Abstract
The paper outlines a coherent development of factorization theory in the framework of polynomial model theory. Starting from the most elementary factorizations of polynomial matrices we build up the connections to invariant subspace theory, factorizations of transfer functions, Wiener-Hopf factorizations. We pass on to spectral factorizations of polynomial matrices and rational functions and the connection with the analysis of the algebraic Riccati equation. Finally we study inner/outer factorizations for a class of transfer functions and the derivation of state space formulas.
The aim throughout is to highlight the logical interconnections and the technique rather than the derivation of the most general results.
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References
[1979] A. C. Antoulas, “A polynomial matrix approach to F mod G-invariant subspaces”, Doctoral Dissertation, Dept. of Mathematics, ETH Zurich.
[1979] II. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Birkhauser, Basel.
[1980] II. Bart, I. Gohberg, M. A. Kaashoek and P. Van Dooren, “Factorizations of transfer functions”, SIAM J. Contr. Optim., 18, 675–696.
[1974] G. Bengtsson, “Minimal system inverses for linear multivariable systems”, J. Math. Anal. Appl. 46, 261–274.
[1972] R.W. Brockett and A. Rahimi, “Lie Algebras and Linear Differential Equations”, in Ordinary Differential Equations, (L. Weiss, Ed.) Academic Press, New York, 1972.
[1988] T. Chen and B. Francis, “Spectral and inner-outer factorizations of rational matrices”, to appear.
[1974a] W. A. Coppel, “Matrix quadratic equations”, Bull. Austr. Math. Soc., 10, 377–401.
[1974b] W. A. Coppel, “Matrices of rational functions”, Bull. Austral. Math. Soc. 11, 89–113.
[1980] E. Emre, “Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces”, SIAM J. Contr. Optimiz. 18,288–296.
[1980] E. Emre and M. L. J. Hautus, “A polynomial characterization of (A,B)-invariant and reachability subspaces”, SIAM J. Contr. Optimiz., 18, 420–436.
[1988] B. Francis, “H ∞ Control Theory”, Springer.
[1976] P. A. Fuhrmann, “Algebraic system theory: An analyst's point of view”, J. Franklin Inst., 301,521–540.
[1977] P. A. Fuhrmann, “On strict system equivalence and similarity”, Int. J. Contr. 25,5–10.
[1978] P. A. Fuhrmann, “Simulation of linear systems and factorization of matrix polynomials”, Int. J. Contr., 28,689–705.
[1979] P. A. Fuhrmann, “Linear feedback via polynomial models”, Int. J. Contr. 30,363–377.
[1981] P. A. Fuhrmann, “Duality in polynomial models with some applications to geometric control theory,” Trans. Aut. Control, AC-26,284–295.
[1983] P. A. Fuhrmann, “On symmetric rational transfer functions”, Linear Algebra and Appl., 50,167–250.
[1983] P. A. Fuhrmann, “A matrix Euclidean algorithm and matrix continued fractions”, Systems and Control Letters, 3, 263–271.
[1984] P. A. Fuhrmann, “On Hamiltonian transfer functions”, Lin. Alg. Appl., 84, 1–93.
[1985] P. A. Fuhrmann, “The algebraic Riccati equation — a polynomial approach”, Systems and Control Letters,, 369–376.
[1979] P. A. Fuhrmann and J. C. Willems, “Factorization indices at infinity for rational matrix functions”, Integral Equat. and Oper. Theory, 2,287–301.
[1980] P. A. Fuhrmann and J. C. Willems, “A study of (A,B)-invariant subspaces via polynomial models”, Int. J. Contr. 31,467–494.
[1959] F. R. Gantmacher, Matrix Theory, Chelsea, New York.
[1982] I. Gohberg, P. Lancaster and L. Rodman, “Factorization of selfadjoint matrix polynomials with constant signature”, Lin. and Multilin. Alg., 11, 209–224.
[1978] M. L. J. Hautus and M. Heymann, “Linear feedback-an algebraic approach”, SIAM J. Control 16,83–105.
[1989] U. Helmke and P. A. Fuhrmann, “Bezoutians”, to appear, Lin. Alg. Appl..
[1970] V. A. Jacubovich, “Factorization of symmetric matrix polynomials”, Soviet Math. Dokl., 11, 1261–1264.
[1982] P. P. Khargonekar and E. Emre, “Further results on polynomial characterization of (F,G)-invariant and reachability subspaces”, IEEE Trans. Aut. Control, 27, 352–366.
[1980] P. Lancaster and L. Rodman, “Existence and uniqueness theorems for the algebraic Riccati equation”, SIAM J. Cont..
[1956] C. C. MacDuffce, The Theory of Matrices, Chelsea, New York.
[1971] K. Martensson, “On the matrix Riccati equation”, Inform. Sci. 3,17–49.
[1973a] B. P. Molinari, “The stabilizing solution of the algebraic Riccati equation”, SIAM J. Cont., 11,262–271.
[1973b] B. P. Molinari, “Equivalence relations for the algebraic Riccati equation”, SIAM J. Cont., 11,272–285.
[1966] J. E. Potter, “Matrix quadratic solutions”, SIAM J. Appl. Math., 14,496–501.
[1985] T. Shamir and P. A. Fuhrmann, “Minimal factorizations of rational matrix functions in terms of polynomial models”,, Lin. Alg. Appl., 68, 67–91.
[1983] M. A. Shayman, “Geometry of the algebraic Riccati equation, Part I”, SIAM J. Cont., 21,375–394.
[1971] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation”, Trans. Automat. Contr., 16,621–634.
[1966] D. C. Youla and P. Tissi, “N-port synthesis via reactance extraction-part I”, IEEE Inter. Convention Record, 183–205.
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This paper is dedicated to Jan C. Willems
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© 1989 Springer-Verlag
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Fuhrmann, P.A. (1989). Elements of factorization theory from a polynomial point of view. In: Nijmeijer, H., Schumacher, J.M. (eds) Three Decades of Mathematical System Theory. Lecture Notes in Control and Information Sciences, vol 135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0008462
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DOI: https://doi.org/10.1007/BFb0008462
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