The Stabilization Of Nonlinear Autonomous Sampled-Data Control Systems: An Approach Of Design Of The Digital Controller Based On Their Approximated Linearization Models

Dynamical systems encountered in engineering usually evolve in continuous time. Contrasting this, in feedback control the most majority of complex systems are controlled using sampled observations of system behavior taken at discrete time instants. Thus the resulting controlled systems are hybrid systems, which are called sampled-data systems, involving both continuous-time and discrete-time signals. The investigation of sampled-data systems is motivated primarily due to the widespread use of digitally implemented controller in present-day feedback control of continuous time systems. Linear sampled-data control theory does not suffice to deal with nonlinear sampled-data systems. Hence analysis and design of nonlinear sampled-data systems has been, in recent years, a subject of growing interest in the international community of control research.When a continuous-time plant is controlled using a digitally implemented controller, it is often faced that a continuous-time plant is transformed into its equivalent discrete-time model. A digital controller is usually designed on the base of approximate discrete-time models of the continuous-time nonlinear systems because it is difficult to obtain the closed form of the exact discrete-time model for nonlinear control systems. Approximation usually makes signal distortion. A natural question is raised that the controllers based on the approximate models are still working for the original continuous-time systems. This is a question that must be seriously investigated. A.Teel, D.Nesic etc. initially studied ISS stabilization of nonlinear sample-data systems of this type in 1998. Practical stabilization of nonlinear sampled-data systems based on discrete-time approximation with sufficiently small sampling periods has been studied in recent years. The sufficient conditions have been given for the practical stabilization of the above case. Often in practice, however response frequency of implemented devices is limited and a sampling period could not be sufficiently small, and furthermore a sampling period T is usually fixed prior to the design of the controller in engineering. This is a common situation for many nonlinear sampled-data systems. A fundamental design question is therefore left open for the case when the sampling period T is fixed first. Prof. Zheng Yufan was first to investigate the case of a fixed sampling period. He studied exponential stabilization using continuous prototype approach. And the distinguished scientist P. Kokotovic also proposed such kind of question in his paper "Constructive Nonlinear Control: A Historical Perspective", Automatica 37, 2001,637-662.In this paper, we shall restrict ourselves to the subject of stabilization of nonlinear sample-data systems in the case of sampled period T fixed prior to die design of the stable controller which is get from the approximated linear discrete-time model. The paper is organized as follows:In the part one, we introduce a background and currant research situation of nonlinear sampled-data systems using digital controller designed on the base of the approximation and brief a main idea of the method I am going to take to solve the problems. We propose two basic assumptions that guarantee the validity of the method.In the part two, the sufficient condition of stability of linear sampled-data systems was discussed. After analyzing the solution of linear sampled-data system in details, we find that increasing the order of approximate discrete time system will obtain and improve the stability of linear sampled-date system. To any sample period, there is one smallest order of approximate discrete time system. And the existence of sample controller is ensured by the controllability of the original systems. Then we analyze the stability of time-varying linear sampled-data control systems.In the part three, we study the sufficient conditions for stability of general sampled-data nonlinear system. Based on the value of state at sampling point, we give the design method of digital controller,which stabilize...