2. Some Standard Problems Involving LMIs
Society for Industrial and Applied Mathematics eBooks · 1994
2.1 Linear Matrix Inequalities A linear matrix inequality (LMI) has the form F (x) ≜ F0 + ∑ i=1 m xi Fi >0, 2.1 where is the variable and the symmetric matrices , , are given. The inequality symbol in (2.1) means that F(x) is positive-definite, i.e., for all nonzero . Of course, the LMI (2.1) is equivalent to a set of n polynomial inequalities in x, i.e., the leading principal minors of F(x) must be positive. We will also encounter nonstrict LMIs, which have the form F (x) ≥0. 2.2 The strict LMI (2.1) and the nonstrict LMI (2.2) are closely related, but a precise statement of the relation is a bit involved, so we defer it to §2.5. In the next few sections we consider strict LMIs. The LMI (2.1) is a convex constraint on x, i.e., the set is convex. Although the LMI (2.1) may seem to have a specialized form, it can represent a wide variety of convex constraints on x. In particular, linear inequalities, (convex) quadratic inequalities, matrix norm inequalities, and constraints that arise in control theory, such as Lyapunov and convex quadratic matrix inequalities, can all be cast in the form of an LMI.