Observer-Based Robust Control of Uncertain Switched Fuzzy Systems with Combined Switching Controller

Abstract

The observer-based robust control for a class of switched fuzzy (SF) time-delay systems involving uncertainties and external disturbances is investigated in this paper. A switched fuzzy system, which differs from existing ones, is firstly employed to describe a nonlinear system. Next, a combined switching controller is proposed. The designed controller based on the observer instead of the state information integrates the advantages of both the switching controllers and the supplementary controllers but eliminates their disadvantages. The proposed controller provides good performance during the transient period, and the chattering effect is removed when the system state approaches the origin. Sufficient condition for the solvability of the robust control problem is given for the case that the state of system is not available. Since convex combination techniques are used to derive the delay-independent criteria, some subsystems are allowed to be unstable. Finally, various comparisons of the elaborated examples are conducted to demonstrate the effectiveness of the proposed control design approach.

1. Introduction

In practical control systems, the nonlinear property, uncertainty, and time delay exist widely, because of the complexity of control procedure, disturbances, and the constraint of system. Therefore, the stabilization and control issues for nonlinear time-delay systems with uncertainties are very important issues, and they are challenging and important in both theory and practical applications [1–5].

Also recently, the switched systems, which are an important class of hybrid systems and have wide background and technological applications, have also been one of the main research focuses. In turn, a considerable number of fruitful results in analysis and design of switching systems have been derived too; for example, see [6–13]. On the other hand, the research activities on the synergy of fuzzy-system-based controls, as an important intelligent control approach, combined with some of math-analytical control theories have attracted great attention. In particular, the class of Takagi-Sugeno (T-S) fuzzy models has been found to be most effective as system model in various fuzzy-system-based methods. Based on T-S fuzzy model representations and the feedback control strategy, stability and robust analysis and design as well as handling parameter uncertainties for fuzzy systems have acquired a considerable number of fruitful results; for example, see [14–26].

Recently, the stabilizability conditions and smoothness conditions for fuzzy switching control systems were reported. Based on the T-S fuzzy systems, Tanaka et al. [16–19] introduced new switching fuzzy systems for more complicated real systems such as multiple nonlinear systems, switched nonlinear hybrid systems, and second-order nonholonomic systems. This class of the model has two levels of structures. It will switch between the second-level region rules according to the first-level region rules. In fact, it is a type of switching for the same premise variable.

To effectively achieve nonlinear control, we propose an SF model in this paper [27]. Here, by assumption, Takagi-Sugeno fuzzy systems, which are also denoted as T-S fuzzy systems, are considered as terms of plant representation models. These fuzzy systems belong to the category of data-driven fuzzy systems, in contrast to the more traditional category of semantic-driven fuzzy models. The proposed corresponding model of SF systems, presented below, differs from the existing ones cited in the literature in the fact that each subsystem is a T-S fuzzy system, hence defining a class of SF systems. It should be noted that this class inherits some essential features of hybrid systems while retaining all of the information and knowledge representation capacity of fuzzy systems. Furthermore, this model is used for the extra design of the switching law based on the previous works [16–19].

Comparing with the previous works [16–19], this model of SF systems does not have two levels of structures, but it is switching between each of the sub fuzzy systems, not depending on region fuzzy rules. In this case, stability analysis is often facilitated by the fact that properties of each individual sub fuzzy system are of concern only in the regions where this subsystem is active, and the behavior of the sub fuzzy systems in other parts of the state space has no influence on the SF system. It is possible to find a switching law that renders the SF systems stable for unstable sub fuzzy systems. Particularly, when the parameters suddenly change or discontinuously change, the switching rules can be designed as any combinational function of variables, which make up the insufficiency of the switching depending on the single variable of the switching fuzzy model [16–19].

Control of SF systems is a challenging task because no systematic mathematical tools exist to help find necessary and sufficient conditions to guarantee the stability and performance. The problem becomes more complex if some of the system’s parameters are uncertain, especially observer-based control. The study of observer-based control, which has been well addressed for switched systems with delay [28–30] and fuzzy systems with delay [31–33], has been rarely investigated for SF systems. In [34], although the observer-based control of SF systems has been investigated, the external disturbance is not considered. However, to our knowledge, there are no papers considering observer-based control of SF time-delay systems with the external disturbance. Observer-based control for the SF system which is based on designed switching method is still an open and interesting issue.

It is well known that parallel distributed compensation (PDC) control has been found to be a particular means for fuzzy systems due to its universal property [16–19, 34]. However, an undesired chattering effect will occur. This paper presents a combined switching controller, which has nonfuzzy-rules-based combination of switching controller and supplementary controllers, for SF systems subject to parameter uncertainties. The switching controller consists of several linear controllers. One of the switching controllers is employed at each moment according to a switching scheme, which is derived based on the Lyapunov stability theory. The supplementary controller is employed to handle the external disturbance. The smooth transition acted on the switching controller and the supplementary controllers is governed. As a result, the combined switching controller provides good performance during the transient period, and the chattering effect is removed when the system state approaches the origin.

In this paper, we investigate the problem of observer-based robust control for SF systems involving uncertainties and external disturbances with time delay. Sufficient condition for the solvability of the robust control problem is given for the case that the state of a system is unmeasurable. The design of a switching control law based on observer state instead of the original state information is considered. We use the multiple Lyapunov function technique to design a combined switching controller and a switching law such that the observer performance is satisfied. The feasibility of the problem can be realized by convex combination techniques and linear matrix inequalities. Finally, the simulation examples show the validity of the proposed design method.

This paper is organized as follows. Section 2 introduces a switching fuzzy system and an SF system. In Section 3, we describe the model of an SF system of time-delay case involving uncertainties and external disturbances. In Section 4, state observer and the combined switching controller are derived. Two compared example simulations will be presented in Section 5. Finally, a conclusion will be drawn in Section 6.

2. Preliminaries

We compare a switching fuzzy control system and an SF control system.

2.1. Switching Fuzzy Control System

Consider a switching fuzzy model defined by Tanaka et al. as follows [16–19].

The Region Rule 𝑗.

If 𝑧₁(𝑡) is 𝑁_𝑗1 and ⋯ and 𝑧_𝑝(𝑡) is 𝑁_𝑗𝑝:

then the following is given.

