Modeling and solution of two-dimensional input time-delay system
A 2D linear continuous-time input-delay system is considered, having the Roesser type representation \[ \begin{bmatrix}\frac{\partial x_{c}^{h}(x,t)}{\partial x}\\\frac{\partial x_{c}^{v}(x,t)}{\partial t}\end{bmatrix}= \begin{bmatrix} E_{11}&E_{12}\\E_{21}&E_{22}\end{bmatrix} \begin{bmatrix} x_{c}^{h}(x,t)\\x_{c}^{v}(x,t)\end{bmatrix}+ \begin{bmatrix} F_{1}\\F_{2}\end{bmatrix} f_{c}(x-\tau_{1},t-\tau_{2}) \] \[ y_{c}(x,t)=\begin{bmatrix} G_{1}&G_{2}\end{bmatrix} \begin{bmatrix} x_{c}^{h}(x,t)\\x_{c}^{v}(x,t)\end{bmatrix} +Hf_{c}(x-\tau_{1},t-\tau_{2}) \] where \(x_{c}^{h}(x,t)\in \mathbb{R}^{n_{1}}\) is the horizontal state vector, \(x_{c}^{v}(x,t)\in \mathbb{R}^{n_{2}}\) is the vertical state vector, \(f_{c}(x,t)\in \mathbb{R}^{m}\) is the input vector and \(y_{c}(x,t)\in \mathbb{R}^{l}\) is the output vector. For piecewise constant input vectors \(f_{c}(x,t)=f_{c}(k_{1}T_{1},k_{2}T_{2})\) for \((k_{1}T_{1}+\tau_{1},k_{2}T_{2}+\tau_{2})\leq(x,t)<(k_{1}T_{1}+T_{1}+\tau_{1},k_{2}T_{2}+T_{2}+\tau_{2}), k_{1},k_{2}\in\mathbb{N}, T_{1},T_{2},\tau_{1},\tau_{2}\in\mathbb{R}^{+}\), a sampling procedure is applied and a 2D discrete-time Roesser type input-delay system is obtained, having the state space representation \[ \begin{aligned} \begin{bmatrix} x^{h}(i+1,j)\\x^{v}(i,j+1)\end{bmatrix} & = \begin{bmatrix} A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix} \begin{bmatrix} x^{h}(i,j)\\x^{v}(i,j)\end{bmatrix}\\ &+ \begin{bmatrix} B_{11}\\B_{21}\end{bmatrix} f(i-1,j-1)+\begin{bmatrix} 0\\B_{22}\end{bmatrix} f(i-1,j) \\ &+\begin{bmatrix} B_{12}\\0\end{bmatrix} f(i,j-1)\end{aligned} \] \[ y(i,j) =\begin{bmatrix} C_{1}&C_{2}\end{bmatrix} \begin{bmatrix} x^{h}(i,j)\\x^{v}(i,j)\end{bmatrix} +Df(i-1,j-1). \] By using Roesser's state-transition matrix, the formula of the state of the discrete-time system is derived as well as its general response formula. The authors state that the presented discrete-time model significantly facilitates analysis and design of a system when it faces the unavoidable inherent time delay and the computation time delay which can be modeled as a part of delay at the system input.
scientific article; zbMATH DE number 1912418
scientific article