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

We review a technique for the design of controllers for linear, time-periodic systems. A major appeal of the technique, first proposed by Sinha and Joseph, is the use of Floquet-Lyapunov theory to transform the periodic system to a form where classical control strategies for time-invariant systems may be employed. However, it is normally impossible to find a completely time-invariant control system that is equivalent to the original time-varying system: Application of the Floquet-Lyapunov transformation in fact yields a time-varying control system that the technique makes equivalent to a time-invariant one in the least-squares sense, in order to subsequently synthesize 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-equivalent, time-invariant system always matches the behaviour of the original timeperiodic system. Using an example found in the original paper, we provide a simple counter-example that illustrates the failure of the proposed strategy and an analysis of the reasons for its failure.

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