Linear Matrix Inequality Method for Periodic Systems

For homogenous systems with periodic coefficients, the existence of a quadratic Lyapunov function has been studied, and the Floquet theory has been applied to provide the necessary and sufficient condition for the stability of the system. However, determining the transformation that reduces a nonautonomous linear periodic system to an autonomous linear system (having constant coefficients) is in general a difficult problem which requires series methods and the theory of special functions. In this thesis, I analyze the stability of the system through linear matrix inequalities by restricting Lyapunov function to a piecewise linear function. This method does not distinguish the values of the system parameters with one discretization interval. However, it is possible to provide more information of the system in order to increase the accuracy of the result without finer discretization of the Lyapunov function. I also discretized the linear periodic system with delay and reformulate the criteria of the stability in the form of linear matrix inequalities