Nonlinear Delay Systems Suboptimal Control Successive Approximation Method

Time-delay and nonlinear exist widely in many systems such as industry, energy, traffic, aerospace, irrigation, economy management and society. The analysis of control systems with time-delay and nonlinear are significant both in theoretics and practice. For a nonlinear system with time-delay, the optimal control problem based on quadratic performance indexes is usually transformed to a nonlinear two-point boundary value problem with not only time-delay terms but also time-advance terms. Therefore, this kind of optimal control problem is especially difficult to be solved, no matter exact solutions or numeral solutions.This paper firstly introduce the practical and theoretical background of research of nonlinear systems and time-delay systems, then the history of the development in optimal control and sub-optimal control for nonlinear systems and time-delay systems is reviewed, The latest research tendency in this field and the main methods of optimal control and sub-optimal control problem for nonlinear systems and time-delay systems are also presented. We compare the strength and lacks of all these methods.Basing on the current status of the research work, the sub-optimal control for a class of nonlinear systems with time-delay in state variables is considered in this paper, and a successive approximation approach is proposed. We construct a iterative system with known nonlinear and time-delay stimulation basing on the original system model, and prove that the solution sequence of the iterative system uniformly converge to the optimal solution of the original system. For the nonlinear two-point boundary value problem with both time-delay terms and time-advanced terms driven by the optimal control theory, we build a sequence of nonhomogeneous linear two-point boundary value problems according to the successive approximation approach of constant differential equation. A finite iterative result of the two-point boundary value problem sequence is taken as a sub-optimal control law of the original system. The Matlab simulation results show that the design for the sub-optimal controller of nonlinear system with time-delay by using presented algorithm is effective.In the conclusion part of this paper, we summarize the main work of this paper, and point out the problem which should be considered in the future study of the design for the nonlinear system with time-delay. The direction for the future research work is indicated.