The Local Region Rule 𝑖.

If 𝑧₁(𝑡) is 𝑀_𝑗𝑖1 and ⋯ and 𝑧_𝑝(𝑡) is 𝑀_𝑗𝑖𝑝:

then 𝑥̇(𝑡)=𝐴_𝑗𝑖𝑥(𝑡)+𝐵_𝑗𝑖𝑢(𝑡),𝑦(𝑡)=𝐶_𝑗𝑖𝑥(𝑡),𝑖=1,2,…,𝑟,𝑗=1,2,…,𝑠. (1)

𝑧₁(𝑡),…,𝑧_𝑝(𝑡) are the premise variables. 𝑀_𝑗𝑖1,…,𝑀_𝑗𝑖𝑝 are the fuzzy sets. 𝑟 is the number of the if-then rules. This switching fuzzy model in [16–19] has two levels of structures: the region rule level and the local fuzzy rule level. The region rule is crisply switched according to the premise variables. 𝑠 is the number of regions partitioned on the premise variables space. Note that 𝑁_𝑗𝑘(𝑧(𝑡)) is a crisp set.

Where 𝑁_𝑗𝑘(𝑧(𝑡=(𝑡)))1,𝑧∈𝑁_𝑗𝑘,0,otherwise. (2)

When the variables 𝑧₁(𝑡),…,𝑧_𝑝(𝑡) satisfy the condition of the region rule, the fuzzy model which belongs to the local region rule below the region rule is active.

2.2. SF Control System

Comparing with the previous works [16–19], the SF model breaks through two levels of structures, but it is switching between each of the sub fuzzy models directly. The SF system is a more general system than the model of [16–19]. Now, we introduce the SF systems.

When subsystems of the switched systems are T-S fuzzy systems, the systems are switched fuzzy systems. Now, we define SF model including 𝑁_𝜎(𝑡) pieces of rules as follows: 𝑅^𝑙_𝜎(𝑡)∶if𝑥_𝜎(𝑡)1(𝑡)is𝑀^𝑙_𝜎(𝑡)1,…,and𝑥_{𝜎(𝑡)𝑝}(𝑡)is𝑀^𝑙_{𝜎(𝑡)𝑝},𝑥̇(𝑡)then=𝐴_{𝜎(𝑡)𝑙}𝑥(𝑡)+𝐵_{𝜎(𝑡)𝑙}𝑢_𝜎(𝑡)(𝑡),𝑙=1,2,…,𝑁_𝜎(𝑡), (3) with 𝜎(𝑡)∶𝑀={1,2,…,𝑚}, (4) which is a piecewise constant function, called a switching signal.

𝑅^𝑙_𝜎(𝑡) denotes the 𝑙th fuzzy inference rule, 𝑁_𝜎(𝑡) is the number of inference rules, 𝑢_𝜎(𝑡)(𝑡) is the input variable, 𝑥(𝑡) is the state variable vector, 𝐴_{𝜎(𝑡)𝑙} and 𝐵_{𝜎(𝑡)𝑙} are matrices of appropriate dimensions and 𝑥_𝜎(𝑡)1,𝑥_𝜎(𝑡)2,…,𝑥_{𝜎(𝑡)𝑝} are the vectors of premise variables.

Sketch map of the switched fuzzy systems is depicted in Figure 1; Ω_𝑖 denotes the state area of the 𝑖th switched subsystem; Ω_𝑖𝑙 denote the 𝑙th sub fuzzy area in Ω_𝑖. In fact, the switched fuzzy systems again partition the Ω_𝑖 subarea into 𝑙 sub fuzzy areas Ω_𝑖1,…,Ω_𝑖𝑙,…,Ω_𝑖ℓ. There is local linear model in every sub fuzzy area; namely, local linear model in Ω_𝑖𝑙 is 𝑥̇(𝑡)=𝐴_𝑖𝑙𝑥(𝑡)+𝐵_𝑖𝑙𝑢_𝑖(𝑡). The model of every switched subarea Ω₁,…,Ω_𝑚 is composed of local linear model which is linked by fuzzy membership function. We design the switching law for sub fuzzy area model to ensure stability of the switched fuzzy system. When local model in sub fuzzy area satisfies the switching law, we switch to the Ω_𝑖th subsystem to ensure stability of the switched fuzzy system.

Figure 1 

Sketch map of the switched fuzzy systems.

Also, switching law can be designed by arbitrary form, not only depending on fuzzy rules as in [16–19]. That is, the SF systems are extended further for the switching fuzzy systems (1) and (2).

3. Novel Models of Uncertain Switched Fuzzy Time-Delay Systems

So in here, we will introduce an innovated representation modeling of uncertain SF time-delay systems. In this model, each subsystem is an uncertain fuzzy time-delay system, namely, uncertain sub fuzzy time-delay system.

Consider the uncertain SF time-delay model including 𝑁_𝜎(𝑡) pieces of rules as follows: 𝑅^𝑙_𝜎(𝑡)∶if𝑥_𝜎(𝑡)1(𝑡)is𝑀^𝑙_𝜎(𝑡)1,…,and𝑥_{𝜎(𝑡)𝑝}(𝑡)is𝑀^𝑙_{𝜎(𝑡)𝑝},𝑥̇(𝑡)=𝐴then_{𝜎(𝑡)𝑙}+Δ𝐴_{𝜎(𝑡)𝑙}𝑥(𝑡)+𝐷_{𝜎(𝑡)𝑙}𝑥(𝑡−𝜏)+𝐵_{𝜎(𝑡)𝑙}𝑢_{𝜎(𝑡)𝑙}(𝑡)+𝐷_𝜎(𝑡)𝜔(𝑡),𝑦(𝑡)=𝐶_{𝜎(𝑡)𝑙}𝑥(𝑡)[)(0)𝜑,𝑡∈0,∞,𝑥=0,_𝜎(𝑡)(𝜃)=𝑥(𝑡+𝜃),𝜃∈𝑡_𝑗−𝜏,𝑡_𝑗,𝑗=0,1,…,𝑙=1,2,…,𝑁_𝜎(𝑡), (5) with 𝜎(𝑡)∶𝑀={1,2,…,𝑚}, (6) which is a piecewise constant function, called a switching signal. It can be characterized by the switching sequence ∑={𝑥₀;(𝑖₀,𝑡₀),(𝑖₁,𝑡₁),…,(𝑖_𝑗,𝑡_𝑗)∣𝑖_𝑗∈𝑀,𝑗=0,1,…}. Moreover, 𝜎(𝑡)=𝑖 implies that the ith subsystem (𝐴_𝑖,𝐷_𝑖,𝐵_𝑖,𝐶_𝑖) is active.

