Decentralized Control of Uncertain Fuzzy Large-Scale System with Time Delay and Optimization

This paper studies the decentralized stabilization problem for an uncertain fuzzy large-scale system with time delays. The considered large-scale system is composed of several T-S fuzzy subsystems. The decentralized parallel distributed compensation PDC fuzzy control for each subsystem is designed to stabilize the whole system. Based on Lyapunov criterion, some sufficient conditions are proposed. Moreover, the positive definite matrices Pi and PDC gain Kij can be solved by linear matrix inequality LMI toolbox of Matlab. Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index. Finally, numerical examples are given and compared with those of Zhang et al., 2004 to illustrate the effectiveness and less conservativeness of our method.


Introduction
Many real-life problems, such as power system, economic systems, societal system, and nuclear system are frequently of high dimension.Such systems are regarded as large-scale system.They consist of a number of subsystems which serve particular functions, share resources, and are governed by a set of interrelated goals and constrains 1 .Over the past decades, many methods have been to investigate the stability and stabilization of large-scale system 2-10 .
Fuzzy systems of Takagi-Sugeno T-S models 11 have become an effective method to represent complex nonlinear dynamic system by fuzzy sets and fuzzy reasoning.The method of T-S model is feasible since, in many situations, human experts can provide linguistic descriptions of local systems in terms of IF-THEN rules 12-14 .Reference 15 proposed a control concept "parallel distributed compensation" PDC for fuzzy controller design of T-S fuzzy model.Under some conditions, PDC can stabilize the closed-loop fuzzy system ẋi t r i j 1 where for for all t ≥ 0, j 1, 2, . . ., r i , k 1, 2, . . ., J, x i t is the state vector, u i t is the control input, A ij , B ij , G ij , H ij , E ij , C ki , D ki , E ki , M ki , L ki are known constant matrices with appropriate dimensions.τ ki is the time delay.h ij t is the normalized weigh in 2.6 .F ij t and F ki t are time-varying matrices with appropriate dimensions satisfying Each isolated subsystem S i is represented by a T-S fuzzy model.The jth rule of this T-S fuzzy model is represented as follows.
Rule j.If z 1i t is M j1i and . . .and where z i t z 1i t , z 2i t , . . ., z pi t T , z 1i , . . ., z pi t are premise variables, and M jli l 1, 2, . . ., p are fuzzy sets.By "fuzzy blending," the final output of the ith fuzzy subsystem is described as follows: with where M jil z il t is the grade of membership of z il t in M jil , and r i is the number of fuzzy rules of subsystem S i .We assumed that ω ij t ≥ 0 for all t, j 1, 2, . . ., r i .Therefore

2.7
Define a quadratic performance index where Q i and R i are positive matrices.
The main propose of this paper is to synthesize a decentralized PDC fuzzy controller u i t for each subsystem such that the closed-loop large-scale T-S fuzzy systems 2.1 is asymptotically stable and the optimization design method for decentralized control respect to the quadratic performance index.
Before starting the main results, we need the following lemmas.

Stabilization and PDC Synthesis of Fuzzy Large-Scale System
In this section, the decentralized concept and PDC approach are applied to synthesize a local feedback controller for each local subsystem.Let the fuzzy controller be as the PDC form: Rule j.If z 1i t is M j1i and... and z pi t is M jpi , then where i 1, 2, . . ., J, j 1, 2, . . ., r i .The overall state feedback fuzzy control law is represented by:

3.3
Now, our work is to determine the local feedback gains K ij such that the whole fuzzy large-scale system 3.3 is asymptotically stable.
Theorem 3.1.The fuzzy large-scale system 2.1 can be asymptotically stabilized by the decentralized PDC fuzzy control 3.2 , if there exist matrices F ij , positive definite matrices X i to satisfy the following LMIs: for i 1, 2, . . ., J, j 1, 2, . . ., r i , n 1, 2, . . ., r i , where Proof.Let the Lyapunov functional be where P i > 0 is to be selected.It is obviously that there exist σ 1 and σ 2 such that Taking the derivative of the V i t along the trajectories of 3.3 , Vi t

