Convergent LMI relaxations for non-convex optimization over polynomials in control

It is well known that most analysis and design problems in robust and nonlinear control can be formulated as global optimization problems with polynomial objective functions and constraints [1]. Typical examples include robust stability analysis for characteristic polynomials with parametric uncertainty, simultaneous stabilization of linear systems, pole assignment by static output feedback, and stability analysis for polynomial systems by Lyapunov’s second approach. In some specific cases, there exist computationally efficient techniques for solving these problems. For example, vertex or extremal results such as Kharitonov’s Theorem or the Edge Theorem can be used to perform robust stability analysis without optimization [2, 3, 4]. Some of these results have been extended to robust design of fixed-order or fixed-structure controllers, for example, PID design [5]. In the same vein, static state-feedback design, or design of a controller of the same order as the plant, can be formulated as a convex linear matrix inequality (LMI) optimization problem [6], for which polynomial-time interior point methods are available.