On Formalisation of Martin Distance for Linear Dynamical Systems

The Martin distance was first developed to compare Autoregressive-Moving-Average (ARMA) models and has since been extended to compare linear dynamical systems (LDS). In this paper, we formalise the concept of Martin distance and put it on a firm mathematical footing. In particular, we prove that the Martin distance is not a metric on the space of LDSs. Furthermore, we show that unlike ARMA processes, where one can use the Martin distance to compare LDSs with different dimensions, this is not the case for LDSs. Moreover, we show that even on the space of LDSs with
$n$
states,
$p$
inputs, and
$q$
outputs, the Martin distance is only a pseudo-metric. We use this to construct a full-fledged metric on an appropriate quotient space. Finally, we extend the Martin distance to compare control-affine nonlinear systems.