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A convex characterization of gain-scheduled ${ℋ}_{\infty }$ controllers. (English) Zbl 0826.93028
Summary: 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 $\theta$. 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 $\theta$ undergo large variations during system operation. In general, higher performance can be achieved by control laws that incorporate available measurements of $\theta$ and therefore “adjust” to the current plant dynamics. This paper discusses extensions of ${ℋ}_{\infty }$ 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 ${ℋ}_{\infty }$ 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 considering parametric uncertainty are discussed. In particular, simple heuristics are proposed to compute such controllers.
##### MSC:
 93B36 ${H}^{\infty }$-control 93B50 Synthesis problems 93C25 Control systems in abstract spaces 93B51 Design techniques in systems theory 93B35 Sensitivity (robustness) of control systems