Reliable Control for Time-Varying Delay Switched Fuzzy Systems with Faulty Actuators Based on Observers Switching Method

Abstract

This paper deals with the reliable control problem of nonlinear systems represented by switched fuzzy systems (SFS) with time-varying delay, where each subsystem of switched system is a time-varying delay fuzzy system. A switched fuzzy system with a Takagi and Sugeno (T-S) fuzzy model, which differs from existing ones, is firstly employed to describe a nonlinear system. When the actuators are serious failure, the residual part of actuators cannot make original system stability, using switching technique depends on the states of observers, and the fuzzy reliable controller based on measured observers states instead of the original system states information is built. The stabilization criterion of the reliable control problem is given for the case that the state of original system is unmeasurable. The multi-Lyapunov functions method is utilized to the stability analysis and controller design for time-varying delay switched fuzzy systems with faulty actuators. Moreover, observers switching strategy achieving estimation errors decreasing uniformly asymptotically to zero of the switched fuzzy systems is considered. Finally, the stabilization criterion is transformed into the solvability of sufficient linear matrix inequality (LMI) conditions. To illustrate the effectiveness of the proposed stabilization criterion and controller design approaches, a designed numerical example is studied, and some simulations are provided.

1. Introduction

With the growing complexity of some control engineering problems, control techniques drawn from linear theory have shown their limits. Among nonlinear theory, new control approaches have appeared in the last decades such as fuzzy or hybrid techniques.

The fuzzy model-based control has been rapidly developed in recent years [1–7]. This approach is completely different from typical fuzzy controls [8–10]. By employing the Takagi-Sugeno (T-S) fuzzy model [11], which utilizes local linear system description for each rule, we can devise a control methodology to fully take advantages of linear control theory. A typical design procedure of fuzzy model-based control consists of two steps:(i) fuzzy model construction for a nonlinear system; (ii) fuzzy controller design.Nevertheless, an inherent drawback remains since the number of fuzzy rules of a T-S model increase exponentially with the number of nonlinearities constituting the matched nonlinear system. This makes controller design and implementation difficult as the complexity of the nonlinear system to be controlled increases [12], especially for reliable controller design. To outline the problem of rules explosion in T-S model, some authors have proposed to combine the merit of switched systems with T-S models to deal with nonlinear control problems.

A combination of hybrid systems and fuzzy multiple model systems for the continuous-time case was described, and an idea of the fuzzy switched hybrid control was put forward [13]. The new switching fuzzy systems model, which more approaches the complicated real systems such as multiple nonlinear systems, switched nonlinear hybrid systems, and second-order nonholomonic systems, was introduced based on the T-S fuzzy model [14–17]. This model is switching in local fuzzy rule level of the second level according to the premise variable in region rule level of the first level. So, as stated in [18], it is switching according to the same premise variable.

To obtain less conservative stability analysis results than the T-S fuzzy model-based control, in [19], we proposed a switched fuzzy (SF) model constructed by the idea of multidimensional sector nonlinearity [20] and a switching Lyapunov function [14]. A switched system is said to be a switched fuzzy system (SFS) if all of its subsystems are fuzzy systems. It may well be found that the results on switched fuzzy systems are very few. Differing from the conventional switching fuzzy systems which are all of two-level architecture, the new system consists of several subsystems, and they switch on/off each other. The resulting switched fuzzy system inherits some essential features of hybrid systems [21, 22] and maintains all the information and knowledge representation capacity of fuzzy systems. Continuous controllers for all switched subsystems and a switching law are designed to give robust asymptotic stability [19]. The method provides a kind of different premise variable switching directly. Yang et al. have compared the different features of switching fuzzy systems with switched fuzzy systems and introduced the sketch map to illustrate the construction of switched fuzzy system [18]. Among the earliest applications of switched fuzzy systems is the design problem of fuzzy switching control law for a helicopter, in the whole flight envelope [23]. Here 𝐻_∞ control approach is considered. In [24] the switching method for the fuzzy control of the overhead cranes is presented. One uses the information of the trolley position, load swing, and their differentiations to derive the proper control signal. The switching fuzzy algorithm to overcome the condition of dead-zone is especially investigated. Authors in [25] propose a switching control method based on fuzzy energy regions for this kind of systems, and the control method is applied to control a two degrees-of-freedom (dofs) under actuated manipulator with one active dof and one passive dof. From previously mentioned application areas of switched fuzzy systems, it can be concluded that these systems are ubiquitous and of significant practical application.

It should be noted that, since faults and even failures of control components often occur in real-world control systems, classical robust control methods may not provide satisfactory performance always and even drive closed-loop system unstable. A specific control design strategy named fault tolerant control (FTC) is the most popular method applied to systems in the case of unexpected faults, and passive and active approaches to FTC are available. The active FTC approach requires a fault detection and diagnosis (FDD) mechanism to detect and identify the faults in real time and then a mechanism to reconfigure the controllers according to the online fault information from the FDD. In particular, Wu and Ho propose a methodology for the design of both the fuzzy-rule independent and the fuzzy-rule dependent fault detection filters by using a general observer-based fault detection filter as a residual generator [26]. Compared to the passive approach, the active one needs more significantly computational power to implement. In [27], motivations to study controller failures are generalized. The first reason is that the signals are not transmitted perfectly or the controller itself is not available, for example, as in the case of packet dropout phenomenon in networked control systems (NCSs) [28]. The other is for a positive reason: for an economic or system life consideration, where the controller is purposefully suspended from time to time.

