New Approach on Robust and Reliable Decentralized H ∞ Tracking Control for Fuzzy Interconnected Systems with Time-Varying Delay

This paper investigates the robust and reliable decentralized H ∞ tracking control issue for the fuzzy large-scale interconnected systems with time-varying delay, which are composed of a number of T-S fuzzy subsystems with interconnections. Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then, according to the Lyapunov direct method and the decentralized control theory of large-scale interconnected systems, the new linear matrix inequalities(LMIs-) based conditions with some free variables are derived to guarantee the H ∞ tracking performance not only when all control components are operating well, but also in the presence of some possible actuator failures. Moreover, there is no need for the precise failure parameters of the actuators, rather than the lower and upper bound. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.


Introduction
Large-scale interconnected systems, such as electrical power systems, computer communication systems, economic systems, and process control systems, have attracted great interests from many researchers in recent years.Takagi-Sugeno (T-S) fuzzy model has become a popular and effective approach to control complex systems, and a lot of significant results on stabilization and  ∞ control via linear matrix inequality (LMI) approach have been reported; see [1][2][3][4].Compared with the centralized control, the decentralized scheme is preferred in the control design issue of the largescale interconnected systems [5].Recently, there are some works about stability and stabilization of fuzzy large-scale systems [6][7][8][9].It is well known that delays appear in many dynamic systems, which are potential causes of system instability [10,11].The tasks of stabilization and tracking are two typical control problems.In general, tracking problems are more difficult than stabilization problems especially for nonlinear systems [12].Reference [13] has given decentralized  ∞ fuzzy model reference tracking control design, and the stable conditions in the sense of Lyapunov are given.The technology of descriptor model transformation is used in [14,15].A T-S fuzzy descriptor tracking control design for nonlinear systems with a guaranteed  ∞ model reference tracking performance is discussed [16].However, in practical situations, failure of actuators often occurs.Thus, an important requirement is to design a reliable controller such that the stability and performance of the closed-loop system can tolerate actuator failures [17][18][19][20].
In this paper, we investigate the robust and reliable decentralized  ∞ tracking control issue for the fuzzy interconnected systems with time delay, which are composed of a number of T-S fuzzy subsystems with interconnections.Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then, according to the Lyapunov direct method and the decentralized control theory of large-scale interconnected systems, the new LMIs-based conditions with some free variables are derived to guarantee the  ∞ tracking performance not only when all control components are operating well, but also in the presence of some possible actuator failures.Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.
The innovation of this paper can be summarized as follows: (1) the more practical  ∞ performance index is used, 2 ISRN Applied Mathematics which considers not only the effect of tracking error () but also the effect of control (); (2) utilizing the descriptor model transformation, the new LMIs-based reliable  ∞ performance conditions with some free variables are derived; (3) there is no need for the precise failure parameters of the actuators, rather than the lower and upper bound of failure parameters.
In the following sections, the identity matrices and zero matrices are denoted by  and 0, respectively.  denotes the transpose of matrix .R  denotes the -dimensional Euclidean space.The standard notation > (<) is used to denote the positive (negative-) definite ordering of matrices.Inequality  >  shows that the matrix  −  is positively definite.The symbol of * denotes the transposed element in the symmetric position.

Systems Description
Suppose there are the interconnected systems consisting of  interconnected subsystems   ,  = 1, . . ., .Each rule of the subsystem   is represented by a T-S fuzzy model as follows: where   () ∈ where where  1 , . . ., where   () denote the reference states,   denote the specific asymptotically stable matrices,   denote the system matrices with appropriate dimensions, and V  () denotes the bounded reference input.

𝐻 ∞ Tracking Control Design
According to the conventional parallel distributed compensation (PDC) concept, the fuzzy controllers corresponding to   are used as follows: where    are the controllers gains of the th rule for subsystem   .
And the final output of the fuzzy controllers for each subsystem   is Instead of actuator outage, a more general actuator failure model is adopted in this paper.Let    () be the control input vector after failures have occurred.The following actuator failure model is adopted: where   = diag( 1 , . . .,    ) with 0 ≤   ≤   ≤   ≤ 1,  = 1, . . ., ,  = 1, . . .,   .Note that the parameters   and   characterize the admissible failures of the th actuator in the th subsystem.
Let the   no failure actuators are the last   actuators of th subsystem, according to Lemma 4, one has where

Denote the tracking error by
Then, the whole closed-loop fuzzy interconnected systems become where Equation ( 13) can be transformed to be the descriptor system form as follows: where Definition 5.The  ∞ tracking control problem for the interconnected systems ( 13) is to design the controllers to minimize the prescribed level of disturbance attenuation ∑  =1   > 0,  = 1, . . ., , if the following two conditions are satisfied.
(2) For the zero initially condition Remark 6.From Definition 5, we can see that, compared with the condition commonly used for the  ∞ tracking control problem, the practical  ∞ performance index considers not only the effect of () but also the effect of ().That is to say, the condition that the reducing of the tracking error () needs to cost the much bigger gain of the controller can be avoided effectively.
ISRN Applied Mathematics where Proof.We choose the following Lyapunov function for the whole interconnected system (13): where   ,   are positive definite matrices,   P = P   ≥ 0, and denote Computing the time derivative of (  ), we have From the Leibniz-Newton formula, the following equation is considered: ẋ  () ) = 0. (24) From (24), we have where where According to Lemma 3, we have Letting  1 =    2 and using (15), we have where From (29), we can see that, if Π  < 0, then Π   < 0, where If then the whole nonlinear interconnected systems are asymptotically stable.

By Schur complement and premultiplying and postmultiplying to (29) by positive-define matrix diag[𝑃
where Denoting ], using (13), and by Schur complement, the proof is completed.
Remark 8. Firstly, utilizing the descriptor model transformation, LMIs-based conditions with some free variables are derived.Secondly, when   () = 0, compared with the stable conditions in the sense of Lyapunov in [13], we obtain the asymptotically stable conditions.Finally, compared with [16], the time-varying delay fuzzy large systems are considered, and a more practical  ∞ performance index is considered.Therefore, obtained results are new and less conservative.
If  = 1, then the T-S model-based interconnected systems (13) can be transformed into a common T-S system as follows: where where (41)

Illustrative Examples
Example 1.Consider the two-machine interconnected systems which are composed of two subsystems as follows [13]: We assume the two-machine interconnected systems' parameters as follows: The systems are approximated by the following nine-rule fuzzy model which is the same as [13], and the details of the rules are omitted here.
Let  [16], the gain of controller is much smaller, and the control performances are nearly the same.For the different , , the trajectories of  2 are shown in Figure 7.

Conclusion
This paper investigates the robust and reliable decentralized  ∞ fuzzy tracking control issue for the fuzzy interconnected systems with time-varying delay, which consist of a number of T-S fuzzy subsystems with interconnections.Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then the new LMIs-based conditions with some free variables are derived to guarantee the  ∞ tracking performance not only when all control components are operating well, but also in the presence of some possible actuator failures.Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.