Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma

The purpose of this paper is twofold. The first is to make explicit the relationship between sum of squares (SOS) decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov (KYP) lemma. The second is to present an efficient algorithm for explicitly finding an SOS decomposition of such matrices, inspired by the Hamiltonian-type methods for the solution of Riccati equations.

[1]  Shinji Hara,et al.  Generalized KYP lemma: unified frequency domain inequalities with design applications , 2005, IEEE Transactions on Automatic Control.

[2]  James Rovnyak,et al.  The factorization problem for nonnegative operator valued functions , 1971 .

[3]  Harry L. Trentelman,et al.  New Algorithms for Polynomial J-Spectral Factorization , 1999, Math. Control. Signals Syst..

[4]  D. Djoković.  Hermitian matrices over polynomial rings , 1976 .

[5]  Joseph F. Traub,et al.  The Algebraic Theory of Matrix Polynomials , 1976 .

[6]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[7]  Ali H. Sayed,et al.  Linear Estimation (Information and System Sciences Series) , 2000 .

[8]  H. Trentelman,et al.  The Kalman-Yakubovich-Popov lemma in a behavioural framework , 1997 .

[9]  A. Rantzer.  On the Kalman-Yakubovich-Popov lemma , 1996 .

[10]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[11]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[12]  Andrew Packard,et al.  The complex structured singular value , 1993, Autom..

[13]  Bruce Reznick,et al.  Real zeros of positive semidefinite forms. I , 1980 .

[14]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[15]  D. Kamenetsky.  Symmetry Groups , 2003 .

[16]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

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

[18]  G. Dullerud,et al.  A Course in Robust Control Theory: A Convex Approach , 2005 .

[19]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[20]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[21]  A. Garulli,et al.  Positive Polynomials in Control , 2005 .

[22]  M. Kojima.  Sums of Squares Relaxations of Polynomial Semidefinite Programs , 2003 .