Pitfalls of a least-squares-equivalent controller for linear, time-periodic systems

We review a technique for the design of controllers for linear, time-period ic systems. A major appeal of the technique, first proposed by Sinha and Jo seph, is the use of Floquet-Lyapunov theory to transform the periodic syste m to a form where classical control strategies for time-invariant systems m ay be employed. However, it is normally impossible to rnd a completely time -invariant control system that is equivalent to the original time-varying s ystem: Application of the Floquet-Lyapunov transformation in fact yields a time-varying control system that the technique makes equivalent to a time-i nvariant one in the least-squares sense, in order to subsequently synthesiz e the controller via pole placement using a constant feedback matrix. However, classical control and Floquet-Lyapunov theory clearly show that it is erroneous to conclude that the behaviour of the least-squares-equivalen t, time-invariant system always matches the behaviour of the original time- periodic system. Using an example found in the original paper, we provide a simple counter-example that illustrates the failure of the proposed strate gy and an analysis of the reasons for its failure.