Improper ${cal L}_{\infty }$ Optimal/Suboptimal Controllers

We consider L_∞-control of MIMO systems and address solvability of the problem over all finite dimensional LTI controllers: i.e., controllers whose transfer functions can be proper or improper. We show that improper controllers are easily dealt with using the behavioral approach, unlike the standard state-space/transfer-matrix methods, and argue that there are cases where an improper controller can outperform a proper controller. In this setting, we next formulate and prove necessary and sufficient conditions for suboptimal L_∞-control problem solvability and relate this to existing results about system invariant zeros. Further, we infer that in our formulation, assuming suboptimal solvability conditions on the system, an optimal controller always exists, possibly with an improper transfer function. In other words, the infimum L_∞-norm of the closed loop system is achievable when dealing with both proper and improper controller transfer functions. We illustrate these results through an example for which the optimal L_∞-controller has an improper transfer function.