REDUCED ORDER MODELLING AND CONTROLLER DESIGN

Abstract

The complexity of physical systems makes their exact analysis difficult and possibly a non-desirable task, mainly due to the difficult economic and computational considerations involved. To deal with the physical systems, first step is to develop a mathematical model. This quest for mathematical models is justified because the use of such models facilitates the analysis of the systems they describe. In many practical situations a fairly complex and high order model is obtained in modelling different components/subsystems of a system. The analysis of such high order system is not only tedious but also not cost effective for on line implementation. In manycontexts of control and system theory, it is advisable or even necessary to refer to simplified models of complex processes both for simulation and for control system synthesis. Many reduction methods have been introduced in last decades for high order state space models or high degree transfer function/transfer function matrices of large scale linear time invariant single input single output and multi input multi output systems. The model reduction methods can be applied directly to either state space model formulation of the system known as 'time domain order reduction methods' or to transfer function model formulations of the systems called 'frequency domain order reduction methods'. So far no universally accepted model order reduction method has been developed which can be applied to all systems. Each method is best applied in a specific situation and has its own advantages and disadvantages. One method may produce a model that approximates low frequency behavior well, whereas others produce good approximation to impulse or step responses. The choice of the particular technique to be used depends on the specific problem at hand. In general, the retention of the stability of the original system and a "good" approximation of its response are considered to be essential features of a reduction method. The aim of this thesis is to develop some new methods of model order reduction for large scale linear time invariant systems and to design a controller using the proposed method. The model order reduction methods are developed both in frequency domain and time domain. The new developed order reduction methods guarantee the stability of the reduced order model if the original high order system is stable. Some of them are utilized for controller design. Some mixed methods of model order reduction are proposed in frequency domain, using the advantages of reciprocal transformation, in which the existing techniques of model order reduction have been modified. The new developed methods in frequency domain are extended for the reduction of multi input multi output systems. In some of the proposed methods based on error minimization technique, the error function formulated is the weighted squared error sum of difference of time moments and Markov parameters of original and reduced system while in others the error function is integral square error (ISE) between transient parts of step responses of original and reduced systems. This error function is minimized using Genetic Algorithm (GA). A program has been made in MATLAB to minimize this error function using GA. The results are compared on the basis of unit step response, impulse response and frequency response along with impulse response energy (IRE) and integral square error (ISE). Some time domain order reduction methods are also explored and a modified Hankel norm approximation (HNA) method is proposed for the model order reduction of linear high order dynamic systems. The proposed method avoids the drawback of mismatch of steady state responses of original and reduced systems, which is a serious flaw in some cases. The frequency domain and time domain characteristics of original system are preserved in the reduced order models. This new proposed method in time domain is extended to multi input multi output systems and discrete time systems. A method for reduction of unstable systems is also proposed by combining hankel norm approximation method with linear transformation. The response of reduced order model obtained is compared on the basis of unit step response and frequency response. In addition to it frequency domain computations are also carried outto validate the proposed method. The method is applied on various practical examples of power system and control system. The new proposed methods remove some of the inherent difficulties associatedwith present day established techniques of order reduction. The controller is designed on the basis of approximate model matching, with both the approaches, direct and indirect, using the modified HNA method. The desired ii performance specifications of the plant are translated into a specification/reference model transfer function. In direct approach reduced order model is obtained for the original high order system and a controller is designed for low order model. In indirect approach of controller design, a controller is designed for the high order system and the closed loop response of original high order system and high order controller with unity feedback is reduced and compared with the reference model. An alternative approach of controller design is proposed, in which a high order controller designed for original high order plant is reduced and the closed loop response of reduced order controller and original high order plant with unity feedback is compared with the response of reference model. The performance comparison of various models has been carried out using MATLAB software package.