Study Of Approximate Optimal Control For Discrete Time-Delay And Nonlinear Systems With Disturbances

Almost all practical control systems are affected by external disturbances. For instances, fully-submerged hydrofoil craft, off-shore platform and ship locate in hostile marine environment and undergo excessive vibrations due to wave loads and wind. The vibrations derived from the external disturbances make the system performances and working environment bad to some extent. On the other hand, time-delay is a common phenomenon in real control system. The phenomenon of time delay not only degrades the system performance, but also causes instability of the system. Although discrete time-delay systems can be transformed into systems without time-delay by expanding the system's dimension, systems of high dimension will suffer from a"dimension disaster", which can burden computer with geometric progression. Actual systems are nonlinear inherently. The optimal control law for general nonlinear system with quadratic index is given in terms of solution to the Hamilton-Jocabi-Bellman (HJB) equation or the nonlinear Two Point Boundary Value (TPBV) problem. But for general nonlinear systems with exception of the simplest cases, the analytical solution does not exist. This has inspired researchers to present some approaches to obtain an approximate solution to the HJB equation or the nonlinear TPBV problem as well as obtaining an approximate optimal control for the nonlinear systems. Moreover, there are few results on the optimal tracking control (OTC) problem of nonlinear systems. In conclusion, optimal disturbance rejection and output tracking for time delay and nonlinear systems are important and meaningful in theory and practice.The dissertation first reviews the history of the development in optimal disturbance rejection and optimal output tracking problem of time-delay and nonlinear systems. The latest research tendency and the main methods are also introduced. The major results are shown as follows:1. An optimal damping control (ODC) law for discrete time-delay systems with sinusoidal disturbances with respect to the quadratic performance index is discussed. An approximation approach is applied to design the ODC law. The algorithm to obtain an approximate ODC law is proposed. Simulation examples indicate that the ODC method proposed in this paper is easy to implement and more robust with respect to external sinusoidal disturbances than that of classical feedback optimal control laws.2.Optimal disturbance rejection control (ODRC) problem for discrete time-delay systems whose additive external disturbance is denoted by exosystem is discussed. An approximation approach is presented to design the ODRC law. By introducing a sensitivity parameter, the original ODRC problem is transformed into a series of difference equations without time-advance and time-delay terms. The ODRC law obtained consists of analytic feedback and feedforward terms and a compensation term, which is the sum of the infinite series of the adjoint vector. The existence and uniqueness of the ODRC law are proved. The algorithm to obtain an approximate ODRC law is proposed. When considering process and measurement noise, the optimal estimator of the disturbance state can be obtained by a kalman filter, which makes the approximate ODRC law to be physically realizable. Numerical examples show the algorithm is robust with respect to the additive persistent disturbances and effective under different time-delays, especially for systems with long time-delay.3.Optimal tracking control (OTC) for discrete time-delay systems affected by persistent disturbances with infinite and finite horizon quadratic performance indexes is considered. For OTC problem of the discrete time-delay system whose reference input is generally produced by an exosystem, the state of the exosystem is used as the information of feedforward control action, instead of constructing an augment system. Because a feedforward control action is added, the tracking error can be reduced. This method is easy to be implemented and avoids expanding the system's dimension. By designing two external disturbances observers, the physically realizable problem of the OTC law is solved. As we mentioned above, discrete time-delay systems can be transformed into systems without time-delay by expanding the system's dimension and designed the controller by using the classical optimal control theory. In this case we compared simulation results by the method proposed in this paper and that based on classical optimal control theory.4.OTC for discrete nolinnear systems with infinite and finite horizon quadratic performance indexes is considered. The optimal control law for general nonlinear system with quadratic index is given in terms of solution to the HJB equation or the nonlinear TPBV problem. The analytical solutions of the nonlinear TPBV problem or the HJB equation do not generally exist, with exception of the simplest cases. An approximation approach is presented instead of the method based on"state-dependent Riccati equation". By introducing a sensitivity parameterεinto the variables of the systems and expanding the Maclaurin series around it atε= 0, the original OTC problem is transformed into a series of nonhomogeneous linear TPBV problems. The OTC law obtained consists of linear analytic functions and a compensation term which is a series sum of adjoint vectors. By using a finite sum of the series of the adjoint vectors, we can obtain a suboptimal output tracking control (SOTC) law.5.Problem of optimal disturbance rejection with zero steady-state error for discrete-time systems affected by external persistent disturbance is considered. Here the disturbance is neither stable nor asymptotically stable. Based on the internal model principle (IMP), external persistent disturbances (sinusoidal) are compensated by constructing a disturbance compensator. Then, the optimal controller is designed by using classical optimal control theory, and the convergence of the cost functional can be guaranteed. The augmented system is proved to be controllable and observable under the assumption. A dynamic optimal feedback loop controller with zero steady-state error is obtained. When considering process and measurement noise, a dynamic LQG controller is designed by stochastic maximum principle. Then, the proposed optimal dynamic controller is applied to the offshore structure subjected to wave force. The effectiveness of the presented law is demonstrated by a numerical example of jacket-type offshore structure. The vibration of the platform can be reduced significantly with low cost.