi ðtÞ þ ri Bmi vi ðtÞ 

N X

ri Aij ðtÞxj ðt  sij ðtÞÞ  fi ðxi ; ui ; pi ; tÞ.

ð34Þ

j6¼i j¼1

It can be easily shown that the equivalent error dynamic equation for each subsystem in the sliding mode is in the form 2 3 N X 6 7 e_ i ðtÞ ¼ ðAmi þ Bi K i Þei ðtÞ þ ½I ni  Bi ri 6 ðA  A Þx ðtÞ þ B v ðtÞ  Aij ðtÞxj ðt  sij ðtÞÞ7 mi i i mi i 4 5.

ð35Þ

j6¼i j¼1

Furthermore, substituting matching condition (13) and ri Bi ¼ I mi into (35), (35) can be simplified as e_ i ðtÞ ¼ ðAmi þ Bi K i Þei ðtÞ.

ð36Þ

Observing (36), it reveals that when the controlled system is in the sliding mode, we can assure the stability of ei(t) only by selecting an appropriate matrix Ki using any pole assignment method such that kmax(Ami + BiKi) < 0. Furthermore, by (36), it shows that when the controlled system is in the sliding mode, the error dynamics in the sliding mode is independent of the interconnected delayed states and insensitive to parameter perturbation. In other words, the invariance condition [3] also holds even though the composite system with input nonlinearity.

4. An illustrative example Consider a decentralized control system with nonlinear input, which is of the form







0 1 0 0 0 0 x1 ðtÞ þ /1 ðu1 ðtÞÞ þ x2 ðt  s12 ðtÞÞ þ f1 ðx1 ; u1 ; p1 ; tÞ; X1 : x_ 1 ðtÞ ¼ 2 1 1 sinðtÞ 0:3 cosð2tÞ 1







0 1 0 0 0 0 X2 : x_ 2 ðtÞ ¼ x2 ðtÞ þ /2 ðu2 ðtÞÞ þ x1 ðt  s21 ðtÞÞ þ f2 ðx2 ; u2 ; p2 ; tÞ; 3 2 1 1 þ sinðtÞ cosð2tÞ 1 where /1 ðu1 ðtÞÞ ¼ ½0:6 þ 0:3 sinðu1 ðtÞÞu1 ðtÞ; f1 ðx1 ; u1 ; p1 ; tÞ ¼ ð0:5 þ 0:2 sinðtÞÞkx1 ðtÞk þ 0:2 cosð2tÞu1 ðtÞ þ 0:3 cosðtÞ; /2 ðu2 ðtÞÞ ¼ ½0:3 þ 0:5ej1þcosðu2 ðtÞÞj u2 ðtÞ; f2 ðx2 ; u2 ; p2 ; tÞ ¼ ð1  0:3 cosðtÞÞkx2 ðtÞk þ 0:2ecosðtÞ u2 ðtÞ þ 0:7 sinðtÞ; s12 ðtÞ ¼ j0:4  0:1 sinðtÞj;

s21 ðtÞ ¼ j0:3  0:2 cosð2tÞj;

smax ¼ 0:5.

M.-L. Hung, J.-J. Yan / Chaos, Solitons and Fractals 33 (2007) 1558–1568

The state vector of composite system is defined as follows: X ðtÞ ¼ ½ xT1 ðtÞ xT2 ðtÞ T ¼ ½ x11 ðtÞ x12 ðtÞ x21 ðtÞ

x22 ðtÞ T .

And the reference model is as shown below: R1 : y_ 1 ðtÞ ¼



Am1

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{



y 11 ðtÞ 0 1

y_ 11 ðtÞ ¼ y_ 12 ðtÞ 1 2

y 12 ðtÞ

Am2

Bm1

þ

zffl}|ffl{

0 1

v1 ðtÞ;

Bm2

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{



zffl}|ffl{

y_ 21 ðtÞ y 21 ðtÞ 0 1 0 R2 : y_ 2 ðtÞ ¼ v2 ðtÞ; ¼ þ y_ 22 ðtÞ y 22 ðtÞ 3 1 1

v1 ðtÞ ¼ 2 sinðtÞ;

v2 ðtÞ ¼ cosðtÞ.

