- Email: [email protected]
Robust decentralized adaptive control for uncertain large-scale delayed systems with input nonlinearities
Chaos, Solitons and Fractals 39 (2009) 1515–1521 www.elsevier.com/locate/chaos
Robust decentralized adaptive control for uncertain large-scale delayed systems with input nonlinearities Her-Terng Yau a
a,*
, Jun-Juh Yan
b
Department of Electrical Engineering, Far-East University, 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 1 June 2007
Communicated by Prof. Ji-Huan He
Abstract The paper is concerned with the problem of robust stabilization for uncertain large-scale time-varying delayed systems with input nonlinearities. Based on the sliding mode control, a memoryless decentralized adaptive sliding mode controller (DASMC) is developed. The proposed controller ensures the occurrence of the sliding manifold of the composite system even subjected to input nonlinearity. It shows that the uncertain nonlinear large-scale system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. A numerical example is given to verify the validity of the developed memoryless DASMC. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Large-scale systems, such as electrical power networks, computer communication networks, and transportation networks, exist naturally in physical systems and engineering systems. Time delays, due to the information transmission between subsystems, also naturally exist in large-scale systems and hence the control problem becomes more important than those in systems without time delays. In a robust control system, sliding mode control (SMC) is frequently adopted due to its inherent advantages of easy realization, fast response, good transient performance and insensitive to variation in plant parameters or external disturbances. So far, many works [1–3,5,6,8–11,15] have successfully applied the concept of sliding mode control to deal with the robust stabilization problem for uncertain systems. However, in the above works, the control schemes in [6,10,13] were only derived for single time-delay systems. Additionally, the controllers in [3,5,9,11,15] were synthesized for large-scale systems. Unfortunately, these types of controllers in [3,9,11] 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. To overcome the drawback, Xu et al. [15] proposed a new approach to deal with the interconnected terms. However, Xu et al. only coped with no delayed large-scale systems and in his approach, the global reaching condition might fail as
*
Corresponding author. Fax: +886 6 5977570. E-mail addresses: [email protected], [email protected] (H.-T. Yau), [email protected] (J.-J. Yan).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.035
1516
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
one of the switching modes is attained [5]. Moreover, in the above works [3,9–11,13,15] the control schemes are all under the assumption of linear input. Hsu [5] proposed robust variable structure controllers for uncertain systems with nonlinear inputs. However, they were only coped 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. In this paper, a memoryless decentralized adaptive sliding mode control (DASMC) is designed to stabilize the uncertain large-scale time-varying delayed dynamical systems with input nonlinearities. A local proportional-integral (PI) switching surface, which makes it easy to determine the performance of the subsystems in the sliding mode motion, is first proposed. And then, a memoryless DASMC is proposed to guarantee the existence of the composite sliding mode 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 DASMC. For simplicity, in the following section, kWk represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. k(W) denotes an eigenvalue of W and kmax(W) represents the eigenvalue of W with the maximum real part.
2. Problem formulation Consider an uncertain composite system X defined by N interconnected subsystems Xi, i = 1, . . . , N. Each subsystem Xi can be described by the following equation X N x_ i ðtÞ ¼ Ai xi ðtÞ þ 9 Aij ðtÞxj ðt sij ðtÞÞ þ Bi /i ðui ðtÞÞ þ Bi fi ðxi ; ui ; pi ; tÞ; i ¼ 1; 2; . . . ; N ; ð1Þ j–i 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 N N X X ni ¼ n; mi ¼ m; ð2Þ i¼1
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, Bi), i = 1, 2, . . . , N are completely controllable. 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 : Throughout this paper, the following assumptions are taken: Assumption 1. There exists matrix function Hij(t) satisfying the following matching condition, as in the work of Xu et al. [15]: Aij ðtÞ ¼ Bi H ij ðtÞ;
kH ij ðtÞk 6 cij ;
i; j ¼ 1; 2; . . . ; N ;
ð4Þ
where cij are positive constants. Assumption 2. For the lumped uncertainty f(xi, ui, pi; t), non-negative constants k1,i, k2,i and k3,i satisfy: 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:
ð5Þ
Assumption 3. The nonlinear input /i(ui(t)) in (1) applied to the subsystem Xi satisfies uTi ðtÞ/i ðui ðtÞÞ P hi uTi ðtÞui ðtÞ;
ð6Þ
where hi is a positive non-zero constant, and /i(0) = 0. The parameter hi is often called as the gain reduction tolerance. Define the local PI switching surface for Xi as Z t S i ðtÞ ¼ ri xi ðtÞ ðri Ai þ K i Þxi ðkÞ dk;
ð7Þ
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
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1517
kmax ðAi þ Bi K i Þ < 0:
ð8Þ
For this type of large-scale system, the sliding manifold can be described as T SðtÞ ¼ S T1 S T2 . . . S TN :
ð9Þ
To achieve our control goal, there exist two major problems to be well solved. First, it needs to select an appropriate switching surface to ensure the sliding motion on the switching surface has the desired properties. Second, it needs to determine a memoryless adaptive decentralized control such that the existence of the sliding mode can be guaranteed even with the input nonlinearity. These problems are well solved in the following section.
