The ρ performance measure: a new tool for controller design with multiple frequency domain specifications

Multiobjective frequency domain design aims to find a controller that gives desirable properties to several closed loop frequency responses simultaneously. This problem may be cast as a problem of finding a controller such that the worst-case H_∞ norm, i.e. the largest among several H_∞ norms, satisfies a prespecified bound. The authors define an upper bound, the ρ measure, for the worst-case H_∞, norm in terms of a scaled H_∞ cost function. The authors show that the problem of finding a controller that minimizes ρ can be reduced to a finite dimensional convex program for the unknown scales and a standard H_∞ problem. The authors give affine matrix inequalities for controller synthesis when the scales are restricted to constant matrices. A numerical example is used to illustrate the advantages of the authors\´ formulation, over the single objective H_∞, formulation, for solving multiobjective H_∞ problems.