Reduction of robust stabilization problems to standard H∞ problems for classes of systems with structured uncertainty

Abstract

We consider the problem of robustly stabilizing a family of linear time-invariant systems with a linear time-invariant controller. For classes of robust stabilization problems with unmodelled dynamics, recent breakthroughs in H ∞ theory lead to solution via two Riccati equations. For systems with structured real uncertainty, however, a similar comprehensive theory does not exist. The objective of this paper is to delineate classes of linear time-invariant systems with structured uncertainty (either real or complex) for which a similar solution to the robust stabilization problem is possible. The theory has the following salient features. First, for classes of uncertainty structures, no conservatism is introduced. For structured real parametric uncertainty, we make a connection with H ∞ theory that does not amount to any crude overbounding via complex discs; the stabilizability conditions we obtain are not only sufficient but also necessary. The second salient feature in the theory is a special weighting function which we construct. For example, for the case when the structured uncertainty enters only in the denominator, the robust stabilization problem is replaced by a standard H ∞ sensitivity minimization problem with our special weighting function. A similar H ∞ complementary sensitivity problem is associated with numerator uncertainty.