Optimal Control For Discrete Nonlinear Time-delay Systems With Applications: A Successive Approximation Approach

It is well known that computer control is the most typical discrete-time control system. In recent years, the rapid development of computer technology has greatly facilitated the applications of microprocessors and microcomputers in control systems. On the one hand, the emergences of the high efficiency microprocessor make the application of the nonlinear time-delay controller become simple. On the other hand, the modern techniques request stricter controllers. Thus the analysis and synthesis of the discrete-time systems have become an important component in the control theory.The optimal control of nonlinear time-delay systems exists in life science, engineering science, computer science and economics etc. It is an optimal functional problems which subject to nonlinear time-delay dynamic systems constraints. The research of numerical optimization methods and algorithms in this field is not only the front subjects of nonlinear control, but also the leading edges of cross subjects. At present the results are mostly focus on the qualitative analysis, and it is imperious to study the numerical optimization methods and algorithms.The dissertation first reviews the relative studies on the optimal control and optimal output tracking control (OOTC) problem for nonlinear time-delay systems up to now in detail. The latest research tendency and the main methods are also introduced. The major results of this dissertation are summarized as follows.1. The optimal control problem for discrete-time nonlinear systems with a quadratic performance index is studied. A successive approximation approach (SAA) is proposed to find a solution sequence of the nonlinear two-point boundary value (TPBV) problem, which is obtained from the necessary optimality conditions. We take the nonlinear term as an additional disturbance of the system and turn the system model into an iterative form. By introducing a nonlinear compensation vector and using the SAA, we transform the nonlinear TPBV problem into a sequence of nonhomogeneous linear TPBV problems. By taking the finite iterative value, we obtain a suboptimal control law. The conditions of existence and uniqueness of the optimal control law are presented for infinite-time horizon problems. Simulation examples are employed to test the validity of the optimal control algorithm.2. The optimal control problem of finite-time and infinite-time nonlinear discrete systems with state delay is developed. A sequence of non-delay discrete systems is constructed, which uniformly converges to the original discrete system with time-delay. Then the optimal control for the original discrete nonlinear system is transformed into an optimal control sequence for non-delay linear systems. By truncating a finite term of the optimal sequence, a suboptimal control law is obtained. The suboptimal laws consist of linear analytic terms and a time-delay compensation term, and the time-delay compensation term is described by a limit of the solution sequence of the adjoint state vector equations. Simulations show the algorithm has lower computation complexity and can be easily implemented. Moreover, even there is large delay in the systems, the accuracy of the optimal solution and the computation speed is satisfied.3. Systems subject to an input delay or measurement delay are more common. The so-called"Artstein model reduction"is often involved when one considers systems with input delay. By introducing the new variable the original system is reduced to a system free-of-delay. The optimal control problem of a class of discrete-time nonliear system with input delay is considered based on the thoughts of Artstein model reduction of the continuous systems. Then the optimal controller is designed by SAA.4. The OOTC problem of the discrete-time system with multiple state and input delays whose reference input is generally produced by an exosystem is addressed. The state of the exosystem is introduced into the feedforward control, instead of constructing an augment system as the classical optimal control theory. Thus the tracking error can be reduced and the feedback control effort can be decreased. The SAA is then applied to the OOTC problem. The existence and uniqueness of the optimal control law in infinite-time horizon is proved and the detail design process is proposed. The obtained optimal control laws consist of linear analytic terms and time-delay compensation terms. The linear analytic terms can be found by solving a Riccati equation and a Stein equation respectively, and the time-delay compensation term is described by the limit of the solution sequence of the adjoint state vector. In this case, the OOTC law obtained contains the state variable of the Exoystem which is unrealizable in physical. In order to solve this problem, a reduced observer is introduced. 5. The OOTC problem of the discrete-time system with input delays is addressed. Firstly the system is transformed into a nonlinear system without delay in state equations by using the model reduction in discrete-time systems. Then an adjoint vector is introduced to compensate the input delays in output equation and the nonlinear part in state equation. Using SAA, the original nonlinear TPBV problem is transformed into a sequence of nonhomogeneous linear TPBV problems without unknown time-delay terms and nonlinear terms and then an approximate OOTC law is obtained. Simulation results show that the proposed algorithm is effective and has better convergence properties at different time-delays.Finally, the conclusions are given, and a proposition is indicated on the research work in the future.