3. Stability of the sliding mode It is well known that when the system operates in the sliding manifold S(t) = 0, it satisfies the equations [7,14] S_ i ðtÞ ¼ 0 and S i ðtÞ ¼ 0;
i ¼ 1; 2; . . . ; N :
ð10Þ
Therefore, the local equivalent control /eq,i(ui(t)) in the sliding manifold is obtained by differentiating (7) with respect to time and substituting from (1) S_ i ðtÞ ¼ K i xi ðtÞ þ
N X
ri Aij ðtÞxj ðt sij ðtÞÞ þ fi ðxi ; ui ; pi ; tÞ þ /eq;i ðui ðtÞÞ ¼ 0;
ð11Þ
j–i j¼1
where ri Bi ¼ Bgi Bi ¼ I mi has introduced. From (11), the equivalent control /eq,i(ui(t)) in the sliding mode is given by /eq;i ðui ðtÞÞ ¼ K i xi ðtÞ
N X
ri Aij ðtÞxj ðt sij ðtÞÞ fi ðxi ; ui ; pi ; tÞ:
ð12Þ
j–i j¼1
It can be easily shown that the equivalent dynamic equation for each subsystem in the sliding mode is in the form 3 2 N 7 6X x_ i ðtÞ ¼ ðAi þ Bi K i Þxi ðtÞ þ ½I ni Bi ri 6 Aij ðtÞxj ðt sij ðtÞÞT 7 5: 4
ð13Þ
j–i j¼1
Furthermore, substituting matching condition (4) and ri Bi ¼ I mi into (13), (13) can be simplified as x_ i ðtÞ ¼ ðAi þ Bi K i Þxi ðtÞ:
ð14Þ
Observing (14), it reveals that when the controlled system is in the sliding mode, the subsystem dynamics in the sliding mode is independent of the interconnected delayed states and insensitive to external perturbation. In other words, the invariance condition [4] also holds even though the composite system with input nonlinearity. Furthermore, we can easily assign the system performance in the sliding mode just by selecting an appropriate matrix Ki using any pole assignment method. 4. Decentralized adaptive sliding mode control (DASMC) deign Once proper switching surfaces have been decided with appropriate matrices Ki. All one has to do next is to propose a memoryless DASMC to derive the composite system trajectories onto the sliding surfaces. To ensure the occurrence of the sliding motion, a memoryless DASMC is proposed as ^i ; tÞ ui ðtÞ ¼ Gi ðxi ; b where
S i ðtÞ ; kS i ðtÞk 2
^i ; tÞ ¼ Gi ðxi ; b
i ¼ 1; 2; . . . ; N;
ð15Þ 1
0 N BX
3
C 7 ni 6 ^ 7 6ðkK i k þ k 1;i Þkxi ðtÞk þ B cij C Abi ðtÞ þ k 3;i 5; @ hi k 2;i 4 j–i j¼1
ni > 1; hi > k 2;i :
ð16Þ
1518
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
The adaptive law is 1 0 N C BX ^_ i ðtÞ ¼ B cij C b AkS i ðtÞk; @
^ i ð0Þ ¼ b ^i0 ; b
ð17Þ
j–i j¼1
^i ðtÞ. It is obvious b ^i ðtÞ > 0 for all t > 0: ^i0 is the positive and bounded initial value of b where b Theorem 1. Consider the uncertain large-scale time-varying delayed system (1) with input nonlinearity and this system is controlled by ui(t) in (15) with adaptation law (17). 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 e2i ðtÞ ; 2 i¼1
ð18Þ
where ei(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 (1) into the differentiation of Si(t) as (7), one has 8 9 3 2 > > > > < = N N X 7 6X T 7 6 ð19Þ V_ ðtÞ ¼ S i ðtÞ4 ri Aij ðtÞxj ðt sij ðtÞÞ þ ri Bi /i ðui ðtÞÞ þ ri Bi fi ðxi ; ui ; pi ; tÞ K i xi ðtÞ5 þ ei ðtÞ_ei ðtÞ : > > > j–i i¼1 > : ; j¼1
However, since we have chosen ri ¼ Bgi to result in ri Bi ¼ I mi , the above equation can be rewritten as 8 9 3 2 > > > > N N < = X 7 6X T T 7 6 _ S i ðtÞ4 H ij ðtÞxj ðt sij ðtÞÞ þ fi ðxi ;ui ; pi ; tÞ K i xi ðtÞ5 þ S i ðtÞ/i ðui ðtÞÞ þ ei ðtÞ_ei ðtÞ V ðtÞ ¼ > > > j–i i¼1 > : ; j¼1 ( " # ) N X X N T kS i ðtÞk j ¼ 1 cij kxj ðt sij ðtÞÞk þ ðkK i k þ k 1;i Þkxi ðtÞk þ k 3;i þ k 2;i kui ðtÞk þ S i ðtÞ/ðui ðtÞÞ þ ei ðtÞ_ei ðtÞ : 6 j–i