Motivated by the aforementioned progress, reliable control has acquired wide attention, and considerable developments were achieved [29–31]. Different from traditional controller design, the objective of this study is to design an appropriate controller by using the information of residual actuators or sensors such that the closed-loop system can tolerate the abnormal operation of some specific control components and retain overall system stability with acceptable system performance. An abnormal operation may include degradation, amplification, and partial outage. In [32], the reliable controller design with actuator fault for the attitude control of an orbiting spacecraft equipped with six thrusters is considered. The application of the reliable control with actuator fault control schemes to the vertical takeoff and landing (VTOL) aircraft system is presented [33]. In the aforementioned manuscripts, the following two methods are widely used.(a) The fault model is depicted as a scaling factor. The outputs of controller without actuator failure are used as the inputs of controller in which actuator failures occurred. It is worth noting that these reliable control methods are all based on a basic assumption that the never failed actuators must stabilize the given system, which is obviously somehow unpractical. (b) Decompose the matrix B (parameters of the control input) into two sets: the set of actuators that are susceptible to failures and the set of actuators that are robust to failure. In other words, actuators may suffer “serious faults” and even failures in some of them, yet the system can remain stable [34]. Motivated by this method, this paper studies the problem of reliable control where actuators suffer “serious faults” that tend towards failures and presents a more general reliable control method.

On the other hand, in almost all engineering control systems such as chemical processes and communication systems, delays are frequently encountered, which affect the performance of the system adversely, including causing instability. Hence, researchers have been paying remarkable attention to the problems of analysis and synthesis for time delay systems [4, 6, 35–42]. All the existing literature for stability analysis can be roughly divided into two types: time constant delay results [35] and time-varying delay ones [4, 6, 36–42]. It is obviously important to establish under what conditions the system is still stable subject to the “serious faults” and the time-varying delay. It is believed, in here, that a more general method that combines reliable control and switched fuzzy control of nonlinear systems, whose states are not available, in the sense of employing observers switching is addressed.

In this paper, the reliable control problem of switched fuzzy systems is considered for the case that the state of system is unmeasurable. The problem begins with the representation models for time-varying delay switched fuzzy system. The derivative of time-varying delay is allowed to be bounded with an unknown constant. Both fuzzy reliable controllers for subsystems and observers switching laws which make switched fuzzy system asymptotic stability are designed. Sufficient conditions are derived via multiple Lyapunov function method. The remainder of this paper is organized as follows. Section 2 presents an outline for switched fuzzy systems model and preliminaries. In Section 3, stabilization and the reliable controller are derived. An illustrative example is given in Section 4. Finally, some concluding remarks are included in Section 5.

2. Model Description and Preliminaries

In this section, we present a time-varying delay switched fuzzy model with faulty actuators. The main novelty is the combination of reliable control and switched fuzzy control. Consider the continuous time-varying delay switched fuzzy model in which the delays are changed depending on time, namely, every subsystem of switched systems is time-varying delay fuzzy system: 𝑅^𝑙_𝜎(𝑡)∶if𝜉₁is𝑀^𝑙_𝜎(𝑡)1⋯and𝜉_𝑝is𝑀^𝑙_{𝜎(𝑡)𝑝},then (1)𝑥̇(𝑡)=𝐴_{𝜎(𝑡)𝑙}𝑥(𝑡)+𝐴_{ℎ𝜎(𝑡)𝑙}𝑥𝑡−𝑑_𝜎(𝑡)(𝑡)+𝐵_{𝜎(𝑡)𝑙}𝑢_𝜎(𝑡)(𝑡)𝑥(𝑡)(𝑡)[],𝑦(𝑡),=𝜑,𝑡∈−𝜏,0=𝐶_{𝜎(𝑡)𝑙}𝑥(𝑡),𝑙=1,2,…,𝑁_𝜎(𝑡). (2a)(2b)(2c)

In here the symbols represent the following: 𝜎∶𝑀={1,2,…,𝑚} is the switching signal to be designed. 𝜉₁,𝜉₂,…,𝜉_𝑝 are the premise variables. 𝑀^𝑙_𝜎(𝑡)1,…,𝑀^𝑙_{𝜎(𝑡)𝑝} denote fuzzy sets in the 𝜎th switched subsystem. 𝑅^𝑙_𝜎(𝑡) denote the 𝑙th fuzzy inference rule in the 𝜎th switched subsystem. 𝑁_𝜎(𝑡) is the number of inference rules in the 𝜎th switched subsystem, and fuzzy rules are selected in every switched subsystem. 𝑢_𝜎(𝑡)(𝑡) is the input variable of the 𝜎th switched subsystem. 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)⋯𝑥_𝑛(𝑡)^𝑇∈ℝ^𝑛 is the state variable vector. 𝐴_{𝜎(𝑡)𝑙}, 𝐴_{ℎ𝜎(𝑡)𝑙}, 𝐵_{𝜎(𝑡)𝑙}, and 𝐶_{𝜎(𝑡)𝑙} are known constant matrices of appropriate dimensions of the 𝜎th switched subsystem. 𝜑(𝑡) is a continuously differentiable vector-valued initial function, and 𝜏>0. 𝑑_𝜎(𝑡)(𝑡) denote the time-varying delay of the 𝜎th switched subsystem satisfying Assumption 1.

