Guaranteed Cost Control For Uncertain Nonlinear Time-Delay Systems

Recently, the problem of designing robust controllers to make uncertain systems not only be stable but also guarantees an adequate level of performance has drawn considerable attention. One approach to this problem is the guaranteed cost control approach first introduced by Chang and Peng. This approach is to design a controller such that the closed-loop system is asymptotically stable for all admissible uncertainties and the closed-loop cost function value is not more than a specified upper bound. It is well known that time-delays as well as parameter uncertainties are frequently the main cause of deterioration of systems performance and instability of systems. Therefore, there has been increasing interest in the guaranteed cost control of uncertain time-delay systems, and many significant results have been presented. However, there have been few results for nonlinear systems, owing to their particularity and complexity.There are lots of uncertain nonlinear time-delay systems in the ocean engineering, for instances, offshore platform vibration control system, control systems for the tracking and positioning of towed vehicles, ship motion control systems, and so on. How to design controllers which make these systems not only be stable but also guarantee an adequate level of performance is a hot research field. Therefore, it' s of important theoretical and practical meaning to study the problems of guaranteed cost control for the uncertain nonlinear time-delay systems.Based on Lyapunov stability theory and linear matrix inequality approach, the problems of guaranteed cost control for nonlinear time-delay systems with norm-bounded uncertainties are considered. The main contents in this paper are as follows:1. There are few results for the discrete-time guaranteed cost control problem, compared with the existed results for continuous-time systems. Firstly, the guaranteed cost control problem for uncertain nonlinear discrete-time systems with state-delay is considered. Several criteria for the existence of the guaranteed cost control law and design method of the controller are derived in terms of LM1. An upper bound on the guaranteed cost is minimized by solving a convex optimization problem with LMIs.2. Although the controllers yielded by using the existed methods are robust with regard to system uncertainties, their robustness with regard to controller uncertainties has not been considered. The controller may be very sensitive or fragile, with respect to perturbations in the controller coefficients. In this paper, using the LMI method, the nonfragile guaranteed cost control problem is discussed for uncertain nonlinear time-delay systems under two classes of controller gain perturbations, namely, additive form and multiplicative form. The existence conditions and design method of the nonfragile guaranteed cost controller are given in terms of LMI. Solving a convex optimization problem gives the optimal guaranteed cost value of the system.3. In some real plants, the system delay varies with the time going. Therefore, the guaranteed cost control problem for the systems with time-varying delay needs to be considered. In this paper, the nonfragile guaranteed cost control problem for the uncertain nonlinear systems with time-varying delay is Investigated. Using the Lyapunov functional method combined with LMI technique, some delay-dependent conditions for the existence of such controller are derived. By solving the specified LMIs,the design method for the guaranteed cost controller is given. The solutions of the optimal problem give the minimal upper bound on the cost function.4. There are lots of real control systems with multiple time delays in the ocean engineering, and the results on guaranteed cost control for systems with multiple time-delays are seldom seen. The problem of nonfragile guaranteed cost control for a class of nonlinear systems with multiple time-delays is studied in this paper. A state feedback nonfragile guaranteed cost control law is given in terms of LMI. An upper bound on the guaranteed cost is minimized by solving a convex optimization problem.5. The problem of nonfragile guaranteed cost control is discussed for the uncertain nonlinear discrete systems with time-varying delay. By representing the time-delay system in the descriptor form and using the new bounding method on the cross term, new delay-dependent sufficient condition for the existence of the guaranteed cost controller is given in terms of LMI. A convex optimization problem is introduced to find the optimal nonfragile guaranteed cost controller which minimizes the upper bound on the cost function.The last section summarizes the main work in this paper and prospects the research of uncertain nonlinear guaranteed cost control problems in the future.