𝑅^𝑙_𝜎(𝑡) denotes the 𝑙th fuzzy inference rule, 𝑁_𝜎(𝑡) is the number of inference rules, 𝑢_{𝜎(𝑡)𝑙}(𝑡)∈ℝ^𝑞 is the control input, 𝑥(𝑡)∈ℝ^𝑛 and 𝑦(𝑡)∈ℝ^𝑠 denote the state vector and the output vector, respectively, 𝜔(𝑡)∈ℝ^𝑛 is the bounded exogenous disturbance, 𝐷_{𝜎(𝑡)𝑙} are known constant matrices of appropriate dimensions, 𝐴_{𝜎(𝑡)𝑙} and 𝐷_{𝜎(𝑡)𝑙}∈ℝ^𝑛×𝑛are known system matrices, and 𝐵_{𝜎(𝑡)𝑙}∈ℝ^𝑛×𝑞 is the input matrix. Moreover; 𝐶_{𝜎(𝑡)𝑙}∈ℝ^𝑠×𝑛 is the output matrix, 𝜏 is the constant-bounded time delay in the state, and 𝜑_𝜎(𝑡)(𝑡) is the continuous vector-valued function.

It is readily seen that the 𝑖th sub fuzzy system can be represented as follows: 𝑅^𝑙_𝑖∶if𝑥_𝑖1(𝑡)is𝑀^𝑙_𝑖1,…,and𝑥_𝑖𝑝(𝑡)is𝑀^𝑙_𝑖𝑝,𝑥̇(𝑡)=𝐴then_𝑖𝑙+Δ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔(𝑡),𝑦(𝑡)=𝐶_𝑖𝑙𝑥(𝑡),𝑡∈[0,∞),𝑥(0)=0,𝑙=1,2,…,𝑁_𝑖,𝑖=1,2,…,𝑚. (7)

Therefore, the global model of the 𝑖th sub fuzzy system is described by means of the following equation: 𝑥̇(𝑡)=^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝑥(𝑡))𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔(𝑡,𝑦(𝑡)=_{𝑁𝑖𝑙=1})𝜂_𝑖𝑙𝐶_𝑖𝑙𝑥(𝑡), (8) along with 0≤𝜂_𝑖𝑙(𝑥(𝑡))≤1,^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝑥(𝑡𝑤))=1,_𝑖𝑙(𝑥(𝑡))=^𝑝_𝜌=1𝑀^𝑙_𝑖𝜌𝑥_𝜌(𝑡),𝜂_𝑖𝑙(𝑥(𝑡=𝑤))^𝑖𝑙(𝑥(𝑡))∑^𝑁𝑖_𝑙=1𝑤_𝑖𝑙(𝑥(𝑡,)) (9) where 𝑀^𝑙_𝑖𝜌(𝑥_𝜌(𝑡)) denotes the membership function in which 𝑥_𝜌(𝑡) belongs to the fuzzy set 𝑀^𝑙_𝑖𝜌.

4. New Stabilization Based on State Observer and Combined Switching Controller

Suppose that the state observer is given by the following: 𝑅^𝑙_𝜎(𝑡)∶if𝑥_𝜎(𝑡)1(𝑡)is𝑀^𝑙_𝜎(𝑡)1,…,and𝑥_{𝜎(𝑡)𝑝}(𝑡)is𝑀^𝑙_{𝜎(𝑡)𝑝},𝑥̇(𝑡)then=𝐴_{𝜎(𝑡)𝑙}𝑥(𝑡)+𝐷_{𝜎(𝑡)𝑙}𝑥(𝑡−𝜏)+𝐵_{𝜎(𝑡)𝑙}𝑢_{𝜎(𝑡)𝑙}(𝑡)+𝐷_𝜎(𝑡)𝜔(𝑡)+𝐿_{𝜎(𝑡)𝑙}𝑦(𝑡)−𝑦(𝑡),𝑦(𝑡)=𝐶_{𝜎(𝑡)𝑙}𝑥(𝑡) (10) in which 𝑦(𝑡) and 𝜎(𝑡) are measurable output and switching signal of systems (5), respectively. The matrices 𝐿_{𝜎(𝑡)𝑙}∈ℝ^𝑛×𝑠 are to be determined later.

Then, the global model of the 𝑖th sub fuzzy observer is described as follows: 𝑥̇(𝑡)=^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝑥(𝑡))𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔(𝑡)+𝐿_𝑖𝑙𝑦(𝑡)−𝑦(𝑡),𝑦(𝑡)=^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝑥(𝑡))𝐶_𝑖𝑙𝑥(𝑡). (11)

Define the difference between the real state and the observer state as follows: 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡). (12)

Subtracting (11) from (8) gives the following error switched system: 𝑒̇(𝑡)=^𝑁𝑖_𝑙=1𝜂^𝑁_𝑖𝑙^𝑖_ℎ=1𝜂_𝑖ℎ𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+Δ𝐴_𝑖𝑙𝑥+𝐷_𝑖𝑙𝑒(𝑡−𝜏). (13)

Our purpose is to design a controller 𝑢=𝑢(𝑡) and a switching law 𝜎=𝜎(𝑡) such that (5) is asymptotically stable for the observer-based robust control.

Now, a combined switching controller is employed to control the SF time-delay model of (5). The combined switching controller, which has nonfuzzy rules, integrates the advantages of both switching controllers and supplementary controllers but eliminates their disadvantages. The switching controllers consist of some simple subcontrollers. These subcontrollers will switch among each other to control the system of (5) according to an appropriate switching scheme. The supplementary controller is employed to handle the external disturbance. It is shown in the sequel how to design controllers to achieve asymptotical stability in the closed loop and under the switching law.