Assumption 1. 𝑑_𝜎(𝑡)(𝑡) is a differentiable function, satisfying 0≤𝑑_𝜎(𝑡)(𝑡)≤𝜏 and 𝑑̇_𝜎(𝑡)(𝑡)≤𝑑<1 for some known constant 𝑑.
The 𝑖th switched subsystem is 𝑅^𝑙_𝑖∶if𝜉₁is𝑀^𝑙_𝑖1⋯and𝜉_𝑝is𝑀^𝑙_𝑖𝑝,then (3)𝑥̇(𝑡)=𝐴_𝑖𝑙𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐵_𝑖𝑙𝑢_𝑖(𝑡),𝑥(𝑡)(𝑡)[],𝑦(𝑡)=𝜑,𝑡∈−𝜏,0=𝐶_𝑖𝑙𝑥(𝑡),𝑖=1,…,𝑚,𝑙=1,2,…,𝑁_𝑖. (4a)(4b)(4c)Therefore, the global model of the 𝑖th switched subsystem is described by𝑥̇(𝑡)=^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝜉(𝑡))𝐴_𝑖𝑙𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐵_𝑖𝑙𝑢_𝑖(𝑡,𝑥(𝑡)(𝑡)[],𝑦(𝑡)=)=𝜑,𝑡∈−𝜏,0_{𝑁𝑖𝑙=1}𝜂_𝑖𝑙(𝜉(𝑡))𝐶_𝑖𝑙𝑥(𝑡),𝑖=1,…,𝑚, (5a)(5b)(5c)along with 0≤𝜂_𝑖𝑙(𝜉(𝑡))≤1,^𝑁𝑖_𝑙=1𝜂_𝑖𝑙(𝜉(𝑡))=1, (6) in which 𝜔_𝑖𝑙(𝜉(𝑡))=^𝑝_𝜌=1𝑀^𝑙_𝑖𝜌𝜉_𝜌(𝑡),𝜂_𝑖𝑙(𝜉(𝑡=𝜔))^𝑖𝑙(𝜉(𝑡))∑^𝑁𝑖_𝑙=1𝜔_𝑖𝑙(𝜉(𝑡,)) (7) where 𝑀^𝑙_𝑖𝜌(𝜉_𝜌(𝑡)) denotes the membership function and 𝜉_𝜌(𝑡) belongs to the fuzzy set 𝑀^𝑙_𝑖𝜌.
Since the system states often are not directly measurable, we consider the switching signal 𝜎=𝜎(𝑥(𝑡)), which depends on observer states. 𝑥(𝑡) is the observers signal of 𝑥(𝑡). Suppose Ω₁,Ω₂,…,Ω_𝑚 is a partition of ℝ^𝑛, that is, ⋃^𝑚_𝑖=1Ω_𝑖=ℝ^𝑛⧵{0} and Ω_𝑖⋂Ω_𝑗=Φ,𝑖≠𝑗. The design of switching signal is 𝜎=𝜎(𝑥(𝑡))=𝑖∶[0,+∞)⟶𝑀={1,2,…,𝑚} (8) which depends on Ω₁,Ω₂,…,Ω_𝑚. When 𝑥(𝑡)∈Ω_𝑖, the switching signal subjects to 𝑣_𝑖(𝑥(𝑡=1𝑥(𝑡)∈))Ω_𝑖,0𝑥(𝑡)∉Ω_𝑖,𝑖∈𝑀. (9) That is, if and only if 𝜎=𝜎(𝑥(𝑡))=𝑖, then 𝑣_𝑖(𝑥(𝑡))=1.
From 𝑣_𝑖(𝑥(𝑡)), the system (2a), (2b), and (2c) is represented by

𝑥̇(𝑡)=^𝑚𝑁_𝑖=1^𝑖_𝑙=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡×𝐴))_𝑖𝑙𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐵_𝑖𝑙𝑢_𝑖(𝑡,𝑥(𝑡)(𝑡)[],𝑦(𝑡)=)=𝜑,𝑡∈−𝜏,0_{𝑚𝑁𝑖=1𝑖𝑙=1}𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝐶_𝑖𝑙𝑥(𝑡). (10a)(10b)(10c)

In order to investigate the reliable control problem, we classify actuators of 𝑖th switched subsystem (5a), (5b), and (5c) into two groups [34]. One is a set of actuators susceptible to failures, denoted by Θ_𝑖={1,2,…,𝑘_𝑖}, (𝑖=1,2,…,𝑚). The other is a set of actuators robust to failures, denoted by Θ_𝑖={1,2,…,𝑘_𝑖}−Θ_𝑖, (𝑖=1,2,…,𝑚). Therefore, introduce the decomposition of 𝐵_𝑖𝑙 as 𝐵_𝑖𝑙=𝐵_Θ𝑖𝑙+𝐵_Θ𝑖𝑙,𝑖=1,2,…,𝑚,𝑙=1,2,…,𝑁_𝑖, (11) where 𝐵_Θ𝑖𝑙 and 𝐵_Θ𝑖𝑙 are gained by making the column elements of 𝐵_𝑖𝑙 which correspond to Θ_𝑖𝑙 and Θ_𝑖𝑙, respectively, zero. Chose 𝜔_𝑖 as set of actuators susceptible to failures in practice, 𝜔_𝑖⊆Θ_𝑖, then 𝐵_𝑖=𝐵_𝜔𝑖+𝐵_𝜔𝑖,𝑖=1,2,…,𝑚, (12) where 𝐵_𝜔𝑖 and 𝐵_𝜔𝑖 are gained by making the column elements of 𝐵_𝑖 which correspond to 𝜔_𝑖 and 𝜔_𝑖, respectively, zero; the following is obvious 𝐵_Θ𝑖𝐵^𝑇_Θ𝑖≤𝐵_𝜔𝑖𝐵^𝑇_𝜔𝑖,𝐵_𝜔𝑖𝐵^𝑇_𝜔𝑖≤𝐵_Θ𝑖𝐵^𝑇_Θ𝑖𝐵_𝑖𝑙𝐵^𝑇_𝑖𝑙=𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑙+𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑙,𝑖=1,2,…,𝑚. (13)

Remark 2. In the existing standard reliable control problem, the condition (𝐴_𝑖𝑙,𝐵_Θ𝑖𝑙) must be a stable pair requisite. This strong condition is no longer needed here for switched fuzzy systems because we can design the switching law to make the origin system stabilizable. In fact, if (𝐴_𝑗𝑙,𝐵_Θ𝑗𝑙) is a stable pair for any 𝑗∈𝑀, then we can design state feedback controller for the 𝑗th subsystem that makes the system (10a), (10b), and (10c) stabilizable.

