Weighting function selection in H^∞ design

We examine an H$^∞ mixed sensitivity design problem formulated to achieve good disturbance rejection and tracking properties as well as robust stability. A two degree-of-freedom controller is used, then, as is well known, the tracking objectives can be considered separately. A major concern in H^∞ design is the selection of weighting functions to satisfy the design specifications. In this paper we give conditions on the weights which guarantee the optimization problem is well-defined; describe how a constrained optimization can be posed so that improper and unstable weights can be used; describe the effect the weights have on various design objectives; show how an appropriate selection of weights can reduce a 2-block optimization problem (requiring an iterative solution) to a 1-block problem (with a closed-form solution); give an illustrative example. In particular we show how the pole-zero cancellation phenomenon in H^∞ design can be exploited or prevented by a suitable choice of weight, and how an optimal controller with a given frequency roll-off can be obtained.