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 transformation 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 transformation 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 root locus, pole placement and/or optimal control theory. The effectiveness of the proposed technique is demonstrated through three examples. The first example is a single mass inverted pendulum while in the second example a triple inverted pendulum subjected to a periodic follower load is considered. The third example chosen is a simply supported smart Euler-Bernoulli elastic column subjected to a periodic loading. It is found that time-invariant techniques can be employed effectively to achieve vibration control of parametrically excited linear systems.