i¼1
ð20Þ From (6) and (15), we have T ^i ; tÞ S i ðtÞ / ðui ðtÞÞ P hi uT ðtÞui ðtÞ ¼ hi G2 ðxi ; b ^i ; tÞ: uTi ðtÞ/i ðui ðtÞÞ ¼ Gi ðxi ; b i i kS i ðtÞk i
ð21Þ
Therefore, we have the following inequality ^i ; tÞkS i ðtÞk: S Ti ðtÞ/i ðui ðtÞÞ 6 hi Gi ðxi ; b
ð22Þ
Now define some unknown but always existing constants bi satisfying bi P kxj ðtÞk;
j–i; for all 1 > t P smax :
ð23Þ
^ i ðtÞ b denote the adaptation error. Because bi is assumed to be constant, then the following expression Let ei ðtÞ ¼ b i keeps valid ^_ i ðtÞ: e_ i ðtÞ ¼ b
ð24Þ
Inserting (16), (22) and (24) into the right hand of inequality (20), this yields
9 > > = C 7 _ ^ ^ 6 B C 7 _V ðtÞ6 kS i ðtÞk4@ cij Abi þðkK i kþk 1;i Þkxi ðtÞkþk 3;i 5 ðhi k 2;i ÞGi ðxi ; bi ;tÞkS i ðtÞkþei ðtÞbi ðtÞ > > > j–i i¼1 > ; : j¼1 9 8 0 1 0 1 > > > = < N > N N X X X B C B C ^i ðtÞe Þþð1ni ÞkS i ðtÞkB c Cb ^_ i ðtÞ : ^i ðtÞþei ðtÞb C ¼ ð1ni ÞkS i ðtÞk½ðkK i kþk 1;i Þkxi ðtÞkþk 3;i þkS i ðtÞk B c b ðb ij A i ij A i @ @ > > |fflfflfflfflffl{zfflfflfflfflffl} > j–i j–i i¼1 > ; : 8 >
< N > X
20
1
3
N 6BX
j¼1
j¼1
ð25Þ
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1519
By Eq. (17) and ni > 1,one can obtain 9 8 1 0 > > > > N < N = X C BX ^i ðtÞkS i ðtÞk : C ð26Þ b ðni 1ÞB c V_ ðtÞ 6 ij A @ > > > j–i i¼1 > ; : j¼1 PN PN ^ Now if we define wðtÞ ¼ i¼1 ðni 1Þ j–i cij bi ðtÞkS i ðtÞk , and integrating the above equation from zero to t, it j¼1 yields Z t Z t wðkÞ dk P wðkÞ dk: ð27Þ V ð0Þ P V ðtÞ þ 0
0
Taking the limit as t!1 on both side of (27) gives Z t wðkÞ dk: 1 > V ð0Þ P lim t!1
ð28Þ
0
Since V(0) is bounded, thus according to Barbalat lemma [12], we obtain 8 9 1 0 > > > > N < N = X C BX ^i ðtÞkS i ðtÞk ¼ 0: C b ðni 1ÞB c lim wðtÞ ¼ lim ij A @ t!1 t!1 > > > j–i i¼1 > : ;
ð29Þ
j¼1
^i ðtÞ > 0 for all t > 0 and ni > 1 is chosen. Thus Eq. (29) implies Si(t) ! 0 as t ! 1. Hence the proof Furthermore cij ; b is achieved completely. h
5. An illustrative example Consider a decentralized control system X with nonlinear input, which is of the form
0 1 0 0 0 0 x1 ðtÞ þ /1 ðu1 Þ þ x2 ðt s12 ðtÞÞ þ f1 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 Þ þ x1 ðt s21 ðtÞÞ þ f2 ; 3 2 1 1 þ sinðtÞ 1 þ 0:3 cosð2tÞ 1 where /1 ðu1 Þ ¼ ½0:6 þ 0:3 sinðu1 Þu1 f1 ¼ ð0:5 þ 0:2 sinðtÞÞkx1 k þ 0:2 cosð2tÞu1 þ 0:3 cosðtÞ /2 ðu2 Þ ¼ 0:3 þ 0:5ej1þcosðu2 Þj u2 f2 ¼ ð1 0:3 cosðtÞÞkx2 k þ 0:2ecosðtÞ u2 þ 0:7 sinðtÞ s12 ðtÞ ¼ j0:4 0:1 sinðtÞj; s21 ðtÞ ¼ j0:3 0:2 cosð2tÞj;
smax ¼ 0:5:
Based on (5) and (6), the following coefficients can be obtained h1 ¼ 0:3; k 1;2 ¼ 1:3;
c12 ¼ 1:044;
k 1;1 ¼ 0:7;
k2;2 ¼ 0:5437;
k 2;1 ¼ 0:2;
k 3;1 ¼ 0:3;
h2 ¼ 0:8;
c21 ¼ 2:3845;
k 3;2 ¼ 0:7:
We choose r1 ¼ r2 ¼ ½ 0 1 , and K 1 ¼ ½ 2 5 ! kmax ðA1 þ B1 K 1 Þ ¼ 2 < 0. K 2 ¼ ½ 6 8 ! kmax ðA2 þ B2 K 2 Þ ¼ 3 < 0: For numerical simulation, the following parameters for DASMC (15) is selected as n1 ¼ n2 ¼ 1:1;
^10 ¼ 1; b
^20 ¼ 0:5: b
The simulation results with initial value X ðhÞ ¼ xT10 xT20 ¼ ½ 2 3 1 3 T ; 0 P h > smax are shown in Figs. 1–3. From the simulation result, it shows that the proposed memoryless DASMC works well. In particular, it is worthy of note that no information of time-varying delays and delayed states is used in our control design.
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521 7 6 5
S1(t) + S2 (t)
