Lower Bounds on the Infima in Some ${\cal H}_{\infty}$ Optimization Problems

We consider three optimization problems: system inversion, model matching with probably unstable plant, and full information control. A common theme in numerical solution of these problems is that the infimal performance among solutions has to be approximated before a suboptimal solution can be found. Recently, a sequence of lower bounds that converges monotonously towards the exact infimum has been established for the system inversion problem. The goal of this technical note is twofold. First, we show that except for some rare cases these lower bounds converge at least root-exponentially fast, i.e., we establish the good-naturedness of this approximation method. Second, we show how the approximation method can be extended such that also arbitrarily good lower bounds on the infima in the model matching problem and the full information control problem can be obtained.