Lur’e equations and even matrix pencils

Abstract In this work, we consider the so-called Lur’e matrix equations that arise e.g. in model reduction and linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils.

[1]  Michael G. Safonov,et al.  A generalized eigenproblem approach to singular control problems. II. H/sup infinity / problems , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[2]  C. Schröder.  Palindromic and Even Eigenvalue Problems - Analysis and Numerical Methods , 2008 .

[3]  Romeo Ortega,et al.  On the solvability of extended Riccati equations , 2004, IEEE Transactions on Automatic Control.

[4]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[5]  Louis Weinberg,et al.  Network Analysis and Synthesis , 1962 .

[6]  Brian D. O. Anderson,et al.  Matrix inequality solution to linear-quadratic singular control problems , 1977 .

[7]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[8]  R. C. Thompson,et al.  Pencils of complex and real symmetric and skew matrices , 1991 .

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

[10]  R. Byers.  Solving the algebraic Riccati equation with the matrix sign function , 1987 .

[11]  Jason L. Speyer,et al.  Necessary and sufficient conditions for optimally for singular control problems; a transformation approach , 1971 .

[12]  C. Oara,et al.  Generalized Riccati Theory and Robust Control. A Popov Function Approach , 1999 .

[13]  JUDITH D. GARDINER,et al.  A Stabilized Matrix Sign Function Algorithm for Solving Algebraic Riccati Equations , 1997, SIAM J. Sci. Comput..

[14]  Vlad Ionescu,et al.  General matrix pencil techniques for the solution of algebraic Riccati equations: a unified approach , 1997, IEEE Trans. Autom. Control..

[15]  Harry L. Trentelman,et al.  Families of linear-quadratic problems: Continuity properties , 1987 .

[16]  Keith Glover,et al.  Spectral factorization via Hermitian pencils , 1989 .

[17]  Brian D. O. Anderson,et al.  Spectral factorization with imaginary-axis zeros☆ , 1997 .

[18]  B. Anderson,et al.  Transformational solution of singular linear-quadratic control problems , 1977 .

[19]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[20]  D. J. Clements,et al.  A State-Space Approach to Indefinite Spectral Factorization , 2000, SIAM J. Matrix Anal. Appl..

[21]  E. Jonckheere,et al.  A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds , 1988 .

[22]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[23]  B. Anderson,et al.  Impedance synthesis via state-space techniques , 1968 .

[24]  Luís Miguel Silveira,et al.  Guaranteed passive balancing transformations for model order reduction , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[25]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[26]  C. Scherer.  The solution set of the algebraic Riccati equation and the algebraic Riccati inequality , 1991 .

[27]  V. A. Yakubovich,et al.  Singular problem in optimal control of linear stationary system with quadratic functional , 1985 .

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

[29]  Hung-Yuan Fan,et al.  A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations , 2005 .

[30]  Cornelius T. Leondes,et al.  Robust control system techniques and applications , 1992 .

[31]  Antonie Arij Stoorvogel.  The singular H2 control problem , 1992, Autom..

[32]  D. Jacobson,et al.  Necessary and sufficient conditions for optimality for singular control problems - A limit approach , 1971 .

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

[34]  Chun-Hua Guo,et al.  Analysis and modificaton of Newton's method for algebraic Riccati equations , 1998, Math. Comput..

[35]  M. Safonov,et al.  A generalized eigenproblem approach to singular control problems. I. LQG problems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[36]  Athanasios C. Antoulas,et al.  A new result on passivity preserving model reduction , 2005, Syst. Control. Lett..

[37]  Brian D. O. Anderson,et al.  Algebraic description of bounded real matrixes , 1966 .

[38]  Timo Reis,et al.  Positive real and bounded real balancing for model reduction of descriptor systems , 2010, Int. J. Control.

[39]  J. Willems.  Dissipative dynamical systems Part II: Linear systems with quadratic supply rates , 1972 .

[40]  N. Barabanov,et al.  Kalman–Yakubovich lemma in general finite dimensional case , 2007 .

[41]  N. Barabanov,et al.  On the existence of a complete set of solutions of the Lur’e equation , 2007 .

[42]  B. Anderson.  A SYSTEM THEORY CRITERION FOR POSITIVE REAL MATRICES , 1967 .

[43]  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.

[44]  Jason L. Speyer,et al.  System characterization of positive real conditions , 1994, IEEE Trans. Autom. Control..

[45]  R. C. Thompson,et al.  The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil , 1976 .

[46]  Brian D. O. Anderson,et al.  Singular Optimal Control: The Linear-Quadratic Problem , 1978 .