On Design of Robust Linear Quadratic Regulators

Closed-loop stability of uncertain linear systems is studied under the state feedback realized by a linear quadratic regulator (LQR). Sufficient conditions are presented that ensure the closed-loop stability in the presence of uncertainty, initially for the case of a non-robust LQR designed for a nominal model not reflecting the system uncertainty. Since these conditions are usually violated for a large uncertainty, a procedure is offered to redesign such a non-robust LQR into a robust one that ensures closed-loop stability under a predefined level of uncertainty. The analysis of this paper largely relies on the concept of inverse optimal control to construct suitable performance measures for uncertain linear systems, which are non-quadratic in structure but yield optimal controls in the form of LQR. The relationship between robust LQR and zero-sum linear quadratic dynamic games is established.