Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 2597071, 13 pages
https://doi.org/10.1155/2017/2597071

Research Article

Control for T-S Fuzzy Singularly Perturbed Switched Systems

Qianjin Wang, Linna Zhou, Wei Dai, and Xiaoping Ma

School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, China

Correspondence should be addressed to Linna Zhou; linnazhou@cumt.edu.cn

Received 20 March 2017; Revised 4 August 2017; Accepted 12 September 2017; Published 15 October 2017

Academic Editor: Asier Ibeas

Copyright © 2017 Qianjin Wang 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.

Abstract.

This paper is concerned with the design of fuzzy controller with guaranteed performance for a class of Takagi-Sugeno (T-S) fuzzy singularly perturbed switched systems. First, by using the average dwell time approach together with the piecewise Lyapunov function technique, a state feedback controller that depends on the singular perturbation parameter is developed. This controller is shown to work well for all . Then, for sufficiently small an -independent controller design method is proposed. Furthermore, under the -independent controller, the -bound estimation problem of the overall switched closed-loop system is solved. Finally, an inverted pendulum system is used to evaluate the feasibility and effectiveness of the obtained results.

1. Introduction

Many practical systems possess multiple time scale characteristics [1–4]. It is well known that control methods for normal systems can not be directly applied to this class of systems, since these methods may cause ill-conditioned numerical problems. To conquer these problems, singular perturbation methods have been widely used in control design of multiple time scale systems (see [5–9] and the references therein).

Singularly perturbed systems (SPSs), whose partial time derivatives involve a small singular perturbation parameter , have been widely investigated by many researchers. It is important to obtain the -bound such that stability and other performances of SPSs can be ensured. Studies on -bound can be mainly divided into two types: one only presents sufficient conditions for the existence of the -bound [10, 11], and the other proposes methods to estimate the -bound [9, 12].

Singularly perturbed switched systems (SPSSs) consist of a group of SPSs and a certain switching law, which specifies the active SPS at the switching instance. In practical processes, a great number of control systems, whose behavior is simultaneously determined by multiple time scales and switching, can be modeled as SPSSs [13–15]. The control model of the hot strip mill was treated as a SPSS and an robust controller was designed in [15]. Recently, SPSSs have attained much attention in the literature (see [16–20]). It has been shown in [17–20] that the stability of fast/slow switched subsystems can not guarantee the stability of the original switched systems. In order to guarantee the system stability under an arbitrary switching signal, a common Lyapunov function for individual SPSs or a dwell time scheme has to be considered. In [19], by combining the multiple Lyapunov functions method with the dwell time scheme, sufficient conditions for ensuring exponential stability of time-delay SPSSs with stable fast switched subsystems were derived. These conditions were described by a few -dependent algebraic inequalities, which led to a heavy and tedious calculation. The proposed method in [19] was further extended to time-delay SPSSs with impulsive effects [20]. However, few results on the -bound estimation are available for SPSSs. The exceptions were given by [21, 22]. A convex optimization based method was presented to make the best estimate of -bound for SPSSs whose fast switched subsystems can not depend on the switching signal in [21]. The -dependent stabilization and -bound problems were addressed by adopting dwell time switching signal and constructing -dependent multiple Lyapunov functions for SPSSs in [22].

Fuzzy control has been widely used for engineering practice [23, 24]. The Takagi-Sugeno (T-S) model, which has elegant ability to approximate a certain class of complex nonlinear functions, was extensively utilized in control system design [25–27]. Based on T-S fuzzy control, many LMI-based fuzzy control design methods have been developed for SPSs. The stabilization problem was addressed for fuzzy SPSs in [28], and some LMI-based control algorithms were proposed. Over past a few years, control for fuzzy SPSs has attained a lot of attention. control was addressed for fuzzy SPSs in [11, 29–31], and sufficient conditions independent of for the existence of controller were presented. The resulting conditions in [11, 28–31] are only applicable to stabilizing the system and being with an performance for sufficiently small . Hence, some researchers have concentrated on the -bound design problem for fuzzy SPSs [32–34].

As an important class of hybrid systems, switched fuzzy systems have been a hot spot of present research. Recently, a great number of theoretical results are available for switched fuzzy systems. By using a common or multiple Lyapunov functions method, stability issues for switched fuzzy systems were considered in [35–39]. However, the existing literature on fuzzy SPSSs is rather limited. An LMI-based dynamic state feedback control method was given and the -bound problem of system was solved for nonaffine-in-control SPSSs [40]. This method decomposed the switched system into fast and slow subsystems and can not be suitable for nonstandard fuzzy SPSSs. In our previous paper [41], where the stabilization and -bound problems were addressed for T-S fuzzy SPSSs without the external disturbance input by constructing the -dependent piecewise Lyapunov function. Moreover, to the authors’ best knowledge, control for fuzzy SPSSs has not been addressed yet.

