Guaranteed Cost Controls

Abstract

This chapter considers so called guaranteed cost controls (GCC) which stabilize the closed-loop systems and guarantee some upper bounds of given costs, where control structures are given a priori in feedback forms of single state and input delays. These guaranteed cost controls are in between non-optimal stabilizing controls of previous chapters and optimal stabilizing controls of subsequent chapters. Guaranteed cost controls for systems with time-varying delays are not discussed due to the space limit. For input delayed systems, state feedback guaranteed LQ controls for a predictive LQ cost and also for a standard LQ cost are obtained with guaranteed LQ performance bounds. The guaranteed LQ control for the predictive cost is given in a feedback form of memoryless state and distributed input delays, while those for the standard cost in a state feedback form of single state and input delays. Then an output feedback guaranteed LQ control for the standard LQ cost is obtained with a guaranteed LQ performance bound in a dynamic output feedback form via the cascaded-delay system approach. For state delayed systems, state feedback guaranteed LQ controls for a standard LQ costs, either delay-independent or delay-dependent, are obtained with guaranteed LQ performance bounds in feedback forms of single state and input delays. A robust guaranteed LQ control for state delayed systems with model uncertainties is also explored for the standard LQ cost. Then an output feedback guaranteed LQ control for the standard LQ cost is obtained with a guaranteed LQ performance bound in a dynamic output feedback form via the cascaded-delay system approach. For input delayed systems, state feedback guaranteed \(\mathcal {H}_\infty \) controls for a predictive \(\mathcal {H}_\infty \) cost and for a standard \(\mathcal {H}_\infty \) cost are obtained with guaranteed \(\mathcal {H}_\infty \) performance bounds. The guaranteed \(\mathcal {H}_\infty \) control for the predictive cost is given in a feedback form of memoryless state and distributed input delays, while that for the standard cost in a feedback form of single state and input delays. Then an output feedback guaranteed control for the standard \(\mathcal {H}_\infty \) cost is obtained with a guaranteed \(\mathcal {H}_\infty \) performance bound in a dynamic output feedback form via the cascaded-delay system approach. For state delayed systems, state feedback guaranteed \(\mathcal {H}_\infty \) controls for standard \(\mathcal {H}_\infty \) costs, either delay-independent or delay-dependent, are obtained with guaranteed \(\mathcal {H}_\infty \) performance bounds in feedback forms of single state and input delays. A robust guaranteed \(\mathcal {H}_\infty \) control for state delayed systems with model uncertainties is also explored for the standard \(\mathcal {H}_\infty \) cost. Then an output feedback feedback guaranteed \(\mathcal {H}_\infty \) control for the standard \(\mathcal {H}_\infty \) cost is obtained with a guaranteed \(\mathcal {H}_\infty \) performance bound in a dynamic output feedback form via the cascaded-delay system approach.