Complex Semi-Definite Programming And Its Applications In System And Control Theory

Over the past few years, convex optimization, and semidefinite programming in particular, have come to be recognized as a valuable tool for control system analysis and design. A number of important problems in system and control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality constraints. From semidefinite programming duality theory, conditions for infeasibility of the linear matrix inequalitys as well as dual optimization problems can be formulated. These dual problems can in turn be re-interpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. Moreover, the most efficient algorithms for convex optimization solve the primal and dual problems simultaneously. Insight into the system-theoretic meaning of the dual problem can therefore be very helpful in developing efficient algorithms.In this paper, we propose the use of complex semidefinite programming, i.e., the extension of semidefinite programming in which one replaces the real symmetric matrices by complex Hermitian matrices. In the first chapter, we summarily introduce the development of linear matrix inequality and semidefinite programming as well as the main achievements of this thesis are summarized. In the second chapter, we extend the results of semidefinite programming to the Hermitian complex form. Both the weak and strong duality theories corresponding to Lagrange problems and optimality conditions are estableshed. Moreover, we propose a primal-dual central path algorithm for the solution of large-scale complex semidefinite programming problems arising in control. Since complex semidefinite programming is reducible to semidefinite programming. The polynomial-time solvability of semidefinite programmings implies that complex semidefinite programmings are also solvable in polynomial time. In the third chapter, we present two new proofs for existing results from system and control theory by means of establishing severl alternative theorems on complex linear matrix...