An LMI-based Method of Periodic Time-varying State Feedback Stabilizing Controller Design for Periodic Linear Time-varying Systems

A necessary and sufficient condition for linear time-varying (LTV) systems to be exponentially stable has been given based on differential Lyapunov inequalities (DLIs). The exponential stability of an LTV system is guaranteed if the DLI for the system has a solution, however, few systematic methods have been reported for finding a solution of DLIs. We have proposed a method to search solutions of DLIs using linear matrix inequalities (LMIs) for recent several years, especially when the system matrices of LTV systems are periodic. In this paper, we propose a method to design periodic time-varying state feedback controllers that stabilize periodic LTV systems by utilizing our DLI solutions search method. Moreover, we add new variables to the LMIs that use the addition theorems of trigonometric functions to reduce the conservatism in stability analysis and controller design. An example shows the process of stabilizing a periodic LTV system using our proposed design method.