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Two-dimensional pheromone propagation controller applied to run-to-run control for semiconductor manufacturing

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Abstract

This paper presents a new perspective on process control, called the two-dimensional pheromone propagation controller (2D-PPC), which considers the spatial information about disturbances of the process within a wafer to generate new predicted intercepts of the models for the subsequent use in time–effect controller (the exponentially weighted moving average, EWMA, in this study). The 2D-PPC assumes that the disturbances have their own behavior and affect others nearby in a wafer at a run; thus, it involves the “space-effect” among disturbances of the process at measurement positions within a wafer. The framework of the space–time controller (STC), which interlaces the time–effect controller and the space–effect 2D-PPC is constructed, and the stability analysis and intrinsic characteristics of the STC are discussed. Simulations are conducted using two-dimensional anthropogenic disturbances generated from fabrication data. The results show that the STC has better performance as compared to the conventional time–effect controllers. From implementation view point, since STC does not change the original code of time–effect controller, it can be easily implemented in the current process control loop by only adding an additional space-effect controller.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu City, Taiwan, People’s Republic of China

    Der-Shui Lee & An-Chen Lee

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  1. Der-Shui Lee
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  2. An-Chen Lee
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Correspondence to An-Chen Lee.

Appendices

Appendix A

To overcome the end effect of the transition function [22], we modify the propagation-out ratio at the frontier points from \( {{{{F_2}}} \left/ {{\left( {2 - {F_2}} \right)}} \right.} \) [22] to Γ. Without loss of generality, this study takes the shape of the two-dimensional pheromone basket is shown as Fig. 2 and R 2 (k, 0) is the 12 × 1 matrix of 1s as an example. The transition functions are

$$ {q_2}\left( {k,{b_{{{m_2}}}},i + 1} \right) = \left\{ {\matrix{ {\frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{{m_2} + 6}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} + 6}}},i} \right)} \right),} \hfill &{{\text{if}}\,{m_2} = 1,2, \ldots, 6} \hfill \\ {\Gamma \left( {{r_2}\left( {k,{b_{{{m_2} - 6}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} - 6}}},i} \right)} \right) + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{{m_2} + 1}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} + 1}}},i} \right)} \right)} \hfill &{} \hfill \\ { + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{{m_2} - 1}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} - 1}}},i} \right)} \right),} \hfill &{{\text{if}}\,{m_2} = 8,9, \ldots, 11} \hfill \\ {\Gamma \left( {{r_2}\left( {k,{b_{{{m_2} - 6}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} - 6}}},i} \right)} \right) + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{{m_2} + 1}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} + 1}}},i} \right)} \right)} \hfill &{} \hfill \\ { + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{12}}},i} \right) + {q_2}\left( {k,{b_{{12}}},i} \right)} \right),} \hfill &{{\text{if}}\,{m_2} = 7} \hfill \\ {\Gamma \left( {{r_2}\left( {k,{b_{{{m_2} - 6}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} - 6}}},i} \right)} \right) + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_7},i} \right) + {q_2}\left( {k,{b_7},i} \right)} \right)} \hfill &{} \hfill \\ { + \frac{{{F_2}}}{3}\left( {{r_2}\left( {k,{b_{{{m_2} - 1}}},i} \right) + {q_2}\left( {k,{b_{{{m_2} - 1}}},i} \right)} \right),} \hfill &{{\text{if}}\,{m_2} = 12} \hfill \\ }<!end array> } \right. $$
(A.1)
$$ {s_2}\left( {k,{b_{{{m_2}}}},i + 1} \right) = \left\{ {\matrix{ {{s_2}\left( {k,{b_{{{m_2}}}},i} \right) + \left( {1 - \Gamma } \right)\left( {{r_2}\left( {k,{b_{{{m_2}}}},i} \right) + {q_2}\left( {k,{b_{{{m_2}}}},i} \right)} \right),} \hfill &{{\text{if }}{m_2} = 1,{2,}...{6}} \hfill \\ {{s_2}\left( {k,{b_{{{m_2}}}},i} \right) + \left( {1 - {F_2}} \right)\left( {{r_2}\left( {k,{b_{{{m_2}}}},i} \right) + {q_2}\left( {k,{b_{{{m_2}}}},i} \right)} \right),} \hfill &{{\text{if }}{m_2} = 7,8,...,12} \hfill \\ }<!end array> } \right. $$
(A.2)

