<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Control for T-S Fuzzy Singularly Perturbed Switched Systems

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

doi:https://doi.org/10.1155/2017/2597071

Mathematical Problems in Engineering

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Advanced Control for Singular Systems with Applications

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Research Article | Open Access

Volume 2017 | Article ID 2597071 | https://doi.org/10.1155/2017/2597071

Control for T-S Fuzzy Singularly Perturbed Switched Systems

Qianjin Wang,¹Linna Zhou,¹Wei Dai,¹and Xiaoping Ma¹

Academic Editor: Asier Ibeas

Received20 Mar 2017

Revised04 Aug 2017

Accepted12 Sept 2017

Published15 Oct 2017

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 iswhere , 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 ifholds, 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 ifholds 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 havefor 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 inequalitieshold, then

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

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 thatwhere , , , , 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 thatBy the Schur complement, inequality (29) is equivalent toPre- and postmultiplying (30) by and its transpose, respectively, we obtainwhere and .
Using Lemma 6 again, it follows from (24) and (25) thatwhere
By the Schur complement, inequality (32) can be replaced by the following inequality:By using Lemma 5, we get from inequality (33) thatPre- and postmultiplying (34) by and its transpose, respectively, we haveBy Lemma 7, LMI conditions (20) and (21) guarantee that the inequalityholds, which implies Define the piecewise Lyapunov functionwhere 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 haveUsing Lemma 5 again, we obtainFrom equality (39) and inequality (40), it follows thatIt follows from (31), (35), and (41) thatFurthermore, by Lemma 6, LMIs (26) and (27) imply thatApplying the Schur complement to (43) shows thatPre- and postmultiplying (44) by and its transpose, respectively, and taking into account the fact that , inequality (44) is equivalent toBy the Schur complement, it follows from (45) thatThen, 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 thatwhere
It follows from (48) thatThe 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 obtainwhere , and one can see from [45] thatwhich 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 inequalityholds.
Taking into account the fact that and , it follows from (49) thatIt follows from (53) and (54) thatwhich 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 thatwhere , , , , 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 thatwhere , , , , 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 bywhere 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.

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).

