The H-infinity norm calculation for large sparse systems

In this paper, we describe an algorithm for estimating the H∞-norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm is designed to be efficient for state-space models defined by {A,B,C,D} where A is a large sparse matrix of order n which is assumed very large relative to the input and output dimensions of the system.

[1]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

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

[3]  T. Kailath,et al.  Properties of the system matrix of a generalized state-space system , 1980, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[4]  T. Kailath,et al.  Properties of the system matrix of a generalized state-space system , 1978 .

[5]  P. Dooren.  The generalized eigenstructure problem in linear system theory , 1981 .

[6]  Karl Johan Åström,et al.  Computer-Controlled Systems: Theory and Design , 1984 .

[7]  P. Dooren,et al.  Numerical aspects of different Kalman filter implementations , 1986 .

[8]  Babak Hassibi,et al.  Indefinite-Quadratic Estimation And Control , 1987 .

[9]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

[10]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[11]  S. Boyd,et al.  A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L ∞ -norm , 1990 .

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

[13]  L. Ghaoui,et al.  History of linear matrix inequalities in control theory , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[14]  Paul Van Dooren,et al.  Convergence of the calculation of Hoo norms and related questions , 1998 .

[15]  T. Kailath,et al.  Indefinite-quadratic estimation and control: a unified approach to H 2 and H ∞ theories , 1999 .

[16]  X. Rao.  LARGE SCALE STABILIZATION WITH LINEAR FEEDBACK , 1999 .

[17]  P. Van Dooren,et al.  Real and complex stability radii of polynomial matrices , 2002 .

[18]  Kyle A. Gallivan,et al.  Singular Riccati equations stabilizing large-scale systems , 2006 .