The combined switching controller for the sub fuzzy time-delay system is described by 𝑢_𝑖𝑙=𝑢¹_𝑖𝑙+𝑢²_𝑖𝑙, (14) where 𝑢¹_𝑖𝑙 is the switching controller and 𝑢²_𝑖𝑙 is the supplementary controller. Consider the following: 𝑢¹_𝑖𝑙=−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎(𝑥(𝑡))𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖𝑥(𝑡),𝑢²_𝑖𝑙=0,if𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡Θ)=0,_𝑖𝑙Θ,otherwise,_𝑖𝑙𝜃(𝑡)=−⋅𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡−_{𝑁𝑖𝛽=1})𝛾_𝛽𝜃(𝑡)⋅𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)²𝐵^𝑇_𝑖𝛽𝑃_𝑖𝑥(𝑡), (15)(16) where 𝑅 is symmetric positive definite.

Now, we will make the following standard assumptions for the uncertain SF system (5).

Assumption 1. The admissible parameter uncertainties are of the norm-bounded form Δ𝐴_𝑖𝑙=𝑀_𝑖𝑙𝐹_𝑖𝑙(𝑡)𝑁_𝑖𝑙, (17) where 𝑀_𝑖𝑙 and 𝑁_𝑖𝑙 are known real constant matrices of proper dimensions which represent the structure of the uncertainties and 𝐹_𝑖𝑙(𝑡) is unknown time-varying matrix satisfying 𝐹^𝑇_𝑖𝑙(𝑡)𝐹_𝑖𝑙(𝑡)≤𝐼. (18)

Assumption 2. There exists constant 𝛾_𝑙 (𝑙=1,2,…𝑁_𝑖), such that 𝐷_𝑖=^𝑁𝑖_𝑙=1𝛾_𝑙𝐵_𝑖𝑙. (19)

Assumption 3. There exists known continuous nonnegative function 𝜃(𝑡), such that max‖𝜔(𝑡)‖≤𝜃(𝑡). (20)

The following theorem provides a condition for asymptotic stability of the system (5) and gives the design of state-dependent switching laws.

Theorem 4. Suppose that there exist constants 𝛽_𝑖𝑗(𝑖,𝑗∈𝑀) (either all nonnegative or all nonpositive) and positive definite symmetric matrices 𝑃_𝑖, 𝑄_𝑖 such that the following matrix inequalities are satisfied. Then, the system (5) along with fuzzy observer (10) is asymptotical stable via the combined switching controller (14): Υ_𝑖𝑎𝑙+^𝑚_𝑗=1𝛽_𝑖𝑗𝑃_𝑖−𝑃_𝑗𝑃_𝑖𝐷_𝑖𝑙𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ0𝐷^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐶00_𝑇𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖0Ψ_𝑖𝑙ℎ𝑄_𝑖𝐷_𝑖𝑙00𝐷^𝑇_𝑖𝑙𝑄_𝑖−𝑄_𝑖<0,𝑖=1,2,…,𝑚,𝑙,𝑎,ℎ=1,2,…𝑁_𝑖, (21) where Υ_𝑖𝑎𝑙=−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙+𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙+𝑃_𝑖, Ψ_𝑖𝑙ℎ=𝑄_𝑖+𝐴^𝑇_𝑖𝑙−𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑄_𝑖+𝑄_𝑖𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ+2𝑄_𝑖𝑀_𝑖𝑙𝑀^𝑇_𝑖𝑙𝑄_𝑖+𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙. (22) And, the switching law is designed as follows: 𝜎=𝜎(𝑥(𝑡))=𝑖∶[0,+∞)⟶𝑀={1,2,…,𝑚}. (23)

