Research On Stabilization Of A Class Of Time-varying Nonlinear Sampled-data Systems

Sampled-data systems are widely used in real applications. In a sampled-data system, the plant is usually continuous-time but the controller is discrete-time. When the sampled-data systems are nonlinear, the most common method is to design the controller based on their approximate models and the controllers can stabilize their exact models. By this way, we can void the high complexity of exact models of nonlinear sampled-data systems. The research work of this method has two parts which are focus on the systems modeled as differential equations and differential inclusions, respectively. For the method that design controllers based on approximate models, much work regarding time-invariant nonlinear systems have done but the research on the situation where the nonlinear systems are time-varying are still rare. Because the character of time-varying systems, the results about time-invariant sampled-data systems are not suitable to solve the problem of time-varying sampled-data systems and consider that time-varying systems are so widespread, research on the problem of stabilizing time-varying sampled-data systems via their approximate models is important. Therefore, we will research the problem of stabilizing a class of time-varying nonlinear systems via their approximate models in this article.According to the time-varying sampled-data system is modeled as differential equations or differential inclusions, we apart our research into two parts. Firstly, we will research the problem when systems are modeled as differential equations. Above all, we analyze their model and provide the structure of their approximate and exact models and the assumption on the systems. Then we will give the concept and property of bounded consistency for time-varying nonlinear systems. The bounded consistency reflects the error between approximate models and exact models. Next, we introduce a basic concept of stability for time-varying discrete-time systems and its corresponding equivalent Lyapunov formulation. At last, we not only provide the method to judge the bounded consistency without the knowlegde of exact models but also provide the sufficient conditions under which the controllers designed based on the approximate models can stabilize the exact models of the systems. What’s more, some simple examples are given and simulated by Matlab to verify our theories.For the time-varying nonlinear systems modeled as differential inclusions, we research the problem of designing controllers via approximate models in the similar way that we research the problem for time-invariant systems. According to the charater of time-varying nonlinear sampled-data systems, we analyze their models and provide the concept of bounded upper-semi consisitency which guarantees the difference between the sets of solutions of approximate and exact models is bounded. And the property of bounded upper-semi consistency is also discussed. Then we discuss some basic concepts of stability for time-varying discrete-time differential inclusion systems in non-Lyapunov-based way. At last, we provide the sufficient conditions under which the controllers designed based on the approximate models can stabilize the exact models of the systems and verify it by Matlab.