4 3 2 1 0 -1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
Time (sec.)
Fig. 1. Time response of jS1(t)j + jS2(t)j. 7 6 5
x1(t) + x2 (t)
1520
4 3 2 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec.)
Fig. 2. Time responses of kx1(t)k + kx2(t)k. 1.4
βˆ1
1.2 1 0.8
βˆ2 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec.)
^1 ðtÞ and b ^2 ðtÞ. Fig. 3. Time responses of adaptation parameters b
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1521
6. Conclusions A robust decentralized adaptive sliding mode control (DASMC) is developed for uncertain large-scale time-varying delayed systems with input nonlinearities. The proposed work in this paper offers the following results. These are (i) The DASMC can be implemented without using any delayed state or coupling state, i.e. it is decentralized. (ii) The DASMC does not involve any information of time-varying delays, i.e. the proposed controller is memoryless. (iii) The DASMC can ensure the occurrence of the sliding mode even under the effect of input nonlinearity. (iv) In the sliding mode, the uncertain composite systems with nonlinear inputs still preserve the invariance property of sliding mode control system as that of a system with linear input.
Acknowledgement The work is supported by the National Science Council of Republic of China under contract NSC 95-2221-E-269 007.
References [1] Kwon OM, Park Ju H. Guaranteed cost control for uncertain large-scale systems with time-delays via delayed feedback. Chaos, Solitons & Fractals 2006;27:800–12. [2] Liu X, Zhang H, Zhang F. Delay-dependent stability of uncertain fuzzy large-scale systems with time delays. Chaos, Solitons & Fractals 2005;26:147–58. [3] Corfmat J, Morse S. On the decentralized control of linear multivariable systems. Automatica 1989;12:475–9. [4] Drazenovic B. The invariance condition in variable structure systems. Automatica 1986;5:287–95. [5] Hsu KC. Decentralized variable-structure control design for uncertain large-scale systems with series nonlinearities. Int J Control 1997;68:1231–40. [6] Hsu KC. Adaptive variable control design for uncertain time-delay systems with nonlinear input. Dynam Control 1998;8:341–54. [7] Utkin VI. Sliding mode and their applications in variable structure systems. Moscow: Mir Editors; 1978. [8] Kwon WH, Pearson AE. A note on feedback stabilization of a differential-difference system. IEEE Trans Automat Control 1977;June:468–70. [9] Matthews G, Decarlo R. Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica 1988;24:187–93. [10] Oucheriah S. Dynamic compensation of uncertain time-delay systems using variable approach. IEEE Trans Circuits Syst-I 1995;42:466–70. [11] Chou CH, Cheng CC. A decentralized model reference adaptive variable structure controller for large-scale time-varying delay systems. IEEE Trans Automat Control 2003;48:1213–7. [12] Popov VM. Hyperstability of control system. Berlin: Springer; 1973. [13] Shyu KK, Chen SR. Estimation of asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers. Automatica 1996;32:797–800. [14] Utkin VI. Sliding mode in control optimization. Berlin: Springer; 1992. [15] Xu X, Wu Y, Huang W. Variable-structure control of decentralized model-reference adaptive systems. In: Proceedings of the institution of electrical engineers, Pt D, vol. 137; 1990. p. 302–6.