3.8
Using Lemma 2.1, we get

3.9
Noticing that the facts as follows: x T i t M ik M T ik x i t

3.11
From Schur complement, we know where So we have V t < 0 while x i t i 1, 2, . . ., J are not all zero vectors.Note that the matrix inequalities in 3.13 can be transformed into certain forms of linear matrix inequalities LMIs .Therefore, multiplying both sides of matrix inequalities 3.13 by diag{P −1 i , I, I, I, I} and applying the change of variables such that P i X −1 i , K in F in X −1 i , i 1, 2, . . ., J, n 1, 2, . . ., r i , then 3.4 is obtained.
With the similar proof of Theorem 3.1, the stabilization criterion of large-scale system 2.1 without uncertainties is also discussed.The result is presented as follows.
Corollary 3.2.The fuzzy large-scale system 2.1 without uncertainties can be asymptotically stabilized by the decentralized PDC fuzzy control 3.2 , if there exist matrices F ij , positive definite matrices X i to satisfy the following LMIs: for i 1, 2, . . ., J, j 1, 2, . . ., r i , n 1, 2, . . ., r i , where Proof.The proof is similar with that of Theorem 3.1; therefore details are omitted.
Remark 3.3.This corollary is similar with Theorem 1 in 10 , so the theorem in this paper is more general.
Theorem 3.4.If there exist matrices F ij , positive definite matrices X i to satisfy the following LMIs:

3.15
the fuzzy large-scale system 2.1 can be asymptotically stabilized by the decentralized PDC fuzzy control 3.2 , and the performance index 2.8 is satisfied the following inequality: Proof.From the proof of Theorem 3.1, we know if L ik L T ik < 0 3.17 then V t < 0. Noticing that Q i and R i are positive matrices, if inequality holds, we obtain V t < 0: h in t K in x i t dt.

3.19
Using Lemma 2.3, we get

3.20
If 3.18 holds, we have

3.21
Therefore, multiply both sides of 3.18 by P −1 i , and let X i P −1 i , F in K in P −1 i .From Schur complement, the proof is completed.

Numerical Examples
In this section, some numerical simulations for uncertain fuzzy large-scale system will be given to illustrate the effectiveness of the proposed stabilization criteria and also compared with the existing results.
Example 4.1.Consider an uncertain FLSS time delays S composed of three fuzzy subsystems S i , i 1, 2, 3 by the following equations: It is noted that the above large-scale system without control u i t 0 has unstable responses with initial conditions x 1 t −3, 3 T , x 2 t −2, 2 T , x 3 t 1, −1 T as shown in Figure 1.
In order to stabilize the large-scale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.2. It is obvious that they are stabilized asymptotically.

The complete simulation results with initial conditions
Example 4.2.Consider the fuzzy large-scale system S composed of two subsystems S i as follows 25 :   Remark 4.3.From Figure 3, we can see that the system can be stabilized through appropriate decentralized control.Obviously, in Table 1, we can see the method of this paper has smaller gain matrices and performance index, so it has better control effectiveness.

Conclusions
In this paper we explore the stabilization problems for uncertain fuzzy large-scale system with time delays.The decentralized PDC fuzzy controller has been designed under some conditions such that the whole closed-loop large-scale fuzzy system is asymptotically stable.Then, the optimization design method for decentralized control is also considered with respect to a quadratic performance index.Finally, numerical examples are provided to demonstrate the correctness and less conservativeness of the theoretical results.However, there are still some other problems to be addressed, such that time-varying delays and delaydependent stability and stabilization of fuzzy large-scale system and the results developed in the paper can be extended to the case that the underlying systems are invovled with any switching dynamics.

Figure 2 :
Figure 2: The state response with fuzzy control by Theorem 3.1.

Figure 3 :
Figure 3: The state response with fuzzy control by Theorem 3.4.
Controller C 1 Rule 1.If x 11 t is M 111 then Rule 1.If x 12 t is M 112 then , M 111 x 11 t exp −x 2 11 t , M 211 x 11 t 1 − M 111 x 11 t , M 112 x 21 t 1/ 1 exp −x 21 t , M 212 x 21 t 1 − M 112 x 21 t .In order to stabilize the large-sale fuzzy system, three decentralized PDC fuzzy controllers are designed in the following.Rule 1.If x 11 t is M 111 then The control gain K ij for subsystem S 1 , S 2 is compared with 24 in Table 1.The complete simulation results with initial conditions x 1 t 1, −1 T , x 2 t 2, −2 T are shown in Figure 3.

Table 1 :
Comparison of fuzzy controller, for example, with Zhang et al. 25 .