A factorization principle for stabilization of linear control systems

Abstract

By introducing a fictitious signal yo if necessary we define a transform of a given linear control system which generalizes the passage from the scattering to the chain formalism in circuit theory. Given a factorization b = 0 R of $where R is a block matrix function with a certain key block equal to a minimal phase (or outer) matrix function, we show that a given compensator U = Ky is internally stabilizing for the system B if and only if a related compensator K' is stabilizing for 0. Factorizations 9'= 0 R with 0 having a certain block upper triangular form lead to an alternative derivation of the Youla parametrization of stabilizing compensators. Factorizations with 0 equal to a J-inner matrix function (in a precise weak sense) lead to a parametrization of all solutions K of the H" problem associated with 9. This gives a new solution of the H" problem completely in the transfer function domain. Computation of the needed factorization B= 0 R in terms of a state-space realization of 9 ' leads to the state-space formulas for the solution of the H" problem recently obtained in the literature. KEY WORDS Feedback stabilization H" control J-inner-outer factorization J-spectral factorization 0. INTRODUCTION A standard general feedback configuration in terms of which many problems of interest in

Nonlinear localJ-lossless conjugation and factorization

“…Suppose that the system G given by (1) and (2) is invertible and satisfies conditions 2 and 3 of Theorem 3. Let O be a system defined by equations (26), (27).…”

“…Suppose that a system G with realization (1) and (2) is SO stable with an invariant attractive manifold M given in (21) and there exist positive constants k,, k,, 6, and r, c such that…”

Inner–Outer Factorization for Nonlinear Noninvertible Systems

“…For example, in the theory of -optimal control ( [7], [22], [23], [28], [29] for linear; [4]- [6], [9]- [11], [42], [44] for nonlinear) and nonlinear chemical process control ([16], [31], [32], [49]), this type of factorization is used extensively. In our paper, we will mainly discuss nonlinear inner-outer factorization and its application in the former two settings.…”

Internal stability and loop-transformations: an overview on LFTs, Möbius transforms and chain scattering

“…This question has already got a partial answer in Ball et al [1991] based on the scattering approach through the Potapov-Ginsburg transformationP of the generalized plant P . However, that method should assume a left or right invertibility of P and does not provide an explicit formula for the transformed configuration.…”