Robust decentralized adaptive control for uncertain large-scale delayed systems with input nonlinearities Her-Terng Yau a
a,*
, Jun-Juh Yan
b
Department of Electrical Engineering, Far-East University, 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 1 June 2007
Communicated by Prof. Ji-Huan He
Abstract The paper is concerned with the problem of robust stabilization for uncertain large-scale time-varying delayed systems with input nonlinearities. Based on the sliding mode control, a memoryless decentralized adaptive sliding mode controller (DASMC) is developed. The proposed controller ensures the occurrence of the sliding manifold of the composite system even subjected to input nonlinearity. It shows that the uncertain nonlinear large-scale system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. A numerical example is given to verify the validity of the developed memoryless DASMC. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Large-scale systems, such as electrical power networks, computer communication networks, and transportation networks, exist naturally in physical systems and engineering systems. Time delays, due to the information transmission between subsystems, also naturally exist in large-scale systems and hence the control problem becomes more important than those in systems without time delays. In a robust control system, sliding mode control (SMC) is frequently adopted due to its inherent advantages of easy realization, fast response, good transient performance and insensitive to variation in plant parameters or external disturbances. So far, many works [1–3,5,6,8–11,15] have successfully applied the concept of sliding mode control to deal with the robust stabilization problem for uncertain systems. However, in the above works, the control schemes in [6,10,13] were only derived for single time-delay systems. Additionally, the controllers in [3,5,9,11,15] were synthesized for large-scale systems. Unfortunately, these types of controllers in [3,9,11] 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. To overcome the drawback, Xu et al. [15] proposed a new approach to deal with the interconnected terms. However, Xu et al. only coped with no delayed large-scale systems and in his approach, the global reaching condition might fail as
*
Corresponding author. Fax: +886 6 5977570. E-mail addresses: [email protected], [email protected] (H.-T. Yau), [email protected] (J.-J. Yan).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.035
1516
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
one of the switching modes is attained [5]. Moreover, in the above works [3,9–11,13,15] the control schemes are all under the assumption of linear input. Hsu [5] proposed robust variable structure controllers for uncertain systems with nonlinear inputs. However, they were only coped 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. In this paper, a memoryless decentralized adaptive sliding mode control (DASMC) is designed to stabilize the uncertain large-scale time-varying delayed dynamical systems with input nonlinearities. A local proportional-integral (PI) switching surface, which makes it easy to determine the performance of the subsystems in the sliding mode motion, is first proposed. And then, a memoryless DASMC is proposed to guarantee the existence of the composite sliding mode 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 DASMC. For simplicity, in the following section, kWk represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. k(W) denotes an eigenvalue of W and kmax(W) represents the eigenvalue of W with the maximum real part.
2. Problem formulation Consider an uncertain composite system X defined by N interconnected subsystems Xi, i = 1, . . . , N. Each subsystem Xi can be described by the following equation X N x_ i ðtÞ ¼ Ai xi ðtÞ þ 9 Aij ðtÞxj ðt sij ðtÞÞ þ Bi /i ðui ðtÞÞ þ Bi fi ðxi ; ui ; pi ; tÞ; i ¼ 1; 2; . . . ; N ; ð1Þ j–i 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 N N X X ni ¼ n; mi ¼ m; ð2Þ i¼1
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, Bi), i = 1, 2, . . . , N are completely controllable. 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 : Throughout this paper, the following assumptions are taken: Assumption 1. There exists matrix function Hij(t) satisfying the following matching condition, as in the work of Xu et al. [15]: Aij ðtÞ ¼ Bi H ij ðtÞ;
kH ij ðtÞk 6 cij ;
i; j ¼ 1; 2; . . . ; N ;
ð4Þ
where cij are positive constants. Assumption 2. For the lumped uncertainty f(xi, ui, pi; t), non-negative constants k1,i, k2,i and k3,i satisfy: 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:
ð5Þ
Assumption 3. The nonlinear input /i(ui(t)) in (1) applied to the subsystem Xi satisfies uTi ðtÞ/i ðui ðtÞÞ P hi uTi ðtÞui ðtÞ;
ð6Þ
where hi is a positive non-zero constant, and /i(0) = 0. The parameter hi is often called as the gain reduction tolerance. Define the local PI switching surface for Xi as Z t S i ðtÞ ¼ ri xi ðtÞ ðri Ai þ K i Þxi ðkÞ dk;
ð7Þ
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
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1517
kmax ðAi þ Bi K i Þ < 0:
ð8Þ
For this type of large-scale system, the sliding manifold can be described as T SðtÞ ¼ S T1 S T2 . . . S TN :
ð9Þ
To achieve our control goal, there exist two major problems to be well solved. First, it needs to select an appropriate switching surface to ensure the sliding motion on the switching surface has the desired properties. Second, it needs to determine a memoryless adaptive decentralized control such that the existence of the sliding mode can be guaranteed even with the input nonlinearity. These problems are well solved in the following section.
