REDUCED ORDER MODELLING FOR CONTROL SYSTEM DESIGN

Abstract

The physical system can be represented in mathematical models. The mathematical procedure of system modelling often leads to a comprehensive description of a process in the form of higher order ordinary differential equations or partial differential equations which are difficult to use and sometimes necessary to find the possibility of some equations of the same type but of lower order that may adequately reflect all essential characteristics of the original system. Hence a systematic approximation of the original model is required which results in a reduced order model. The systematic procedure that leads to reduced order model is termed as model order reduction (MOR), which tries to quickly capture the essential features of an original system. A large number of order reduction techniques have been suggested by several authors in the literature. These are broadly categorized as time and frequency domain reduction techniques. The frequency domain reduction methods also utilized to reduce the order of interval systems based on Kharitonov’s theorem and interval arithmetic operation (IAO). Furthermore, combined methods have been developed by several authors in which denominator polynomials are determined by one method and numerator terms are determined by another method. In spite of many existing reduction techniques, there is always a scope of developing new techniques. Therefore, the model order reduction of original higher order systems is in demand in the field of system and control due to the various issues like good time/frequency response matching, stability and realizability etc. So, it is of great interest to investigate the efficacy of new algorithms. The initial aim of this thesis is to highlight the frequency domain and interval domain order reduction methods available in the literature. This lead to motivate to develop some new algorithm for order reduction of linear time invariant single input single output (SISO) and multi input multi output (MIMO) systems. The work represented in this thesis involves the use of both conventional and interval approach for order reduction of continuous and discrete time systems. In addition, the other objective is to ensure the superiority of the new reduction methods by comparing i with other well-known reduction methods available in the literature. Lastly, to solve the problem of designing the controller both in direct and indirect approaches by using proposed reduced order methods. The introduction followed by importance and application of model order reduction is presented, subsequently followed by the mathematical preliminaries, then the concept of interval systems is introduced. Besides a brief overview of the development that have taken place in the area of model order reduction, various existing reduction methods and their associated qualities/ drawbacks are also reflected. New composite reduction methods are developed for reduction of higher order linear time invariant systems. Time moment matching method, factor division algorithm, Pade approximation method and differentiation method are employed to propose composite MOR methods. These methods are applicable to SISO/MIMO systems taken from the literature and the results are compared with the some available reduction models. The comparative analysis has been done on the basis of their performance indices which justify the proposed methods. New composite reduction methods are developed for reduction of higher-order linear-time invariant (LTI) interval systems using differentiation method, stability equation method and time moment matching method based on Kharitonov’s theorem. Further, based on interval arithmetic operations new mixed methods have also been proposed by using Pade approximation method, factor division algorithm and differentiation method. To show the efficacy and powerfulness of the proposed reduction methods the popular numerical examples available in the literature are considered. Some of these methods are also extended to model reduction of discrete time systems. The controller is designed on the basis of approximate model matching, with both the direct and indirect approaches, using the proposed reduction methods. The desired performance specifications of the plant are translated into a specification/reference model transfer function. In direct approach the original higher order plant is reduced and the controller designed for reduced order model. In indirect approach of controller design, a controller is designed for original plant ii transfer function and the higher order closed loop transfer function is obtained with unity feedback. Then this higher order closed loop transfer function is reduced to lower order model and performance is compared with that of the reference model. The performance comparison of various models has been carried out using MATLAB software package