LMIs in Control/Matrix and LMI Properties and Tools/Schur Complement Lemma-Based Properties

LMI Condition Consider , , , , , and . There exists such that: 1 if and only if : 2Any matrix satisfying + is a solution to (1) Consider , , , ,, and . There exists such that: 3 if and only if : , and 4 If two inequalities in (4) hold, then a solution to (3) is given by Consider , , , and , where . There exists such that: 5 if and only if: 6 Consider , , , and , where .: 7implies Consider , , , , , , and LMI gives: , and 8if and only if: 9 Consider , , , , , , and . LMI gives: , and 10are satisfied if and only if: 11 Consider , , , , and , , where , , , and . There exists , , , and such that: , and = 12if and only if: , and 13 asas Proof Proof for (3) Necessity ((3) (4)) comes from the requirement that the submatrices corresponding to the principle minors of (3) are negative definite Sufficiency ((4) (3)) is shown by rewriting the matrix inequalities of (4) in the equivalent form , and Concatenating the two matrices and choosing gives the equivalent matrix inequality : 3-1or: 3-2which is equivalent to (3) using the Schur complement lemma. Proof for (6) the LMI in (5) can be written using the Schur complement lemma as: 5-1 : 5-2 : 5-3 Proof for (7) Using the Schur complement lemma on (7) for Using the property or equivalent gives 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.