SIMPLIFYING THE H-INFINITY THEORY VIA LOOP-SHIFTING, MATRIX-PENCIL AND DESCRIPTOR CONCEPTS - Safonov, M; Limebeer, D; Chiang, R (1989)
The 2-Riccati H∞ controller formulas and derivations are simplified via various 'loop-shifting' transformations that are naturally expressed in terms of a degree-one polynomial system matrix (PSM) closely related to the Luenberger descriptor form of a system. The technique enables one without loss of generality to restrict attention to the simple case in which D11 = 0, D22 = 0, D12T = [0 I], D21 = [0 I], D12TC1 = 0 and B1D21T = 0. Matrix-fraction descriptions (MFDs) for the algebraic Riccati equation solutions afford another change of variables, which brings the 2-Riccati H∞ controller formulas into a cleaner, more symmetric descriptor form having the important practical advantage that it eliminates the numerical difficulties that can occur in cases where one or both of the Riccati solutions, P and Q, blow up and in cases where I - QP is nearly singular. Numerical difficulties previously associated with verifying the existence conditions P≥0, Q≥0, and λmax (QP) <1 are largely eliminated by equivalent alternative conditions.