Robust Output Feedback Controller Design: Lmi Approach

Robustness has been recognized as a key issue in the analysis and design of control systems for the last two decades. During the last decades numerous papers dealing with the design of static robust output feedback control schemes to stabilize uncertain systems have been published ([1], [3], [4], [6], [8], [10], [12], [13], [15], [19], [21]). Various approaches have been used to study the two aspects of the robust stabilization problem, namely conditions under which the linear system described in state space can be stabilized via output feedback and the respective procedure to obtain a stabilizing or robustly stabilizing control law. The necessary and sufficient conditions to stabilize the linear continuous time invariant system via static output feedback can be found in [11], [20]. In the above and other papers, the authors basically conclude that despite the availability of many approaches and numerical algorithms the static output feedback problem is still open. Recently, it has been shown that an extremely wide array of robust controller design problems can be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI has been introduced in [7]. In this paper, the BMI problem of robust controller design with output feedback is reduced to a LMI problem [2]. The theory of Linear Matrix Inequalities has been used to design robust output feedback controllers in [1], [3], [4], [8], [12], [19] and [20]. Most of the above works present iterative algorithms in which a set of LMI problems are repeated until certain convergence criteria are met. The V-K iteration algorithm proposed in [4] is based on an alternative solution of two convex LMI optimization problems obtained by fixing the Lyapunov matrix or the gain controller matrix. This algorithm is guaranteed to converge, but not necessarily, to the global optimum of the problem depending on the starting conditions. The main criticism formulated by control engineers against modern robust analysis and design methods for linear systems concerns the lack of efficient easy to use and systematic numerical tools. This is especially true when analyzing robust stability as affected by highly structured uncertainty with BMI, for which no polynomial-time algorithm has been proposed so far [9]. This paper is concerned with the class of uncertain linear systems that can be described as