REDUCED ORDER MODELLING IN CONTROL SYSTEM

Abstract

In the present digital world, the advancement of modern day technology towards miniaturization associated is with societal and environmental processes which are contributing to the soaring system complexities, thereby resulting in outsized systems. It's a known fact, that the behavioral study of any system starts with building up of a mathematical model based on theoretical considerations. Accordingly, a set of ordinary differential equations (ODE) or partial differential equations (PDE) are derived by applying physical laws, signifying a mathematical model. In nature and industry, most of these mathematical models turn out to be of higher order, hereafter called as "original model". The direct simulation or design of such models is neither computationally desirable nor physically convenient to handle. Additionally, such models pose difficulty during analysis, control, synthesis and identification as the said tasks are not so easy as they seem to be. It is really grueling, sometimes not feasible and also prove to be a costly affair because of what it may be called as "the curse of dimensionality". In engineering and science, it is often desirable to use the simplest mathematical model that “does the job”. Hence a systematic approximation of the original model is very much in need which results in a reduced order model. The systematic procedure that ends up in reduced order model is called Model Order Reduction (MOR). Hence MOR has born out of the necessity to provide simplified/reduced models, that address the ill effects of higher dimensional models. The order reduction phase consists of reducing the number of ODE's appropriately, using model reduction technique, to form a reduced model. But, the derived reduced model should provide a good approximation for the original model by preserving some vital features viz. stability, realizibility, good time/frequency response matching etc. It is therefore desirable that the original model can be replaced by the reduced model enabling easy analysis, design, simulation, control and cost effective on line implementation, apart from ensuring the following qualities (a) Simplify the understanding of the system. (b) Reduction of computational and hardware complexity. (c) Reduction of storage requirements. (d) Ease of efficient controller design and implementation. (e) Cost effectiveness. Abstract ii Consequently, order reduction ends up as a necessary procedure for simulating large complex systems; the same is generally practiced in systems and control engineering in spite of having high speed processors and is active area of research. In the existing literature, abundant order reduction methods have been developed by several authors and are mainly categorized as time and frequency domain reduction methods. Normally, the time domain methods start with a state space description whereas the latter rely on the transfer function model. However, the reduced models obtained from different reduction techniques are unique and the quality is ultimately judged by the way it is utilized. But, none can be judged as the universally best reduction method as these methods depend upon various reason. The best method is one, which shields the vital dynamics of the system under consideration; how well it satisfies the application specifications with reasonable error/computational efforts, in addition to storage. Consequently, the need for better approximation techniques persist. An effort is being made in the present research work to address this issue. The initial objective of this thesis is to recapitulate most of the model reduction methods available in the research literature. This is succeeded by the purpose to promote some new model order reduction methods applicable to SISO/MIMO time-invariant continuous time systems. The work presented here is confined to linear systems/models and the examples therein. The task mentioned involves the use of both conventional and evolutionary strategies. The considered systems may be represented in frequency domain or time domain. In addition, the other objective is to ensure the superiority of the new reduction methods by comparing with other well known methods, beside checking its validity for LTI discrete system. Lastly, to solve the problem of designing a suitable PID controller for the higher ordered model, utilizing the newly developed method is being considered. In addition, direct and indirect approaches of controller design are dealt with apart from alternative approach to check its applicability for the original model. At the outset, introduction followed by importance and applications of MOR is presented, subsequently followed by statement of MOR problem in both time and frequency domain for continuous time systems (SISO and MIMO). Besides brief overview about the developments that have taken place in the area of MOR, various existing reduction methods and their associated qualities/drawbacks are also reflected. Composite reduction methods are developed for reduction of higher ordered LTI continuous systems. Stability Equation (SE), Eigen Spectrum Analysis (ESA), Dominant Pole (DP), Modified Pole Clustering (MPC) are Abstract iii employed to propose composite methods. These methods are applicable for SISO/MIMO systems taken from the available literature and are comparable to the available reduced models. The same proposed methods are also extended for SISO/MIMO discrete systems. Comparison of responses to step input and their associated performance indices justifies the proposed methods. Evolutionary schemes including the recently introduced Big Bang Big Crunch(BBBC) optimization technique are adopted to float new reduction methods. Mixed methods using BBBC in combination with Routh Approximation (RA) and Stability Equation (SE) method yield good results. In addition, BBBC also plays an important role in optimizing the linear shift point 'a' for order reduction in least square sense. Further, systems of higher order represented in both time and frequency domain are considered for reduction using BBBC and the same is extended for discrete systems as well. Original models having the order upto 200, are considered for reduction. The proposed methods are applied on SISO/MIMO systems and are justified by considering the available bench mark examples. The TMS320C54X processor, grouped under a fixed point DSP is a low-cost, comprehensive development tool that allows new DSP designers to explore the TMS320C5000 DSP architecture and begin developing DSP based applications. It has functional adaptability to a great extent and processing speed. BBBC is roped in to do the required task. The order of Butterworth and Chebyshev filters are designed, and their order is reduced and implemented on TMS320C54x processor. Simulations are carried out in MATLAB and Code Composer Studio (CCS). The input/output waveforms obtained are compared and substantiated. In addition, the frequency response and FFT power spectrum of the input/ output signals are also plotted. The design of controller for the original models representing practical systems are also dealt to ensure the suitability of developed MOR methods. Further, fractional order PID controller is discussed and shown to perform better than the integer order PID controller using an example. Both direct and indirect approach of controller design are employed in addition to an alternative approach for controller design. The design examples are confined to frequency domain. The unit step response of closed loop transfer functions obtained from the original and reduced plant transfer function are compared with the unit step response of the reference model. Overall the viability/validity and use of the MOR techniques developed are conclusively established through several numerical examples.