Linear Matrix Inequalities for Systems and Control (LMI)

Many challenges in control engineering can be efficiently handled and solved using linear matrix inequalities (LMIs). Examples include stability analysis, robust controller design, or constrained handling. For instance, Lyapunov inequalities ensuring stability can be easily formulated as LMIs. In general, LMIs provide a mathematical framework to express constraints on decision variables using linear combinations of symmetric matrices. If constrained to the set of positive (or negative) (semi-) definite matrices, a convex constraint on the decision variables results. Solving such LMIs then leads to a so-called semi-definite program (SDP), which is a convex optimization problem and can hence be efficiently solved using appropriate software.

The course covers the following topics:

• Introduction to LMIs and SDPs.
• Lyapunov stability for linear systems via LMIs.
• The bounded real lemma and its relation to the existence of a stabilizing controller and the feasibility of a certain LMI.
• The design of robust H2 and H-infinity controllers.
• Flexible pole placement using LMIs.
• The design of general linear dynamic controllers and the separation principle,
• Observer design including Luenberger observer and Kalman filter,
• The numerical solution of LMIs and SPDs using Matlab and Yalmip.

Upon completion of the course, students are able to

• name, explain, apply and asses LMIs and SDPs,
• perform stability analysis and design controllers using LMI-based techniques,
• design robust H2 and H-infinity controllers, accounting for uncertainties and disturbances,
• apply appropriate software tools and algorithms to numerically solve LMIs and SDPs.

Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics (SIAM), 1994.