PHYSCON 2011, Le ´ on, Spain, September, 5–September, 8 2011 GENERALIZED CONTROLLABILITY SUBSPACES FOR TIME-INVARIANT SINGULAR LINEAR SYSTEMS M. I. Garc´ ıa-Planas Departament de Matem` atica Aplicada I Universitat Polit` ecnica de Catalunya, Spain maria.isabel.garcia@upc.edu Abstract Controllability subspaces play an important role in geometric control theory for proper linear systems (A, B). In this paper we attempt to extend this con- cept to singular systems constructing generalized in- variant subspaces of controllability for triples of ma- trices (E,A,B) representing singular systems. Key words Singular systems, feedback proportional and deriva- tive, invariant subspaces. 1 Introduction Let us consider a finite-dimensional singular linear time-invariant system E ˙ x = Ax + Bu, where E,A ∈ M n (C), B ∈ M n×m (C). For simplicity, we denote the systems as a triples of matrices (E,A,B) and we denote by M = {(E,A,B) | E,A ∈ M n (C),B ∈ M n×m (C)} the set of singular systems. the set of this kind of sys- tems. In the case where E = I n the system is standard and we denote merely, as a pair (A, B). For simplicity but without loss of generality, we con- sider that matrix B has column full rank: 0 < rank B = m ≤ n. Invariant subspaces for transformations from C m+n into C n was introduced by Gohberg, Lancaster, Rod- man [I. Gohberg, P. Lancaster, L. Rodman, (1986)], as a generalization of similarity called block-similarity. Our objective is to develop a generalization of the con- cept of invariant subspace for triples of matrices as gen- eralized linear maps defined modulo a subspace. Remember that a subspace G ⊂ C n , is invariant un- der (A, B) as a map from C n+m into C n if and only if there exists a subspace ¯ G of C n+m where the canoni- cal projection of ¯ G over C n is G,(π |C n ¯ G = G), and ( AB ) ¯ G ⊂ π |C n ¯ G = G. Equivalently (see [I. Gohberg, P. Lancaster, L. Rod- man, (1986)] for a proof), a subspace G ⊂ C n is in- variant under (A, B) if and only if AG ⊂ G + Im B (1) In this paper, we consider triples of matrices (E,A,B), that we can see as a pair of maps (E,B), (A, B) defined modulo a subspace (see [M a ¯ I. Garc´ ıa- Planas, (2006)]), classified under the following equiv- alence relation: two triples (E,A,B), (E 0 ,A 0 ,B 0 ) are equivalent if and only if the following equality holds: ( E 0 A 0 B 0 ) = Q ( EAB )   P P F E F A R   , (2) where Q, P ∈ Gl(n; C), R ∈ Gl(m; C), F E ,F A ∈ M m×n (C). Analyzing the definition of invariant subspace for standard systems, we extend this concept to singular systems. The main objective of the paper is to charac- terize invariant subspaces for singular linear systems, and in particular to study some special invariant sub- spaces as are the controllability subspaces. 2 Preliminaries Some basics facts about group action, Grassmannian manifold and controllability character are as follows. 2.1 Equivalence relation as a Lie group action The equivalence relation defined in (2), can be see as an action over M under a Lie group action. Let us consider the following Lie group G = Gl(n; C) × Gl(n; C) × Gl(m; C) × M m×n (C) × M m×n (C). The product ? in G is given by (Q1,P1,R1,F E 1 ,F A 1 ) ? (Q2,P2,R2,F E 2 ,F A 2 )= (Q 2 Q 1 ,P 1 P 2 ,R 1 R 2 ,F A 1 P 2 + R 1 F E 2 ,F E 1 P 2 + R 1 F A 2 ) (3)