Proof. Without loss of generality, suppose that 𝛽_𝑖𝑗≥0. For any 𝑖∈𝑀, if 𝑥^𝑇(𝑡)(𝑃_𝑖−𝑃_𝑗)𝑥(𝑡)≥0, forall𝑗∈𝑀, then from (21), we have Υ_𝑖𝑎𝑙𝑃_𝑖𝐷_𝑖𝑙𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ0𝐷^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐶00_𝑇𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖0Ψ_𝑖𝑙ℎ𝑄_𝑖𝐷_𝑖𝑙00𝐷^𝑇_𝑖𝑙𝑄_𝑖−𝑄_𝑖<0. (24)
Obviously, for all 𝑥(𝑡)∈ℝ^𝑛⧵{0}, there is certainly an 𝑖∈𝑀 such that 𝑥^𝑇(𝑡)𝑃_𝑖−𝑃_𝑗𝑥(𝑡)≥0,∀𝑗∈𝑀. (25)
For arbitrary 𝑖∈𝑀, let Ω_𝑖=𝑥(𝑡)∈𝑅^𝑛⧵{0}∣𝑥^𝑇(𝑡)𝑃_𝑖−𝑃_𝑗𝑥(𝑡)≥0,∀𝑗∈𝑀. (26)
Then, ⋃^𝑚_𝑖=1Ω_𝑖=ℝ^𝑛⧵{0}. Furthermore, let us construct the sets Ω₁=Ω₁,…,Ω_𝑖=Ω_𝑖−⋃^𝑖−1_𝑗=1Ω_𝑗,…, Ω_𝑚=Ω_𝑚−⋃^𝑚−1_𝑗=1Ω_𝑗. Obviously, we have ⋃^𝑚_𝑖=1Ω_𝑖=ℝ^𝑛⧵{0} and Ω_𝑖∩Ω_𝑗=Φ, 𝑖≠𝑗.
Note that (24) holds; then there exists Ξ^𝑇(𝑡)Υ_𝑖𝑎𝑙𝑃_𝑖𝐷_𝑖𝑙𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ0𝐷^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐶00_𝑇𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖0Ψ_𝑖𝑙ℎ𝑄_𝑖𝐷_𝑖𝑙00𝐷^𝑇_𝑖𝑙𝑄_𝑖−𝑄_𝑖Ξ(𝑡)<0, (27) where Ξ(𝑡)≜𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏)𝑒^𝑇(𝑡)𝑒^𝑇(𝑡−𝜏)^𝑇.
When 𝑥(𝑡)∈Ω_𝑖,𝜎(𝑡)=𝑖, then choose the multiple Lyapunov functional as follows: 𝑉_𝑖(𝑡)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝜃)𝑃_𝑖𝑥(𝜃)𝑑𝜃+^𝑡_𝑡−𝜏𝑒^𝑇(𝜃)𝑄_𝑖𝑒(𝜃)𝑑𝜃, (28)
which is positive definite since 𝑃_𝑖 and 𝑄_𝑖 are positive definite matrices.
The time derivative of 𝑉_𝑖(𝑡) is given as follows: 𝑉̇_𝑖(𝑡)=𝑥̇^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑥̇(𝑡)+𝑒^𝑇̇(𝑡)𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒̇(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃_𝑖𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑄_𝑖𝑒()=𝑡−𝜏_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1}𝜂_𝑖𝑙𝜂_𝑖ℎ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔+𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)^𝑇𝑃_𝑖𝑥(𝑡)+^𝑁𝑖𝑁_𝑙=1^𝑖_ℎ=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔+𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡+_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1})𝜂_𝑖𝑙𝜂_𝑖ℎ𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+Δ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑒(𝑡−𝜏)^𝑇𝑄_𝑖𝑒(𝑡)+^𝑁𝑖𝑁_𝑙=1^𝑖_ℎ=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝑒^𝑇(𝑡)𝑄_𝑖𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+Δ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑒(+𝑥𝑡−𝜏)_𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃_𝑖𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑄_𝑖𝑒()=𝑡−𝜏_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1}𝜂_𝑖𝑙𝜂_𝑖ℎ𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑙+𝑥^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙+𝑢^𝑇_𝑖𝑙(𝑡)𝐵^𝑇_𝑖𝑙+𝜔^𝑇𝐷^𝑇_𝑖+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+^𝑁𝑖𝑁_𝑙=1^𝑖_ℎ=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝐷_𝑖𝜔+𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡+_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1})𝜂_𝑖𝑙𝜂_𝑖ℎ𝑒^𝑇(𝑡)𝐴^𝑇_𝑖𝑙+Δ𝐴^𝑇_𝑖𝑙−𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙+𝑥^𝑇(𝑡)Δ𝐴^𝑇_𝑖𝑙+𝑒^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+^𝑁𝑖𝑁_𝑙=1^𝑖_ℎ=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝑒^𝑇(𝑡)𝑄_𝑖𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+Δ𝐴_𝑖𝑙𝑥(𝑡)+𝐷_𝑖𝑙𝑒(+𝑥𝑡−𝜏)_𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃_𝑖𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑄_𝑖𝑒()=𝑡−𝜏_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1}𝜂_𝑖𝑙𝜂_𝑖ℎ𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝑢^𝑇_𝑖𝑙(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑢_𝑖𝑙(𝑡)+𝑥^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝜔^𝑇𝐷^𝑇_𝑖𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝜔+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡+_{𝑁𝑖𝑁𝑙=1𝑖ℎ=1})𝜂_𝑖𝑙𝜂_𝑖ℎ𝑥^𝑇(𝑡)Δ𝐴^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖Δ𝐴_𝑖𝑙𝑥(𝑡)+𝑒^𝑇(𝑡)𝐴^𝑇_𝑖𝑙+Δ𝐴^𝑇_𝑖𝑙−𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝐴_𝑖𝑙+Δ𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝐷_𝑖𝑙𝑒(+𝑥𝑡−𝜏)_𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃_𝑖𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑄_𝑖𝑒(𝑡−𝜏). (29)

(I) When 𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)≠0, then we have 𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝑢¹_𝑖𝑙^𝑇(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑢¹_𝑖𝑙(𝑡)=𝑥^𝑇(𝑡)−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝐵^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝑙−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)=𝑥^𝑇(𝑡)−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝐵^𝑇_𝑖𝑙𝑃_𝑖−^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝑃_𝑖+^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝑙−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖−^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)=𝑥^𝑇(𝑡)−^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝑃_𝑖−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎−𝜂_𝑖𝑎×𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝑃_𝑖−^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖−^𝑁𝑖_𝑎=1𝜌_𝑖𝑙𝑎−𝜂_𝑖𝑎×𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡). (30)

Let 𝜌_𝑖𝑙𝑎=1+sgn𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖𝑥(𝑡)2,sgn(𝑧)=1,if𝑧>0,−1,otherwise. (31)(32)

From (30), we obtain =𝑥^𝑇(𝑡)^𝑁𝑖_𝑎=1𝜂_𝑖𝑎−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙−2^𝑁𝑖_𝑎=1sgn𝑒^𝑇_𝑟(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖𝑒_𝑟(𝑡)2−𝜂_𝑖𝑎−12𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖^𝑇𝑃_𝑖𝑥(𝑡)≤^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝑥^𝑇(𝑡)−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)−2^𝑁𝑖_𝑎=112−𝜂_𝑖𝑎−12𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖𝑥(𝑡). (33)

As 𝜂_𝑖𝑎−(1/2)∈[−1/21/2], due to the property of the switched fuzzy model with time delay, it can be shown that (33) satisfies the following inequality: 𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝑢¹_𝑖𝑙^𝑇(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑢¹_𝑖𝑙(𝑡)≤^𝑁𝑖_𝑎=1𝜂_𝑖𝑎𝑥^𝑇(𝑡)−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡). (34) Considering Assumptions 2 and 3, we have 𝜔^𝑇𝐷^𝑇_𝑖𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝐷_𝑖𝜔+𝑢²_𝑖𝑙^𝑇(𝑡)𝐵^𝑇_𝑖𝑙𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝐵_𝑖𝑙𝑢²_𝑖𝑙(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙𝑢²_𝑖𝑙+𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝜔=2𝑥^𝑇(𝑡)𝑃_𝑖𝐵_𝑖𝑙−𝜃(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡−_{𝑁𝑖𝛽=1})𝛾_𝛽𝜃(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)²⋅𝐵^𝑇_𝑖𝛽𝑃_𝑖𝑥(𝑡𝑥)+2^𝑇(𝑡)𝑃_𝑖^𝑁𝑖_𝛽=1𝛾_𝛽𝐵_𝑖𝛽𝜔≤2−𝜃(𝑡)𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)−^𝑁𝑖_𝛽=1𝛾_𝛽𝜃(𝑡)𝐵^𝑇_𝑖𝛽𝑃_𝑖𝑥(𝑡)+2^𝑁𝑖_𝛽=1𝛾_𝛽𝜃(𝑡)𝐵^𝑇_𝑖𝛽𝑃_𝑖𝑥(𝑡(𝑡𝐵)=2−𝜃)^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡). (35)

