A Polynomial-Based Approach of Reachable Set Estimation for Periodic Time-Varying Systems

This article is focused on estimating reachable sets for periodic time-varying systems, whose dynamics are represented by trigonometric series through Fourier theory. Given the difficulty of dealing with coupled time-varying coefficients in reachable set estimation, a constructive polynomial-based approach is utilized to transform time-varying nonconvex constraints into constant vertex constraints through the property of multiaffine matrix polynomials. Based on that, when the decay rate parameters for Lyapunov functions are known, the reachable set bounding region can be determined by solving vertex constraints in linear matrix inequality forms. In cases where the decay rates are unknown, a simulated annealing algorithm and a Boundary Intersection algorithm are proposed to provide a tractable solution for parameter searching, and to achieve lower conservatism in the reachable set bounding regions. The effectiveness of our polynomial-based approach is illustrated through simulations using a single-mesh gear system. Moreover, the efficiency of the two algorithms can be assessed by contrasting the conservatism reduced in measurements of bounding regions.