3. Main New Results

In particular, the main attention of this paper is concentrated on actuators of switched fuzzy system suffering serious failures, which has great significance in theoretical study and engineering applications. Moreover, according to PDC (Parallel Distributed Compensation), the observers and fuzzy output feedback controllers are designed.𝑥̇(𝑡)=^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝐴))_𝑖𝑙𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐵_𝑖𝑙𝑢_𝑖(𝑡)+𝐿_𝑖𝑙𝑦(𝑡)−𝐶_𝑖𝑔𝑥(𝑡,𝑢_𝑖)(𝑡)=−^𝑚𝑁_𝑖=1^𝑖_𝑙=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝐾_𝑖𝑙𝑥(𝑡). (14a)(14b)

It ensures that the system (2a), (2b), and (2c) is global asymptotic stability, where 𝑥(𝑡)∈ℝ^𝑛 is state vectors of fuzzy observer, and 𝐿_𝑖𝑙 represents observer gain matrix for the 𝑙th fuzzy rule of the 𝑖th switching subsystem. 𝐿_𝑖𝑙 can be determined by solving conditions (18). 𝐾_𝑖𝑙 denote the controllers feedback gain matrix, and 𝐾_𝑖𝑙 can be determined by solving controller design conditions (19).

In this section, we derive the sufficient conditions for global asymptotic stability of the time-varying delay switched fuzzy system (2a), (2b), and (2c). Choosing observer error 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡), we can obtain𝑥̇(𝑡)=^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖_𝑗=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡×𝐴))_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐵_𝑖𝑙𝐾_𝑖𝑗𝑒(𝑡,𝑥̇(𝑡)=_{𝑚𝑁𝑖=1𝑖𝑁𝑙=1𝑖𝑁𝑗=1𝑖𝑔=1})𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝐴))_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡,𝑒̇(𝑡)=_{𝑚𝑁𝑖=1𝑖𝑁𝑙=1𝑖𝑔=1})𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝐴))_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝐴_ℎ𝑖𝑙𝑒𝑡−𝑑_𝑖(𝑡). (15a)(15b)(15c)

Theorem 3. Suppose that the switched fuzzy system (10a), (10b), and (10c) satisfies the Assumption 1. If there exist constants 𝛽_𝑖𝑟(𝑖,𝑟∈𝑀) (either all nonnegative or all nonpositive), positive definite matrixes 𝑃_𝑖, 𝑅_𝑖, 𝑄_𝑖, 𝑃_𝑒, 𝑅_𝑒, 𝑄_𝑒, and a group of positive scalars constants 𝛼_𝑖𝑗, 𝜉_𝑖𝑙, 𝜀₁, 𝜀₂, 𝜀₃, satisfying the inequalities 𝜋₁₁+∑^𝑚_𝑟=1𝛽_𝑖𝑟𝑃_𝑖−𝑃_𝑟𝜋₁₂𝜋₁₃0∗𝜋₂₂00∗∗𝜋₃₃𝜋₃₄∗∗∗𝜋₄₄<0,𝑖=1,…,𝑚,𝑙,𝑗,𝑔=1,2,…,𝑁_𝑖, (16) where ∗ denotes the symmetric term, where 𝜋₁₁=𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙−𝛼_𝑖𝑙𝑃_𝑖𝐵_Θ𝑖𝑗𝐵^𝑇_Θ𝑖𝑙𝑃_𝑖−𝛼_𝑖𝑗𝑃_𝑖𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝜏𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇𝑅_𝑖𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝜏𝜀₁+𝜀₂𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇×𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝑄_𝑖,𝜋₁₂=𝑃_𝑖𝐴_ℎ𝑖𝑙,𝜋₁₃=𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔,𝜋₂₂=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₁⁻¹𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₃𝐴^𝑇_ℎ𝑖𝑙𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑖,𝜋₃₃=𝜏𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜀₂⁻¹+𝜀₃⁻¹𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑒+𝑃_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝑄_𝑒,𝜋₃₄=𝑃_𝑒𝐴_ℎ𝑖𝑙+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_ℎ𝑖𝑙,𝜋₄₄=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑒, (17) then the closed-loop system (10a), (10b), and (10c) is asymptotically stable in zero by the reliable controllers (14b). The parameters of formula (14a), (14b) satisfy 𝐿_𝑖𝑙=𝜉_𝑖𝑙𝑃_𝑒⁻¹𝐶^𝑇_𝑖𝑙,𝑖=1,…,𝑚,𝑙=1,2,…,𝑁_𝑖,𝐾_𝑖𝑗=𝛼_𝑖𝑗𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖,𝑖=1,…,𝑚,𝑗=1,2,…,𝑁_𝑖. (18)(19)

Remark 4. The controllers are reliable controller because of the controllers feedback gain matrix 𝐾_𝑖𝑙 that are the functions of the actuators failures matrix 𝐵_Θ𝑖𝑙. Moreover, differing from traditional switching system, the switching signal (8) in this paper is dependeny on the state of observer.

