Optimal Tracking And Disturbance Rejection With Zero Steady-State Error For Time-Delay Systems

Output regulation is a classical problem in control theory, which aims to design a feedback control law to impose a prescribed response subject to every external command in a prescribed family. This includes the problem of tracking a prescribed reference input signal and rejecting undesired disturbances, meanwhile ensuring the asymptotic stability of the closed-loop systems. Both reference signals and disturbances are generally described by autonomous differential equations, which are referred as exosystems. Early in the 1970s, with the introduction of the famous internal model principle, the problem of output regulation of linear systems was posed and perfectly solved. Recently, the output regulation theory has been extended to the case of nonlinear systems and uncertain systems. The issue of tracking control and disturbance rejection is also an active research subject in control engineering, which can find its wide applications in oceanics, aeronautics and astronautics, process control, and robotic control, etc.In aeronautics and astronautics, process control, and networked control systems, due to the delay in signal transmission and computation time, time delays are commonly encountered. The presence of time delays could seriouly degrade performances of the control system, or even drive the system to instability. From the theoretical point of view, time-delay systems are infinite dimensional systems described by functional differential equations. Therefore, the issue concerning time-delay systems is rather difficult to solve. Stability analysis and control algorithm design of time-delay systems are important research subjects. On the other hand, practical control systems are frequently subjected to various external disturbances. These disturbances can not only lead to working-position offset, but worsen the dynamic and stable characteristics of the system, or even destabilize the system and result in control failure. So it is of great significance to cancel or attenuate the effect of exogenous disturbances, both in theory and practice.This dissertation investigates the problem of optimal tracking and disturbance rejection with zero steady-state error for a class of time-delay systems, respectively. For a class of reference input signals with known generating dynamics, an optimal tracking control algorithm is designed to control the output of the master system to track the exosystem while optimizing the performance index at the same time. For a class of known disturbances, an optimal disturbance rejection algorithm is designed to guarantee the closed-loop system with zero steady-state error. Finally, a reduce-order reference-input observer and disturbance observer are constructed to make the state variables of the exosystem in the controller physically realizable. The contents of this dissertation are as follows:1. The preface gives an overview of the optimal control theory, the approximate approaches of optimal control for nonlinear and time-delay systems, the state-of-the-art of tracking control and disturbance rejection, time-delay systems and their relevant control problems. The research subject and significance of this dissertation are also given.2. The problem of optimal tracking control is considered for a class of linear systems with state time delay based on a quadratic performance index. For the two-point boundary value (TPBV) problem with both time-delay and time-advance terms deduced from the maximum principle, by using the successive approximation approach (SAA), two iterative sequences of differential equations with known initial and terminal conditions are first constructed, respectively. Their solution sequences are then proven to uniformly converge to the solution of the original TPBV problem. For the case of infinite time horizon, the condition of existence and uniqueness of the optimal solution is given and proven. Furthermore, by finite iteration of the solution sequences, an approximate solution of the optimal tracking control problem is obtained. Finally a reduced-order reference-input observer is designed to solve the physically realizable problem with the feedforward term in the optimal tracking controller. Simulation examples illustrate that this approach is effective and simple to implement.3. An observer-based optimal tracking control problem is considered for a class of time-delay systems with exogenous disturbances. Based on the SAA, two iterative sequences of nonhomogenous linear vector differential equations are first constructed, whose solution sequences are then proven to be uniformly convergent. The condition of existence and uniqueness of the optimal tracking control law is given and proven for the case of infinite time horizon. Hence, the original problem is transformed into a problem of iterating a sequence of adjoint vector differential equations. Furthermore, by finite iterations of the solution sequence, an approximate optimal tracking control law is obtained, which consists of an analytical state feedforward and feedback term and a time-delay compensation term in the limit of adjoint vector sequence. A reduced-order disturbance observer and a reference-input observer are constructed respectively to estimate the unknown states of the exosystems. Simulation examples illustrate that this algorithm is effective, and robust with respect to some exogenous disurbances.4. An optimal tracking control problem is considered for a class of interconnected large-scale systems with state time delays, which are composed of N coupled time-delay subsystems. First, two iterative sequences of differential equations are constructed based on the SAA and the uniform convergence of their solution sequences are then proven. Thus, the interconnected large-scale systems are decomposed into N decoupled subsystems. A feedforward and feedback optimal tracking control law is obtained. The condition of existence and uniqueness of the optimal solution is derived and proven for the infinite-time horizon case. To solve the physically realizable problem with the feedforward term, N reduced-order reference-input observers are constructed respectively to estimate the unknown states of the exosystems. Simulation results show that the proposed algorithm is effective and has a light on-line computation load.5. An optimal disturbance rejection problem with zero steady-state error is considered for a class of time-delay systems. Based on the internal model principle, a disturbance servo compensator is constructed to neutralize the effect of the exogenous persistent disurbances. An augmented system is obtained by connecting the disturbance compensator in series with the original control system. By selecting a quadratic performance index, an optimal control law is designed based on the optimal regulator theory. By using the SAA, the optimal control problem of time-delay systems is reduced to solving a sequence of nonhomogenous linear TPBV problems without time-delay or time-advance terms, whose solution sequences are proven to uniformly converge to the solutions of the original problem. The obtained control law consists of an analytical feedforward and feedback term, and a time-delay compensation term in the limit of adjoint vector sequence. This controller can guarantee the closed-loop system without steady-state error meanwhile optimizing the quadratic performance index. Simulation results illustrate that this algorithm is effective, and has a fast convergence rate and a high iteration precision. Furthermore, this approach is extended to the optimal zero steady-state control of nonlinear systems with persistent disturbances. Based on the SAA, new theoretical results and simulation results are obtained by treating the nonlinear term of the system as additive disturbances.6. The last chapter summarizes the research work of this dissertation and gives an outlook on tracking control and disturbance rejection for time-delay systems.