On optimization with competing performance criteria

In control system design one typically optimizes a performance function over the space of frequency response functions of all designable controllers. In H^∞ control one typically optimizes a sup norm type of performance function and it has been known for many years that at optimum this performance function is frequency independent (flat). In this paper the authors study simultaneous (Pareto) optimization of several competing performances Γ₁ ,...Γ_l. The authors find that under strong assumptions on the performance functions that if there are N designable subsystems (f₁,...,f_N) and l performance measures with l⩽N, then at a nondegenerate Pareto optimum (f₁*,...,f_N*) every performance is flat. It is common in control to have many designable subsystems so it is natural to look for ways to use the substantial freedom this gives profitably. The implication here is that if one wants many performance functions to be flat then there are circumstances where this might be achievable. Besides flatness there are other “gradient alignment” conditions which must hold at an optimum, The article presents these and thus gives the precise “first derivative” test for a natural class of H^∞ Pareto optima. In current jargon the gradient alignment would be called a primal-dual type optimality condition. Such optimality conditions are valuable for assessing how iterations in a computer run are progressing. Also in the traditional case the optimality conditions have been the base of highly successful computer algorithms