Periodic Lyapunov Equation Based Approaches to the Stabilization of Continuous-Time Periodic Linear Systems

This note is concerned with stabilization of continuous-time periodic linear (CPL) systems with state feedback. The design is based on solutions to a class of parametric periodic Lyapunov differential equations (PLDEs) resulting from the problem of minimal energy control with guaranteed convergence rate. By carefully studying the properties of the PLDEs and their solutions, a continuous periodic state feedback is designed. The PLDE based approach is effective in designing stabilizing controller for CPL systems as the designers need only to solve a linear differential equation whose solution can be obtained analytically if the system is relative simple and can be computed numerically by integration in general cases. Necessary and sufficient conditions on the free parameter in the PLDE are proposed to guarantee the stability of the closed-loop system whose characteristic multiplier set can even be exactly computed accordingly. A numerical example is worked out to show the effectiveness of the proposed approach.