In this paper, we will investigate the design of fuzzy controller with guaranteed performance for a class of T-S fuzzy SPSSs. The problem is composed of stabilization control, control, and -bound design. First, for a given upper bound for and a prespecified performance bound , an -dependent controller is developed, such that, for any , the switched system is asymptotically stable and the -gain from the disturbance input to the controlled output is less than or equal to . This controller is shown to work well for all . Then, an -independent controller design method is proposed in terms of LMIs. Furthermore, under the -independent controller, the -bound estimation approach is given. Finally, an inverted pendulum system is used to evaluate the feasibility and effectiveness of the obtained results.

The rest of this paper is organized as follows. In Section 2, the problems to be considered are formulated and preliminaries are presented. The main results are given in Section 3. An example is given in Section 4 to illustrate the obtained methods. And Section 5 concludes the paper. The notations used in this paper are standard. The notations and stand for the matrix transpose and the transpose of the off diagonal element of the LMI, respectively. and denote the maximal and minimal eigenvalues of a symmetric matrix , respectively. denotes Euclidean norm for vectors or the spectral norm of matrices. is defined as for a square matrix .

2. Problem Formulation

Consider a T-S fuzzy SPSS, which involves rules of the following form.

The th rule is

where , is a small positive scalar which represents the singular perturbation parameter. is a piecewise constant function with respect to time, referred as to a switching signal, which takes its values in the finite set , represents the number of individual subsystems. are fuzzy sets, is the number of fuzzy rules, is the premise vector that may depend on states in many cases, is the number of premise variables, is the state vector, is the control input, is the disturbance input that belongs to , and is the controlled output. For any given , a time sequence of is labeled as the switching instants over the interval . It means that the th T-S fuzzy subsystem is active as . And , , , and are of the following form:

Denote

where is the grade of membership of in

It is assumed in this paper that

Let

Then

For the convenience of notations, we denote ,  

Then, the th T-S fuzzy subsystem can be inferred as

Throughout the paper, it is assumed that the singular perturbation parameter is available for feedback. By using the concept of parallel distributed compensation (PDC), the state feedback fuzzy controller can be described by the following.

The th rule is

Because the controller rules are the same as the plant rules, the state feedback controller is given as follows:

Remark 1. In many SPSs, is usually a known physical parameter. Based on the fact that is available for feedback control, some synthesis problems are considered in [19, 20, 22]. In [22], for a given -bound , under the dwell time switching law, the -dependent multiple Lyapunov function method was proposed to ensure exponential stability of the original switched system.

Substituting (9) into (7) yields the closed-loop system

Upon introducing the indicator function

where if the switching system is in mode and if it is in a different mode, and the overall switched closed-loop system can be expressed as follows:

We now recall standard notations and preliminaries, which will help formulate our main results.

Definition 2 (see [42]). For , and the initial condition , the equilibrium of system (7) is said to be asymptotically stable under certain switching signal if there exist constants , such that the solution of the system satisfies

Definition 3 (see [43]). For any switching signal and any , let denote the number of discontinuities in the interval . We say that has the average dwell time property if

holds, where and are called the chatter bound and average dwell time, respectively. As commonly used in the literature, we choose .

Definition 4 (see [30]). Given , a system of the form (7) is said to be with an -norm less than or equal to if

holds for Where is the terminal time of control and denotes the initial condition of system (7).

Lemma 5 (see [44]). Given any constant and any matrices , , of compatible dimensions, then we have

for all , where is an uncertain matric satisfying

Lemma 6 (see [34]). For a positive scalar and the symmetric matrices and of compatible dimensions, if the inequalities

hold, then

Lemma 7 (see [34]). If there exist matrices with satisfying

then

where

The problems under consideration are formulated as follows.

Problem 8. Given an performance bound and an upper bound for the singular perturbation parameter , under admissible switching signals with ADT property, determine a state feedback controller of form (9), such that, for all , the overall switched closed-loop system (12) is asymptotically stable and with an -norm less than or equal to .

