Performance Bounds in ${cal H}_{\infty }$ Optimal Control for Stable SISO Plants With Arbitrary Relative Degree

This note deals with performance bounds for the H _infin-optimal control of discrete-time LTI plants. The case studied corresponds to stable scalar plants with arbitrary relative degree but no finite non-minimum phase zero. By using Nehari´s Theorem and a reformulation of the standard Youla Parameterization a closed-form expression for the characteristic polynomial of the associated eigenvalue problem is obtained. Also, we derive an analytic expression for the optimal H _infin cost as a function of the plant relative degree.