Proof. Without loss of generality, suppose 𝛽_𝑖𝑟≥0. For any 𝑖∈𝑀, if 𝑥^𝑇(𝑡)𝑃_𝑖−𝑃_𝑟𝑥(𝑡)≥0,∀𝑟∈𝑀. (20) From (16), we can gain ^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡))×Ξ^𝑇(𝑡)𝜋₁₁𝜋₁₂𝜋₁₃0∗𝜋₂₂00∗∗𝜋₃₃𝜋₃₄∗∗∗𝜋₄₄Ξ(𝑡)<0, (21) where Ξ(𝑡)=𝑥(𝑡)𝑥𝑡−𝑑_𝑖(𝑡)𝑒(𝑡)𝑒𝑡−𝑑_𝑖(𝑡). (22)
Obviously, for ∀𝑥(𝑡)∈ℝ^𝑛⧵{0}, there certainly is an 𝑖∈𝑀 such that 𝑥^𝑇(𝑡)𝑃_𝑖−𝑃_𝑟𝑥(𝑡)≥0,∀𝑟∈𝑀. (23) For arbitrary 𝑖∈𝑀, let Ω_𝑖=𝑥(𝑡)∈𝑅^𝑛⧵{0}∣𝑥^𝑇(𝑡)𝑃_𝑖−𝑃_𝑟𝑥(𝑡)≥0,∀𝑟∈𝑀, (24) then ⋃^𝑚_𝑖=1Ω_𝑖=ℝ^𝑛⧵{0}. Further, let 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.
To proceed further, choose the Lyapunov functional candidate of the following form: 𝑉_𝑖(𝑥(𝑡),𝑒(𝑡))=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡)+𝑉₅(𝑡)+𝑉₆(𝑡), (25) where 𝑉₁(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉₂(𝑥(𝑡))=⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₃(𝑥(𝑡))=^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑉𝑑𝑠,₄(𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃_𝑒𝑒(𝑡),𝑉₅(𝑒(𝑡))=⁰_−𝜏^𝑡_𝑡+𝜃𝑒^𝑇̇(𝑠)𝑅_𝑒𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₆(𝑒(𝑡))=^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑒^𝑇(𝑠)𝑄_𝑒𝑒(𝑠)𝑑𝑠, (26) where 𝑃_𝑖,𝑅_𝑖,𝑄_𝑖,𝑃_𝑒,𝑅_𝑒, and 𝑄_𝑒 are positive definite matrixes satisfying inequalities (16).
Thereafter, we will show that under the switching law (8), along any nonzero solutions of system (2a), (2b), and (2c), 𝑉̇_𝑖(𝑥(𝑡),𝑒(𝑡))<0. When 𝑥(𝑡)∈Ω_𝑖, from switching law (8), it is known that the 𝑖th subsystem of system (10a), (10b), and (10c) is active. Thus, we have 𝑉̇₁(𝑥(𝑡=𝑥̇))_𝑇(𝑡)𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝑥̇(𝑡)=^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑥))_𝑇(𝑡)𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝑒^𝑇(𝑡)𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)≤^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑥))_𝑇(𝑡)𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇𝑃_𝑖+𝑃_𝑖×𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖𝑥(𝑡)+𝑒^𝑇(𝑡)𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇×𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙×𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡). (27) Thus, it follows further that 𝑉̇₂(𝑥(𝑡𝑥̇))=𝜏^𝑇(𝑡)𝑅_𝑖𝑥̇(𝑡)−^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)𝑥̇𝑑𝑠≤𝜏^𝑇(𝑡)𝑅_𝑖𝑥̇(𝑡)−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)𝑑𝑠≤𝜏^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑥))_𝑇(𝑡)𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗^𝑇𝑅_𝑖𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗^𝑇𝑅_𝑖𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗^𝑇𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐴_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝑒^𝑇(𝑡)𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)𝑑𝑠≤𝜏^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑥))_𝑇(𝑡)𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇𝑅_𝑖𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝜀₁+𝜀₂𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇×𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖𝑥(𝑡)+𝑥^𝑇𝑡−𝑑_𝑖(𝑡)×𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₁⁻¹𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₃𝐴^𝑇_ℎ𝑖𝑙𝐴_ℎ𝑖𝑙×𝑥𝑡−𝑑_𝑖(𝑡)+𝑒^𝑇(𝑡)×𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐿_𝑖𝑔𝐶_𝑖𝑗+𝜀₂⁻¹+𝜀₃⁻¹𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔×𝑒(𝑡)−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)𝑑𝑠. (28) Notice that from Assumption 1, the following can be obtained: 𝑉̇₃(𝑥(𝑡=𝑥))_𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−1−𝑑̇_𝑖(𝑡)𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝑑_𝑖(𝑡)≤^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑥))_𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−1−𝑑_𝑖𝑥^𝑇𝑡−𝑑_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝑑_𝑖(𝑡). (29) It holds that 𝑉̇₄(𝑒(𝑡=))_{𝑚𝑁𝑖=1𝑖𝑁𝑙=1𝑖𝑁𝑗=1𝑖𝑔=1}𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑒))_𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑃_𝑒𝑒(𝑡)+𝑒^𝑇(𝑡)𝑃_𝑒𝐴_ℎ𝑖𝑙𝑒𝑡−𝑑_𝑖(𝑡)+𝑒^𝑇(𝑡)𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑒+𝑃_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡). (30) From 0≤𝑑_𝜎(𝑡)≤𝜏, it is clear that 𝑉̇₅(𝑒(𝑡𝑒̇))=𝜏^𝑇(𝑡)𝑅_𝑒𝑒̇(𝑡)−^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑅_𝑒𝑒̇(𝑠)𝑒̇𝑑𝑠≤𝜏^𝑇(𝑡)𝑅_𝑒𝑒̇(𝑡)−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑒^𝑇̇(𝑠)𝑅_𝑒𝑒̇(𝑠)𝑑𝑠≤𝜏^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑒))_𝑇(𝑡)𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑒^𝑇(𝑡)𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_ℎ𝑖𝑙𝑒𝑡−𝑑_𝑖(𝑡)+𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡)+𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_ℎ𝑖𝑙𝑒𝑡−𝑑_𝑖(𝑡)−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑒^𝑇̇(𝑠)𝑅_𝑒𝑒̇(𝑠)𝑑𝑠. (31) It is easy to derive the following: 𝑉̇₆(𝑒(𝑡))=𝑒^𝑇(𝑡)𝑄_𝑒𝑒(𝑡)−1−𝑑̇_𝑖(𝑡)𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝑄_𝑒𝑒𝑡−𝑑_𝑖(𝑡)≤^𝑚𝑁_𝑖=1^𝑖𝑁_𝑙=1^𝑖𝑁_𝑗=1^𝑖_𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝑒))_𝑇(𝑡)𝑄_𝑒𝑒(𝑡)−1−𝑑_𝑖𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝑄_𝑒𝑒𝑡−𝑑_𝑖(𝑡). (32)
At this point of the argument, we should note that the output of the failure actuators is zero for 𝑤_𝑖⊆Θ_𝑖. Namely, only the residual normal actuators contribute to the time-delay switched fuzzy system. It holds true that 𝐵_𝑖𝑙𝐾_𝑖𝑗=𝐵_𝑤𝑖𝑙𝐾_𝑖𝑗,𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖≤𝐵_𝑤𝑖𝑙𝐾_𝑖𝑗. (33) Thus, when 𝑥(𝑡)∈Ω_𝑖, we have 𝑉̇_𝑖(𝑥(𝑡),𝑒(𝑡))=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡)+𝑉̇₄(𝑡)+𝑉̇₅(𝑡)+𝑉̇₆(𝑡). (34) Substituting (27), (28), (29), (30), (31), and (32) into (34) yields 𝑉̇_𝑖(𝑥(𝑡),𝑒(𝑡))≤Ψ_𝑖−^𝑡_{𝑡−𝑑𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑅_𝑖𝑥̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝑑𝑖(𝑡)}𝑒^𝑇̇(𝑠)𝑅_𝑒𝑒̇(𝑠)𝑑𝑠, (35) where Ψ_𝑖=Ξ^𝑇(𝑡)Π_𝑖Ξ(𝑡),Π_𝑖=𝜋₁₁𝜋₁₂𝜋₁₃0∗𝜋₂₂00∗∗𝜋₃₃𝜋₃₄∗∗∗𝜋₄₄,𝜋₁₁=𝐴^𝑇_𝑖𝑙𝑃_𝑖+𝑃_𝑖𝐴_𝑖𝑙−𝛼_𝑖𝑙𝑃_𝑖𝐵_Θ𝑖𝑗𝐵^𝑇_Θ𝑖𝑙𝑃_𝑖−𝛼_𝑖𝑗𝑃_𝑖𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝜏𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇𝑅_𝑖𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝜏𝜀₁+𝜀₂𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖^𝑇×𝐴_𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗𝑃_𝑖+𝑄_𝑖,𝜋₁₂=𝑃_𝑖𝐴_ℎ𝑖𝑙,𝜋₁₃=𝑃_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔,𝜋₂₂=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₁⁻¹𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₃𝐴^𝑇_ℎ𝑖𝑙𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑖,𝜋₃₃=𝜏𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜀₂⁻¹+𝜀₃⁻¹𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑒+𝑃_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝑄_𝑒,𝜋₃₄=𝑃_𝑒𝐴_ℎ𝑖𝑙+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_ℎ𝑖𝑙,𝜋₄₄=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑒. (36)(37)
From (16) and (35), we know that 𝑉̇_𝑖(𝑥(𝑡),𝑒(𝑡))<0 are well defined under the switching law (8) for arbitrary 𝑥(𝑡)≠0 and 𝑒(𝑡)≠0. According to (35), we can conclude that the closed-loop system of (10a), (10b), and (10c) is asymptotically stable via the state-feedback controller (14b) under the switching law (8) when the actuators are serious failure, and the observer error 𝑒(𝑡) asymptotically converges to zero. This ends the proof.

