Fuzzy Controllers for Nonaffine-in-Control Singularly Perturbed Switched Systems

This paper investigates the problem of fuzzy controller design for nonaffine-in-control singularly perturbed switched systems (NCSPSSs). First, the NCSPSS is approximated by Takagi-Sugeno (T-S) models which include not only state but also control variables in the premise part of the rules. Then, a dynamic state feedback controller design method is proposed in terms of linear matrix inequalities. Under the controller, stability bound estimation problemof the closed-loop system is solved. Finally, an example is given to show the feasibility and effectiveness of the obtained methods.


Introduction
Switched systems, which consist of a finite number of subsystems and a logical rule governing the switching among the subsystems, are widely encountered in mechanical systems, power systems, and aircraft [1][2][3].One of the basic problems of switched systems is the stability analysis of switched systems under arbitrary switching signals [1].For this problem, it is necessary to assume that all the subsystems are asymptotically stable [4].But this assumption is not sufficient for stability of the switched systems [4].To assure stability under arbitrary switching signals, a minimal interval of time between two successive switchings (called dwell time) or a common Lyapunov function for all the subsystems is required [4][5][6][7][8].
On the other hand, many practical systems exhibit multiple time scale behavior, which can lead to high dimensionality and ill-conditioned numerical issues in the analysis and design problems [9,10].Singular perturbation theory has been developed to deal with these problems [11,12].In this framework, the system analysis and synthesis problems are based on decomposing the system into fast and slow subsystems [9].In the past several decades, the theory of singularly perturbed systems has attracted much attention and been applied to chemical processes [13], power systems [14], electromechanical systems [15], and so forth.On stability analysis and stabilization problems of SPSs, there exist two kinds of approaches.One is to present a condition for the existence of an upper bound  max for the singular perturbation parameter , such that the stability of the SPS is ensured for all  ∈ (0,  max ] [16,17].The other is to propose a method to compute the stability bound  max [18,19]. Singularly perturbed switched systems (SPSSs) whose subsystems are SPSs are of practical interest in many industry processes [20,21].An example is given in [22], where the tail end phase of the rolling process in a hot strip mill was modeled as a SPSS.The classical theory for SPSs is based on the Levinson-Tikhonov theorem [23] which shows, under the assumption that the SPS can be decomposed into slow and fast subsystems, that asymptotic stability of the subsystems is a sufficient condition for the stability of the SPS [16,24].However, this principle does not hold for SPSSs.It was shown that stability of the slow and fast switched subsystems is not sufficient for stability of the original SPSS and a dwell-time condition [25,26] or a constraint taking into account the coupling between slow and fast subsystems has to be considered [20,21,27].Although there have been plenty of methods to estimate the stability bound of SPSs, few approaches were proposed for SPSSs.The exception is given by [28], where a stability bound estimation method was proposed for linear SPSSs whose switching subsystems have the same fast subsystems.

Mathematical Problems in Engineering
Fuzzy control has found a great variety of applications in control engineering [29,30].Takagi-Sugeno (T-S) model based fuzzy control has become one of the most successful approaches since T-S model is a universal approximator for a wide class of nonlinear systems [31][32][33][34].Recently, many researchers have focused on the analysis and design of T-S fuzzy SPSs.Stability analysis and stabilization problems for both continuous-and discrete-time T-S fuzzy SPSs were investigated in [17,35], and some LMI-based approaches were derived.To get a satisfactory transient behavior,  ∞ control for T-S fuzzy SPSs with pole placement constraints was considered in [36].Using the results in [17,35,36], the stability of the resulting closed-loop systems is only ensured for sufficiently small singular perturbation parameter .The problem of the stability bound estimation for T-S fuzzy systems was considered in [37,38].Since the T-S models used in [17,[35][36][37][38] are only able to approximate affine nonlinear systems, the proposed methods can not be applied to nonaffine systems.However, there are many nonaffinein-control systems rising from practical applications, such as magnetic servo levitation control system [39], pendulum control systems [40], and chemical reactions [41].
Motivated by [42,43], where a class of generalized T-S fuzzy dynamic models were proposed and shown to be universal approximators to general nonlinear systems, this paper investigates the problem of fuzzy controller design for nonaffine-in-control singularly perturbed switched systems (NCSPSSs).T-S models which include not only state but also control variables in the premise part of the rules are established to approximate the NCSPSS.A fuzzy dynamic state feedback controller is constructed.Under the assumption that the fast subsystems are stable, a controller design method is proposed.Then, the stability bound of the closed-loop system is addressed and an LMI-based method is established.Finally, a numerical example is included to illustrate the proposed results.
The rest of this paper is organized as follows.In Section 2, the problems under consideration are defined.The controller design method which is reduced to the feasibility of a set of -independent LMIs is given in Section 3. In Section 4, the estimation problem of stability bound is considered.An academic example is given in Section 5 to show the effectiveness and advantage of the obtained methods.Section 6 concludes the paper.
By using the algorithm in [42], each nonlinear subsystem  ( = 1, . . ., ) could be represented by a T-S fuzzy model described by   rules of the following form.
The th rule is Plant Rule : The scalars ℎ   are the membership functions.By using the usual center-average defuzzifier, product inference, and singleton fuzzifier, the T-S fuzzy system can be inferred as where Δ  (, , ) denotes the approximation error.According to [42], the error can be made arbitrarily small by choosing large enough number of fuzzy rules.Thus we omit the approximation error Δ  (, , ) in the rest of the paper.[17,[35][36][37][38] are only able to approximate affine nonlinear SPSs, while the T-S model in (4a) and (4b) can represent nonaffine-in-control systems given as in (1a) and (1b), since the control variables are included in the premise part of the rules.As shown in the previous section, there are various examples of nonaffine-incontrol systems and SPSSs in practical applications.Therefore, the T-S fuzzy model in (4a) and (4b) will lay a foundation for fuzzy control of nonaffine-in-control SPSSs and can be applied to various applications such as aircraft control, electrical and electromechanical systems, and chemical reactions.

