The control of linear time-periodic systems using Floquet-Lyapunov theory

In this paper we use Floquet-Lyapunov theory to derive the Floquet factors of the state-transition matrix of a given linear time-periodic system. We show how the periodicity of one of the factors can be determined a priori using a constant matrix, which we call the Yakubovich matrix, based upon the signs of the eigenvalues of the monodromy matrix. We then describe a method for the numerical computation of the Floquet factors, relying upon a boundary-value problem formulation and the Yakubovich matrix. Further, we show how the invertibility of the controllability Gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed-loop system. The controller can be full-state or observer-based. It also allows the engineer to assign all the invariants of the system; i.e. the full monodromy matrix. Deriving the feedback matrix requires solving a matrix integral equation for the periodic Floquet factor of the new state-transition matrix of the closed-loop system. This is achieved via a spectral method, with further refinement possible through a boundary-value problem formulation. The computational efficiency of the scheme may be further improved by performing the controller synthesis on the transformed system obtained from the Lyapunov reducibility theorem. The effectiveness of the method is illustrated with an application to a quick-return mechanism using a software toolbox developed for MATLAB(TM).