Moreover, we reformulate (36) into LMI problem. Considering (36) and multiplying both sides of (36) by the matrix diag{𝑃_𝑖⁻¹,𝐼,𝐼,𝐼}, (36) can be reformulated as follows: Π_𝑖=𝛾₁₁𝛾₁₂𝛾₁₃0∗𝛾₂₂00∗∗𝛾₃₃𝛾₃₄∗∗∗𝛾₄₄𝛾₁₁,=𝑃_𝑖⁻¹𝐴^𝑇_𝑖𝑙+𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑙𝐵_Θ𝑖𝑗𝐵^𝑇_Θ𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗+𝜏𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗^𝑇𝑅_𝑖𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗+𝜏𝜀₁+𝜀₂𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗^𝑇×𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗+𝑃_𝑖⁻¹𝑄_𝑖𝑃_𝑖⁻¹,𝛾₁₂=𝐴_ℎ𝑖𝑙,𝛾₁₃=𝐿_𝑖𝑙𝐶_𝑖𝑔,𝛾₂₂=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₁⁻¹𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₃𝐴^𝑇_ℎ𝑖𝑙𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑖,𝛾₃₃=𝜏𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜀₂⁻¹+𝜀₃⁻¹𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖𝑅_𝑖𝐿_𝑖𝑙𝐶_𝑖𝑔+𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑒+𝑃_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝑄_𝑒,𝛾₃₄=𝑃_𝑒𝐴_ℎ𝑖𝑙+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_ℎ𝑖𝑙,𝛾₄₄=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑒. (38)(39) We restate as follows.

