## New Developments in Mathematical Control and Information for Fuzzy Systems

View this Special IssueResearch Article | Open Access

Yang Cui, Kaiqing Liu, Yang Zhao, Xue Wang, "Robust Control for a Class of Uncertain Switched Fuzzy Time-Delay Systems Based on T-S Models", *Mathematical Problems in Engineering*, vol. 2013, Article ID 234612, 8 pages, 2013. https://doi.org/10.1155/2013/234612

# Robust Control for a Class of Uncertain Switched Fuzzy Time-Delay Systems Based on T-S Models

**Academic Editor:**Peng Shi

#### Abstract

The problem of robust control for a class of uncertain switched fuzzy time-delay systems is discussed for system described by T-S fuzzy model with Lyapunov stable theory and linear matrix inequality approach. A sufficient condition in terms of the LMI is derived such that the stability of the closed-loop systems is guaranteed. The continuous state feedback controller is built to ensure the asymptotically stable closed-loop system for all allowable uncertainties, with the switching law designed to implement the global asymptotic stability of uncertain switched fuzzy time-delay systems. In this model, each and every subsystem of the switched systems is an uncertain fuzzy one to which the parallel distributed compensation (PDC) controller of each sub fuzzy system system is proposed with its main condition given in a more solvable form of convex combinations. Such a switched control system is highly robust to varying parameters. A simulation shows the feasibility and effectiveness of the design method.

#### 1. Introduction

In recent years, as intelligent control method, the research of the fuzzy system has been paid extensive attention [1–3]. The T-S model which is a fuzzy system is the most effective system model. This paper is focused on a class of uncertain switched time-delay systems, in which each subsystem is T-S fuzzy model. The T-S fuzzy model is a kind of fuzzy system proposed by Sugeno et al. [4, 5], which is described by a set of fuzzy IF-THEN rules representing local linear input-output relations of a nonlinear system. The main idea of T-S fuzzy model is to express the local dynamics of each fuzzy rule by a linear system model and to express the overall system by fuzzy “blending” of the local linear system models. The stability studies based on this model fuzzy system have had yielded fruitful results [6–10]. Recently, considerable effort [11–13] has been contributed to the problem of robust fuzzy control for a class of nonlinear systems that can be represented by T-S fuzzy models.

On the other hand, switched systems are an important class of hybrid systems. A switched system consists of a number of subsystems, both continuous time and discrete time dynamic systems, and a switching law, which orchestrates the switching between the subsystems. The applications in robot control systems, computer disc drives, and other engineering systems indicate that switched systems have extensive practice background. Therefore, it has the important theoretical significance and practical value, which has yielded fruitful research results [14–20]. A switched system is called a switched fuzzy system if all subsystems are fuzzy systems. This class of systems can often more precisely describe continuous dynamics and discrete dynamics as well as their interactions in actual systems. Compared with the results on stability of switched systems and those of fuzzy control systems, the results on switched fuzzy systems are very few. Reference [21] presents a novel switched T-S fuzzy control design approach based on control Lyapunov function. Reference [22] advances a fuzzy-basis-dependent and mode-dependent Lyapunov function to study the problems of stability analysis and controller design for discrete-time switched fuzzy systems.

This paper will study the problem of designing state feedback controllers for continuous-time T-S switched fuzzy systems. A new type of state feedback controllers, namely, switched parallel distributed compensation (PDC) controllers, are proposed, which are switched based on the values of membership functions. The problem begins with the representation models for switched systems. The design method inherits some hybrid features and presents the information of fuzzy system. The delay-independent sufficient condition in terms of the LMI is derived such that the quadratic stability of the closed-loop systems is guaranteed. The state feedback controller is built to ensure the asymptotically stable closed-loop system for all allowable uncertainties, with the switching law designed to implement the global asymptotic stability of uncertain fuzzy time-delay switched systems. In this model, each and every subsystem of the switched systems is an uncertain fuzzy one to which the PDC controller of each sub fuzzy system system is proposed with its main condition given in a more solvable form of convex combinations. Numerical example is given to illustrate the effectiveness of the proposed method.

#### 2. Problem Formulation

In this section, we consider the continuous uncertain switched fuzzy time-delay model; namely, every subsystem of switched systems is uncertain fuzzy time-delay system. Consider the following:

: if is and is , then where is the switching signal to be designed (Figure 1).

denote fuzzy sets in the switched subsystem. denotes the fuzzy inference rule in the switched subsystem. is the number of inference rules in the switched subsystem, and fuzzy rules are selected in every switched subsystem. is the state variable vector, is the input variable, is output variable vector, is external disturbance of the switched systems, and satisfies . is the delay constant, and satisfies . , , , , and are known constant matrices of appropriate dimensions of the switched subsystem. and are the uncertain matrices corresponding to the switched subsystem with appropriate dimensions. denotes a differentiable vector-valued initial function. are the premise variables.

