13. Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints
Karolos M. Grigoriadis, Eric Beran · Society for Industrial and Applied Mathematics eBooks · 1999
13.1 Introduction A large number of controller synthesis problems, such as full-order stabilization, , μ-synthesis with constant scaling, robust gain-scheduling, and other full-order control design problems, have been formulated as convex feasibility or optimization problems involving linear matrix inequalities (LMIs) [315, 64, 216, 215, 148, 19, 43]. The book by Skelton, Iwasaki, and Grigoriadis [377] presents a collection and a unified formulation of many of these problems. Recently developed interior-point algorithms provide efficient computational tools for numerical solution [401, 404, 296, 403] and newly developed computer-aided control design software packages have been based on these algorithms; see, for example, the LMI Control Toolbox for Matlab [153, 150] and the Semidefinite Programming Package [401, 402] with user friendly interfaces [68, 122]. These algorithms converge in polynomial time, and the problem structure is exploited to increase computational efficiency. The above LMI control design problems provide controllers of order equal to the order of the generalized plant. However, controller implementation constraints often dictate the use of low-order controllers because of simplicity, hardware limitations, or computational reasons; e.g., [448] discusses the controller-order limitations of the Hubble Space Telescope pointing control system. Many control design problems where the order of the controller is less than the order of the generalized plant can be formulated as LMI problems with an additional matrix coupling rank constraint that destroys the convexity of the optimization problem. Hence, interior-point algorithms with guaranteed convergence cannot be used to obtain a solution. Heuristic algorithms to address rank-constraint problems for low-order control synthesis have been proposed recently [94, 124, 217, 128, 282], but convergence of these algorithms to a solution is not guaranteed.