Studies Of Nonlinear Sampled-Data Systems Based On Approximate Models

Computer control is a key technique of automation. Sampled-data control theory offers theory guidance for computer control and design of sampled-data controller is a core problem of sampled-data control theory.The approaches to design sampled-data controllers can be set to continuous-time design, discrete-time design and sampled-data design while discrete-time design is the main method to design sampled-data controllers for linear systems. But in the discrete-time design of the nonlinear system there is an intrinsic obstacle. That is, it is impossible to obtain a exact discrete-time model generally, and to analyze or design controller based on the model. To settle this difficulty, some researchers try to analyze and design sampled-data controllers based on the approximate models of nonlinear systems, which is called Approximate Discrete-Time Design. In approximate discrete-time design a approximate discrete-time model of the system is obtained and the controller is designed based on the model. Finally a set of conditions are used to guarantee the controller meet the requirement.After nearly ten years' development, approximate discrete-time design makes some progress and may yield better results than the continuous-time design for some specific systems. But there still exists some problems in the theory of approximate discrete-time design, especially in the stability theory. To emphasize the uniformity of mathematical description, the design is based on the most general approximate model so that the consistency conditions of the closed-loop approximate model are irrational and the necessary conditions for the stability of the closed-loop approximate model are not strong enough. In order to ensure the stability of the closed-loop exact discrete-time model, the condition Lyapunov function is globally Lipschitz is required. However, this condition is unnatural.For the problems mentioned above, the research is focused on the Euler approximation and exponential stability in this dissertation, for which more rational consistency condition can be presented. The consistency condition is that the absolute value of the error of the Euler approximation is no more than some coefficient times the product of the square of sample period and the norm of states. This condition make it easier to judge the stability of the exact discrete-time model with the closed-loop Euler approximation, which just requires the closed-loop vector filed of the system to meet the Lipschitz condition. And with the advantage of exponential stability the necessary conditions of Lyapunov function are strengthen without any additional assumption.In this dissertation three aspects of nonlinear sampled-data control systems are studied:(1) The exponential stability beyond state feedback;(2) The error of nonlinear sampled-data observer;(3) The synchronization error of nonlinear systems with sampled-data control.In (1), it is proved that if the sampled-data controller makes the closed-loop vectorfield globally Lipschitz, then the closed-loop Euler approximation exponential stable is the sufficient condition for exponentially stability of the closed-loop exact discrete-time model. The assumption Lyapunov function is globally Lipschitz is unnecessary.In (2), it is presented that the error of sampled-data observer can not be kept zero while the system is not at the equilibrium point. The error's order of magnitude is analyzed which shows that a more precise approximate scheme is helpful in decreasing the error of the sampled-data observer while the sampling period is fixed.In (3), it is illustrated that the error of sampled-data controlled synchronization system may be unable to converge to zero because of the error would be imported from the mismatch between the master and the slave and the error of sampled-data observer. The sufficient conditions under which the synchronization error will converge to zero are developed. The error's order of magnitude is estimated while it can not converge to zero. It is derived that decreasing the mismatch and increasing the precision of the observer is helpful with diminishing the synchronization error, while the former is more important.In addition, a sampled-data observer for the inverted pendulum is designed with the approach of approximate discrete-time design and the precision of the observer is enhanced. The results of (2) and (3) are applied to the research of chaotic systems synchronization. The concept observer sampled-data synchronization is presented and a observer sampled-data synchronization system for Lorenz family is designed. The synchronization beyond sampled-data control is investigated and the concept practical sampled-data synchronization is presented. The effect of sampled-data observer error is analyzed and the design principles of chaotic synchronization systems are developed.