The Robust Stabilization Of Interval Time-varying Delay Systems Control Research

Systems and control theory has been widely used in plenty of fields in human society. It is known that time-delay and uncertainty often exist in practical systems and lead to instability, severely affecting the realization of system performance index. Many scholars all over the world keep focusing on the study of stability and stabilization problem.This paper investigates the state-feedback stabilization problem for linear systems with interval time-varying delays. The systems contain both time-varying state delay and control input delay with certain relationship between the delays. The aim is to design robust state-feedback controllers to stabilize the systems based on the Lyapunov theory. We construct suitable Lyapunov-Krasovskii functionals and give LMI-based results for the stability and stabilization problem by using Jensen inequality method, Leibniz-Newton formula, and Schur complement theorem. The present results are demonstrated to be useful and effective by some numerical examples.The main contents of this paper are as follows:(1) We study the stabilization problem for linear systems with interval time-varying delays. The system considered has time delays in both the state and the control input. Firstly, we introduce a new operator and make a model transformation which turns the system into one containing only one delay. Consequently, we analyze and deal with the studied problem using Lyapunov-Krasovskii method combined with Jensen inequality method, Leibniz-Newton formula, Schur complement theorem and LMI processing method. Then, sufficient conditions are given in terms of LMIs, and the results are verified to be useful and effective by several numerical examples.(2) We also study the robust stabilization problem for uncertain systems with interval time-varying delays. The uncertainties are satisfied with norm-bounded constraints. We use Lyapunov method and give LMI-based conditions for the controller design. Numerical examples are also given to illustrate the present results.