References

G. Peponides, P. Kokotovic, and J. Chow, “Singular perturbations and time scales in nonlinear models of power systems,” IEEE Transactions on Circuits and Systems, vol. 29, no. 11, pp. 758–767, 1982.
View at: Google Scholar
W. W. Chen, M. Niepel, and P. K. Sorger, “Classic and contemporary approaches to modeling biochemical reactions,” Genes & Development, vol. 24, no. 17, pp. 1861–1875, 2010.
View at: Publisher Site | Google Scholar
P. Kokotović, H. K. Khalil, and J. O'reilly, Singular Perturbation Methods in Control: Analysis and Design, Siam, 1999.
S. Bellew and E. O'Riordan, “A parameter robust numerical method for a system of two singularly perturbed convection-diffusion equations,” Applied Numerical Mathematics, vol. 51, no. 2-3, pp. 171–186, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
P. V. Kokotovic, R. E. Omalley, and P. Sannuti, “Singular perturbations and order reduction in control theory-an overview,” Automatica, vol. 12, no. 2, pp. 123–132, 1976.
View at: Google Scholar
A. Saberi and H. Khalil, “Quadratic-type Lyapunov functions for singularly perturbed systems,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 29, no. 6, pp. 542–550, 1984.
View at: Publisher Site | Google Scholar | MathSciNet
E. Fridman, “Effects of small delays on stability of singularly perturbed systems,” Automatica, vol. 38, no. 5, pp. 897–902, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
F. Sun, L. Zhou, Q. Zhang, and Y. Shen, “Stability bound analysis and synthesis for singularly perturbed systems with time-varying delay,” Mathematical Problems in Engineering, vol. 2013, Article ID 517258, 8 pages, 2013.
View at: Publisher Site | Google Scholar
L. Saydy, “New stability/performance results for singularly perturbed systems,” Automatica, vol. 32, no. 6, pp. 807–818, 1996.
View at: Publisher Site | Google Scholar | MathSciNet
H. D. Tuan and S. Hosoe, “Multivariable circle criteria for multiparameter singularly perturbed systems,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 45, no. 4, pp. 720–725, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
H. Liu, F. Sun, and Y. Hu, “H∞ control for fuzzy singularly perturbed systems,” Fuzzy Sets and Systems, vol. 155, no. 2, pp. 272–291, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
L. Cao and H. M. Schwartz, “Complementary results on the stability bounds of singularly perturbed systems,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 49, no. 11, pp. 2017–2021, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
I. Malloci, Z. Lin, and G. Yan, “Stability of interconnected impulsive switched systems subject to state dimension variation,” Nonlinear Analysis: Hybrid Systems, vol. 6, no. 4, pp. 960–971, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
I. Malloci, J. Daafouz, C. Iung, R. Bonidal, and P. Szczepanski, “Switched system modeling and robust steering control of the tail end phase in a hot strip mill,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 239–250, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
I. Malloci, J. Daafouz, C. Iung, R. Bonidal, and P. Szczepanski, “Robust steering control of hot strip mill,” IEEE Transactions on Control Systems Technology, vol. 18, no. 4, pp. 908–917, 2010.
View at: Publisher Site | Google Scholar
F. El Hachemi, M. Sigalotti, and J. Daafouz, “Stability analysis of singularly perturbed switched linear systems,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 57, no. 8, pp. 2116–2121, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
I. Malloci, J. Daafouz, and C. Iung, “Stabilization of continuous-time singularly perturbed switched systems,” in Proceedings of the Joint 48th IEEE Conference on Decision and Control Held Jointly with the 28th Chinese Control Conference (CDC/CCC '09), pp. 6371–6376, Shanghai, China, December 2009.
View at: Publisher Site | Google Scholar
I. Malloci, J. Daafouz, and C. Tung, “Stability and stabilization of two time scale switched systems in discrete time,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 55, no. 6, pp. 1434–1438, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. Alwan, X. Liu, and B. Ingalls, “Exponential stability of singularly perturbed switched systems with time delay,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 3, pp. 913–921, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. S. Alwan and X. Liu, “Stability of singularly perturbed switched systems with time delay and impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4297–4308, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
G. S. Deaecto, J. Daafouz, and J. C. Geromel, “H2 and H∞ performance optimization of singularly perturbed switched systems,” SIAM Journal on Control and Optimization, vol. 50, no. 3, pp. 1597–1615, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
J. Lian and X. Wang, “Exponential stabilization of singularly perturbed switched systems subject to actuator saturation,” Information Sciences, vol. 320, pp. 235–243, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 14–23, 1996.
View at: Publisher Site | Google Scholar
G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676–697, 2006.
View at: Publisher Site | Google Scholar
X. Liu and Q. Zhang, “New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,” Automatica, vol. 39, no. 9, pp. 1571–1582, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