According to Assumption 1, we have 𝑥^𝑇(𝑡)Δ𝐴^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖Δ𝐴_𝑖𝑙𝑥(𝑡)+𝑒^𝑇(𝑡)Δ𝐴^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖Δ𝐴_𝑖𝑙𝑒(𝑡)≤𝑥^𝑇(𝑡)𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙𝑥(𝑡)+2𝑒^𝑇(𝑡)𝑄_𝑖𝑀_𝑖𝑙𝑀^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙𝑒(𝑡). (36)

Then, from (34)–(36), it holds true that 𝑉̇(𝑡)≤^𝑁𝑖𝑁_𝑙=1^𝑖𝑁_ℎ=1^𝑖_𝑎=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝜂_𝑖𝑎×𝑥^𝑇(𝑡)−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑙𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+𝑥^𝑇(𝑡)𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙𝑥(𝑡)+2𝑒^𝑇(𝑡)𝑄_𝑖𝑀_𝑖𝑙𝑀^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙𝑒(𝑡)+𝑒^𝑇(𝑡)𝐴^𝑇_𝑖𝑙−𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏)𝐷^𝑇_𝑖𝑙𝑄_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)𝑄_𝑖𝐷_𝑖𝑙𝑒()+𝑥𝑡−𝜏_𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑃_𝑖𝑥(𝑡−𝜏)+𝑒^𝑇(𝑡)𝑄_𝑖𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑄_𝑖𝑒(=𝑡−𝜏)_{𝑁𝑖𝑁𝑙=1𝑖𝑁ℎ=1𝑖𝑎=1}𝜂_𝑖𝑙𝜂_𝑖ℎ𝜂_𝑖𝑎𝑥(𝑡)𝑥()𝑒(𝑡)𝑒()𝑡−𝜏𝑡−𝜏^𝑇×Υ_𝑖𝑎𝑙𝑃_𝑖𝐷_𝑖𝑙𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ0𝐷^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐶00_𝑇𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖0Ψ_𝑖𝑙ℎ𝑄_𝑖𝐷_𝑖𝑙00𝐷^𝑇_𝑖𝑙𝑄_𝑖−𝑄_𝑖×𝑥(𝑡)𝑥()𝑒(𝑡)𝑒(),𝑡−𝜏𝑡−𝜏 (37) where Υ_𝑖𝑎𝑙=−𝑃_𝑖𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖+𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙+𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙+𝑃_𝑖,Ψ_𝑖𝑙ℎ=𝑄_𝑖+𝐴^𝑇_𝑖𝑙−𝐶^𝑇_𝑖ℎ𝐿^𝑇_𝑖𝑙𝑄_𝑖+𝑄_𝑖𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖ℎ+2𝑄_𝑖𝑀_𝑖𝑙𝑀^𝑇_𝑖𝑙𝑄_𝑖+𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙, (38)

in which 𝑖=𝜎(𝑡) given by (23). Taking (9), (27), and (37) into account, we deduce that (𝑑/𝑑𝑡)𝑉(𝑡)<0,Ξ(𝑡)≠0. Therefore, the system (5) is asymptotical stable under the switching law (23).

(II) When 𝐵^𝑇_𝑖𝑙𝑃_𝑖𝑥(𝑡)=0, then we have 𝑉̇_𝑖(𝑡)≤^𝑁𝑖𝑁_𝑙=1^𝑖𝑁_ℎ=1^𝑖_𝑎=1𝜂_𝑖𝑙𝜂_𝑖ℎ𝜂_𝑖𝑎𝑥(𝑡)𝑥()𝑒(𝑡)𝑒()𝑡−𝜏𝑡−𝜏^𝑇×Υ_𝑖𝑎𝑙𝑃_𝑖𝐷_𝑖𝑙𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖ℎ0𝐷^𝑇_𝑖𝑙𝑃_𝑖−𝑃_𝑖𝐶00_𝑇𝑖ℎ𝐿^𝑇_𝑖𝑙𝑃_𝑖0Ψ_𝑖𝑙ℎ𝑄_𝑖𝐷_𝑖𝑙00𝐷^𝑇_𝑖𝑙𝑄_𝑖−𝑄_𝑖×𝑥(𝑡)𝑥()𝑒(𝑡)𝑒().𝑡−𝜏𝑡−𝜏 (39)

Taking (9) and (21) into account, we deduce that (𝑑/𝑑𝑡)𝑉_𝑖(𝑥(𝑡))<0, 𝑥(𝑡)≠0. Therefore, asymptotic stability follows.

Remark 5. Although any individual subsystem of (5) alone usually cannot achieve the observer-based robust control, a suitable switching may solve the robust control problem, as shown in Theorem 4.

Remark 6. (i) This controller structure consists of two parts: the switching controller 𝑢¹_𝑖𝑙 to stabilize the nominal system and the supplementary controller 𝑢²_𝑖𝑙 to handle the external disturbance. Hence, in practice, this controller is a hybrid switching-robust controller. (ii) It can be seen from (15) that the switching controller 𝑢¹_𝑖𝑙 consists of 2^𝑁𝑖 controllers which are linear combinations of −𝑅𝐵^𝑇_𝑖𝑎𝑃_𝑖𝑒_𝑟(𝑡), 𝑎=1,2,…,𝑁_𝑖.
In other words, we formulate the finding of 𝑃_𝑖 and 𝑄_𝑖 of (21) into LMI problem. Considering (21) and multiplying both sides of (21) by the matrix diag{𝑃_𝑖⁻¹,𝐼,𝐼,𝐼}, we restate as follows.

