Development of Model Order Reduction Methods for Linear System Control Design

Abstract

Mathematical modeling and simulation are very important in science and engineering applications at the beginning of design stage for analysis, design, and control. They also allow virtual experiments when a practical experiment is either time-consuming, too costly, impractical or difficult to execute. Several practical systems are with large dimensionality. Model reduction is an important robust tool allowing for fast numerical simulation of complicated models. It aims to replace a large scale system by an approximate system of lower order, which not only preserves the basic input-output behaviour of the original system but also requires significantly less effort. Low order models result in several advantages such as they are easier to understand; computational requirement is less and reduces hardware complexity, and help to make a feasible controller design, etc. The basic motivation behind various model reduction methods is to find an appropriate low-order model, such that it preserves input-output behavior and essential properties of the original system with minimum error. The development of reduction methods for the analysis and synthesis of high order systems has been an area of active research during the last few decades. Various investigations have been performed and number of methods are suggested for order reduction in both time and frequency domain in the field of control system. However, none of order reduction methods have been universally accepted which can be applied to each system. Every method has its own advantages and limitations and applicable only for specific applications. In this work, order reduction methods are presented for the large-scale LTI control systems in the frequency and time domain. Their basic fundamental, properties, benefits, and limitations are discussed along with the relationship among different approaches to identify the appropriate methods for the given application. The main objective of this thesis is to develop model reduction methods in both frequency and time domain. In the frequency domain, three reduction techniques are presented, whose central objective focuses not only to retain the dominant modes of high order model but also include the effect of each less dominant pole on the system behavior. Further, the accuracy of the proposed method is validated using benchmark examples which exits in the literature and also the proposed method is compared with the state of art methods. In the first proposed algorithm for order reduction, the problem of selecting poles is removed by applying the advantages of time moment and Markov parameters which are specified as a subroutine of the poles and residues of original model having real and complex poles. The coefficients of the numerator polynomial for reduced model are obtained using a factor division algorithm. In the second and third proposed method, the denominator of reduced model is constructed from the new clustering method based on modified Lehmer measure in order to improve dominant pole cluster centers. This method uses the concept of a dominant pole algorithm for the selection of poles for the clustering. The selection of poles is important because it determines both steady-state and transient information of the dynamical system. Pade approximation and frequency response matching technique are used to find the parameters of the numerator polynomial for second and third proposed method respectively. Time domain reduction method employ the idea of balancing for both stable and unstable large-scale continuous-time systems. The method uses a quantitative measure criterion for the selection of dominant eigenvalues. These dominant eigenvalues are used to form a new substructure matrix that retains the dominant modes (or may desirable mode) of the original system. Keeping the dominant eigenvalues in a reduced model allows greater accuracy since the retained eigenvalues provide a physical link to the high order model and also guarantees stability if the original model is stable. The proposed method is accomplished by system transformation using the Sylvester equation, taking into account to preserve the dominant eigenvalues of the high order model. Having obtained a transformed model, the reduced model is achieved by truncating the non-dominant eigenvalues. This work also presents the low order controller design of the higher order plant/process using indirect methods. The implementational issues and aspects of the model order reduction methods have been tested through controller design. The step responses of the closed-loop system obtained from the original and the reduced model are compared with the step response of the reference model. The step response of the overall controlled system for the original system and the reduced model are found in agreement to that of reference model. Order reduction gives benefits such as i) more numerically efficient controller design process, ii) able to calculate simple control laws, and iii) minimum computational efforts required in simulation problems. The accuracy and effectiveness of the proposed method is measure through numerical test examples and compared with state of art methods. The comparison is made by calculating the error index (ISE) between the transient parts of the original and various reduced models. All the numerical computations have been performed with Matlab software package.