Sampled-Data Stabilization Controller Design Of Continuous-time Nonlinear Control Systems: An Approach Based On Their Approximate Discrete-time 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 T fixed prior to the design of the controller. We study stabilization between the controlled continuous-time plant and its equivalent exact discrete-time model using the feedback controller designed on its approximate discrete-time models and stabilization between the exact discrete-time model and the approximate discrete-time models using the same controller. The approach we use for the former is similar to the one for linear sampled-data systems because the exact discrete-time model just coincides a state of the continuous-time plant at sampling instants. The latter is brought into the framework of pure discrete-time investigation. So a discrete Lyapunov direct method and its modern theory with k class function is applicable to the case of the latter. In this paper, various kinds of stabilization are investigated for the cases of exponential stability, asymptotic stability and their relevant practical stability. Meanwhile, various sufficient conditions are given. Furthermore, Various tools of nonlinear control design are introduced Thepaper 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 ide...