Theorem 7. Suppose that there exist constants 𝛽_𝑖𝑗(𝑖,𝑗∈𝑀) (either all nonnegative or all nonpositive) and positive definite symmetric matrices 𝑃_𝑖, 𝑄_𝑖 and matrices 𝐺_𝑖𝑙 such that the following matrix inequalities are satisfied. Then, the system (5) along with the fuzzy observer (10) is asymptotically stable via the combined switching controller (14), and the switching law is designed as (23): Υ_𝑖𝑎𝑙+^𝑚_𝑗=1𝛽_𝑖𝑗𝑃_𝑖−𝑃_𝑗𝑃_𝑖⁻¹𝑁^𝑇_𝑖𝑙𝐷_𝑖𝑙𝐿_𝑖𝑙𝐶_𝑖ℎ00∗−𝐼0000∗∗−𝑃_𝑖000∗∗∗Ψ_𝑖𝑙ℎ𝑄_𝑖𝑀_𝑖𝑙𝑄_𝑖𝐷_𝑖𝑙∗∗∗∗−2𝐼0∗∗∗∗∗−𝑄_𝑖𝑙,𝑎,ℎ=1,2,…,𝑁_𝑖,𝑖=1,2,…,𝑚, (40) where Υ_𝑖𝑎𝑙=−𝐵_𝑖𝑎𝑅𝐵^𝑇_𝑖𝑙−𝐵_𝑖𝑙𝑅𝐵^𝑇_𝑖𝑎+𝑃_𝑖⁻¹𝐴^𝑇_𝑖𝑙+𝐴_𝑖𝑙𝑃_𝑖⁻¹+𝑃_𝑖⁻¹,Ψ_𝑖𝑙ℎ=𝑄_𝑖+𝐴^𝑇_𝑖𝑙𝑄_𝑖−𝐶^𝑇_𝑖ℎ𝐺^𝑇_𝑖𝑙+𝑄_𝑖𝐴_𝑖𝑙−𝐺_𝑖𝑙𝐶_𝑖ℎ+𝑁^𝑇_𝑖𝑙𝑁_𝑖𝑙,𝐺_𝑖𝑙=𝑄_𝑖𝐿_𝑖𝑙. (41)
Once one has 𝑃_𝑖, 𝑄_𝑖 from (40), then the controllers (14) can be constructed.

Remark 8. The design procedure can be summarized by the following steps.
Step 1. Design the combined switching controller (14) by choosing the symmetric matrix 𝑅>0.
Step 2. Check the parameters of the combined switching controller (14) by solving the LMIs in Theorem 7.
Step 3. Design the law of (31) according to Theorem 7.

Remark 9. If (40) holds for some 𝑖, which is the condition for the observer-based robust control problem of nonswitched system to be solvable, then no switching is needed. Obviously, (40) is much weaker than the condition for nonswitched systems because (40) allows the case that the observer-based robust control problem is solvable for no subsystems.

5. Illustrative Examples and Results

In order to demonstrate the efficiency and feasible performance of the proposed control synthesis, now we consider the model as follows: 𝑅¹₁∶if𝑧₂(𝑡)is𝑀¹₁₁,𝑥̇(𝑡)then=𝐴₁₁𝑥(𝑡)+𝐷₁₁𝑥(𝑡−𝜏)+𝐵₁₁𝑢₁₁(𝑡)+𝐷₁𝜔(𝑡),𝑦(𝑡)=𝐶₁₁𝑥(𝑡);𝑅²₁∶if𝑧₂(𝑡)is𝑀²₁₁,𝑥̇(𝑡)then=𝐴₁₂𝑥(𝑡)+𝐷₁₂𝑥(𝑡−𝜏)+𝐵₁₂𝑢₁₂(𝑡)+𝐷₁𝜔(𝑡),𝑦(𝑡)=𝐶₁₂𝑥(𝑡);𝑅¹₂∶if𝑧₂(𝑡)is𝑀¹₂₁,𝑥̇(𝑡)then=𝐴₂₁𝑥(𝑡)+𝐷₂₁𝑥(𝑡−𝜏)+𝐵₂₁𝑢₂₁(𝑡)+𝐷₂𝜔(𝑡),𝑦(𝑡)=𝐶₂₁𝑥(𝑡);𝑅²₂∶if𝑧₂(𝑡)is𝑀²₂₁,𝑥̇(𝑡)then=𝐴₂₂𝑥(𝑡)+𝐷₂₂𝑥(𝑡−𝜏)+𝐵₂₂𝑢₂₂(𝑡)+𝐷₂𝜔(𝑡),𝑦(𝑡)=𝐶₂₂𝑥(𝑡). (42)

Also, this model is used for the extra design of the switching law (23) of the state-dependent form. For the purpose of comparing with SF system, Example 1 in [16] can be described for the following.

Region Rule  1. If𝑧₂(𝑡)≥1, then the following is given. LocalRegionRule1.If𝑧₂(𝑡)isℎ₁₁𝑥₂(𝑡),𝑥̇(𝑡)then=𝐴₁₁𝑥(𝑡)+𝐷₁₁𝑥(𝑡−𝜏)+𝐵₁₁𝑢(𝑡)+𝐷₁𝜔(𝑡),𝑦(𝑡)=𝐶₁₁𝑥(𝑡).LocalRegionRule2.If𝑧₂(𝑡)isℎ₁₂𝑥₂(𝑡),𝑥̇(𝑡)then=𝐴₁₂𝑥(𝑡)+𝐷₁₂𝑥(𝑡−𝜏)+𝐵₁₂𝑢(𝑡)+𝐷₁𝜔(𝑡),𝑦(𝑡)=𝐶₁₂𝑥(𝑡). (43)(44)

Region Rule  2. If𝑧₂(𝑡)<1, then the following is given. LocalRegionRule1.If𝑧₂(𝑡)isℎ₂₁𝑥₂(𝑡),𝑥̇(𝑡)then=𝐴₂₁𝑥(𝑡)+𝐷₂₁𝑥(𝑡−𝜏)+𝐵₂₁𝑢(𝑡)+𝐷₂𝜔(𝑡),𝑦(𝑡)=𝐶₂₁𝑥(𝑡).LocalRegionRule2.If𝑧₂(𝑡)isℎ₂₂𝑥₂(𝑡),𝑥̇(𝑡)then=𝐴₂₂𝑥(𝑡)+𝐷₂₂𝑥(𝑡−𝜏)+𝐵₂₂𝑢(𝑡)+𝐷₂𝜔(𝑡),𝑦(𝑡)=𝐶₂₂𝑥(𝑡). (45)(46)(47)

Obviously, switching fuzzy system of Example 1 in [16] is switching between each of the sub fuzzy systems and depending on region fuzzy rules.

