Stabilizing uncertain LTI systems using linear matrix inequalities: a relaxation algorithm

Although the problem of finding a stabilizing state feedback for an LTI system has been solved, it is still not easy to control an uncertain LTI system by a single state feedback gain which stabilizes a finite number of plant configurations, each representing a particular operating point. Researchers have focused on finding sufficient conditions in order to obtain convex problems which are readily solvable by standard convex optimization methods. One approach uses the quadratic stability condition, i.e. one looks for a single quadratic Lyapunov function which proves the stability of all closed loop models. Although this problem can be easily solved, the resulting sufficient condition is conservative and is more suitable for systems which (rapidly) switch among a family of linear systems. Another sufficient condition, proposed by El Ghaoui, is also a convex one and can be written in terms of LMIs (1993). This condition does not require quadratic stability. In this paper, the authors propose a relaxation algorithm for finding one stabilizing state feedback gain by recasting the whole problem into two LMIs. When all the states are not available, this algorithm can also handle output feedback. Since seeking a dynamic feedback compensator is equivalent to looking for an output feedback for an augmented system, the proposed method can be used for dynamic compensation as well. Due to the flexibility of convex optimization methods, several constraints on the closed loop system performance, such as H₂, H_∞ etc, can be incorporated in the design process.