On discrete, finite-dimensional approximation of linear, infinite dimensional systems

Abstract

Many phenomena are naturally described in terms of dynamical systems of infinite order. Such phenomena cannot be adequately described by an interconnection of a finite number of accumulating elements, i.e., by means of differential or difference equations of finite order. Among the well-known examples are distributed parameter systems, which are usually described by partial differential equations, and fractional order systems, which are described by fractional differential equations. In order for an infinite-dimensional system to be simulated or implemented using a digital computer, it must be approximated by a finitedimensional model. Numerous methods for finite-dimensional approximations of infinite dimensional systems have been considered in literature. If spatial distribution of variables is of interest, distributed parameter systems are often simulated by means of the finite elements method (FEM). If the spatial distribution of variables is not of interest, as it is the case with fractional order models, an approximating ordinary differential equation of sufficiently high order is used for approximation. A novel, flexible and numerically efficient method for rational, finite-dimensional approximation of linear, infinite-dimensional systems is presented in the current chapter. The proposed method uses the least-squares (LS) procedure to interpolate frequency domain response of a fractional order system using a finite number of incident frequencies. An adequate comparative analysis has also been carried out through corresponding examples by applying several other known approximation methods.