Remark 1. The T-S models used in
The design of state feedback stabilizing fuzzy controllers for the fuzzy system (4a) and (4b) is based on the dynamic state feedback control method.The fuzzy controller is described by the following.
The th rule is Plant Rule : Because the controller rules are the same as the plant rules, the controller is given as follows: Then we have the closed-loop system: ż () = which can be rewritten in the following compact form: where Upon introducing the indicator function where   () = 1 if the switching system is in mode  and   () = 0 if it is in a different mode, one can write the whole closed-loop system corresponding to (1a) and (1b) as follows: Assume that   22 ,  = 1, . . ., ,  = 1, . . .,   , are Hurwitz matrices, which is a common assumption for investigations on standard SPSs.Then the corresponding slow and fast subsystems of the SPF (11a) and (11b) are obtained as follows: where

Controller Design
This section will present a controller design method.

Mathematical Problems in Engineering
Proof.Let It follows from ( 15) that Using the Schur complement, inequality (18) implies that which is equivalent to Then we have System (8a) and (8b) can be rewritten as where Let By simple calculation, we have LMI conditions ( 13) and ( 14) imply that which indicates that there exists a positive scalar  0 > 0, such that, for any  ∈ (0,  0 ], it holds that Then it follows from ( 28) and ( 29 Let where .Thus, for sufficiently small , the controller gains can be chosen as Then, by ( 30), (31), and ( 32), there exists a positive scalar  max <  0 , such that Computing the derivative of (()) along the trajectories of system (22) and taking into account (34), we have Therefore, the closed-loop system (11a) and (11b) is stable for all  ∈ (0,  max ].This completes the proof. Remark 3. LMIs ( 13), (14), and ( 15) are independent of the singular perturbation parameter  and thus well-defined.
The feasibility of the LMIs can be checked by the existing algorithms [44].If the LMIs are feasible, an -independent stabilization controller can be obtained when  is small enough.
Remark 4. In [17,[35][36][37][38], some T-S model based control approaches have been proposed for nonlinear SPSs.The results are based on the parallel distributed compensation (PDC) scheme and can design state feedback controllers for affine-in-control nonlinear systems.In contrast, this paper employed a dynamic state feedback fuzzy controller and can be applied to nonaffine-in-control singularly perturbed switched systems.
Remark 5. LMIs (14) imply that the controller design method is valid only if the fast subsystem is open-loop stable, which limits the method to standard NCSPSS.Control of nonstandard NCSPSS is still an open problem.
Remark 6. T-S models, which use a set of fuzzy rules to describe a nonlinear system in terms of a set of local linear models, offer an efficient approach to stability analysis and controller design of complex nonlinear systems.In this framework, most of the stability analysis and controller design problems can be reduced to solve LMI problems.A larger number of individual subsystems or fuzzy rules will lead to larger computational burden.Fortunately, there have been some efficient algorithms to deal with LMI problems with reasonable large dimensions [44].

Stability Bound Analysis
Stability bound is a key stability index of SPSs.Theorem 2 guarantees the existence of the stability bound  max .This section will propose a method to estimate the stability bound of the closed-loop system.To begin with, the closed-loop system is written as ż () = where [ [ are verified for  = 1, . . ., ,  = 1, . . .,   with Then, the closed-loop system is stable for ∀ ∈ (0,  max ]. It follows from (40) that that is, We can rewrite (37a) and (37b) as where Let Define which are equivalent to From LMIs (39), it follows that which is equivalent to which implies that (57) Thus, the closed-loop system is stable for ∀ ∈ (0,  max ].This completes the proof.
which can be effectively solved by a one-dimensional search algorithm with the aid of LMI Control Toolbox in Matlab 7.0.
Remark 9.It is known that LMI-based stability conditions for T-S fuzzy systems are usually sufficient conditions [30].Theorem 2 is derived by using common Lyapunov function, which may lead to conservatism in some cases.Our future work will try to reduce the conservatism by using piecewise quadratic Lyapunov functions and fuzzy Lyapunov functions, which have been demonstrated to be less conservative under certain assumptions [31].

A Numerical Example
To illustrate the proposed results, we consider the following singularly perturbed switched system composed of two modes.