Example for determining the input variables based on the reachable abundance

Example 1 (Placing problem of the gas nozzles in the Ingot heating furnace). It is assumed that the ingot heating problem with two placing-nozzle schemes can be regarded as a distributing and heating problem in the 2-D plane, shown in Fig. 1. Which scheme is more suitable for the ingot heating problem with the greatest ability and efficiency of the fuel nozzles?

Fig. 1 The schemes placing the nozzles

By discretizing the 2-D plane as a $2\times6$ unit array, after considering the heat exchange and heat dissipate among the units, the system models for two schemes in the discrete-time case are modeled as follows

$x_{k+1}=Ax_{k}+B_{i}u_{k}^{(i)},\qquad u_{k}^{(i)}\in[0,1]^{r},i=1,2$

where

        $A=[a_{ij}],\qquad B_{1},B_{2}=[b_{ij}]$

        $a_{ii}=0.95-0.03g_{i}^{x}-0.04g_{i}^{y}-\sum_{\begin{array}{c} j=1\\ j\neq i \end{array}}^{n}\left(0.15g_{i,j}^{x}+0.18g_{i.j}^{y}\right)$

        $a_{ij}=0.15g_{i,j}^{x}+0.18g_{i.j}^{y},\qquad i\neq j$

        $b_{ij}=\left\{ \begin{array}{cc} f_{s}=10 & \mathrm{the\:unit\:is\:to\:nozzle\:\mathit{s}}\\ 0 & \mathrm{others} \end{array}\right.$

For elements $a_{ij}$ and $b_{ij}$ , if the $i$ -th unit is on the outside in X/Y direction or not, $g_{i}^{x}/g_{i}^{y}$ is 1 or 0, and if the $i$ -th and $j$ -th units are adjacent in X/Y direction or not, $g_{i,j}^{x}/g_{i.j}^{y}$ is 1 or 0. Considered the temperatures of the ingot is up to the expected temperatures and then is transferred to the rolling mill immediately, the heating problem can be regarded as a reaching control problem. The better of the two schemes can be determined on computing of the reachable abundances.

The computing results of the reachable abundances as Table 1 and then the scheme 1 is with the better reachable abundance, i.e., the better reach ability and efficiency.

Table 1. The reachable abundance of two schemes