Next, Eq. (A.2) can be rewrote as

$$ \left[ {\matrix{ {{{{\bf Q}}_{{{\bf 2}}}}\left( {k,i + {1}} \right)} \\ {{{{\bf S}}_{{{\bf 2}}}}\left( {k,i + {1}} \right)} \\ }<!end array> } \right] = \left[ {\matrix{ {{{{\bf T}}_{{{{\bf 11}}}}}} &{{{{\bf T}}_{{{{\bf 12}}}}}} \\ {{{{\bf T}}_{{{{\bf 21}}}}}} &{{{{\bf T}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right]\left[ {\matrix{ {{{{\bf Q}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ {{{{\bf S}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ }<!end array> } \right] + \left[ {\matrix{ {{{{\bf U}}_{{{{\bf 11}}}}}} &{{{{\bf U}}_{{{{\bf 12}}}}}} \\ {{{{\bf U}}_{{{{\bf 21}}}}}} &{{{{\bf U}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right]\left[ {\matrix{ {{{{\bf R}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ {{{{\bf R}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ }<!end array> } \right], $$
(A.3)

where

$$ \begin{array}{*{20}{c}} {{{{\mathbf{Q}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{q}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{q}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{q}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right],} \hfill & {{{{\mathbf{S}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{s}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{s}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{s}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right],} \hfill & {{{{\mathbf{R}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{r}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{r}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{r}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right]} \hfill \\ \end{array} , $$
$$ {{{\mathbf{T}}}_{{11}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} \\ \Gamma & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} \\ 0 & \Gamma & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 \\ 0 & 0 & \Gamma & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 \\ 0 & 0 & 0 & \Gamma & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ 0 & 0 & 0 & 0 & \Gamma & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} \\ 0 & 0 & 0 & 0 & 0 & \Gamma & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ \end{array} } \right],\,{{{\mathbf{T}}}_{{12}}} = {{0}_{{12 \times 12}}}, $$
$$ \begin{array}{*{20}{c}} {{{{\mathbf{T}}}_{{21}}} = \left[ {\begin{array}{*{20}{c}} {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {1 - \Gamma } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & {} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} \\ \end{array} } \right],} \\ {{{{\mathbf{T}}}_{{22}}} = {{{\mathbf{I}}}_{{12 \times 12}}},\,{{{\mathbf{U}}}_{{11}}} = {{{\mathbf{T}}}_{{11}}},\,{{{\mathbf{U}}}_{{12}}} = {{0}_{{12 \times 12}}},\,{{{\mathbf{U}}}_{{21}}} = {{0}_{{12 \times 12}}},\,{\text{and}}\,{{{\mathbf{U}}}_{{22}}} = {{{\mathbf{T}}}_{{21}}}.} \\ \end{array} $$

The final propagation result of V 12×1 (k) obtained from Eq. (A.8) in the Appendix B must be the 12 × 1 matrix of ones.

$$ {{{\bf V}}_{{{{\bf 12}} \times {{\bf 1}}}}}(k) = \left[ {\matrix{ {{v_1}(k)} \\ {{v_2}(k)} \\ {{v_3}(k)} \\ {{v_4}(k)} \\ {{v_5}(k)} \\ {{v_6}(k)} \\ {{v_7}(k)} \\ {{v_8}(k)} \\ {{v_9}(k)} \\ {{v_{{10}}}(k)} \\ {{v_{{11}}}(k)} \\ {{v_{{12}}}(k)} \\ }<!end array> } \right] = \frac{1}{{2{F_2} + \Gamma {F_2} - 3}}\left[ {\matrix{ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {\left( {1 - \Gamma } \right)\left( {{F_2} - 3} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ {3\left( {1 + \Gamma } \right)\left( {{F_2} - 1} \right)} \\ }<!end array> } \right] = \left[ {\matrix{ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ }<!end array> } \right] $$
(A.4)

