LMI Approach

Abstract

In Chaps. 4, 5, and 6, we discussed how a robust controller design problem can be cast as an optimization problem, in $\mathcal{H}_{\infty}$ or μ-synthesis formulations. Optimal or sub-optimal solutions can be found by following some formulas which are derived using functional analysis or operator theories. There is actually another way to consider the robust design problem, and control system design in general, as an optimization problem, solutions to which can be directly computed by convex computational procedures. That is the so called Linear Matrix Inequality (LMI) approach. In this chapter, basic concepts of LMI and a few applications of LMI in robust and other control system design problems will be introduced. The major impact behind the wide use of LMIs in control systems analysis and design in the last decade or so is due to the breakthrough of efficient numerical algorithms of interior-point methods in convex optimization. Such a development makes it practically possible to find solutions to LMI’s representing many control systems problems which we will introduce next.