PHYSCON 2011, Le ´ on, Spain, September, 5–September, 8 2011 OUTPUT OBSERVABILITY OF 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 S. Tarragona Departamento de Matem´ aticas Universidad de Le´ on Spain sonia.tarragona@unileon.es Abstract In this paper finite-dimensional singular linear discrete-time-invariant systems in the form Ex(k + 1) = Ax(k)+ Bu(k), y(k)= Cx(k) where E,A ∈ M = M n (C), B ∈ M n×m (C), C ∈ M p×n (C), de- scribing convolutional codes are considered and the no- tion of output observability is analyzed. Key words Singular systems, feedback proportional and deriva- tive, output injection proportional and derivative, out- put observability. 1 Introduction Let us consider a finite-dimensional singular linear discrete-time-invariant system Ex(k + 1) = Ax(k)+ Bu(k), y(k)= Cx(k) where E,A ∈ M n (C), B ∈ M n×m (C), C ∈ M p×n (C), describing convolutional codes. For simplicity, we denote the systems as a quadruples of matrices (E,A,B,C) and we denote by M the set of this kind of systems. In the case where E = I n the system is standard and we denote as a triple (A,B,C). For simplicity but without loss of generality, we con- sider that matrix B has column full rank and rank B = m and C has row full rank and rank C = p, so 0 < p, m ≤ n. It is well known that there is a close connection be- tween linear systems and convolutional codes and there is a large literature about that as for example [F. R. Gantmacher, (1959), M. Kuijper, R. Pinto, (2009), J.L Massey, M.K. Sain, (1967), J. Rosenthal, J.M. Schu- macher, E.V. York, (1996)]. Ch. Fragouli and R. D. Wesel [Ch. Fragouli, R.D. Wesel, (1999)] give the following definition of output observable for standard systems. Definition 1.1. The standard system (A,B,C) is said to be output observable if the state sequence {x 0 ,x 1 ,...,x n-1 } is uniquely determined by the knowledge of the output sequence {y 0 ,y 1 ,...,y n-1 } for a finite number of steps n - 1. Taking into account that y k = CA k x 0 +CA k-1 Bu 0 +...+CABu k-2 +CBu k-1 the output observability is characterized by the follow- ing proposition. Proposition 1.1 ([Ch. Fragouli, R.D. Wesel, (1999)]). The system (A,B,C) ∈M is output observable if and only if the following matrix M=          C 0 0 ... 0 CA CB 0 . . . . . . CA 2 CAB CB . . . . . . . . . . . . CA n-1 CA n-2 B CA n-3 B ... CB          ∈ M pn×(n+(n-1)m) (C) has full rank. In this paper, we generalize the notion of output ob- servability given for standard linear systems to the sin- gular linear systems, and we characterize the set of out- put observable systems. We remark that the observability of singular systems has been widely discussed by T. Kaczorek in [T. Kac- zorek, (1992)]. 2 Preliminaries We consider quadruples of matrices (E,A,B,C) ∈ M, representing singular discrete time invariant lin- ear systems, a manner to understand the properties of the system is treating it by purely algebraic techniques. The main aspect of this approach is defining an equiv- alence relation preserving these properties, many inter- esting and useful equivalence relations between singu- lar systems have been defined. We deal with the equiv- alence relation accepting one or more, of the following transformations: basis change in the state space, input