MODEL ORDER REDUCTION AND ITS APPLICATION IN CONTROL SYSTEM DESIGN

Abstract

In order to carry on the analysis, synthesis and design of real life problems, usually the first step is the development of a 'mathematical model' that can be substituted for the real system. Once the mathematical description ofa real life system is obtained, all the analysis can be done on this mathematical description, called mathematical model of the real system. This mathematical model can take different shapes depending on the system as well as the approach used for modelling. Depending on whether the model is in terms of a single high order differential equation or in the form of a set of first order differential equations, the order of the system is decided by the order of the single differential equation or the number of first order differential equations used inthe model, as the case may be. Whether existing or to be designed, when a system is mathematically modelled for analysis and improvement, initially a complex model of high order is obtained. Simplification of the model is a must to make the model useful. Depending on the application of the model, the method of simplification varies. One such method of simplification is known as order reduction i.e. obtaining the low order model of the existing high order model such that both are equivalent in terms ofresponse. Model order reduction of linear time-invariant systems had been a topic of interest to alarge number ofresearch workers all over the world. It is interesting to note that different methodologies developed for order reduction can be put broadly under two groups, like, time and frequency domain, being obviously the two accepted methods ofsystem modelling. The reasons behind interest in model order reduction are many, but can be precisely put as (i) To have less computation time for analysis and Abstract design of system and (ii) To have economy in hardware requirements for on line simulation of a system. The present attempt is towards development of some new methods for model order reduction of linear time-invariant continuous systems and control system design. The work presented herein deals with the frequency domain model order reduction techniques and control system design based on a transfer function/transfer function matrix description of the original high order systems. The new developed order reduction methods guarantee the stability of the reduced order model if the original high order system is stable and overcome some of the inherent drawbacks associated with the other existing methods. Some of these order reduction methods are completely new, which are based on minimization of the integral square error (ISE) criterion by Genetic algorithm (GA) and Particle swarm optimization (PSO) evolutionary techniques while in the other cases several mixed methods have been suggested in which the existing techniques have been modified for removing their drawbacks. Some of the model reduction methods developed herein have also been extended for the order reduction of linear multivariable systems, which is a direct application of the singleinput single-output (SISO) methods on the elements of the transfer function matrix of multi-input multi-output (MIMO) systems. The numerical/simulation examples chosen to demonstrate the methods are of wide variety including a heating plant and a practical power system. Further, the methods developed have also been illustrated with the help of similar type of examples so that a comparison can be made between the proposed and the other well-known existing order reduction techniques. The comparisons between the proposed and the other well-known existing order reduction techniques have been shown by comparing the ISE, impulse response energy (IRE), relative integral square errors (/ and S), unit step and impulse responses. 11 Abstract The determination ofthe mapping error oflinear time-invariant dynamic system by asimplified model, as one of the applications of reduced order modelling, has also been explored in the current literature. In this thesis, both the relative mapping errors expressed by means of the relative integral square errors /and S, have been determined, plotted with respect to time for both the unit step and impulse inputs and are compared in tabular forms. There are two approaches for designing the low order controllers for high order plants, direct and indirect. The first one consists of designing directly the parameters of alow order controller by using some optimization or other procedures while the second one consists ofdesigning first a high order controller and, in a second step, applying a reduction procedure for simplifying it. In this thesis, both the approaches have been used for designing the low order controllers. The desired performance specifications of the plant are translated into aspecification/reference/model transfer function M(s) and then the ISE between this reference model and the closed loop control system of the plant is minimized, to obtain the unknown parameters ofthe controller. Finally, the contributions made in the thesis are highlighted in the end. Based on the experience of work reported in this thesis, the future scope of research work in this area is also included.