Theorem 5. Suppose that the switched fuzzy system (10a), (10b), and (10c) satisfies Assumption 1. If there exist constants 𝛽_𝑖𝑟(𝑖,𝑟∈𝑀) (either all nonnegative or all non-positive), positive definite matrixes 𝑃_𝑖, 𝑅_𝑖, 𝑄_𝑖, 𝑃_𝑒, 𝑅_𝑒, 𝑄_𝑒, and a group of positive scalars constants 𝛼_𝑖𝑗, 𝜉_𝑖𝑙, 𝜀₁, 𝜀₂, 𝜀₃, satisfying the inequalities 𝛾₁₁+∑^𝑚_𝑟=1𝛽_𝑖𝑟𝑃_𝑖−𝑃_𝑟𝛾₁₂𝛾₁₃0∗𝛾₂₂00∗∗𝛾₃₃𝛾₃₄∗∗∗𝛾₄₄<0,𝑖=1,…,𝑚,𝑙,𝑗,𝑔=1,2,…,𝑁_𝑖, (40) where ∗ denotes the symmetric term, then the closed-loop system (10a), (10b), and (10c) is asymptotically stable in zero by the reliable controllers (14b). The parameters of formulae (14a) and (14b) satisfy (18) and (19).
It should be noted that the stability conditions (40) can be transformed into the linear matrix inequalities by using Schur’s complement. Consider the following:𝜎₁₁+∑^𝑚_𝑟=1𝛽_𝑖𝑟𝑃_𝑖−𝑃_𝑟𝜎₁₂𝜎₁₃𝜎₁₄𝜎₁₅0𝜎₁₇0000∗𝜎₂₂000000000∗∗𝜎₃₃00000000∗∗∗𝜎₄₄0000000∗∗∗∗𝜎₅₅𝜎₅₆00000∗∗∗∗∗𝜎₆₆00000∗∗∗∗∗∗𝜎₇₇𝜎₇₈000∗∗∗∗∗∗∗𝜎₈₈𝜎₈₉00∗∗∗∗∗∗∗∗𝜎₉₉𝜎₉₁₀0∗∗∗∗∗∗∗∗∗𝜎₁₀₁₀𝜎₁₀₁₁∗∗∗∗∗∗∗∗∗∗𝜎₁₁₁₁<0,𝑖=1,…,𝑚,𝑙,𝑗,𝑔=1,2,…,𝑁_𝑖, (41)where ∗ denotes the symmetric term, where 𝜎₁₁=𝑃_𝑖⁻¹𝐴^𝑇_𝑖𝑙+𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑙𝐵_Θ𝑖𝑗𝐵^𝑇_Θ𝑖𝑙−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗,𝜎₁₂=𝑃_𝑖⁻¹,𝜎₁₃=𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗^𝑇,𝜎₁₄=𝐴_𝑖𝑙𝑃_𝑖⁻¹−𝛼_𝑖𝑗𝐵_Θ𝑖𝑙𝐵^𝑇_Θ𝑖𝑗^𝑇,𝜎₁₅=𝐴_ℎ𝑖𝑙,𝜎₅₆=𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖,𝜎₁₇=𝐿_𝑖𝑙𝐶_𝑖𝑔,𝜎₇₈=𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙,𝜎₈₉=𝐶^𝑇_𝑖𝑔𝐿^𝑇_𝑖𝑙𝑅_𝑖,𝜎₉₁₀=𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇,𝜎₂₂=−𝑄_𝑖⁻¹,𝜎₃₃=−𝜏⁻¹𝑅_𝑖⁻¹,𝜎₄₄𝐼=−𝜏𝜀₁+𝜀₂,𝜎₅₅=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑖𝐴_ℎ𝑖𝑙+𝜀₃𝐴^𝑇_ℎ𝑖𝑙𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑖,𝜎₆₆=−𝜏⁻¹𝜀₁𝜎𝐼,₇₇=𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑃_𝑒+𝑃_𝑒𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔+𝑄_𝑒,𝜎₈₈=−𝜏⁻¹𝑅_𝑖⁻¹,𝜎₉₉=−𝜏⁻¹𝜀₂⁻¹+𝜀₃⁻¹⁻¹𝜎𝐼,₁₀₁₀=−𝜏⁻¹𝑅_𝑒⁻¹,𝜎₁₀₁₁=𝑃_𝑒𝐴_ℎ𝑖𝑙+𝜏𝐴_𝑖𝑙−𝐿_𝑖𝑙𝐶_𝑖𝑔^𝑇𝑅_𝑒𝐴_ℎ𝑖𝑙,𝜎₁₁₁₁=𝜏𝐴^𝑇_ℎ𝑖𝑙𝑅_𝑒𝐴_ℎ𝑖𝑙−1−𝑑_𝑖𝑄_𝑒. (42)

Remark 6. The Lyapunov function (34) is multi-Lyapunov containing positive definite matrixes 𝑃_𝑖,𝑅_𝑖,𝑄_𝑖,𝑃_𝑒,𝑅_𝑒, and 𝑄_𝑒. There will be some difficulty to seek positive definite matrixes, which induce some possible conservativeness different from the partition definite matrix method in [38]. However, we adopt the following methods to overcome the main sources of conservativeness.(1) Notice that the stable conditions in Theorem 5 are all of LMI form due to (41), we can solve the problem of stabilization converting to the LMIs’ problem, and the method of augmenting matrix norm is avoided.(2) The maximum delay bound is obtained by solving LMIs containing variables 𝑃_𝑖,𝑅_𝑖,𝑄_𝑖,𝑃_𝑒,𝑅_𝑒, and 𝑄_𝑒 instead of preselecting 𝑃_𝑖,𝑅_𝑖,𝑄_𝑖,𝑃_𝑒,𝑅_𝑒, and 𝑄_𝑒. Thus, optimizing positive definite matrices in order to obtain a less conservative delay bound can be easily realized.

4. Illustrative Numerical Example

To illustrate the effectiveness of proposed approach, the following switched fuzzy system with time-varying delay is considered: 𝑅¹₁∶if𝑧is𝑀¹₁₁𝑥̇(𝑡),then=𝐴₁₁𝑥(𝑡)+𝐴_ℎ11𝑥𝑡−𝑑₁(𝑡)+𝐵₁₁𝑢₁(𝑡),𝑦(𝑡)=𝐶₁₁𝑥(𝑡),𝑅²₁∶if𝑧is𝑀²₁₁𝑥̇(𝑡),then=𝐴₁₂𝑥(𝑡)+𝐴_ℎ12𝑥𝑡−𝑑₁(𝑡)+𝐵₁₂𝑢₁(𝑡),𝑦(𝑡)=𝐶₁₂𝑥(𝑡),𝑅¹₂∶if𝛿is𝑀¹₂₁𝑥̇(𝑡),then=𝐴₂₁𝑥(𝑡)+𝐴_ℎ21𝑥𝑡−𝑑₂(𝑡)+𝐵₂₁𝑢₂(𝑡),𝑦(𝑡)=𝐶₂₁𝑥(𝑡),𝑅²₂∶if𝛿is𝑀²₂₁𝑥̇(𝑡),then=𝐴₂₂𝑥(𝑡)+𝐴_ℎ22𝑥𝑡−𝑑₂(𝑡)+𝐵₂₂𝑢₂(𝑡),𝑦(𝑡)=𝐶₂₂𝑥(𝑡) (43) along with system matrices 𝐴₁₁=−4.5302,𝐴₁₂=,𝐴−182.50−11₂₁=1.501−4.5,𝐴₂₂=,𝐴−120−4−5_ℎ11=𝐴_ℎ12=0.60.500.2,𝐴_ℎ21=𝐴_ℎ22=,𝐵0.10.60.40.3₁₁=𝐵₁₂=1202,𝐵₂₁=𝐵₂₂=,𝐵2025_Θ11=𝐵_Θ12=1000,𝐵_Θ21=𝐵_Θ22=,𝐶0005₁₁=𝐶₁₂=𝐶₂₁=𝐶₂₂=20. (44)

