Premium
A factorization principle for stabilization of linear control systems
Author(s) -
Ball Joseph A.,
Helton J. William,
Verma Madanpal
Publication year - 1991
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.4590010403
Subject(s) - factorization , mathematics , parametrization (atmospheric modeling) , transfer function , linear system , matrix (chemical analysis) , matrix decomposition , transfer matrix , computation , realization (probability) , pure mathematics , control theory (sociology) , mathematical analysis , computer science , algorithm , eigenvalues and eigenvectors , control (management) , physics , quantum mechanics , artificial intelligence , materials science , statistics , engineering , electrical engineering , composite material , computer vision , radiative transfer
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.