MODEL ORDER REDUCTION FOR LINEAR DYNAMIC SYSTEMS AND CONTROLLER DESIGN

Abstract

The physical system can be represented in mathematical models. The mathematical modelling often leads to a comprehensive system description of a process in the form of higherorder ordinary differential equations or partial differential equations which are difficult to use when the order is higher, therefore it is sometimes essential to find the possibility of some equations of the same type but of lower order that may efficiently reflect all necessary characteristics of the original system. Hence a systematic approximation of the original model is required which results in a reduced order model. The systematic mechanism that leads to reduced order approximated model is termed as Model Order Reduction (MOR), which attempts to quickly retain the important features of the original system. In the past several decades, various researchers in the field of control systems, power systems, chemical and mechanical engineering etc. have proposed numerous reduction techniques. These are broadly categorized as time domain and frequency domain reduction techniques. However, every work/method has some advantages as well as some shortcomings. Therefore, there is always some scope to develop new methods that produce improved results. The model order reduction of original higher-order systems is in demand in the field of system and control owing to the various advantages like good response matching, stability and realizability etc. So, it is of great interest to explore the improved new algorithms. The main objective of the present work is to develop some new reduction techniques which produce better results than some well-known existing techniques in the literature. Mixed approaches of model order reduction have been proposed in the frequency domain to reduce the higher-order system into various lower order models. In mixed approaches, the denominator polynomial of the reduced order model is obtained by one of the stability preserving methods such as Routh approximation, Routh stability Criterion, stability equation, Mihailov stability Criterion, Modified pole clustering etc. whereas the numerator polynomial of the reduced model is obtained with other methods such as Padé approximation, Time moment matching method, Factor division, algorithm etc. The proposed methods retain the steady state value and stability of the original system in the reduced order models for stable systems. In recent trends, the mixed methods are being developed with the help of nature-inspired optimization techniques to obtain the reduced order models for a higher-order system. The nature-inspired evolutionary optimization methods are Genetic Algorithm (GA), Harmony Search Algorithm (HSA), Particle Swarm Optimization (PSO), Big Bang-Big Crunch (BB-BC) and Cuckoo Search Algorithm (CSA). GA is inspired by modern evolutionary biology synthesis ii such as mutation, crossover and selection. PSO is inspired by a swarm of birds, fish, etc. CSA is inspired by an observation of the reproduction strategy of cuckoos. With the motivation of the above, some composite reduction techniques have been proposed in the frequency domain for linear time invariant (LTI) single input single output (SISO) and multiple input multiple output (MIMO) systems. The work presented in this thesis involves the use of both conventional and evolutionary approach for order reduction of continuous time systems in the frequency domain. The introduction followed by the importance and application of model order reduction, subsequently followed by the statement of model order reduction problem in both frequency and time domain for linear time invariant SISO and MIMO systems. Besides a brief overview of the development that has taken place in the area of model order reduction, various existing reduction methods and their associated qualities/drawbacks are also discussed. Composite reduction methods are developed for the reduction of original higher-order linear time invariant systems. Routh approximation (RA), Padé approximation (PA), Mihailov stability criterion (MSC) are employed to propose composite methods. These approaches apply to well-known SISO/MIMO systems available in the literature and the results are compared with some existing reduction models. New composite reduction methods are developed for the reduction of higher-order LTI systems such as evolutionary approaches based on Routh approximation, Padé approximation, Mihailov stability criterion etc. In these methods, Routh approximation, Padé approximation and Mihailov stability criterion are utilized to obtain the coefficients of denominator polynomial whereas evolutionary techniques such as Particle Swarm Optimization and Cuckoo search algorithm are used to obtain the coefficients of numerator polynomial of the reduced model. The numerator polynomial is obtained through minimization of Integral Square Error (ISE). To explore the effectiveness of the proposed methods the standard performance indices such as ISE, Relative Integral Square Error (RISE), Integral Absolute Error (IAE) and Integral Time weighted Absolute Error (ITAE) are compared with renowned methods in the literature for standard examples. In addition, Time response and Frequency response of the original higherorder system and reduced order system obtained with various reduction techniques are compared to show the exactness and superiority of the proposed reduction techniques. The controller is designed based on the reduced order systems using the proposed model reduction methods. In the controller design, it is fundamental to have an approximation measure that guarantees the closed-loop system stability when a low-order compensator is used to control the full-order model. The design of PID controller is carried out with the help of the iii reference model. The desired performance specifications of the plant are translated into a reference model transfer function. In this approach, a reduced model is obtained from the largescale original system and a controller is designed with the utilization of the reduced order model. The controller obtained with a reduced order model is applied to the original higherorder system. The performance comparison of various models has been carried out using MATLAB software package.