Solving Eq. (A.4) yields,

$$ \Gamma = \frac{{{F_2}}}{{3 - 2{F_2}}} $$
(A.5)

Relatively, when R 2 (k, 0) is the 12 × 1 matrix of ones, Eq. (A.8) becomes

$$ \mathop{{\lim }}\limits_{{{\text{z}} \to 1}} \left( {{\text{z}} - 1} \right){\text Z}\left\{ {{{{\bf H}}_{{{\bf 2}}}}\left( {k,i} \right)} \right\} = \mathop{{\lim }}\limits_{{{\text{z}} \to 1}} \left( {{\text{z}} - 1} \right){\left( {{\text{z}}{{\bf I}} - {{\bf T}}} \right)^{{ - 1}}}{{\bf U}}{{{\bf I}}_{{{{\bf 24}} \times {{\bf 1}}}}} = \left[ {\matrix{ {{{{\bf 0}}_{{{{\bf 12}} \times {{\bf 1}}}}}} \\ {{{{\bf 1}}_{{{{\bf 12}} \times {{\bf 1}}}}}} \\ }<!end array> } \right]. $$
(A.6)

Equation (A.6) shows that the final propagation result S 2 (k, ∞) is also a 12 × 1 matrix of ones. Thus, the modified transition functions not only obey the energy balance law but also avoid the end effect.

Appendix B

Because states Q 2 (k, i) and S 2 (k, i) are updated simultaneously, the transition functions can be rewritten in matrix form:

$$ \left[ {\matrix{ {{{{\bf Q}}_{{{\bf 2}}}}\left( {k,i + {1}} \right)} \\ {{{{\bf S}}_{{{\bf 2}}}}\left( {k,i + {1}} \right)} \\ }<!end array> } \right] = \left[ {\matrix{ {{{{\bf T}}_{{{{\bf 11}}}}}} &{{{{\bf T}}_{{{{\bf 12}}}}}} \\ {{{{\bf T}}_{{{{\bf 21}}}}}} &{{{{\bf T}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right]\left[ {\matrix{ {{{{\bf Q}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ {{{{\bf S}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ }<!end array> } \right] + \left[ {\matrix{ {{{{\bf U}}_{{{{\bf 11}}}}}} &{{{{\bf U}}_{{{{\bf 12}}}}}} \\ {{{{\bf U}}_{{{{\bf 21}}}}}} &{{{{\bf U}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right]\left[ {\matrix{ {{{{\bf R}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ {{{{\bf R}}_{{{\bf 2}}}}\left( {k{,}i} \right)} \\ }<!end array> } \right], $$
(A.7)

where

$$ \begin{array}{*{20}{c}} {{{{\mathbf{Q}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{q}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{q}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{q}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right],} \hfill & {{{{\mathbf{S}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{s}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{s}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{s}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right],} \hfill & {{{{\mathbf{R}}}_{2}}\left( {k,i} \right) = \left[ {\begin{array}{*{20}{c}} {{{r}_{2}}\left( {k,{{b}_{1}},i} \right)} \\ {{{r}_{2}}\left( {k,{{b}_{2}},i} \right)} \\ \vdots \\ {{{r}_{2}}\left( {k,{{b}_{{12}}},i} \right)} \\ \end{array} } \right]} \hfill \\ \end{array} , $$
$$ {{{\mathbf{T}}}_{{11}}} = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} \\ {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} \\ 0 & {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 \\ 0 & 0 & {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 \\ 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 & {\frac{{{{F}_{2}}}}{3}} \\ 0 & 0 & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{{3 - 2{{F}_{2}}}}} & {\frac{{{{F}_{2}}}}{3}} & 0 & 0 & 0 & {\frac{{{{F}_{2}}}}{3}} & 0 \\ \end{array} } \right],\,{{{\mathbf{T}}}_{{12}}} = {{0}_{{12 \times 12}}}, $$
$$ \begin{array}{*{20}{c}} {{{{\mathbf{T}}}_{{21}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{{3\left( {1 - {{F}_{2}}} \right)}}{{3 - 2{{F}_{2}}}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {1 - {{F}_{2}}} \\ \end{array} } \right],} \\ {{{{\mathbf{T}}}_{{22}}} = {{{\mathbf{I}}}_{{12 \times 12}}},\,{{{\mathbf{U}}}_{{11}}} = {{{\mathbf{T}}}_{{11}}},\,{{{\mathbf{U}}}_{{12}}} = {{0}_{{12 \times 12}}},\,{{{\mathbf{U}}}_{{21}}} = {{0}_{{12 \times 12}}}\,{\text{and}}\,{{{\mathbf{U}}}_{{22}}} = {{{\mathbf{T}}}_{{21}}}.} \\ \end{array} $$