It should be noted that (𝐴_𝑖𝑙,𝐵_Θ𝑖𝑙) is not a stable pair system matrices. In turn, a subsystem is unstable when failures of actuators occur. Nonetheless, the reliable control problem is solvable via switching between unstable subsystems as proved in the previous section (see Theorem 3). If the time-varying delays are chosen as𝑑₁(𝑡)(𝑡),𝑑=0.003−0.003sin₂(𝑡)=0.01−0.01sin(𝑡), (45a)(45b)

then it is clear that 𝑑₁≤0.003, 𝑑₂≤0.01, and 𝜏≤0.02.

The membership functions, respectively, are chosen as, 𝜇_𝑀111(𝑧(𝑡=1))1+𝑒^{−2𝑧(𝑡)},𝜇_𝑀211(𝑧(𝑡))=1−𝜇_𝑀111,𝜇_𝑀121(𝛿(𝑡=1))1+𝑒^{(−2(𝛿(𝑡)−0.3))},𝜇_𝑀221(𝛿(𝑡))=1−𝜇_𝑀121. (46)

Then the parameters of controllers are defined: 𝜉₁₁=𝜉₁₂=1,𝜉₂₁=𝜉₂₂=0.8,𝜀₁=𝜀₂=𝜀₃=0.01. (47)

Define the state observers as 𝑥̇(𝑡)=²²_𝑖=1²_𝑙=1²_𝑗=1𝑔=1𝑣_𝑖(𝑥(𝑡))𝜂_𝑖𝑙(𝜉(𝑡))𝜂_𝑖𝑗(𝜉(𝑡))𝜂_𝑖𝑔(𝜉(𝑡×𝐴))_𝑖𝑙−𝐵_𝑖𝑙𝐾_𝑖𝑗𝑥(𝑡)+𝐴_ℎ𝑖𝑙𝑥𝑡−𝑑_𝑖(𝑡)+𝐿_𝑖𝑙𝐶_𝑖𝑔𝑒(𝑡), (48)

upon choosing 𝛼₁₁=𝛼₁₂=3,𝛼₂₁=𝛼₂₂𝛽=4,_𝑖𝑟=1(𝑖,𝑟=1,2). (49)

Solving inequality (16), one can find that 𝑃₁=0.2034−0.1141−0.11410.9462,𝑃₂=,𝑅0.5998−0.0412−0.04121.5313₁=0.3148−0.1214−0.12140.9789,𝑅₂=,𝑄0.6112−0.0291−0.02910.9891₁=0.6471−0.2438−0.24381.1231,𝑄₂=,𝑃0.9876−0.1128−0.11280.8532_𝑒=0.4924−0.2089−0.20890.9479,𝑅_𝑒=,𝑄0.5133−0.2087−0.20870.8921_𝑒=.0.6127−0.0921−0.09210.9432 (50)

From (18), the observer gain matrices are derived as follows: 𝐿₁₁=𝐿₁₂=4.48070.9875,𝐿₂₁=𝐿₂₂=3.58450.7900. (51)

Next, define Ω₁=𝑥(𝑡)∈𝑅²∣𝑥(𝑡)^𝑇𝑃₁−𝑃₂𝑥(𝑡)𝑥(𝑡),Ω≥0,≠0₂=𝑥(𝑡)∈𝑅²∣𝑥(𝑡)^𝑇𝑃₂−𝑃₁𝑥(𝑡)≥0,𝑥(𝑡)≠0. (52) Then, Ω₁∪Ω₂=ℝ²⧵{0}. Thus the closed-loop system is asymptotic stability under the following switching law: 𝜎(𝑡)=1𝑥(𝑡)∈Ω₁,2𝑥(𝑡)∈Ω₂⧵Ω₁. (53)

From (19), we obtain output feedback controller 𝐾₁₁=𝐾₁₂=,𝐾0.6102−0.342300₂₁=𝐾₂₂=.00−0.824030.6260 (54)

A simulation has been performed with the initial states 𝑥(0)=[−15,3]^𝑇 and the time-varying delay chosen as (45a), (45b). The simulation results for the observer error and control signal are depicted in Figure 1.

(a)

(b)

Figure 1 

(a) The observer error trajectories of SFS with time-varying delay and (b) control signal of the reliable controller.

5. Conclusion

In this paper, the problem of reliable control for nonlinear systems represented by time-varying delay switched fuzzy systems has been investigated. Firstly, a model of switched fuzzy systems which combines reliable control and switched fuzzy control was presented, and all subsystems of the switching system are time-varying delay fuzzy systems. In particular, attention was focused on actuators that may suffer “serious faults”.

Reliable controllers and switching law are designed such that they make the switched fuzzy systems asymptotic stability, based on the switching strategy, that is, switching between faulty actuators. Moreover, the switching strategy is dependent on the state of observer error. Finally, a numerical example has been provided to illustrate the efficiency of the proposed reliable controller design and observer switching law methodology.

Acknowledgments

This work is supported by National Nature Science Foundation under Grant 61004039, by the Science Foundation of Educational Department of Liaoning Province under Grant L2010374, and also by the key laboratory for manufacturing industrial integrated automation of Liaoning Province. The authors would like to thank Professor Ximing Sun for his valuable comments.