Continuous state representations for AR systems (Q1904430)

The behavioral setting is used to study the subclass of time-invariant, discrete-time, linear, finite-dimensional systems which can be represented by autoregressive (AR) equations over the time-axis \(\mathbb Z\). For these systems, continuity properties of realizations with respect to the parameters of the system are studied. The question investigated is: ``Under which conditions does a convergent sequence of AR-systems given by matrices \(R^{l} \in \mathbb R [s]^{p \times q}, l \in {\mathbb {\overline N}} := {\mathbb N} \cup \{ \infty \}\) have a converging sequence of input-state-output (i/s/o) representations \((A^{l}, B^{l}, C^{l}, D^{l})\)?'' It is shown that, with respect to the convergence of all coefficients in an AR representation, there exist continuously dependent (i/s/o) representations provided a specified condition on the degree of some matrix \(Q\) is satisfied. This continuous (i/s/o) representation is given by the Fuhrmann realization. These results are shown to generalize various known results. The paper is concluded with an example.