3. Stability of the sliding mode It is well known that when the system operates in the sliding manifold S(t) = 0, it satisfies the equations [7,14] S_ i ðtÞ ¼ 0 and S i ðtÞ ¼ 0;
i ¼ 1; 2; . . . ; N :
ð10Þ
Therefore, the local equivalent control /eq,i(ui(t)) in the sliding manifold is obtained by differentiating (7) with respect to time and substituting from (1) S_ i ðtÞ ¼ K i xi ðtÞ þ
N X
ri Aij ðtÞxj ðt sij ðtÞÞ þ fi ðxi ; ui ; pi ; tÞ þ /eq;i ðui ðtÞÞ ¼ 0;
ð11Þ
j–i j¼1
where ri Bi ¼ Bgi Bi ¼ I mi has introduced. From (11), the equivalent control /eq,i(ui(t)) in the sliding mode is given by /eq;i ðui ðtÞÞ ¼ K i xi ðtÞ
N X
ri Aij ðtÞxj ðt sij ðtÞÞ fi ðxi ; ui ; pi ; tÞ:
ð12Þ
j–i j¼1
It can be easily shown that the equivalent dynamic equation for each subsystem in the sliding mode is in the form 3 2 N 7 6X x_ i ðtÞ ¼ ðAi þ Bi K i Þxi ðtÞ þ ½I ni Bi ri 6 Aij ðtÞxj ðt sij ðtÞÞT 7 5: 4
ð13Þ
j–i j¼1
Furthermore, substituting matching condition (4) and ri Bi ¼ I mi into (13), (13) can be simplified as x_ i ðtÞ ¼ ðAi þ Bi K i Þxi ðtÞ:
ð14Þ
Observing (14), it reveals that when the controlled system is in the sliding mode, the subsystem dynamics in the sliding mode is independent of the interconnected delayed states and insensitive to external perturbation. In other words, the invariance condition [4] also holds even though the composite system with input nonlinearity. Furthermore, we can easily assign the system performance in the sliding mode just by selecting an appropriate matrix Ki using any pole assignment method. 4. Decentralized adaptive sliding mode control (DASMC) deign Once proper switching surfaces have been decided with appropriate matrices Ki. All one has to do next is to propose a memoryless DASMC to derive the composite system trajectories onto the sliding surfaces. To ensure the occurrence of the sliding motion, a memoryless DASMC is proposed as ^i ; tÞ ui ðtÞ ¼ Gi ðxi ; b where
S i ðtÞ ; kS i ðtÞk 2
^i ; tÞ ¼ Gi ðxi ; b
i ¼ 1; 2; . . . ; N;
ð15Þ 1
0 N BX
3
C 7 ni 6 ^ 7 6ðkK i k þ k 1;i Þkxi ðtÞk þ B cij C Abi ðtÞ þ k 3;i 5; @ hi k 2;i 4 j–i j¼1
ni > 1; hi > k 2;i :
ð16Þ
1518
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
The adaptive law is 1 0 N C BX ^_ i ðtÞ ¼ B cij C b AkS i ðtÞk; @
^ i ð0Þ ¼ b ^i0 ; b
ð17Þ
j–i j¼1
^i ðtÞ. It is obvious b ^i ðtÞ > 0 for all t > 0: ^i0 is the positive and bounded initial value of b where b Theorem 1. Consider the uncertain large-scale time-varying delayed system (1) with input nonlinearity and this system is controlled by ui(t) in (15) with adaptation law (17). 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 e2i ðtÞ ; 2 i¼1
ð18Þ
where ei(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 (1) into the differentiation of Si(t) as (7), one has 8 9 3 2 > > > > < = N N X 7 6X T 7 6 ð19Þ V_ ðtÞ ¼ S i ðtÞ4 ri Aij ðtÞxj ðt sij ðtÞÞ þ ri Bi /i ðui ðtÞÞ þ ri Bi fi ðxi ; ui ; pi ; tÞ K i xi ðtÞ5 þ ei ðtÞ_ei ðtÞ : > > > j–i i¼1 > : ; j¼1
However, since we have chosen ri ¼ Bgi to result in ri Bi ¼ I mi , the above equation can be rewritten as 8 9 3 2 > > > > N N < = X 7 6X T T 7 6 _ S i ðtÞ4 H ij ðtÞxj ðt sij ðtÞÞ þ fi ðxi ;ui ; pi ; tÞ K i xi ðtÞ5 þ S i ðtÞ/i ðui ðtÞÞ þ ei ðtÞ_ei ðtÞ V ðtÞ ¼ > > > j–i i¼1 > : ; j¼1 ( " # ) N X X N T kS i ðtÞk j ¼ 1 cij kxj ðt sij ðtÞÞk þ ðkK i k þ k 1;i Þkxi ðtÞk þ k 3;i þ k 2;i kui ðtÞk þ S i ðtÞ/ðui ðtÞÞ þ ei ðtÞ_ei ðtÞ : 6 j–i