We have 𝐴₁₁=−11.54−0.843−1.0493,𝐵₁₁=,𝐴−10.01−0.4926₁₂=−4.81−0.232−1.4312,𝐵₁₂=,𝐴−10.2−0.1516₂₁=−100.6−0.6−0.8,𝐵₂₁=,𝐴−3.4−1.7₂₂=−121.2−2−4.529,𝐵₂₂=,𝐷−1.2−0.1765₁₁=0.40.20.10.2,𝐷₁₂=,𝐷−0.50.7−0.1−0.4₂₁=0.20.3−0.2−0.2,𝐷₂₂=,𝐶0.20.30.10.4₁₁=2−1,𝐶₁₂=,𝐶−0.80.1₂₁=5−1,𝐶₂₂=,𝑀1.64₁₁=𝑀₁₂=−0.1125110,𝑀₂₁=𝑀₂₂=,𝑁0.01110₁₁=𝑁₁₂=10.200,𝑁₂₁=𝑁₂₂=,𝐹0.5100₁₁(𝑡)=𝐹₁₂(𝑡)=𝐹₂₁(𝑡)=𝐹₂₂(𝑡)=(𝑡)0(𝑡),𝜔sin0cos₁(𝑡)=𝜔₂(𝑡)=cos𝑡²,sin𝑡𝐷₁=𝐷₂=11^𝑇,𝜃(𝑡)=1. (48)

The fuzzy sets of “𝑀¹₁₁,𝑀²₁₁,𝑀¹₂₁,𝑀²₂₁” are represented by the following membership functions, respectively: 𝑀¹₁₁𝑧₂(𝑡)=ℎ₁₁𝑧₂(𝑡1)=1−1+𝑒^{−2𝑧2(𝑡)},𝑀²₁₁𝑧₂(𝑡)=ℎ₁₂𝑧₂(𝑡)=11+𝑒^{−2𝑧2(𝑡)},𝑀¹₂₁𝑧₂(𝑡)=ℎ₂₁𝑧₂(𝑡)1=1−1+𝑒^{(−2(𝑧2(𝑡)−0.3))},𝑀²₂₁𝑧₂(𝑡)=ℎ₂₂𝑧₂(𝑡)=11+𝑒^{(−2(𝑧2(𝑡)−0.3))}. (49)

Consider the closed-loop systems (13). Choosing 𝑙,𝑎,ℎ=1,2…,𝑁_𝑖, 𝑖=1,2, 𝑅=8.4317, and 𝜏=1 for (40), we can have 𝑃₁=,𝑃524.5761342.5681342.5681578.1421₂=,𝑄631.1245412.7413412.7413514.2356₁=5.31520.29420.29422.0512,𝑄₂=.4.85140.35710.35713.6316 (50)

And we have the candidate observer gains via arbitrary switching as 𝐿₁₁=0.89210.2324,𝐿₁₂=,𝐿5.08210.5973₂₁=0.90750.3032,𝐿₂₂=.0.81320.2019 (51)

Taking the initial condition as 𝑥(0)=[−5,13]^𝑇, then the simulation result for the combined switching controller (14) of SF time-delay system (5) is depicted in Figure 2.

(a)

(b)

(c)

Figure 2 

(a) System state 𝑥1(𝑡) (solid line) and observer state 𝑥1(𝑡) (dotted line) with the combined switching controller. (b) System state 𝑥2(𝑡) (solid line) and observer state 𝑥2(𝑡) (dotted line) with the combined switching controller. (c) Control signal of the combined switching controller 𝑢𝑖𝑙(𝑡).

To investigate the effectiveness of the proposed SF system here, we now compare the switching fuzzy system in [16–19]. The SF model is used for the extra design of the switching law (23), but switching fuzzy system switchs depending on region fuzzy rules. For the same data of Example 1 in [16] and the same initial condition 𝑥(0)=[−5,13]^𝑇, the simulation result is depicted in Figure 3. It can be seen that the proposed SF system provides the best performance in terms of faster transient responses and shorter settling time.

(a)

(b)

(c)

Figure 3 

(a) System state 𝑥1(𝑡) (solid line) and observer state 𝑥1(𝑡) (dotted line) of switching fuzzy system with its switching controller in [16]. (b) System state 𝑥2(𝑡) (solid line) and observer state 𝑥2(𝑡) (dotted line) of switching fuzzy system with its switching controller in [16]. (c) Control signal of the switching controller 𝑢(𝑡) in [16] of switching fuzzy system.

The traditional PDC fuzzy controller is employed to handle the same SF time-delay system in [34]. According to [34], the global control of the traditional PDC fuzzy controller is 𝑢(𝑡)=^𝑁𝑖𝑁_𝑖=1^𝑖_𝑙=1𝜈_𝑖ℎ_𝑖𝑙𝐾_𝑖𝑙𝑥(𝑡), (52) where 𝑣_𝑗(𝑧(𝑡=(𝑡)))1,𝑧∈Region𝑗,0,otherwise. (53)

The state feedback gains of subsystems are obtained as follows: 𝐾₁₁=,𝐾−0.542−0.125₁₂=,𝐾−0.6532−7.4367₂₁=,𝐾−3.4−5.32₂₂=−3.12−4.21. (54)

Figure 4 shows the SF system responses and the control signals under the same initial condition 𝑥(0)=[−5,13]^𝑇. Referring to Figure 2, it can be seen that the performance of the combined switching controller (14) is slightly better in terms of transient response time.

(a)

(b)

(c)

Figure 4 

(a) System state 𝑥1(𝑡) (solid line) and observer state 𝑥1(𝑡) (dotted line) with the traditional PDC fuzzy controller of SF system. (b) System state 𝑥2(𝑡) (solid line) and observer state 𝑥2(𝑡) (dotted line) with the traditional PDC fuzzy controller of SF system. (c) Control signal of the traditional PDC fuzzy controller 𝑢(𝑡) of SF system.

6. Conclusion

In this paper, the observer-based robust control for SF system with time delay is investigated. We used the multiple Lyapunov function technique and a switching law to design a control law such that the system along with fuzzy observer is asymptotically stable. The combined switching controller design problem can be solved efficiently by using linear matrix inequalities and convex optimization techniques. The simulation example shows the validity of the switching control law.

Acknowledgments

This paper is supported by the National Natural Science Foundation under Grant no. 61004039, the Science Foundation of the Educational Department of Liaoning Province under Grant no. 2010375, and the Excellent Talented Person Foundation of the Educational Department of Liaoning Province under Grant no. LJQ2011127. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.