First-order representations of delay-differential systems in a behavioural setting
Consider the behaviour \(B=\ker R\) where the matrix \(R\) of delay-differential operators belongs to \(\mathbb{R} [d/dt,\sigma]^{p \times k}\) and \(\sigma\) denotes the forward shift. The purpose of this paper is to find matrices, \(A\), \(B\), \(C\) and \(D\) of appropriate sizes and a permutation matrix \(\rho\) such that \[ \ker R=B= \left\{\rho {u \choose y} \bigl|\text{ there exists a trajectory } x: \begin{matrix} \dot x & = Ax & + Bu \\ y & = Cx & + Du \end{matrix} \right\}. \] Commensurate point-delays are allowed so that the system is over the ring \(\mathbb{R} [\sigma]\). The matrices \(A\), \(B\), \(C\) and \(D\) have delay operators as entries. Such a system \(\dot x=Ax +Bu\), \(y= Cx+Du\) is called a realization of the behaviour. Necessary and sufficient conditions for the realizability as first-order systems of behaviours are derived. A realization procedure is performed.
scientific article; zbMATH DE number 1063884
scientific article