Problem 9. Given an performance bound , determine a state feedback controller of the form (9), such that, under admissible switching signals with ADT property, the overall switched closed-loop system (12) is asymptotically stable and with an -norm less than or equal to for any sufficiently small .

Problem 10. Given an performance bound and a controller, determine an -bound , as large as possible, such that, for any , under admissible switching signals with ADT property, the overall switched closed-loop system (12) is asymptotically stable and with an -norm less than or equal to .

Remark 11. The synthesis problems for T-S fuzzy SPSs have attracted much attention of many researchers. control and -bound design for T-S fuzzy SPSs with pole placement constraints were considered in [34]. This paper will extend the stability analysis and control methods for normal systems to T-S fuzzy SPSSs. Problem 8 considers the stabilization controller design, -bound design, and control. Problem 9 aims to design a controller without considering the -bound. Problem 10 is used to estimate the -bound of the switched system.

3. Controller Design

This section will present a controller design method to solve Problem 8.

Theorem 12. Give an performance bound , an upper bound , and two constants, and , if there exist matrices , , , and of compatible dimensions with , such that

where , , , , and .
Then, for any , the overall switched closed-loop system (12) with , is asymptotically stable and with an -norm less than or equal to under any switching signal with ADT

Proof. Based on Lemma 6, LMIs (22) and (23) imply that

By the Schur complement, inequality (29) is equivalent to

Pre- and postmultiplying (30) by and its transpose, respectively, we obtain

where and .
Using Lemma 6 again, it follows from (24) and (25) that

where
By the Schur complement, inequality (32) can be replaced by the following inequality:

By using Lemma 5, we get from inequality (33) that

Pre- and postmultiplying (34) by and its transpose, respectively, we have

By Lemma 7, LMI conditions (20) and (21) guarantee that the inequality

holds, which implies

Define the piecewise Lyapunov function

where is switched among , , in accordance with the piecewise constant switching signal
Computing the derivative of with respect to along the trajectories of system (12), we have

Using Lemma 5 again, we obtain

From equality (39) and inequality (40), it follows that

It follows from (31), (35), and (41) that

Furthermore, by Lemma 6, LMIs (26) and (27) imply that

Applying the Schur complement to (43) shows that

Pre- and postmultiplying (44) by and its transpose, respectively, and taking into account the fact that , inequality (44) is equivalent to

By the Schur complement, it follows from (45) that

Then, the following properties are obtained for (38):
() Each is continuous and its derivative along the trajectories of the corresponding subsystem satisfies (42).
() There exist constant scalars , , such that

where , .
() There exists a constant scalar such that (46) holds.
Thus, is piecewise monotonically decreasing and its value at switching instants is nonincreasing.
Using the differential inequality (42), we obtain that

where
It follows from (48) that

The following proof consists of two parts. First, we will show that the overall switched closed-loop system (12) with is asymptotically stable under any switching signal with ADT (28). Then, we will verify that system (12) is with an -norm less than or equal to
Part 1. We consider the following average dwell time scheme: for any and a positive scalar smaller than ,

It follows from (50) that . Then, from (49), we obtain

where , and one can see from [45] that

which indicates that system (12) with is asymptotically stable under any switching signal with ADT (28).
Part 2. Similar to Part , for a positive scalar smaller than , the inequality

holds.
Taking into account the fact that and , it follows from (49) that

It follows from (53) and (54) that

which implies that This completes the proof.

Remark 13. , , where , It follows from LMIs (20) and (21) that and , which imply that the matrices are nonsingular. So, the matrices are nonsingular for all . This nonsingularity can ensure that always work well for all . For sufficiently small , the -dependent controller is reduced to an -independent one, since .

Remark 14. The multiple Lyapunov functions method has been widely used in control design of switched systems [36–39]. By employing the average dwell time scheme, the problem of extended dissipative state estimation for a class of discrete-time Markov jump neural networks with unreliable links was addressed in [36]. In this paper, the -dependent piecewise Lyapunov function will be constructed to solve control problem for T-S fuzzy SPSSs.

Remark 15. Theorem 12 is concerned with the situation that is known according to prior information. Moreover, the bisectional search algorithm developed in [46] can be used to derive the -bound.

Remark 16. -bound, which is an essential index of SPSs, has attained much attention. -bound was considered in [40, 47]. Both results were derived by constructing a common Lyapunov function, which may lead to conservatism in some cases. In [41], the piecewise Lyapunov function was constructed and -bound estimation problem was solved for T-S fuzzy SSPSs without the external disturbance input.

