Model order reduction via real Schur-form decomposition

Abstract A new algorithm is presented for computing Moore's reduced-order transfer-function matrix without calculating the balancing transformation, which tends to be ill-conditioned, especially when the original system is non-minimal or when it has very nearly uncontrollable or unobservable modes. The algorithm is based on finding the eigenspaces associated with large eigenvalues of the cross-gramian matrix Wco using the real Schur-form decomposition. The algorithm does not require a minimal model to start with. The state-space realization obtained by this method is related to the balanced realization by a non-singular matrix. An example is presented to illustrate the proposed algorithm

[1]  D. Wilson,et al.  Model reduction for multivariable systems , 1974 .

[2]  Y. Shamash.  Model reduction using the Routh stability criterion and the Padé approximation technique , 1975 .

[3]  G. W. Stewart,et al.  Algorithm 506: HQR3 and EXCHNG: Fortran Subroutines for Calculating and Ordering the Eigenvalues of a Real Upper Hessenberg Matrix [F2] , 1976, TOMS.

[4]  Edison Tse,et al.  Reduced-order models, canonical forms and observers† , 1979 .

[5]  R. Skelton.  Cost decomposition of linear systems with application to model reduction , 1980 .

[6]  Rein Luus,et al.  Optimization in model reduction , 1980 .

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

[8]  C. P. Kwong,et al.  Optimal chained aggregation for reduced-order modeling† , 1982 .

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

[10]  A. Laub,et al.  A note on cross-Grammians for symmetric realizations , 1983, Proceedings of the IEEE.

[11]  H. Nicholson,et al.  On the structure of balanced and other principal representations of SISO systems , 1983 .

[12]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

[14]  A. Laub,et al.  Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms , 1987 .

[15]  M. Safonov,et al.  A Schur Method for Balanced Model Reduction , 1988, 1988 American Control Conference.

[16]  Hassan K. Khalil,et al.  A real Schur form method for modeling singularly perturbed systems , 1989 .