Inner–outer factorization for differential-algebraic systems

We consider transfer functions of linear time-invariant differential-algebraic systems. Based on the stabilizing solutions of certain differential-algebraic Lur’e equations, we will derive simple formulas for realizations of inner–outer factorizations. We show that the existence of a stabilizing solution only requires behavioral stabilizability of the system. We neither assume properness nor (proper) invertibility of the transfer function. We briefly discuss numerical aspects for the determination of such factorizations.

[1]  Timo Reis,et al.  Lur’e equations and even matrix pencils , 2011 .

[2]  Andras Varga,et al.  Computation of J-innerouter factorizations of rational matrices , 1998 .

[3]  André C. M. Ran.  A course in H∞− control theory , 1988 .

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

[5]  Timo Reis,et al.  Outer transfer functions of differential-algebraic systems , 2017 .

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

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

[8]  Timo Reis,et al.  Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits , 2014, Large-Scale Networks in Engineering and Life Sciences.

[9]  Matthias Voigt,et al.  On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems , 2015 .

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

[11]  Guoxiang Gu,et al.  Inner-outer factorization for strictly proper transfer matrices , 2002, IEEE Trans. Autom. Control..

[12]  Matthias Voigt,et al.  The Kalman-Yakubovich-Popov inequality for differential-algebraic systems: Existence of nonpositive solutions , 2015, Syst. Control. Lett..

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

[14]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

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

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

[17]  Timo Reis,et al.  Numerical solution of singular Lur'e equations , 2014 .

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

[19]  Matthias Voigt,et al.  The Kalman–Yakubovich–Popov inequality for differential-algebraic systems , 2015 .

[20]  Jan C. Willems,et al.  Introduction to mathematical systems theory: a behavioral approach, Texts in Applied Mathematics 26 , 1999 .

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

[22]  Joos Vandewalle,et al.  Realization algorithm for determining generalized state-space representations , 1987 .

[23]  O. Rendel,et al.  Numerical Solution Of Matrix Equations Arising In Linear-Quadratic Optimal Control Of Differential-Algebraic Equations , 2016 .

[24]  Thomas Kailath,et al.  Linear Systems , 1980 .

[25]  T. Berger,et al.  Hamburger Beiträge zur Angewandten Mathematik Controllability of linear differential-algebraic systems-A survey , 2012 .

[26]  M. Weiss.  Spectral and inner-outer factorizations through the constrained Riccati equation , 1994, IEEE Trans. Autom. Control..

[27]  T. Berger,et al.  Observability of Linear Differential-Algebraic Systems: A Survey , 2017 .

[28]  Fabian R. Wirth,et al.  On Minimum Phase , 2013, Autom..

[29]  Jan C. Willems,et al.  Introduction to Mathematical Systems Theory. A Behavioral , 2002 .

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