A parameter optimization approach to solving quasi-LMI problems

Many canonical and modern control problems can be recast into the problem of solving a group of matrix inequalities. Some of them are in the form of linear matrix inequalities (LMIs), which can be solved very efficiently by the powerful LMI toolbox in Matlab, but some others are in the form of quasi LMIs (QLMIs). By quasi, we mean that unknown parameters are involved in the matrix inequalities and these inequalities are LMIs only when the unknown parameters are fixed. Thus how to "guess" the unknown parameters is the key to solve the whole problem. In this paper, we present an optimal estimate for the unknown parameters. We will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems. Numerical examples are provided to show the effectiveness of the proposed method