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

This paper deals with the reliable control problem of nonlinear systems represented by switched fuzzy systems (SFS) with timevarying 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) fuzzymodel, 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.


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][2][3][4][5][6][7]. This approach is completely different from typical fuzzy controls [8][9][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][15][16][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.

Mathematical Problems in Engineering
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 ∞ 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 deadzone 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][30][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][36][37][38][39][40][41][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][37][38][39][40][41][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.

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: In here the symbols represent the following: : = {1, 2, . . . , } is the switching signal to be designed.
Therefore, the global model of the th switched subsystem is described bẏ in which where ( ( )) denotes the membership function and ( ) belongs to the fuzzy set . Since the system states often are not directly measurable, we consider the switching signal = (̂( )), which depends on observer states.̂( ) is the observers signal of ( ). Suppose which depends on Ω 1 , Ω 2 , . . . , Ω . When̂( ) ∈ Ω , the switching signal subjects to That is, if and only if = (̂( )) = , then V (̂( )) = 1.

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.
It ensures that the system (2a), (2b), and (2c) is global asymptotic stability, wherê( ) ∈ R is state vectors of fuzzy observer, and represents observer gain matrix for the th fuzzy rule of the th switching subsystem. can be determined by solving conditions (18). denote the controllers feedback gain matrix, and 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 ( ) = ( ) −̂( ), we can obtaiṅ where * denotes the symmetric term, where 11 Remark 4. The controllers are reliable controller because of the controllers feedback gain matrix that are the functions of the actuators failures matrix Θ . Moreover, differing from traditional switching system, the switching signal (8) in this paper is dependeny on the state of observer.
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: 12 where * denotes the symmetric term, where Remark 6. The Lyapunov function (34) is multi-Lyapunov containing positive definite matrixes , , , , , and . 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 , , , , , and instead of preselecting , , , , , and . Thus, optimizing positive definite matrices in order to obtain a less conservative delay bound can be easily realized.

Illustrative Numerical Example
To illustrate the effectiveness of proposed approach, the following switched fuzzy system with time-varying delay is considered: It should be noted that ( , Θ ) 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 as Then the parameters of controllers are defined: Define the state observers aṡ Then, Ω 1 ∪ Ω 2 = R 2 \ {0}. Thus the closed-loop system is asymptotic stability under the following switching law: From (19), we obtain output feedback controller A simulation has been performed with the initial states (0) = [−15, 3] and the time-varying delay chosen as (45a), (45b). The simulation results for the observer error and control signal are depicted in Figure 1.

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.