A New Square-root Balancing-Free Stochastic Truncation Model Reduction Algorithm

Abstract The paper proposes a new computational approach with enhanced numerical robustness for computing reduced order models of continuous systems by using the balanced stochastic truncation model reduction method. The new approach circumvents the computation of possibly III-conditioned stochastic balancing transformations. Instead, well-conditioned projection matrices are determined for computing directly the state-space representations of the reduced order models. The projection matrices are computed in a numerically reliable way, by using exclusively the Cholesky (square-root) factors of systems Gramians. The proposed algorithm can handle both minimal and non-minimal systems.

[1]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[2]  A. Varga,et al.  On Computing High Accuracy Solutions of a Class of Riccati Equations , 1994 .

[3]  A. Varga,et al.  Coprime factors model reduction based on accuracy enhancing techniques , 1993 .

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

[5]  W. Wang,et al.  A tighter relative-error bound for balanced stochastic truncation , 1990 .

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

[7]  A. Laub,et al.  Computing the singular value decompostion of a product of two matrices , 1986 .

[8]  Michael Green,et al.  Balanced stochastic realizations , 1988 .

[9]  M. Safonov,et al.  A Schur method for balanced-truncation model reduction , 1989 .

[10]  M. Konstantinov,et al.  On the numerical properties of the schur approach for solving the matrix Riccati equation , 1987 .

[11]  Michael G. Safonov,et al.  Model reduction for robust control: A schur relative error method , 1988 .

[12]  Andras Varga,et al.  Balancing free square-root algorithm for computing singular perturbation approximations , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[13]  Keith Glover,et al.  Multiplicative approximation of linear multivariable systems with L∞ error bounds , 1986, 1986 American Control Conference.

[14]  Brian D. O. Anderson,et al.  Singular perturbation approximation of balanced systems , 1989 .

[15]  U. Desai,et al.  A transformation approach to stochastic model reduction , 1984 .