This dissertation provides several new tools which help one to tackle the optimal control system design of linear systems with multiplicative and additive noises. The main concern of this dissertation is the optimal linear control system design 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 (LMIs) with a nonconvex equality constraint. This class of problems include the design of fixed order output feedback control, multi-objective control, robust control, decentralized control, joint plant design and control, and the combinations of these problems in stochastic systems. In order to tackle these problems, new system performance analysis conditions have been derived for many system gains of both continuous and discrete-time stochastic systems using the projection lemma. Newly proposed system performance conditions have several nice features over existing ones.; First, many system performance analysis conditions such as stochastic H2 performance, Hinfinity performance, ℓinfinity performance, etc. can be written in a very similar matrix inequality for both continuous and discrete-time stochastic systems.; Second, the form of complicating matrix terms for both continuous and discrete-time stochastic systems are the same and we need mainly two nonlinear change-of-variables linearizing nonconvex synthesis conditions for state feedback control, dynamic output filter, and deterministic model reduction problems. Furthermore, all synthesis conditions can be extended to the case where system matrices lie in a convex bounded domain as in deterministic system case. However, all synthesis conditions for dynamic output feedback control problems in stochastic systems with the structure of deterministic controllers are nonconvex. To cope with the nonconvexity of control synthesis problems, we propose a form of stochastic controllers which is able to reduce system performance synthesis conditions to be convex.; Third, we can eliminate control gain matrices and thus the problem size (the size of matrix inequality) and the number of variables of the new synthesis conditions are reduced. Necessary and sufficient conditions for the solvability of many system gains for (strictly proper) controllers, filters, and reduced models are obtained in terms of LMIs for both continuous and discrete-time stochastic systems.; In order to tackle nonconvex control system design problems, linearization algorithms are proposed to linearize concave terms so as to generate a sequence of semi-definite programming problems with monotonically decreasing cost, guaranteeing the local optimality conditions. This algorithm has the effect of adding a certain positive potential function to the nonconvex constraints to enforce convexity at each iteration. (Abstract shortened by UMI.)
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