In Eq. (A.7), Q 2 (k, ∞) and S 2 (k, ∞) can be obtained with the z-transform and the final value theorem

$$ \begin{array}{*{20}{c}} {{{{\mathbf{H}}}_{2}}\left( {k,\infty } \right) = \mathop{{\lim }}\limits_{{{\text{z}} \to 1}} \left( {{\text{z}} - 1} \right){\rm Z}\left\{ {{{{\mathbf{H}}}_{2}}\left( {k,i} \right)} \right\}} \\ { = \mathop{{\lim }}\limits_{{{\text{z}} \to 1}} \left( {{\text{z}} - 1} \right){{{\left( {{\text{z}}{\mathbf{I}} - {\mathbf{T}}} \right)}}^{{ - 1}}}{\mathbf{U}}\left[ {\begin{array}{*{20}{c}} {{{{\mathbf{R}}}_{2}}\left( {k,i} \right)} \\ {{{{\mathbf{R}}}_{2}}\left( {k,i} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {{{0}_{{12 \times 1}}}} \\ {{{{\mathbf{V}}}_{{12 \times 1}}}\left( k \right)} \\ \end{array} } \right]} \\ \end{array} . $$
(A.8)

where \( {{{\bf H}}_{{{\bf 2}}}}\left( {k,i} \right) = \left[ {\matrix{ {{{{\bf Q}}_{{{\bf 2}}}}\left( {k,i} \right)} \\ {{{{\bf S}}_{{{\bf 2}}}}\left( {k,i} \right)} \\ }<!end array> } \right] \), \( {{\bf T}} = \left[ {\matrix{ {{{{\bf T}}_{{{{\bf 11}}}}}} &{{{{\bf T}}_{{{{\bf 12}}}}}} \\ {{{{\bf T}}_{{{{\bf 21}}}}}} &{{{{\bf T}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right] \), \( {{\bf U}} = \left[ {\matrix{ {{{{\bf U}}_{{{{\bf 11}}}}}} &{{{{\bf U}}_{{{{\bf 12}}}}}} \\ {{{{\bf U}}_{{{{\bf 21}}}}}} &{{{{\bf U}}_{{{{\bf 22}}}}}} \\ }<!end array> } \right] \) and \( {{{\bf V}}_{{{{\bf 12}} \times {{\bf 1}}}}}(k) = {\left[ {\matrix{ {{v_1}(k)} & \cdots &{{v_{{{12}}}}(k)} \\ }<!end array> } \right]^{\text{T}}} \). Note Q 2 (k, ∞) in Eq. (A.8) will converge to the matrix of 0 and S 2 (k, ∞) will converge to V 12 × 1 (k). In addition, the z-transform of the external input Z{R 2 (k, i)} is equal to R 2 (k, 0) since R 2(k, i) is an impulse at i = 0 by definition in Section 2.1. Thus, V 12×1 (k) is a function of F 2 and R 2 (k, 0) for a specific M 2. The final propagation results can be obtained analytically using Eq. (A.8).

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Lee, DS., Lee, AC. Two-dimensional pheromone propagation controller applied to run-to-run control for semiconductor manufacturing. Int J Adv Manuf Technol 66, 917–936 (2013). https://doi.org/10.1007/s00170-012-4377-8

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