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H∞control is one of the most efficient tools in robust control,while the strong stabilization problem has great significance in both control theory and practice.As neglected initially,it is well known that most of the current state-of-the-art design techniques,including LQG/H∞,can not guarantee the stability of controller.On the other hand,due to different kinds of limitations,e.g.,considering only SISO plants or only special kinds of problems,depending on the solutions of nonlinear optimization problems,most results on strong stabilization to date can not be generalized and efficiently solved.Considering the strong stabilization problem with H∞performance index,we first define strongγk-γcl H∞stabilization and simultaneousγk-γcl H∞control problems,which include strong H∞stabilization,stable H∞stabilization,strong stabilization,simultaneous H∞stabilization and simultaneous stabilization.Then by taking advantages of both ARE and LMI methods and exploring the freedom lying in the matrix inequality constraints,we obtain several results on strongγk-γcl H∞stabilization and some related problems,which can be summarized as follows.In Chapter 3,by introducing the descriptor system method into the H∞stabilization problem for LTI systems,we prove a new necessary and sufficient condition for H∞stabilization, i.e.,the extend bounded real lemma.Then this technique is applied to the H∞analysis and synthesis problems for linear polytopic/affine uncertain systems.By constructing parameter-dependent Lyapunov functions,we obtain the corresponding H∞stabilization conditions,which lead to direct H∞state feedback controller design methods by refraining the slack variables.In Chapter 4,by utilizing coprime factorization and Youla parameterization,we first prove that the simultaneousγk-γcl H∞stabilization problem for r（r≥2） plants is equivalent to stronglyγk-γcl H∞stabilize r-1 associated plants.Then based on the bounded real lemma,a sufficient condition for strongγk-γcl H∞stabilization is proposed,which leads to the design procedures that involve an algebraic Riccati equation and some LMI constraints. We also prove that this condition include the modified ARE method in[1].Based on this result,an improved sufficient condition on strong H∞stabilization as well as its dual form is derived,which extends the other ARE constraint into an inequality constraint.Two path-following algorithms are proposed to solve the associated optimization problems with BMI constraints,which lead to full order controllers.In Chapter 5,by utilizing the extended bounded real lemma,a new sufficient condition is proposed for n-th order strongγk-γcl H∞stabilization.This new condition fully relax the H∞constraint on the controller and greatly improves the direct LMI design results.Then by exploring the parameterization of all suboptimal H∞controllers,it’s also proved that the design of an l×n-th order（l≥2） strongγk-γcl H∞controller can be transformed into that of an n-th order controller for an associated plant,which tackles the possible order problem.Despite the BMI constraints in resulting optimization problems,the path-following and alternative iteration methods are adopted to formulate procedures for both n-th and l×n-th order（l≥2） controller design.In Chapter 6,based on the results of Chapter 4 and 5,a new LMI method is proposed for the strongγk-γcl H∞stabilization problem by modifying the controller form,which leads to a so-called generalized H∞central controller form.It is shown that several recently developed methods can be seen as its special cases.Meanwhile,we analyze the conservatism of the bounded real lemma when additional constraints have to be taken into account,and adopt the extended bounded real lemma to further improve the LMI design results iteratively.For the possible order problem,the nested construction method can also be applied.For the case when r=2,we present an efficient algorithm for designing a simultaneousγk-γcl H∞controller.
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