6

5

y11

4

3

2

x 11 1

0

-1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec.)

Fig. 3. Time responses of x11(t) and y11(t) under memoryless DMRAC.

Fig. 4. Time responses of x12(t) and y12(t) under memoryless DMRAC.

1565

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M.-L. Hung, J.-J. Yan / Chaos, Solitons and Fractals 33 (2007) 1558–1568

We choose r1 ¼ Bg1 ¼ ½ 0

1 ; and K 1 ¼ ½ 3 2  such that kmax ðAm1 þ B1 K 1 Þ ¼ 2 < 0;

r2 ¼ Bg2 ¼ ½ 0

1 ; and K 2 ¼ ½ 6 5  such that kmax ðAm2 þ B2 K 2 Þ ¼ 3 < 0.

The equations of the submanifold are taken as (10). Based on (4), (5) and (13), the following coefficients can be obtained a1 ¼ 3:1623; a2 ¼ 6:7082;

g1 ¼ 1; g2 ¼ 1;

k 2;1 ¼ 0:2; h1 ¼ 0:3 > k 2;1 ; c12 ¼ 1:044; k 1;1 ¼ 0:7; k 3;1 ¼ 0:3; k 2;2 ¼ 0:5437; h2 ¼ 0:8 > k 2;2 ; c21 ¼ 2:2361; k 1;2 ¼ 1:3, k 3;2 ¼ 0:7.

For numerical simulation, the memoryless DMRAC (16) is selected with the following parameters: n1 ¼ n2 ¼ 1:2 > 1;

^10 ¼ 0:5; b

^20 ¼ 1. b

The simulation results with initial value X ðhÞ ¼ ½ xT10 xT20  ¼ ½ 1 2 3 2 T , 0 P h > smax and yð0Þ ¼ ½ y T10 y T20  ¼ ½ 5 1 1 1 T are shown in Figs. 3–6. Figs. 3–6 show the transient responses of x11, y11; x12, y12; x21, y21; and x22, y22,

3.5

3

2.5

2

x 21

1.5

1

0.5

y

21

0

-0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec.)

Fig. 5. Time responses of x21(t) and y21(t) under memoryless DMRAC.

Fig. 6. Time responses of x22(t) and y22(t) under memoryless DMRAC.

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Fig. 7. Time response of kS1(t)k + kS2(t)k under memoryless DMRAC.

1.2

βˆ2

1.1

1

0.9

0.8

0.7

βˆ1

0.6

0.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec.)

^1 ðtÞ and b ^2 ðtÞ under memoryless DMRAC. Fig. 8. Time responses of adaptation parameters b

^i ðtÞ; i ¼ 1; 2 under the proposed respectively. Figs. 7 and 8 show, respectively, the corresponding (jS1(t)j + jS2(t)j) and b memoryless DMRAC in (16) with adaptive law (18). From the simulation result, it shows that the proposed memoryless DMRAC works well for the uncertain large-scale time-varying delayed system with nonlinear input.

5. Conclusions A new scheme of memoryless decentralized model-reference adaptive control (DMRAC) for uncertain large-scale time-varying delayed systems with input nonlinearities is proposed. The control method has successfully applied the technique of variable structure control. Comparing this approach with earlier ones. The proposed work in this paper offers the following results. These are (i) no necessity to solve the Lyapunov or Riccati equation and the DMRAC does not involve any information of time-varying delays, i.e. the proposed controller is memoryless. (ii) The DMRAC is able to guarantee the occurrence of the sliding mode even under the effect of input nonlinearity. (iii) Once the error system is forced onto the switching surfaces, its dynamics is governed only by the local error equation ei(t) = (Ami + BiKi)ei(t), which is independent of the coupling delayed states and insensitive to the system perturbations and external distur-

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bances. In other words, the controlled uncertain composite systems with nonlinear inputs still preserve the invariance property of sliding mode control system as that of a system with linear input. (iv) The memoryless DMRAC can be implemented without using any delayed state or coupling state. It is really decentralized.

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