A convex characterization of gain-scheduled H∞ controllers

An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters θ. Small gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters θ undergo large variations during system operation. In general, higher performance can be achieved by control laws that incorporate available measurements of θ and therefore “adjust” to the current plant dynamics. This paper discusses extensions of H_∞ synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H_∞ controllers is fully characterized in terms of linear matrix inequalities. The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and derivation techniques developed here apply to both the continuous- and discrete-time problems. Existence conditions for robust time-invariant controllers are recovered as a special case, and extensions to gain-scheduling in the face of parametric uncertainty are discussed. In particular, simple heuristics are proposed to compute such controllers