Simplification of High-Order Mechanical Systems Using the Routh Approximation

In many recent applications, the control engineer is faced with the problem of controlling a linear system modeled by a transfer function of high or infinite order. In a majority of these cases it is possible and advantageous to approximate the original transfer function by a transfer function of lower order. The Routh approximation is a novel method for reducing the order based on the idea of truncating the well-known Routh table used to determine stability—hence the name given to the method. The Routh approximants can be computed by a finite recursive algorithm that is suited for programming on a digital computer. The algorithm, flow diagrams, and simple numerical examples are presented. A detailed description of the theoretical background and properties of the method are given in [1] and [2]. This paper focuses on the application of the Routh approximation method and the simplification of two mechanical systems are described. First, a low order transfer function for studying vibration is computed from a finite element model of the structure. Second, the Routh approximation is utilized in the design of a thermal control system where the heat conduction is modeled as a distributed parameter system whose transfer function has infinite order.