A Hybrid Approach for Simplification Using Logarithmic Clustering and Moments Matching

Introduction

The simplification of linear dynamic systems via reducing the order of the transfer function has important applications in control theory. Firstly, the reduced or simplified model of any large-scale dynamic system is used to design an efficient controller for any uncontrolled system as it generates effective control law. Secondly, the analysis of such simplified system becomes very easy. Nowadays, the optimum simplified models are being realised by using optimization techniques. The simplified models are also used in model based control techniques. The simplification of model is an important research area mainly in the field of control theory and computer aided design of systems and controllers.

For analysis and controller design of any dynamic system, the control engineers require mathematical modelling. The mathematical modelling of the system is a procedure to write the mathematical equations which represent the characteristics of the whole system. These equations may be algebraic equations with differential/partial differential equations. Then, in control theory, a relation between output and input variable can be established by taking Laplace transform with zero initial conditions. Using this approach, a transfer function and transfer function matrix may be derived for single-input single output (SISO) and multiple-inputs and multiple-outputs (MIMO) system respectively. Sometimes, the mathematical equations of a dynamic system are not known or difficult to identify. In these conditions, the input and output data are generally collected by performing experiments and then system identification concept is usually applied to get approximate model. In the above procedure, it is quite obvious that the designer may get the complex models in the form of transfer function or state model. But, for analysis and control system design, we need a simple model having reduced order denominator preferably 1^st order or 2^nd order, so that the designer may compute various time and frequency domain specifications. The various approaches for getting simplified models via reducing the order of the complex transfer function have been suggested and available in the literature. The few well-known simplifications methods are briefly discussed as below: