A factorization principle for stabilization of linear control systems

By introducing a fictitious signal y 0 if necessary we define a transformwhich generalizes the passage from the scattering to the chain formalism in circuit theory. Given a factorization ˜ = ⊖ 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 if and only if a related compensator K ′ is stabilizing for ⊖. Factorizations ˜ = ⊖ R with ⊖ having a certain block upper triangular form lead to an alternative derivation of the Youla parametrization of stabilizing compensators. Factorizations with ⊖ 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 . This gives a new solution of the H ∞ problem completely in the transfer function domain. Computation of the needed factorization ˜ = ⊖ R in terms of a state‐space realization of leads to the state‐space formulas for the solution of the H ∞ problem recently obtained in the literature.