In this paper, the switching signal is state dependent; namely, , the switching signal , and the switching signal is totally described by , that is, if and only , . will be designed later.

Through the function , the global model of the switched subsystem is described by where , and .

Also, , and , where denotes the membership function, and belongs to the fuzzy set .

*Definition 1. *The control problem for the switched fuzzy system (1) is stated as follows.

Let a constant be given. Find a continuous state feedback controller for each subsystem and a switching law such that(1)the closed-loop system is asymptotically stable when ,(2)the output satisfies under the zero initial condition.

#### 3. Main Results

*Assumption 2. *The uncertainties can be represented and emulated as
where, , , and are known constant matrices, and is an unknown time-varying matrix satisfying , for all .

Here, the PDC fuzzy controller design method is used for each sub fuzzy system namely, fuzzy controller and system (2) have the same fuzzy inference premise variables. Consider the following:

: if is and is , then

Thus, the global controller is

The global closed-loop system can be expressed as follows:

Lemma 3 (see [23]). *Let and be real matrices of appropriate dimensions, with ; then, one has that for any scalar ,
*

Lemma 4 (Schur complement). *The matrix is symmetrical matrix, and then the following two functions are equivalent:
*

Theorem 5. *Let a constant be given. If there exist the symmetric positive definite matrices , scalar , , and the matrices satisfying **then system (6) is globally asymptotically stable, and robust control problem is solvable under the switching law , where .**So, the gain matrices of switched fuzzy controllers and the switching law are given as follows: **where
*

*Proof. *We define and .

So, choose the Lyapunov function as follows:
We now compute the time derivative of the function (13):

It follows from Lemma 3 that for any of scalars and , we have that
When , the function (14) can be given as follows: That is,
where . So, we need to prove that , the closed-loop system (6) is asymptotically stable when .

Inequalities (9) are pre- and postmultiplied by the transformation , respectively. In turn, we have tha following:

Due to Lemma 4, inequality (18) can be changed as follows:
So, for any ,

Due to the switching law (11), for all , the inequality is tenable as follows:

Let
Constructing the sets , obviously, we have , and .

Now, we focus on designing a switching law as follows:
When , is correct. So, we have that ; so, the system is asymptotically stable.

From the design of switching law, we can obtain that , for all ; that is, the switching fuzzy controllers can make the system (6) asymptotically stable, when .

To demonstrate that the control problem of system is solvable, we define that

Thus, when the initial state is , for any nonzero vector , the function can be written as follows:

Due to the Lemma 3, the function (14) can be changed as follows:

So, the function can be further expressed as follows:

From the design of switching law, we get that ; that is, for any , holds.

#### 4. Simulation Example

The switched systems (1) consists of two fuzzy subsystems, and each subsystem has two fuzzy rules; that is, , and , where . Hence, we approximate the system by the following modes:

: if is , then : if is , then : if is , then : if is , then where

The membership functions are, respectively,

Let , choosing , where .

For the linear matrix inequality (9) of Theorem 5, the following can be obtained with LMI toolbox: Because , we have that Designing the switching law by (11), the system state responses with the initial condition are depicted by Figure 2 which is obtained by the Matlab simulation.

Obviously, the closed-loop switched fuzzy system is stable under the corresponding switching laws and the robust fuzzy controllers in Figure 3, in spite of the influences by parameter uncertainties and time-delay, and possesses the strong robustness. Comparing their results with others, the simulation results indicate that the design controllers can satisfy the robust performance requirements under the corresponding switching laws.

#### 5. Conclusions

In this paper, the problem of state feedback robust control for a class of uncertain switched fuzzy time-delay systems based on T-S models is discussed. The sufficient condition in terms of the LMI is derived such that the quadratic stability of the closed-loop systems is guaranteed. A new type of continuous state feedback PDC controllers are built to ensure the asymptotically stable closed-loop system with disturbance attenuation level , with the switching law designed to implement the global asymptotic stability of uncertain fuzzy time-delay switched systems. The results are also extended to the interval fuzzy time-delay systems. Simulation results demonstrate that a quality control performance has been achieved.

#### Acknowledgment

This work is partially supported by Scientific Research Fund of Heilongjiang Provincial Education Department 2010 (no. 11551092).

