LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Convexity of LMIs

Definition A set, , in a real inner product space is convex if for all and , where , it holds that . Lemma 1.1 The set of solutions to an LMI is convex. That is, the set is a convex set, where is an LMI. Lemma 1.2 An LMI, , in the variable is an expression of the form where and , . Proof Consider and , and suppose that and satisfy Lemma 1.2. The LMI is convex, since Convexity of LMI From Lemma 1.1, it is known that an optimization problem with a convex objective function and LMI constraints is convex. The following is a non-exhaustive list of scalar convex objective functions involving matrix variables that can be minimized in conjunction with LMI constraints to yield a semi-definite programming (SDP) problem. , where , , , and . Special case when and , where, , and . Special case when 2, , and , where . , where , , , , , and . Special case when and , where , , and . Special case when , and , where . Special case when , and , where , . Special case when , , , and , where . , where and . Relative Definition of a Matrix The definiteness of a matrix can be found relative to another matrix. For example, Consider the matrices and . The matrix inequality is equivalent to or . Knowing the relative definiteness of matrices can be useful. For example, If in the previous example we have and also know that , when we know that . This follows from . External Links LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet. LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd. LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.