Optimal approximation of transfer functions of linear dynamical systems

This paper is concerned with the optimal approximation of transfer functions of linear dynamical systems by low-order models. The proposed methods are based on the minimization of the given error criterion function between the original identified system and its low-order model. Three kinds of approximations of transfer functions are presented. First, the techniques of least squares fit and linear programming are used for approximation. The transfer functions of low-order models are obtained by using the input and output data of the original identified systems. Since the sampled input and output data are used, discrete approximate models are obtained by these methods. However, if the sampling period is sufficiently small it is possible to obtain continuous approximate models. Next, a parameter optimization technique is applied to obtain low-order models. The integral-squared error of the transient responses of the original system and its low-order model is minimized by this method. The transfer function of the original system is expressed by a state equation to obtain an optimal solution. Since the input and output relation is considered in this method, the optimal solution is independent of the state equations, provided that they have the same transfer function. The methods for multivariable unconstrained minimization can be applied to obtain an optimal solution, and the optimal low-order model is always stable when the original system is stable. Therefore, the proposed methods can be used also to the case where there are no dominant poles in the transfer function of the original system.