An approach to computing bounds on H∞ performance under hard constraints

A number of standard control problems can be formulated as optimization problems where the goal is to minimize the H_∞ -norm of some transfer function while subject to constraints on, for example, system bandwidth or peak sensitivity or output variance. Such constraints usually limit the achievable H_∞ performance, but, since the constraints are typically included in the H _∞ optimization problem only indirectly through the selection of weights, it is generally not known to what degree the constraints limit the performance. This paper shows how lower bounds on the achievable H_∞ norm can be computed for problems where the objective function and constraints can all be written in terms of the sensitivity function. Both pointwise-in-frequency constraints (such as the requirement that the peak sensitivity be limited) and integral constraints (such as the requirement that the output variance be limited when subject to a stochastic disturbance) are dealt with. All constraints are explicitly included in the problem formulation; the solution approach involves optimal curve shaping and is, at least for the single-input single-output case, computationally straightforward