The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

A method of model reduction for reducing a high-order transfer function to its low-order models is introduced based upon the stability-equation method. The transfer functions of reduced orders are obtained directly from the pole-zero patterns of the stability-equations of the original transfer function. Comparisons with methods in the current literature are made. Extension of the proposed method to discrete systems is given.

Reduction of Transfer Functions by the Stability-Equation Method

A means is provided to assess the value or 'cst' of each component of a large scale system, when the total cost is a quadratic function. Such a 'cost decomposition' of the system has several important uses. When the components represent physical subsystems which can fail, the 'component cost' is useful in failure mode analysis. When the components represent mathematical equations which may be truncated, the 'component cost' becomes a criterion for model truncation. In this latter event component costs provide a mechanism by which the specific control objectives dictate which components should be retained in the model reduction process. This information can be valuable in model reduction and decentralized control problems.

Cost decomposition of linear systems with application to model reduction

Recently published methods for generating stable reduced order models are compared with an old and very simple method of reduction based on the truncation of the high order system transfer function. Numerical examples show the truncation method, which requires no computation, to be as good as the other methods.

Truncation method of reduction: a viable alternative

A combined method making use of the advantages of the stability-equation method and the Pade approximation method for reducing high order transfer functions of single-input/single-output systems and multivariable systems is presented. The reduction procedure is simple and computer-oriented. All the reduced models are guaranteed to be stable if the original system is stable. The proposed method is applied to the investigation of (1) the effect of model reduction on the limit-cycle of non-linear systems and (2) the effect of model reduction on Horowitz compensators. Detailed calculations are given and comparisons with results in the current literature are made.

Model reduction using the stability-equation method and the Padé approximation method

A new approach to stable reduced-order linear model construction is given, based on the concept of Hurwitz polynomial approximation. Pade approximants to the tangent function of a given large-degree Hurwitz polynomial are constructed, such that the corresponding low-degree polynomials are also Hurwitz. We prove a theorem to that effect and give methods for obtaining the reduced Hurwitz polynomials. Model reduction is achieved by first invoking this theorem and completing the reduction using the Pade equations. Two examples are given.

Linear model reduction using Hurwitz polynomial approximation

Two Pade methods are discussed for constructing low-degree Hurwitz polynomials from a given high-degree Hurwitz polynomial to approximate its argument. Using the Hurwitz polynomial approximants as characteristic polynomials, the numerator dynamics of reduced-order (matrix) transfer-function models are then easily determined by partial Pade approximation of a given large-order model. Stability of such reduced models is always assured. By suitable linear fractional transformations the methods are made applicable to discrete-time systems. The methods are compared in simulation examples for both continuous and discrete-time systems.

R. K. Appiah

Papers

Linear model reduction using Hurwitz polynomial approximation

Padé methods of Hurwitz polynomial approximation with application to linear system reduction