An LMI Optimization Approach for Structured Linear Controllers

Abstract— This paper presents a new algorithm for the design of linear controllers with special structural constraints imposed on the control gain matrix. This so called SLC (Structured Linear Control) problem can be formulated with linear matrix inequalities (LMI’s) with a nonconvex equality constraint. This class of problems includes fixed order output feedback control, multi-objective controller design, decentralized controller design, joint plant and controller design, and other interesting control problems. Our approach includes two main contributions. The first is that many design specifications are described by a similar matrix inequality. A new matrix variable is introduced to give more freedom to design the controller. Indeed this new variable helps to find the optimal fixedorder output feedback controller. The second contribution is to propose a linearization algorithm to search for a solution to the nonconvex SLC problems. This has the effect of adding a certain potential function to the nonconvex constraints to make them convex. The convexified matrix inequalities will not bring significant conservatism because they will ultimately go to zero, guaranteeing the feasibility of the original nonconvex problem. Numerical examples demonstrate the performance of the proposed algorithms and provide a comparison with some of the existing methods.