Formal Methods in Manufacturing

2.4 DIOID MODEL OF TIMED EVENT GRAPHS

Recall our example of a manufacturing system introduced in Section 2.1. The corresponding TEG modelling of this system is given in Figure 2.4, and the recursive equations for the earliest possible firing instants have been determined in (2.9) through (2.14). Rewriting these equations in max-plus algebra, that is, in the dioid $({\overline{\mathbb{Z}}}_{\max },\oplus ,\otimes )$ , we get the following:

$\begin{array}{l}{x}_1(k)=u_1(k)\oplus x_2(k-2),\hfill \\ x_2(k)=10x_1(k),\hfill \\ x_3(k)=u_2(k)\oplus x_4(k-1),\hfill \\ x_4(k)=4x_3(k),\hfill \\ x_5(k)=u_3(k)\oplus 1x_2(k)\oplus 1x_4(k)\oplus x_6(k-1),\hfill \\ x_6(k)=3x_5(k),\hfill \end{array}$

and the output $y(k)\in {\overline{\mathbb{Z}}}_{\max }$ is

$y(k)=x_6(k).$

This can be rewritten in matrix–vector form with x(k) = [x₁(k)x₂(k)x₃(k)x₄(k)x₅(k)x₆(k)]^T and u(k) = [u₁(k) u₂(k) u₃(k)]^T:

$x\left(k\right)=A_{0}x(k)\oplus A_{1}x(k-1)\oplus A_{2}x(k-2)\oplus Bu\text{(}k\text{),}$

(2.40)

$y(k)=Cx(k)$