Reliable Guaranteed Cost Control of Uncertain Systems with Nonlinear Perturbations

: Network control system (NCS) is a distributed real-time feedback control system with the continuous development of network technology. It is generally composed of network, controller, actuator and sensor. It has brought great convenience to people in many fields, but it also has many problems, which makes the research of control system more complicated. Recently, there have been some efforts to tackle the reliable guaranteed cost controller design problem, and some good results have also been obtained for the continuous-time and for the discrete-time. However, there have been few results in the literature of an investigation for the reliable guaranteed cost controller design of nonlinear uncertain systems with time-varying state delay and actuator failure. This paper concerns the reliable guaranteed cost control problem of uncertain systems with time-varying state delay and nonlinear perturbations for a given quadratic cost function. The problem is to design a reliable guaranteed cost state feedback control law which can tolerate actuator failures, such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound. Firstly, the existence condition of reliable guaranteed cost control law is given by constructing Lyapunov stability function and using linear matrix inequality (LMI). Secondly, the design method of the optimal reliable guaranteed cost controller is given by solving the convex optimization problem with LMI constraints, which minimizes the upper bound of guaranteed cost for closed-loop systems. In the end, the numerical simulation result illustrates the effectiveness of the proposed method.


Introduction
The problem of designing robust controllers for systems with parameter uncertainties has drawn considerable attention in recent control system literatures. It is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance. One approach to this problem is the so-called guaranteed cost control approach [1]. This approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound. Based on this idea, some significant results have been proposed for the continuous-time case [2,3] and for the discrete-time case [4]. Kao Y. designed a non-fragile H∞ guaranteed cost controller for uncertain long time delay nonlinear network control system with perturbation of controller parameters [5]. Yin combined the robust control theory to model the NCS, introduced performance indicators, established the internal relationship with network factors, and it also reduced the conservativeness of the results [6]. The controller design for Markov time-delay NCS has also achieved good results, and a design method to maintain NCS performance by switching feedback gain is proposed [7].
In practical application, actuators are very important in transforming the controller output to the plant. Actuator failures may be encountered sometimes. Furthermore, how to preserve the closed-loop system performance in the case of actuator failures will be tougher and more meaningful. Recently, there have been some efforts to tackle the reliable guaranteed cost controller design problem [8][9][10]. Yao took the uncertain nonlinear time-delay system as the controlled object, and designed a reliable guaranteed cost controller considering the actuator failure of the NCS [11][12]. Zhang J.-S. presented a guaranteed cost control method for multiple time delays and actuator failures [13]. Zhu considered the simultaneous failure of the sensor and controller, and modeled as two time-varying and bounded parameters [14]. Sun first attempted to implement H∞ guaranteed cost control for switched T-S fuzzy stochastic systems with intermittent actuator [15]. For the same fault model, Zou redesigned the static output feedback controller, so that the system can keep stable and satisfy the original performance indexes no matter the actuator fails or not [16].
In this paper, the problem of reliable guaranteed cost control of uncertain systems with time-varying state delay and nonlinear perturbations is considered. In Section 2, the problem under consideration and some preliminaries are given. In section 3, several stability criteria for the existence of the reliable guaranteed cost controller are derived in terms of LMI, and their solutions provide a parameterized representation of the controller. A numerical example is given in Section 4. Finally, Section 5 concludes the paper.

Problem Statement
Consider the following uncertain systems with time-varying delay and nonlinear perturbations ), , where H is a known constant matrix satisfying Associated with this system is the cost function Where Q and R are given positive-definite matrices.
For the control input ( ) u t denote the signal from the actuator that has failed. The following failure model is adopted in this paper: In the above model of actuator failure, if i , it corresponds to the partial failure case, namely, partial degradation of the actuator. Denote The objective of this paper is to develop a procedure to design a memoryless state feedback control law such that for any admissible uncertain α , the resulting closed-loop system is quadratically stable and the cost function (4) satisfies * J J ≤ , where * J is some specified constant. Definition 1. If there exists a control ( ) ( ) u t Kx t = and a said to be a reliable guaranteed cost control law for system (1) and cost function (4). Define From (7) and (10) , the following holds:

Main Results
Since it holds that ( ) Then, rewrite system (9) in an equivalent form where, The following Lyapunov-Krasovskii functional is applied Then, the following theorem gives the delay-dependent reliable guaranteed cost control for the systems (1) and (4).
= is a reliable guaranteed cost control law if there exist positive-definite matrices P , S , R , matrices Y , Z , and a scalar 1 0 ε > , such that for any admissible α , the following matrix inequalities hold: 11 12 13 22 23 where ( ) * denotes the symmetric element of a matrix, and   T  T  T  T  T  C  C  C  C  T  C  D  D  T  C  T  T  D  D  T   Moreover, the cost function (4) satisfies the following bound: Proof. Taking ( ) ( ) u t Kx t = in the system (1), the resulting closed-loop system is given by (9).

t x t Px t x t Px t A A x t A x s ds f Px t x t P A A x t A x s ds f x t A A P P A A x t x t PA x s ds x t Pf f Px t
we obtain

t Rx t x t Rx t d t x t Rx t x s Rx s ds hx t Rx t x s Rx s ds h A x t A x t t f R A x t A x t t f x s Rx s ds
From the above inequalities, we can obtain Then, the matrix inequality (19) implies that Noting that 0 Q > and 0 R > , this implies that the system (9) is asymptotically stable by Lyapunov stability theory. Moreover, from (28) we have by integrating both sides of (29) from 0 to f T , we obtain As the closed-loop systems (9) is asymptotically stable This completes the proof.
In the sequel, we will show that the criterion for the existence of guaranteed cost controller is equivalent to the feasibility of a LMI.
Theorem 2. For system (1) with cost function (4), if there exist scalars 1 0 R > , such that the following LMI is feasible:  is a reliable guaranteed cost controller of system (1), and the corresponding closed-loop value of the cost function satisfies Proof. Letting 0 Y ≡ and 0 Z > in (20), in light of Lemma 2 and using the Schur complement, we can obtain 11 1 substituting the representation of matrix C A , D A into (35), and using the Schur complement again, we can obtain Using the Schur complement again, and by (12) T  T  T  T  d  T  T  T  d It is assumed that the single input to the system has partial failure as follows:

Conclusions
In this paper, based on the Lyapunov method, we have presented a design method to the reliable guaranteed cost controller via memoryless state feedback control for uncertain systems with time-varying state delay and nonlinear perturbations in an LMI framework. The parameterized representation of a set of the controller, which guaranteed not only the robust stability of the closed-loop system but also the cost function bound constraint, has been provided in terms of the feasible solutions to the LMIs. Furthermore, a convex optimization problem has been introduced to select the optimal reliable guaranteed cost controller. Finally, a numerical example is given for illustration of the controller design.