#### References

- K. Lee, E. T. Jeung, and H. B. Park, “Robust fuzzy ${H}_{\infty}$ control for uncertain nonlinear systems via state feedback: an LMI approach,”
*Fuzzy Sets and Systems*, vol. 120, no. 1, pp. 123–134, 2001. View at: Publisher Site | Google Scholar - M. J. Wang and M. F. Yang, “Application of signed distances method to integrate linear programming for fuzzy multi-objective project management,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 1, pp. 237–252, 2011. View at: Google Scholar - M. H. Bahari, A. Karsaz, and N. Pariz, “High maneuvering target tracking using a novel hybrid Kalman filter-fuzzy logic architecture,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 2, pp. 501–510, 2011. View at: Google Scholar - T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,”
*IEEE Transactions on Systems, Man and Cybernetics*, vol. 15, no. 1, pp. 116–132, 1985. View at: Google Scholar - K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,”
*Fuzzy Sets and Systems*, vol. 45, no. 2, pp. 135–156, 1992. View at: Google Scholar - W. J. Wang and C. H. Sun, “Relaxed stability and stabilization conditions for a T-S fuzzy discrete system,”
*Fuzzy Sets and Systems*, vol. 156, no. 2, pp. 208–225, 2005. View at: Publisher Site | Google Scholar - X. Liu and Q. Zhang, “Approaches to quadratic stability conditions and ${H}_{\infty}$ control designs for T-S fuzzy systems,”
*IEEE Transactions on Fuzzy Systems*, vol. 11, no. 6, pp. 830–839, 2003. View at: Publisher Site | Google Scholar - M. Wang, B. Chen, and S. Tong, “Adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown time delays,”
*International Journal of Innovative Computing, Information and Control*, vol. 4, no. 4, pp. 829–837, 2008. View at: Google Scholar - D. H. Lee, J. B. Park, Y. H. Joo, and H. S. Moon, “Further studies on stabilization conditions for discrete-time Takagi-Sugeno fuzzy systems,” in
*Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '09)*, pp. 831–835, August 2009. View at: Publisher Site | Google Scholar - L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 41, no. 1, pp. 273–286, 2011. View at: Publisher Site | Google Scholar - X. J. Su, L. G. Wu, P. Shi, and Y. D. Song, “${H}_{\infty}$ model reduction of T-S Fuzzy stochastic systems,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 42, no. 2, pp. 1574–1585, 2012. View at: Google Scholar - L. Wu, X. Su, P. Shi, and J. Qiu, “Model approximation for discrete-time state-delay systems in the TS fuzzy framework,”
*IEEE Transactions on Fuzzy Systems*, vol. 19, no. 2, pp. 366–378, 2011. View at: Publisher Site | Google Scholar - W. J. Chang, C. C. Ku, and P. H. Huang, “Robust fuzzy control via observer feedback for passive stochastic fuzzy systems with time-delay and multiplicative noise,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 1, pp. 345–364, 2011. View at: Google Scholar - J. Daafouz, P. Riedinger, and C. Iung, “Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 11, pp. 1883–1887, 2002. View at: Publisher Site | Google Scholar - Z. He, M. Wang, and S. Liu, “Discrete sliding mode prediction control of uncertain switched systems,”
*Journal of Systems Engineering and Electronics*, vol. 20, no. 5, pp. 1065–1071, 2009. View at: Google Scholar - F. Li and X. Zhang, “A delay-dependent bounded real lemma for singular LPV systems with time-variant delay,”
*International Journal of Robust and Nonlinear Control*, vol. 22, no. 5, pp. 559–574, 2012. View at: Publisher Site | Google Scholar - Z. Wu, P. Shi, H. Su, and J. Chu, “Delay-dependent stability analysis for switched neural networks with time-varying delay,”
*IEEE Transactions on Systems, Man and Cybernetics B*, vol. 41, no. 6, pp. 1522–1530, 2011. View at: Publisher Site | Google Scholar - Z. L. He, L. G. Wu, and X. J. Su, “${H}_{\infty}$ control of linear positive systems: continuous-and discrete-time cases,”
*International Journal of Innovative Computing, Information and Control*, vol. 5, no. 6, pp. 1747–1756, 2009. View at: Google Scholar - E. Fridman and U. Shaked, “Delay-dependent stability and ${H}_{\infty}$ control: constant and time-varying delays,”
*International Journal of Control*, vol. 76, no. 1, pp. 48–60, 2003. View at: Publisher Site | Google Scholar - Z. L. He, J. F. Wu, G. H. Sun, and C. Gao, “State estimation and sliding mode control of uncertain switched hybrid systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 20, no. 5, pp. 1065–1071, 2009. View at: Google Scholar - S. S. Chen, Y. Chang, J. L. Wu, W. C. Cheng, and S. F. Su, “Fuzzy control design for switched nonlinear systems,” in
*Proceedings of the SICE Annual Conference of International Conference on Instrumentation, Control and Information Technology*, pp. 352–357, August 2008. View at: Publisher Site | Google Scholar - Y. Chen, L. X. Zhang, H. R. Karimi, and X. D. Zhao, “Stability analysis and controller design of a class of switched discrete-time fuzzy systems,” in
*Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference*, pp. 6159–6164, 2011. View at: Google Scholar - I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,”
*Systems and Control Letters*, vol. 8, no. 4, pp. 351–357, 1987. View at: Google Scholar

#### Copyright

Copyright © 2013 Yang Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.