i¼1
ð20Þ From (6) and (15), we have T ^i ; tÞ S i ðtÞ / ðui ðtÞÞ P hi uT ðtÞui ðtÞ ¼ hi G2 ðxi ; b ^i ; tÞ: uTi ðtÞ/i ðui ðtÞÞ ¼ Gi ðxi ; b i i kS i ðtÞk i
ð21Þ
Therefore, we have the following inequality ^i ; tÞkS i ðtÞk: S Ti ðtÞ/i ðui ðtÞÞ 6 hi Gi ðxi ; b
ð22Þ
Now define some unknown but always existing constants bi satisfying bi P kxj ðtÞk;
j–i; for all 1 > t P smax :
ð23Þ
^ i ðtÞ b denote the adaptation error. Because bi is assumed to be constant, then the following expression Let ei ðtÞ ¼ b i keeps valid ^_ i ðtÞ: e_ i ðtÞ ¼ b
ð24Þ
Inserting (16), (22) and (24) into the right hand of inequality (20), this yields
9 > > = C 7 _ ^ ^ 6 B C 7 _V ðtÞ6 kS i ðtÞk4@ cij Abi þðkK i kþk 1;i Þkxi ðtÞkþk 3;i 5 ðhi k 2;i ÞGi ðxi ; bi ;tÞkS i ðtÞkþei ðtÞbi ðtÞ > > > j–i i¼1 > ; : j¼1 9 8 0 1 0 1 > > > = < N > N N X X X B C B C ^i ðtÞe Þþð1ni ÞkS i ðtÞkB c Cb ^_ i ðtÞ : ^i ðtÞþei ðtÞb C ¼ ð1ni ÞkS i ðtÞk½ðkK i kþk 1;i Þkxi ðtÞkþk 3;i þkS i ðtÞk B c b ðb ij A i ij A i @ @ > > |fflfflfflfflffl{zfflfflfflfflffl} > j–i j–i i¼1 > ; : 8 >
< N > X
20
1
3
N 6BX
j¼1
j¼1
ð25Þ
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1519
By Eq. (17) and ni > 1,one can obtain 9 8 1 0 > > > > N < N = X C BX ^i ðtÞkS i ðtÞk : C ð26Þ b ðni 1ÞB c V_ ðtÞ 6 ij A @ > > > j–i i¼1 > ; : j¼1 PN PN ^ Now if we define wðtÞ ¼ i¼1 ðni 1Þ j–i cij bi ðtÞkS i ðtÞk , and integrating the above equation from zero to t, it j¼1 yields Z t Z t wðkÞ dk P wðkÞ dk: ð27Þ V ð0Þ P V ðtÞ þ 0
0
Taking the limit as t!1 on both side of (27) gives Z t wðkÞ dk: 1 > V ð0Þ P lim t!1
ð28Þ
0
Since V(0) is bounded, thus according to Barbalat lemma [12], we obtain 8 9 1 0 > > > > N < N = X C BX ^i ðtÞkS i ðtÞk ¼ 0: C b ðni 1ÞB c lim wðtÞ ¼ lim ij A @ t!1 t!1 > > > j–i i¼1 > : ;
ð29Þ
j¼1
^i ðtÞ > 0 for all t > 0 and ni > 1 is chosen. Thus Eq. (29) implies Si(t) ! 0 as t ! 1. Hence the proof Furthermore cij ; b is achieved completely. h
5. An illustrative example Consider a decentralized control system X with nonlinear input, which is of the form
0 1 0 0 0 0 x1 ðtÞ þ /1 ðu1 Þ þ x2 ðt s12 ðtÞÞ þ f1 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 Þ þ x1 ðt s21 ðtÞÞ þ f2 ; 3 2 1 1 þ sinðtÞ 1 þ 0:3 cosð2tÞ 1 where /1 ðu1 Þ ¼ ½0:6 þ 0:3 sinðu1 Þu1 f1 ¼ ð0:5 þ 0:2 sinðtÞÞkx1 k þ 0:2 cosð2tÞu1 þ 0:3 cosðtÞ /2 ðu2 Þ ¼ 0:3 þ 0:5ej1þcosðu2 Þj u2 f2 ¼ ð1 0:3 cosðtÞÞkx2 k þ 0:2ecosðtÞ u2 þ 0:7 sinðtÞ s12 ðtÞ ¼ j0:4 0:1 sinðtÞj; s21 ðtÞ ¼ j0:3 0:2 cosð2tÞj;
smax ¼ 0:5:
Based on (5) and (6), the following coefficients can be obtained h1 ¼ 0:3; k 1;2 ¼ 1:3;
c12 ¼ 1:044;
k 1;1 ¼ 0:7;
k2;2 ¼ 0:5437;
k 2;1 ¼ 0:2;
k 3;1 ¼ 0:3;
h2 ¼ 0:8;
c21 ¼ 2:3845;
k 3;2 ¼ 0:7:
We choose r1 ¼ r2 ¼ ½ 0 1 , and K 1 ¼ ½ 2 5 ! kmax ðA1 þ B1 K 1 Þ ¼ 2 < 0. K 2 ¼ ½ 6 8 ! kmax ðA2 þ B2 K 2 Þ ¼ 3 < 0: For numerical simulation, the following parameters for DASMC (15) is selected as n1 ¼ n2 ¼ 1:1;
^10 ¼ 1; b
^20 ¼ 0:5: b
The simulation results with initial value X ðhÞ ¼ xT10 xT20 ¼ ½ 2 3 1 3 T ; 0 P h > smax are shown in Figs. 1–3. From the simulation result, it shows that the proposed memoryless DASMC works well. In particular, it is worthy of note that no information of time-varying delays and delayed states is used in our control design.
