REDUCED ORDER MODELLING FOR LINEAR SYSTEMS AND CONTROLLER DESIGN

Abstract

The physical system can be represented by mathematical models. The mathematical procedure of system modelling often leads to a comprehensive description of a process in the form of higher order ordinary differential equations or partial differential equations which are difficult to use either for analysis or for controller synthesis. It is therefore useful and sometimes necessary to find the possibility of some equations of the same type but of lower order that may adequately reflect all essential characteristics of the original system. Hence a systematic approximation of the original model is required which results in a reduced order model. The systematic procedure that leads to reduced order model is termed as Model Order Reduction (MOR), which tries to quickly capture the essential features of an original system. A large number of order reduction techniques have been suggested by several authors in the literature. These are broadly categorized as time and frequency domain reduction techniques. The concept of optimization techniques have also been utilised to obtain reduced order model, in time as well as in frequency domain. In all such techniques the impulse response error between the transient part of the original and reduced order model has been minimized. Recently, meta-heuristic algorithms inspired by Nature have received considerable attention in the field of optimization. These algorithms include Genetic algorithm (GA), Particle swarm optimization (PSO), Harmony search algorithm (HSA), Big Bang-Big Crunch (BB-BC) algorithm and Cuckoo search (CS) etc. Many such algorithms find application in order reduction of linear time invariant systems. Furthermore, combined methods have been developed by several authors in which denominator polynomials are obtained by one method and numerator terms are achieved by another method. In spite of many existing techniques, there is always a scope of developing new techniques. Therefore, to obtain reduced order model from the higher order model is in demand in the field of control systems due to the various issues like stability, realizability and good time/frequency response matching. So, it is of great interest to investigate the efficacy of new algorithms. The balancing control problem of two wheeled mobile robot is not new in the field of control system/robotics. This type of robot is in great demand now a day because of its simple structure, simpler dynamics and applications in different fields i such as transportation, security, search and rescue, entertainment and reducing the man power. As we know that two wheeled mobile robots are always unstable and also affected by external disturbances, so new control techniques are in demand for proper and smooth balancing control and movement of robots. The initial aim of this thesis is to highlight the frequency and time domain order reduction methods available in the literature. This lead to motivate to develop some new algorithm for order reduction of linear time invariant single input single output (SISO) and multi input multi output (MIMO) systems. The work presented in this thesis involves the use of both conventional and evolutionary approach for order reduction of continuous and discrete time systems. The system under consideration may be represented in the time domain or in frequency domain. In addition, the other objective is to ensure the superiority of the new reduction methods by comparing with other well-known methods available in the literature. Lastly, to solve the problem of designing a suitable controller for balancing control of two wheeled mobile robots via reduced order modelling is being considered. At the start, introduction followed by importance and application of model order reduction is presented, subsequently followed by the statement of model order reduction problem in both time and frequency domain for linear time invariant single input single output (SISO) and multi input multi output (MIMO) continuous/discrete time systems. Besides a brief overview of the developments that have taken place in the area of model order reduction, various existing reduction methods and their associated qualities/ drawbacks are also reflected. Composite reduction methods are developed for reduction of higher order linear time invariant systems. Factor division, pole clustering and stability equation methods are employed to propose composite methods. These methods are applicable to SISO/MIMO systems taken from the literature and the results are comparable to the available reduced models. The comparative analysis has been done on the basis of their performance indices which justify the proposed methods. Cuckoo search algorithm is adopted to float new reduction methods. Composite methods along with cuckoo search in combination with Eigen spectrum analysis, stability equation and pole clustering yields good results. The higher order system represented in time as well as in the frequency domain is considered for reduction using cuckoo search algorithm. Original models having the order up to two hundred is considered for reduction. The proposed methods are applied to SISO/MIMO systems ii Abstract and are justified by considering the benchmark examples available in the literature. The controller design for balancing control of two wheeled mobile robot (TWMR) representing practical system is also dealt to ensure the suitability of the developed model order reduction techniques. The controllers are designed for this practical system using the concept of reduced order modelling by utilizing the proposed reduction technique in this thesis. Further, structure specified controllers are not only designed for TWMR but fractional order controllers are also designed and shown to perform better than other conventional controller available in the literature. The time response specifications and performance indices are compared to show the efficacy of the proposed techniques. iii