25-08-2012, 04:41 PM
A New Mixed Method for Reducing Order of Linear System Using Clustering Method And Chebyshev polynomial Approximation
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ABSTRACT
This Paper presents “A New Mixed Method for Reducing Order of Linear System Using Clustering Method and Chebyshev Polynomial Techniques.” Denominator polynomial of the reduced order model is obtained by forming of the pole clustering group of the original system, the cluster centre indentified by using inverse distance measure (IDM) criterion. Than co-efficiant of numerator are determined by chebyshev polynomial series. It representing the frequency response characteristics of the higher order system, and model representing the approximating low order model. This method is generalized the classical pade approximation (s-plane) carried out by the chebyshev polynomial series expansion, being over a design frequency interval. This new mixed method gives guarantees of stability response reduced order model and also preserve the characteristics of original higher order system is stable. This methodology dealt with s-plane and the squared amplitude of the transfer function is expanded by chebyshev polynomial series and approximated to desired order of the model.
INTRODUCTION
Objective of order reduction:
Model .Order Reduction is a branch of systems and control theory, which studies properties of dynamical systems in application for reducing their complexity, while preserving (to the possible extent) their input-output behavior. This problem applies both to continuous –time and discrete –time systems.
Large engineering systems (e.g. nuclear reactors, steam generators ,chemical plants ,air flight systems ,power systems ,transportation systems etc.) are essential features of our society .Control techniques when applied to such large engineering systems lead to mathematical descriptions which are complex and of high dimensions ,thus making it very difficult to analyze, understand and control the systems performances due to very large dimensions of the model and it requires a large computer storage capacity and considerable simulation time. In addition, the control strategies for such models are costly to develop and sometimes unreliable.
Method based on time moments and Markov parameters matching
Pade-Approximation of s-domain transfer function has been very popular reduction techniques for continuous time system[1, 2]. This is due to its computational simplicity and its ability to automatically match a number of time moments of the impulse response of the simplified system with those of the original one .Like many other simplification techniques , it suffers from occasional failures where Pade' approximant may contain an unstable pole even if the original transfer function is perfectly stable .Moreover, the algorithms for the above techniques are applicable to continuous time system only and cannot be applied directly to z-transfer functions. Improved Pade approximation technique using stability equation method [2] in this method obtained reduced order model approximation non dominant poles thus over comes this draw back.
PROPOSED WORK
My thesis work framed, following objectives
• To study and implement the reduced order modeling (ROM) using clustering method and pade approximation techniques which is my base paper decided in August-2010
• To implement of different combination of clustering pole group and check the stability of the time response of each clustered pole group reduced order model.
• To implement a new techniques of this mixed method which gives better response of the base paper
• My objectives I have checked and implement in MATLAB-7.01 and also trace the each pole group combination of reduced order model. A new techniques is (Chebyshev polynomial approximation) is gives the better response which is mentioned and trace the wave shape in their work
Definition of clustering-
We define the clustering, “the process of organizing objects into groups whose members are similar in some way” .A cluster is therefore a collection of objects which are “similar” between them and are “dissimilar” to the objects belonging to other clusters. In this case we easily identify the 4 clusters into which the data can be divided; the similarity criterion is distance: two or more objects belong to the same cluster if they are “close” according to a given distance (in this case geometrical distance). This is called distance-based clustering .Another kind of clustering is conceptual clustering: two or more objects belong to the same cluster if this one defines a concept common to all that objects. In other words, objects are grouped according to their fit to descriptive concepts, not according to simple similarity measures