Higher-Order Singular Systems and Polynomial Matrices
The Correspondence between Quadruples of Matrices and Polynomial Matrices in Higher-Order Singular Systems
by Praveen Samadhiya*, Dr. K. K. Jain , Dr. Yogesh Sharma,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 1, Issue No. 1, Feb 2011, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
There is aone-to-one correspondence between the set of quadruples of matricesdefiningsingular linear time-invariant dynamical systems and a subset of the set ofpolynomial matrices. This correspondence preserves the equivalence relationsintroduced in both sets (feedback-similarity and strict equivalence): twoquadruples of matrices are feedback-equivalent if, and only if, the polynomial matrices associated to them are alsostrictly equivalent.
KEYWORD
higher-order singular systems, polynomial matrices, quadruples of matrices, linear time-invariant dynamical systems, equivalence relations
inear singular systems (also called descriptor representations or DAEs, differential-algebraic quations) and their control have been widely studied from the end of the 1970s by many authors see, for example, [1], [2], [3], [4], [7], [8], [11]). They arise naturally and frequently when athematically modelling mechanical, electric, economic,... systems. Minimal indices ronecker’s theory of singular matrix pencils has been widely used to obtain a canonical form of he matrices defining a system. he matrix pencils are naturally associated to systems which can be epresented in the form x˙ (t) = Ax(t) + Bu(t). Block-equivalent pairs of matrices (equivalent pairs hen considering basis changes, in the state and input spaces, and state feedback) are haracterized as those whose associated matrix pencils are strictly equivalent. See [9], [10], [12], or example, for further details. t is also known that equivalent triples of matrices defining systems of the form x˙ (t) = Ax(t) + u(t); y(t) = Cx(t), under the equivalence relation derived from: basis changes in the state, input nd output spaces, state feedback and output injection, are those having strictly equivalent ssociated matrix pencils ist-order singular systems may be represented by here . Singular systems are called egular when the matrix pencil jsE ¡ Aj does not vanish identically. Regularity ensures the xistence and uniqueness of a solution to the singular system ([2]). n [5] a special type of polynomial matrices of degree two are associated to them and it is shown hat there is a one-to-one correspondence between quadruples of matrices defining such singular ystems under “feedback-similarity”, the equivalence relation derived when applying one, or ore, of the following elementary transformations: 1) basis similarity for the state space, 2) basis changes for the control space, 3) basis changes for the output space, 4) output injection, 5) state feedback, 6) derivative feedback nd degree two polynomial matrices of the form: nder “strict equivalence”: are said to be strictly equivalent when there exist nvertible matrices L and R such that . In this work we generalize the result bove to the case of higher-order singular systems. Though higher-order singular systems are often tudied reducing them to first-order systems, this is not convenient in some cases (when the ifferentiability of u(t) is limited) since such transformation of the system may lead to a non- quivalent problem: for instance, it is possible to find a continuous solution for the original ystem, but not for the first-order transformed one). This shows the necessity for directly treat igher-order systems. igher-order singular systems and polynomial matrices Let us consider `th-order (` ¸ 1) ifferential-algebraic equations with constant coefficients of the form irst, we will generalize the equivalence relations considered in the Introduction to the set of -tuples of matrices defining singular systems. This equivalence relation will be called eedback-similarity. hen we will associate to each -tuple of matrices describing a singular system a polynomial atrix and recall the notion of strict equivalence in [5], which generalizes strict equivalence for atrix pencils. inally, we will prove that there is a one-to-one correspondence between the set of quadruples of atrices and a subset of the set of polynomial matrices of degree two which preserves the onsidered equivalence relations: feedback-similarity and strict equivalence. That is to say, two uadruples of matrices are feedback-similar if, and only if, the associated matrix polynomials are trictly equivalent. iven a linear time-invariant differential-algebraic equations with constant oefficients, we will consider the following elementary transformations: 1) basis similarity for the state space: hat is to say, two -tuples of matrices are said to be feedback-similar when one can be btained from the other by means of one, or more, of the elementary transformations (1) - (6) bove. et us associate to the the polynomial matrix trict equivalence in the set of matrix pencils may be generalized to the set of polynomial atrices.
efinition 2. We will say that two polynomial matrices and are strictly equivalent
hen there exist constant regular matrices L and R such that e can state now the relationship between high-order singular systems under feedbacksimilarity nd polynomial matrices under strict equivalence. t is straightforward that nd therefore onversely. We assume that quivalently, rom these equalities we conclude that L1 is regular, for and thus herefore and are feedback-similar
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