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The problem of optimal fault-tolerant control for a class of descriptor time-varying systems with nonlinear input is considered. Based on the Lyapunov stability theorem, the sufficient conditions of the stability are obtained when the system is normal and ineffective. Furthermore, the fault-tolerant control of the systems is carried out in two cases, and the state feedback fault-tolerant controller is obtained to satisfy the quadratic performance index and reach the minimum value in order to achieve the optimal fault-tolerant control. Finally, the validity of the proposed approach is illuminated by a numerical example.

Descriptor systems were first studied in the 1970s and are now widely used in control systems, electrical circuits, mechanical systems, economics, and other areas. Because descriptor systems can better describe the actual system, there has been an extensive research in this area, resulting in insubstantial outcomes during the past 40 years. The theory underlying descriptor systems has become a critical area of research in modern control theory. But in the actual control systems, there are usually a series of unforeseen and unavoidable problems, such as the actuator failure and sensor failure in the systems. Therefore, focusing on these problems, the systems should have certain fault-tolerant performance to continue to maintain the original operation ability while the actuators or sensors of the systems are under fault. In addition, in the actual control system, according to the input of the system there will be some nonlinear input and the controller design is more complex; it is necessary to solve the problem of the fault-tolerant control against descriptor time-varying systems with nonlinear input.

In recent years, fault-tolerant control has attracted many scholars’ attention; Zhou and Ding [

In the past literatures, we have had a lot of work on stability analysis and fault-control problems for descriptor systems, but the related results on descriptor time-varying systems with nonlinear input remain undiscovered. Considering the importance of the systems in real world, in this paper, we investigate the optimal fault-tolerant control against time-varying descriptor systems with nonlinear input; the sufficient conditions are obtained by Lyapunov stability theorem when the system is normal and ineffective. Further, based on the stability condition, the optimal fault-tolerant controller for the system is obtained through the boundary conditions, which met the given quadratic performance index. Finally, an example is given to solve the problem by MATLAB.

Consider the following descriptor time-varying system with nonlinear input:

For convenience, let

The fault-tolerant control law for system (

The descriptor time-varying system (

The descriptor time-varying system (

The descriptor time-varying system (

In order to solve the optimal control problem of system (

The descriptor time-varying system (

Consider the control input

Therefore, solving the stability condition of system (

Based on Theorem

The descriptor time-varying system (

When there is no failure in system (

For the descriptor time-varying system with nonlinear inputs described by (

When there is no fault in the system function, according to the optimal control theory, solving the optimal control problem of the descriptor time-varying systems with nonlinear input described by (

For the descriptor time-varying fault system with nonlinear inputs described by (

Similar to the proof of Theorem

Considering system (

Let the nonlinear input in system (

Then

Select the following quadratic performance index:

When there is no fault in the system, we choose

System state under optimal control when the system without fault.

When there is failure in the system, we still choose the initial input

System state without fault-tolerant control.

System state with fault-tolerant control.

This paper considers a class of descriptor time-varying systems with nonlinear input; sufficient conditions are obtained by Lyapunov stability theorem; on the basis of stability conditions, the optimal fault-tolerant controller of the system is obtained by using the optimal control theory, while fault occurs in the system; this method can realize fault tolerance control and reduce the impact of failure; then the example is carried on by MATLAB simulation. The method used in this paper is a general generalization of descriptor system to descriptor time-varying systems, and it has important theoretical significance for further study of some applications of descriptor time-varying systems.

The authors declare that they have no conflicts of interest.

This work was supported by

_{∞}passive fault-tolerant control for uncertain singular systems