H. Shen, J. H. Park, and Z.-G. Wu, “Finite-time reliable L2-L∞/H∞ control for Takagi-Sugeno fuzzy systems with actuator faults,” IET Control Theory & Applications, vol. 8, no. 9, pp. 688–696, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
H. Shen, L. Su, and J. H. Park, “Reliable mixed H∞/passive control for T-S fuzzy delayed systems based on a semi-Markov jump model approach,” Fuzzy Sets and Systems, vol. 314, pp. 79–98, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Liu, F. Sun, and Z. Sun, “Stability analysis and synthesis of fuzzy singularly perturbed systems,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 2, pp. 273–284, 2005.
View at: Publisher Site | Google Scholar
W. Assawinchaichote, S. K. Nguang, and P. Shi, “H∞ output feedback control design for uncertain fuzzy singularly perturbed systems: an LMI approach,” Automatica, vol. 40, no. 12, pp. 2147–2152, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
W. Assawinchaichote and S. K. Nguang, “H∞ fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: an LMI approach,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 34, no. 1, pp. 579–588, 2004.
View at: Publisher Site | Google Scholar
W. Assawinchaichote and S. K. Nguang, “Fuzzy H∞ output feedback control design for singularly perturbed systems with pole placement constraints: An LMI approach,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 361–371, 2006.
View at: Publisher Site | Google Scholar
T.-H. S. Li and K.-J. Lin, “Composite fuzzy control of nonlinear singularly perturbed systems,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 2, pp. 176–187, 2007.
View at: Publisher Site | Google Scholar
G.-H. Yang and J. Dong, “Control synthesis of singularly perturbed fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 3, pp. 615–629, 2008.
View at: Publisher Site | Google Scholar
C. Yang and Q. Zhang, “Multiobjective control for T-S fuzzy singularly perturbed systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 1, pp. 104–115, 2009.
View at: Publisher Site | Google Scholar
K. Tanaka, M. Iwasaki, and H. O. Wang, “Stable switching fuzzy control and its application to a hovercraft type vehicle,” in Proceedings of the FUZZ-IEEE 2000: 9th IEEE International Conference on Fuzzy Systems, vol. 2, pp. 804–809, May 2000.
View at: Google Scholar
H. Shen, Y. Zhu, L. Zhang, and J. H. Park, “Extended dissipative state estimation for Markov jump neural networks with unreliable links,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 346–358, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
C. Qin, Z. Xiang, and H. R. Karimi, “Robust stability and H∞ stabilization of switched systems with time-varying delays using delta operator approach,” Mathematical Problems in Engineering, vol. 2013, Article ID 230576, 11 pages, 2013.
View at: Publisher Site | Google Scholar
W.-J. Wang, Y.-J. Chen, and C.-H. Sun, “Relaxed stabilization criteria for discrete-time T–S fuzzy control systems based on a switching fuzzy model and piecewise Lyapunov function,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 37, no. 3, pp. 551–559, 2007.
View at: Publisher Site | Google Scholar
Y. Liu and J. Zhao, “Nonfragile control for a class of uncertain switching fuzzy time-delay systems,” Control Theory and Technology, vol. 8, no. 2, pp. 229–232, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
L. Zhou, Q. Wang, X. Ma, and C. Yang, “Fuzzy controllers for nonaffine-in-control singularly perturbed switched systems,” Mathematical Problems in Engineering, vol. 2015, Article ID 896413, 11 pages, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
X. Ma, Q. Wang, J. Cheng, Y. Guo, C. Yang, and L. Zhou, “Controller design for T-S fuzzy singularly perturbed switched systems,” in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '16), pp. 1159–1166, 2016.
View at: Publisher Site | Google Scholar
D. Liberzon, Switching in Cystems and Control, Birkhäuser, Boston, Mass, USA, 2003.
View at: Publisher Site | MathSciNet
J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), pp. 2655–2660, Phoenix, Ariz, USA, December 1999.
View at: Google Scholar
I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Systems & Control Letters, vol. 8, no. 4, pp. 351–357, 1987.
View at: Publisher Site | Google Scholar | MathSciNet
X. Li and Z. Chen, “Stability properties for Hopfield neural networks with delays and impulsive perturbations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3253–3265, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
C. Yang, L. Ma, X. Ma, and X. Wang, “Stability analysis of singularly perturbed control systems with actuator saturation,” Journal of The Franklin Institute, vol. 353, no. 6, pp. 1284–1296, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
X. Ma, Q. Wang, L. Zhou, and C. Yang, “Controller design and analysis for singularly perturbed switched systems with actuator saturation,” International Journal of Robust and Nonlinear Control, vol. 26, no. 15, pp. 3404–3420, 2016.
View at: Publisher Site | Google Scholar | MathSciNet

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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.

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