When are Multiplier Relaxations Exact

Abstract We revisit the robust performance analysis problem for structured uncertainties whose frequency responses lie in the rather general class of diagonal matrices with blocks that are full ellipsoidal or repeated and contained in intersections of disks or circles or that are located in finitely generated convex sets. Based on the full block S-procedure we suggested LMI relaxations to verify robust performance. Our main purpose is to prove a general computationally verifiable condition for when these relaxations do not involve any conservatism. This allows to show the general exactness of the suggested relaxations for small block-structures, comprising an elementary proof for the structured singular value being exactly computable by convex optimization for three full complex blocks.

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