Simplifying theH∞ theory via loop-shifting, matrix-pencil and descriptor concepts

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Abstract
The 2-Riccati H∞ controller formulae 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 D 11 =0, D 22 = 0, D T 12= [0 l], D 21 = [0 l],D l 12 C1=0 and B 1 D T 21 =0. Matrix-fraction descriptions (MFDs) for the algebraic Riccati equation solutions afford another change of variables, which brings the 2-Riccati H∞ controller formulae 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, Pi and Q, blow up and in cases where l— QP is nearly singular. Numerical difficulties previously associated with verifying the existence conditions P ≥ 0, Q ≥ 0, and λmaxlpar;QP) < 1 are largely eliminated by equivalent alternative conditions.

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