By Theorem 12, under the assumption that the singular perturbation parameter is known, sufficient conditions for both stability and performance of system (12) are derived. In the following theorem, for sufficiently small and unknown , the above sufficient conditions are generalized to design the -independent controller.

Theorem 17. Given an performance bound , two constants, and , if there exist matrices , , , and of compatible dimensions with , such that

where , , , , and .
Then there exists a positive scalar such that, for all , the overall switched closed-loop system (12) with the controller gains of the form is asymptotically stable and with an -norm less than or equal to under any switching signal with ADT

Proof. For sufficiently small , LMI conditions (20)–(27) in Theorem 12 can be reduced to LMI conditions (56)–(61) in Theorem 17. Thus, we omit the proof of Theorem 17 that can be carried out by referring to the standard techniques used in Theorem 12.

-bound is an essential stability index of SPSs. Theorem 17 ensures the existence for -bound In the following theorem, we will propose a method to estimate the -bound of the closed-loop system with the obtained controllers in Theorem 17.

Theorem 18. Give an performance bound , an upper bound , controller gains , and two constants, and , if there exist matrices , , and of compatible dimensions with , such that

where , , , , and .
Then, for all , the overall switched closed-loop system (12) is asymptotically stable and with an -norm less than or equal to under any switching signal with ADT

4. Example

To illustrate the proposed results, we consider the well-known inverted pendulum system. The equations of motion for the pendulum are given by

where denotes the angle of the pendulum from the vertical upward, , is the gravity acceleration, , and are the masses of the pendulum and the cart, respectively, is the length of the pendulum, is a horizontal force applied to the cart, and is the external disturbance variable, which is a piecewise function of time of the form

is the controlled output. The parameters for the plant are as follows:  m/s²,  Kg,  Kg, and  m.

The angle of the pendulum is divided into two areas and , where is ; is , which results in two fuzzy subsystems [39].

For the individual system, we choose the membership functions of the fuzzy sets as follows.

Mode 1

Mode 2

Then, the dynamics of Mode can be exactly represented by the following T-S fuzzy model under :

where

The dynamics of Mode can be exactly represented by the following T-S fuzzy model under :

where

Choosing , the above switched system can be modeled by a SPSS (1) with the following.

Mode 1

Mode 2

The fuzzy controller is described as follows.

Mode 1

Mode 2

Taking , , and and solving the LMIs in Theorem 12, we obtain the stabilization controller gains:

Taking and and solving the LMIs in Theorem 17, we obtain the stabilization controller gains:

Under the controller obtained by Theorem 17, the -bound of the closed-loop system is by using Theorem 18 and the bisectional search algorithm developed in [46].

To illustrate the proposed method, we first consider the simulation of system (10) without the controller and then apply the designed controller to system (10).

Choosing , , , , , , and any switching signal with ADT (71), the simulation result without the controller is shown in Figure 1. It can be seen from Figure 1 that system (10) is not stable. Applying the fuzzy controller obtained by Theorem 12 to the original system, the sate trajectories of the overall switched closed-loop system are shown in Figure 2 and the ratio of the output energy to the disturbance input energy, that is, , is depicted in Figure 3. It is easy to find that after 5 seconds the ratio of the output energy to the disturbance input energy is fixed at a constant value, which is about . So , which is less than the prescribed value 1.

Figure 1: State trajectories without the controller.

Figure 2: State trajectories.

Figure 3: The ratio of the output energy to the disturbance input energy.

5. Conclusion

In this paper, we are concerned with the design of fuzzy controller with guaranteed performance for T-S fuzzy SPSSs. An LMI-based method of designing an -dependent controller has been proposed. Through this method, the obtained controller can work well for any . This controller guarantees that, for a given upper bound for and a prespecified performance bound , under admissible switching signals, the switched system is asymptotically stable and with an -norm less than or equal to . Then, for sufficiently small , the -independent feedback controller has been developed. Furthermore, under this controller, the -bound estimation problem of the switched system has been solved. The involved example has shown the feasibility and effectiveness of the obtained results.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61374043, 61603392, 61603393, and 61503384), Nature Science Foundation of Jiangsu Province (BK20160275, BK2010275, and BK20150199), the Open Project Foundation of State Key Laboratory of Synthetical Automation for Process Industries under Grant PAL-N201706, Postdoctoral Science Foundation of China (2015M581885), and Ordinary University Graduate Student Scientific Research Innovation Projects of Jiangsu Province (KYLX16−0533).

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