On Some Questions Concerning Controllability and Observability Indices

Abstract

This paper investigates some questions relating to the controllability and observability indices of a linear constant system, particularly with regard to the occurrence of the generic case, in which a set of indices differ by at most unity. It is argued, with the support of examples, that non-generic cases can be expected to arise in practice. It is shown that a non-generic system can be made generic by connecting to it an auxiliary system of a certain order, and the required order is evaluated; the cases of parallel, cascade and feedback connection are considered. The relationship between the indices of a continuous-time system and those of the discretetime system obtained by sampling it, is examined; it is shown that the sampled system tends to be nearer to the generic case than the original system, and it is argued that, in practice, a sampled-data system is almost certain to be generic.