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521 7 6 5
S1(t) + S2 (t)
4 3 2 1 0 -1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
Time (sec.)
Fig. 1. Time response of jS1(t)j + jS2(t)j. 7 6 5
x1(t) + x2 (t)
1520
4 3 2 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec.)
Fig. 2. Time responses of kx1(t)k + kx2(t)k. 1.4
βˆ1
1.2 1 0.8
βˆ2 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec.)
^1 ðtÞ and b ^2 ðtÞ. Fig. 3. Time responses of adaptation parameters b
H.-T. Yau, J.-J. Yan / Chaos, Solitons and Fractals 39 (2009) 1515–1521
1521
6. Conclusions A robust decentralized adaptive sliding mode control (DASMC) is developed for uncertain large-scale time-varying delayed systems with input nonlinearities. The proposed work in this paper offers the following results. These are (i) The DASMC can be implemented without using any delayed state or coupling state, i.e. it is decentralized. (ii) The DASMC does not involve any information of time-varying delays, i.e. the proposed controller is memoryless. (iii) The DASMC can ensure the occurrence of the sliding mode even under the effect of input nonlinearity. (iv) In the sliding mode, the uncertain composite systems with nonlinear inputs still preserve the invariance property of sliding mode control system as that of a system with linear input.
Acknowledgement The work is supported by the National Science Council of Republic of China under contract NSC 95-2221-E-269 007.
References [1] Kwon OM, Park Ju H. Guaranteed cost control for uncertain large-scale systems with time-delays via delayed feedback. Chaos, Solitons & Fractals 2006;27:800–12. [2] Liu X, Zhang H, Zhang F. Delay-dependent stability of uncertain fuzzy large-scale systems with time delays. Chaos, Solitons & Fractals 2005;26:147–58. [3] Corfmat J, Morse S. On the decentralized control of linear multivariable systems. Automatica 1989;12:475–9. [4] Drazenovic B. The invariance condition in variable structure systems. Automatica 1986;5:287–95. [5] Hsu KC. Decentralized variable-structure control design for uncertain large-scale systems with series nonlinearities. Int J Control 1997;68:1231–40. [6] Hsu KC. Adaptive variable control design for uncertain time-delay systems with nonlinear input. Dynam Control 1998;8:341–54. [7] Utkin VI. Sliding mode and their applications in variable structure systems. Moscow: Mir Editors; 1978. [8] Kwon WH, Pearson AE. A note on feedback stabilization of a differential-difference system. IEEE Trans Automat Control 1977;June:468–70. [9] Matthews G, Decarlo R. Decentralized tracking for a class of interconnected nonlinear systems using variable structure control. Automatica 1988;24:187–93. [10] Oucheriah S. Dynamic compensation of uncertain time-delay systems using variable approach. IEEE Trans Circuits Syst-I 1995;42:466–70. [11] Chou CH, Cheng CC. A decentralized model reference adaptive variable structure controller for large-scale time-varying delay systems. IEEE Trans Automat Control 2003;48:1213–7. [12] Popov VM. Hyperstability of control system. Berlin: Springer; 1973. [13] Shyu KK, Chen SR. Estimation of asymptotic stability region and sliding domain of uncertain variable structure systems with bounded controllers. Automatica 1996;32:797–800. [14] Utkin VI. Sliding mode in control optimization. Berlin: Springer; 1992. [15] Xu X, Wu Y, Huang W. Variable-structure control of decentralized model-reference adaptive systems. In: Proceedings of the institution of electrical engineers, Pt D, vol. 137; 1990. p. 302–6.