REDUCTION OF LINEAR DYNAMIC SYSTEMS IN CONTROL ENGINEERING ENVIRONMENT

Abstract

The basic motivation for system approximation is the need for simplified models of dynamical systems, which capture the main features of the original complex model. This need arises from limited computational, accuracy, and storage capabilities about the total behavior of these dynamical systems. Physical as well as artificial processes are described mainly by mathematical models. In this framework of mathematical models, there is an ever-increasing need for improved accuracy, which leads to models of high complexity. The simplified model is used in place of the original complex model, for either simulation or control. In the former case, simulation; one seeks to predict the system behavior, however simulation of the full model is not feasible. Consequently, an appropriate simplification of this model is necessary, resulting in simulation with reduced computational complexity. Prominent examples include weather prediction and air quality simulations. The complexity of models, measured in terms of the number of coupled firstorder differential or difference equations, may reach the tens or hundreds or thousands. In particular, discretization in problems that arise from dynamical partial differential equations (PDEs) which evolve in three spatial dimensions can easily lead to 1 million equations. In such cases, reduced simulation models are essential for the quality and timeliness of the prediction. Other methods for accelerating the simulation time exist, like parallelization of the corresponding algorithm. In many contexts of control and system theory, it is advisable or even necessary to refer to simplified models ofcomplex processes both for simulation and for control system synthesis. Many reduction methods have been introduced in last decades for high order state space models of 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 orderreduction 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 continuous frequency domain and discrete frequency 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 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) as well particle swarm optimization PSO. A program has been made in MATLAB to minimize the error functions using GA. The results are compared on the basis of unit step response, and frequency response along with and integral square error (ISE). A new method, referred to as 'Indirect method' to reduce the higher order system into various lower order models of different orders is introduced. In this method, first the denominator of the reduced order model is obtained by one of the stability based reduction methods and then the numerator of the reduced model is obtained by simultaneously multiplying and dividing the denominator polynomial to the power series expansion of the original system. The method is attractive and can be used as an alternative approach for fitting the initial time moments and/or Markov parameters. The method preserves steady state value and stability of original system in the reduced order models for stable systems. This method can be extended to reduce interval systems and discrete time systems. The response of reduced order model obtained is compared on the basis of unit step response and frequency response. The new proposed methods remove some of the inherent difficulties associated with present day established techniques of order reduction. l] The controller is designed on the basis of approximate model matching, with both the approaches, direct and indirect, using the conventional method as well as genetic algorithm (GA). The desired 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.