A unified matrix inequality approach to linear control design

Tetsuya Iwasaki, Purdue University

Abstract

The objective of this research is to develop a simple unifying methodology for designing linear controllers which meet different specifications. Contributions of this unifying methodology include alleviation of the gap between practical applications and control theories, in the following two ways. Firstly, the single elementary framework allows practicing engineers and students to learn many of modern control design methods efficiently and thoroughly. Secondly, our unifying methodology is naturally suited for computer-aided controller design since it requires only one computer software package supported by powerful convex programming techniques for designing controllers with different specifications. Thus, our result provides control engineers with reliable design tools. Our approach is purely algebraic and based on matrix inequalities. System analysis results are first developed as preliminary materials or cited from the literature to characterize certain control specifications in terms of matrix inequalities. Then, it is shown that many control design problems with stability, performance and robustness specifications can be reduced to a single algebraic problem of solving a matrix inequality. The solution to this problem is derived by using only elementary algebra. With this approach, necessary and sufficient conditions for the existence of a controller which meets a given set of specifications are obtained in terms of linear matrix inequalities (LMIs), and all such controllers are parametrized explicitly in the state space using solutions to the LMIs. If the controller order is not specified a priori, then matrices satisfying the LMI existence condition form a convex set, and can be computed by efficient algorithms. In this case, the resulting controller order can always be chosen to be equal to or less than the plant order. If the controller order is fixed, then we have an additional nonconvex coupling constraint for the existence condition, and the computational problem becomes much harder. We shall propose an algorithm to address this non-convex problem, and numerical examples will illustrate the idea and demonstrate the applicability of our matrix inequality based method for fixed order controller design.

Degree

Ph.D.

Advisors

Skelton, Purdue University.

Subject Area

Aerospace materials

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