Lecture Notes DISC Course on Linear Matrix Inequalities in Control

Preface Objectives In recent years linear matrix inequalities (LMI's) have emerged as a powerful tool to approach control problems that appear hard if not impossible to solve in an analytic fashion. Although the history of LMI's goes back to the fourties with a major emphasis of their role in control in the sixties (Kalman, Yakubovich, Popov, Willems), only recently powerful numerical interior point techniques have been developed to solve LMI's in a practically efficient manner (Nesterov, Nemirovskii 1994). Several Matlab software packages are available that allow a simple coding of general LMI problems and of those that arise in typical control problems (LMI Control Toolbox, LMI-tool). Boosted by the availability of fast LMI solvers, research in robust control has experienced a paradigm shift – instead of arriving at an analytical solution, the intention is to reformulate a given problem to verifying whether an LMI is solvable or to optimizing functionals over LMI constraints. The main emphasis of the course is • to reveal the basic principles of formulating desired properties of a control system in the form of LMI's, • to demonstrate the techniques to reduce the corresponding controller synthesis problem to an LMI problem, • to get familiar with the use of software packages for performance analysis and controller synthesis using LMI tools. The power of this approach is illustrated by several fundamental robustness and performance problems in analysis and design of linear and certain nonlinear control systems. i ii Preface Topics The lecture notes for this course have been written for a course of eight weeks on the subject. Within the DISC graduate program, two class hours are taught once per week during a period of eight weeks. The topics covered for this course are the following. 1. Examples. Facts from convex analysis. Interior point methods in convex programming and their efficiency. Linear Matrix Inequalities: History. The three basic problems and their solution with LMI-Lab. bound of peak-to-peak norm. LMI stability regions. 3. Frequency domain techniques for the robustness analysis of a control system. Integral Quadratic Constraints. Multipliers. Relations to classical tests and to µ-theory. 4. A general technique to proceed from LMI analysis to LMI synthesis. State-feedback and output-feedback synthesis algorithms for robust stability, nominal performance and robust performance using general scalings. 5. Mixed control problems. Multi-model control problems. Lyapunov shaping technique. 6. Extension to linear parametrically varying systems. Gain-scheduling. Examples of occurrence. Solution of design problem …