An extended inner–outer factorisation algorithm based on the structure of a transfer function matrix inverse

In this paper, the structure feature of the inverse of a multi-input/multi-output square transfer function matrix is explored. Instead of complicated advanced mathematical tools, we only use basic results of complex analysis in the analysing procedure. By employing the Laurent expression, an elegant structure form of the expansion is obtained for the transfer function matrix inverse. This expansion form is the key of deriving an analytical solution to the inner–outer factorisation for both stable plants and unstable plants. Different from other computation algorithm, the obtained inner–outer factorisation is given in an analytical form. The solution is exact and without approximation. Numerical examples are provided to verify the correctness of the obtained results.

[1]  D. Youla,et al.  On the factorization of rational matrices , 1961, IRE Trans. Inf. Theory.

[2]  Dante C. Youla.  Correction to 'On the Factorization of Rational Matrices' , 1961, IRE Trans. Inf. Theory.

[3]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[4]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[5]  M. A. Kaashoek,et al.  Constructive Methods of Wiener-Hopf Factorization , 1986 .

[6]  Evanghelos Zafiriou,et al.  Robust process control , 1987 .

[7]  Michael Green.  On inner-outer factorization , 1988 .

[8]  H. Kimura.  Conjugation, interpolation and model-matching in H ∞ , 1989 .

[9]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[10]  B. Francis,et al.  Spectral and inner-outer factorizations of rational matrices , 1989 .

[11]  P. Dooren.  Polynomial matrix factorizations via recursive pole-zero cancellation , 1990 .

[12]  Fang-Bo Yen,et al.  Inner-outer factorizations or right-invertible real-rational matrices , 1990 .

[13]  Jim Freudenberg,et al.  Loop transfer recovery for nonminimum phase plants , 1990 .

[14]  T. Sugie,et al.  Inner—outer factorization for strictly proper functions with j w -axis zeros , 1991 .

[15]  M. Tsai,et al.  A numerical algorithm for inner-outer factorizations of real-rational matrices , 1993 .

[16]  H. Kwakernaak,et al.  Polynomial J-spectral factorization , 1994, IEEE Trans. Autom. Control..

[17]  Mitsumasa Sato,et al.  An L-lossless factorization approach to the positive real control problem , 1996 .

[18]  Cemil Ulus.  Numerical computation of inner-outer factors for a class of retarded delay systems , 1997, Int. J. Syst. Sci..

[19]  T. Mita,et al.  Inner-Outer Factorization for Non-Square Proper Functions with Infinite and Finite jω-Axis Zeros , 1998 .

[20]  A. Varga.  Computation of inner-outer factorization of rational matrices , 1998, IEEE Trans. Autom. Control..

[21]  Pablo A. Iglesias,et al.  An entropy formula for nonlinear systems , 2000, J. Frankl. Inst..

[22]  András Varga,et al.  Computation of general inner-outer and spectral factorizations , 2000, IEEE Trans. Autom. Control..

[23]  C. Oar.  Constructive solutions to spectral and inner-outer factorizations with respect to the disk , 2005 .

[24]  Daniel W. C. Ho,et al.  Computation of the (J, J')-Lossless Factorization for General Rational Matrices , 2005, SIAM J. Control. Optim..

[25]  Cristian Oara.  Constructive solutions to spectral and inner-outer factorizations with respect to the disk , 2005, Autom..

[26]  Charles A. Desoer,et al.  Linear System Theory: The State Space Approach , 2008 .

[27]  A. Varga.  On computing normalized coprime factorizations of periodic systems , 2010 .

[28]  A. Sasane,et al.  Extension of the ν-metric: the H∞ Case , 2010 .

[29]  Mohamed Najim,et al.  Rayleigh fading channel simulator based on inner-outer factorization , 2010, Signal Process..

[30]  Dawu Gu,et al.  No-weight design of H 2 controllers for square plants , 2011 .

[31]  Jerzy Stefan Respondek,et al.  Numerical recipes for the high efficient inverse of the confluent Vandermonde matrices , 2011, Appl. Math. Comput..

[32]  S. Djouadi.  Optimal Disturbance Rejection and Robustness for Infinite Dimensional LTV Systems , 2012, 1207.3120.

[33]  A. Varga.  Chapter 6: Descriptor System Techniques in Solving H2/∞-Optimal Fault Detection and Isolation Problems , 2012 .

[34]  Andreas Varga,et al.  New computational paradigms in solving fault detection and isolation problems , 2013, Annu. Rev. Control..

[35]  Andreas Varga,et al.  On computing inner-outer factorizations of periodic systems , 2013, 2013 European Control Conference (ECC).

[36]  Jerzy S. Respondek.  Recursive numerical recipes for the high efficient inversion of the confluent Vandermonde matrices , 2013, Appl. Math. Comput..

[37]  Wei Xing,et al.  Using Inner-Outer Factorization to Solve the Spectral Factor of Discrete-Time Descriptor Systems , 2013, BIC-TA.

[38]  Clarence W. Rowley,et al.  Normalized Coprime Robust Stability and Performance Guarantees for Reduced-Order Controllers , 2013, IEEE Transactions on Automatic Control.