Near aggregation, the dual GHR and pole-zero cancellation

Abstract

It is known that a linear system will aggregate if and only if it is unobservable. An equivalent condition is that the transfer function has a pole-zero cancellation. In this paper, we consider SISO systems that nearly aggregate and we show that these systems have almost pole-zero cancellations. Defining near unobservability as, roughly, the existence of an invariant subspace near the null space of C, we also show that nearly unobservable systems exhibit almost pole-zero cancellations. The main tool in this analysis is the dual generalized Hessenberg representation (dual GHR). The dual GHR extends the GHR by incorporating the input structure with the output structure of the GHR. The result is a canonical form (the dual GHR) that relates the internal states to the input-output properties of the system. By exploiting this structure, the main results are obtained.