Control of General Dynamic Systems With Periodically Varying Parameters Via Liapunov-Floquet Transformation

Abstract

A new technique in the design of controllers for linear dynamic systems with periodically varying coefficients is presented. The idea is to utilize the well-known Liapunov-Floquet (L-F) transformation such that the original time-varying system can be reduced to a form suitable for the application of standard time-invariant methods of control theory. For this purpose, a procedure for computing the L-F matrices for general linear periodic systems is outlined. In this procedure, the state transition matrices are expressed in terms of Chebyshev polynomials which permits the computation of L-F matrices as explicit functions of time. Further, it is shown that controllers can be designed in the transformed domain via full state or observer based feedback using principles of pole placement and optimal control theory. The effectiveness of the proposed technique is demonstrated through two examples. The first example belongs to the class of commutative systems while in the second example a triple inverted pendulum subjected to a periodic follower load is considered. It is found that both types of controllers can be successfully designed.