Linear Matrix Inequalities in System and Control Theory

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan

Proceedings Allerton Conference on Communication, Control and Computing, pages 237-246, October 1993.

A wide variety of problems in system and control theory can be formulated (or reformulated) as convex optimization problems involving linear matrix inequalities, that is, constraints requiring an affine combination of symmetric matrices to be positive semidefinite. For a few very special cases, there are ‘analytical solutions’ to these problems, but in general they can be solved numerically very efficiently. We introduce the reader to Linear Matrix Inequalities or LMIs, provide a brief history of LMIs in system and control theory, and discuss a few problems from systems and control that can be solved via convex optimization over LMIs.