Robust Exponential Stability Of Uncertain Sampling Systems

With the rapid development of computer technology and digital control techniques, the ways of people’s living and producing have altered significantly in new century. Meanwhile, the field of control engineering is also affected by these techniques and has a great evolution. On the one hand, computer technology is applied to the field of control engineering, and subsequently a new subject called ”Computer Control System” emerges in recent years. On the other hand, digital controllers take places of the traditional analog controllers in many practical applications due to their own advantages over the analog ones. As the mathematical model of the computer control system, sampled-data system is the one that employs the controller of discrete signal to achieve the control objectives of the continuous-time system. On account of the wide application of sampleddata system in the network control system, lots of researchers have devoted much time and energy to studying the theory of sampled-data system. Stability, the basic structural property of system,is an important topic in system theory analysis. Considering that uncertainties can’t be avoided in actual systems, thus it makes a lot of sense to study stability and robust stability for sampled-data systems.This paper investigates the robust exponential stability problem of the linear sampled-data system with polytopic type uncertainty. Firstly, the exponential stability conditions are derived for linear sampled-data system by employing Lyapunov-like functional. Then these conditions are extended to the linear sampled-data system with polytopic type uncertainty, ending with the robust exponential stability conditions. And these conditions can be verified by LMI toolbox.The first chapter in this paper makes a brief introduction to the research background and status of the sampled-data system.In the sequence, Chapter 2 introduces some related concepts and theories of this paper, which lays the foundation for the paper below to run smoothly.Chapter 3 is concerned with the exponential stability problem of the linear sampled-data system under aperiodic samplings. In the framework of discrete time Lyapunov theories, a new Lyapunov-like functional is constructed with the following features. It explicitly depends on time t, contains the integral quadratic term of the states, and finally is not required to be positive definite between sampling instants. Employing this new Lyapunov-like functional, a new lemma is proposed in this paper to study the exponential stability of a class of nonlinear sampled-data system.And then new conditions for the linear sampled-data system with aperiodic sampling to be exponentially and asymptotically stable are given respectively in terms of linear matrix inequalities by taking advantages of the new lemma and the improved Wirtinger integral inequality. Then these conditions are extended to the linear sampled-data system with polytopic type uncertainty. At last,by using the LMI toolbox of MATLAB to make simulation, examples are provided to illustrate that the stability conditions have less conservatism compared with the ones given in some existing literatures.Chapter 4 makes a further improvement about the results obtained in Chapter 3. Compared with the Lyapunov-like functional in Chapter 3, a more general one appears in this chapter by adding a cross-term, which includes the integral term of the state and the sampling instant state.Consequently improved results to guarantee the exponential stability and robust exponential stability are given respectively. At last examples illustrate that the conditions given in this chapter do have an improvement compared with the ones in Chapter 3.At last, Chapter 5 summarizes the main content of this paper, and gives the work that we need to do in the future.