\(z\)-transform and Volterra-operator based approaches to controllability and observability analysis for discrete linear repetitive processes
Discrete linear repetitive processes are considered, having the state space representation \[ \begin{aligned} x(k+1,t+1) &= Ax(k+1,t)+Bu(k+1,t)+B_{0}y(k,t),\\ y(k+1,t) &= Cx(k+1,t)+Du(k+1,t)+D_{0}y(k,t), \end{aligned} \] \(k\in\mathbb{Z}^{+}\), \(t\in[0,\alpha]\subset\mathbb{Z}^{+}\); \(x(k,t)\in\mathbb{R}^{n}\) is the state vector, \(y(k,t)\in\mathbb{R}^{m}\) is the pass profile vector and \(u(k,t)\in\mathbb{R}^{l}\) is the control input vector. The initial (or boundary) conditions have the form \[ x(k,0)=\gamma(k),\;k\geq1;\quad y(0,t)=d_{t},\;0\leq t\leq\alpha \] where \(\gamma(k)\) and \(d_{t}\) are known constant \(n\)-vectors. The repetitive processes are connected with 2D linear systems, but the key difference is that in their case the duration of information propagation in one of the two directions, namely the along-the-pass direction, is finite (i.e. \(t\in[0,\alpha]\)). This difference imposes the necessity to develop a distinct system theory for repetitive processes. The aim of this paper is to introduce and to study different concepts of controllability and observability for discrete linear repetitive processes. A repetitive process is said to be pass controllable if there exists a control input sequence which drives the process to produce any pre-specified pass profile \(y_{k_{f}}=[y_{k_{f}}(0), y_{k_{f}}(1),\dots,y_{k_{f}}(\alpha)]\) on some pass index \(k_{f}\), for any initial conditions. If the pre-specified pass profile occurs on a pre-specified pass index \(k_{f}\) the process is said to be \textit{pass profile controllable}; this property has been studied in a previous paper written by three of the authors. A repetitive process is said to be stongly pass controllable if \(\exists\) a pass index \(k_{f}\) and a moment \(t_{f}\in[0,\alpha]\) such that for any bounded sequence of initial conditions there exists a control input sequence which drives the process to produce any pre-specified bounded state sequences \(x(k,t_{f}),\;k\geq k_{f}+1\) and pass profiles \(y(k_{f},t),\;t\in[t_{f},\alpha]\). Two concepts of pass observability and strong pass observability are also introduced. Necessary and sufficient conditions for pass controllability and observability are given in the form of rank tests on some polynomial matrices. They are obtained by using 1D z-transforms of the pass-to-pass and along-the-pass dynamics. The connection between pass controllability and pass profile controllability is emphasized. A Volterra operator associated with the process dynamics is employed to provide necessary and sufficient conditions for strong pass controllability and observability.
scientific article; zbMATH DE number 1960150