MODEL ORDER REDUCTION OF LINEAR DYNAMIC SYSTEMS FOR CONTROL SYSTEMS DESIGN

Abstract

Every physical system can be translated into mathematical model. The mathematical procedure of system modelling often leads to comprehensive description of a process in the form of high order differential equations which are difficult to use either for analysis or controller design. It is therefore useful, and sometimes necessary to find the possibility of finding some equations of the same type but of lower order that may be considered to adequately reflect the dominant properties of the high order system. Some of the reasons for using reduced order models of high order linear systems could be • To reduce computational complexity. • To reduce hardware complexity. • To have a better understanding of the system. • To make feasible controller design. • To use the reduced order system economically for on-line implementation. The development of the reduced order models for analysis and synthesis of control systems has been an area of active research during the last few decades. A wide variety of model order reduction methods both in time domain and frequency domain have been proposed by several authors in the field of control and system theory. But none of the model order reduction methods has been universally accepted which can be applied to all systems. Almost all the reduction methods are best applied in a specific situation and have own merits and demerits. The selection of the particular reduction technique to be used depends on the specific problem at hand. The essential features of a reduction method are the retention of stability of the original system and a better approximation of its response. The aim of this thesis is to develop some new methods for model order reduction for large scale linear time-invariant systems and to design the low order controllers using the few proposed methods. The model order reduction methods are developed both in frequency and time domain. The proposed methods are compared with the well-known order reduction methods and are found comparable in quality. These methods remove some of the inherent shortcomings associated with some existing methods. The suitability of the few proposed reduction methods is checked for designing of controllers. Some mixed order reduction methods are proposed in frequency domain using the advantages of differentiation method and Mihailov stability criterion. The numerator coefficients are determined by using factor division algorithm, improved Pade approximation, and Cauer second form. The biased reduction method using Mihailov criterion is also proposed. These methods are extended to linear time-invariant multi-input multi-output (MIMO) systems. Some order reduction methods based on pole clustering technique and modified pole clustering are proposed. In these techniques, pole cluster centre is either obtained with the help of inverse distance measure (IDM) criterion or by using the proposed algorithm. These proposed methods are also extended to linear multivariable systems. Further, some methods are developed which are based on the minimization of error function using Genetic Algorithm (GA). The error function is an integral square error (ISE) between the unit step responses of the original and reduced system. The idea of error minimization is only applied to get optimum numerator coefficients while the denominator polynomial is obtained by using differentiation method, Mihailov stability criterion, pole clustering and modified pole clustering technique. The reduced order models obtained by the proposed methods are compared with the original high order system with the help of performance indices and step/frequency responses. Three methods are also proposed for reducing the order of the large- scale time domain models. These methods are applicable to linear time-invariant systems. The first method is based on partial realization using outer products. In this method, the Hankel matrices containing both time-moments and markov parameters are used. The second and third methods are based on minimal realization in which modified Hankel matrices are taken. The proposed methods are also extended to linear time-invariant MIMO systems. The time and frequency responses of the reduced order models are compared with the corresponding responses of the original system. Theperformance indices are used to compare the reduced order models. Thecontrol systems are designed on the basis of approximate model matching using Pade approximation. The direct and indirect approaches have been used for this purpose. Few proposed order reduction methods are used for controller design. In the direct approach, reduced order model of the high order system is obtained and then a controller is designed for low order model. In indirect approach, a controller is designed for the original high order system and then closed loop transfer function ofthe system with unity feedback is reduced by any one ofthe proposed methods and then compared with the reference model. The performance comparison ofthe various reduced order models with the original high order system and error minimization has been carried out using MATLAB software package.