On the stability of projections of balanced realizations

Abstract This paper generalizes a stability property concerning the state matrix of a balanced realization established by Pernebo and Silverman. It is shown that stability is preserved under a general projection of the state matrix provided that the Hankel singular values of the realization are distinct. A necessary and sufficient condition for the case with nondistinct Hankel singular values is also established.

[1]  James Lam,et al.  L 1 impulse response error bound for balanced truncation , 1992 .

[2]  D. Wilson.  Optimum solution of model-reduction problem , 1970 .

[3]  J. L. Goldberg.  Matrix theory with applications , 1991 .

[4]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[5]  M. Aoki.  Some approximation methods for estimation and control of large scale systems , 1978 .

[6]  Keith Glover,et al.  A structured approximation problem with applications to frequency weighted model reduction , 1992 .

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

[8]  R. Curtain,et al.  Realisation and approximation of linear infinite-dimensional systems with error bounds , 1988 .

[9]  M. Aoki.  Control of large-scale dynamic systems by aggregation , 1968 .

[10]  D. Enns.  Model reduction with balanced realizations: An error bound and a frequency weighted generalization , 1984, The 23rd IEEE Conference on Decision and Control.

[11]  D. Bernstein,et al.  The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore , 1985 .

[12]  John T. Spanos,et al.  A new algorithm for L2 optimal model reduction , 1992, Autom..

[13]  Leonard M. Silverman,et al.  Linear time-variable systems: Balancing and model reduction , 1983 .

[14]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[15]  Naresh K. Sinha,et al.  Canonical forms for aggregated models , 1978 .

[16]  W. D. Ray.  Matrices in control theory , 1984 .

[17]  D. Wilson,et al.  A new algorithm for optimal reduction of multivariable systems , 1980 .

[18]  K. Arun,et al.  Balanced approximation of stochastic systems , 1990 .

[19]  Raimund J. Ober,et al.  Balanced realizations: canonical form, parametrization, model reduction , 1987 .

[20]  N. Sinha,et al.  A comparison of the balanced matrix method and the aggregation method of model reduction , 1983, The 22nd IEEE Conference on Decision and Control.

[21]  James Lam,et al.  Balanced realization of Pade approximants of e/sup -sT/ , 1991 .

[22]  Keith Glover,et al.  Optimal Hankel-norm approximation for stable systems with first-order stable weighting functions , 1986 .

[23]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .