Bounding the distance from structurally stable quadruples to non-structurally stable ones M. ISABEL GARC ´ IA-PLANAS AND SONIA TARRAGONA Departament de Matem´atica Aplicada I Universitat Polit` ecnica de Catalunya, C. Miner´ ıa 1, Esc C, 1 o -3 a 08038 Barcelona, Spain E-mail: maria.isabel.garcia@upc.edu Abstract:- Given a quadruple of matrices (E,A,B,C ) defining a generalized linear system E ˙ x(t)= Ax(t)+ Bu(t), y(t)= Cx(t) with E,A ∈ M n (C), B ∈ M n×m (C) and C ∈ M p×n (C), we present a lower bound for the distance between a structurally stable quadruple of matrices and the nearest non-structurally one, in terms of the singular values of a certain matrix associated to the quadruple. Key-Words:- Generalized linear systems, feedback and derivative feedback. 1 Introduction We consider generalized time-invariant lin- ear systems given by the matrix equations E ˙ x(t)= Ax(t)+ Bu(t), y(t)= Cx(t) where E,A ∈ M n (C), B ∈ M n×m (C), C ∈ M p×n (C). We represent this systems by quadruples of matrices (E,A,B,C ). These equations arise in theoretical areas as differential equations on manifolds as well as in applied areas as sys- tems theory and control. We are interested in obtaining lower bounds for the distance between a quadruple of matrices structurally stable under a equiva- lence relation defined in the set of quadruples and the nearest quadruple non structurally stable. The structure of this paper is as follows In Section 2 a equivalence relation is de- fined and a geometric study of the equivalence classes (orbits) and the tangent spaces to the orbits is presented. Section 3 is devoted to recall the matrix norm considered and to obtain a lower bound. 2 Equivalence relation Let us consider the set M = {(E,A,B,C ) | E,A ∈ M n (C),B ∈ M n×m (C),C ∈ M p×n (C)} of quadruples of matrices defining a generalized time-invariant linear system. We consider the standard transformations 1) basis change in the state space x(t)= Px 1 (t), 2) basis change in the input space u(t)= Ru 1 (t) 3) basis change in the output space y 1 (t)= Sy(t) 4) feedback u(t)= u 1 (t) − Ux(t), 5) derivative feedback u(t)= u 1 (t) −V ˙ x(t), 6) output injection x(t)= x 1 (t)+ Wy(t) Proceedings of the 8th WSEAS International Conference on APPLIED MATHEMATICS, Tenerife, Spain, December 16-18, 2005 (pp330-333)