Transfer learning based 3D fuzzy multivariable control for an RTP system

Abstract

Rapid thermal processing (RTP) is an important process in the fabrication of semiconductor devices. It is difficult to achieve temperature uniformity control of the wafer in RTP since the system is a highly nonlinear process with strong spatial distribution. In this study, a transfer learning-based three-dimensional (3D) fuzzy multivariable control scheme is proposed for the temperature uniformity control of an RTP system. In difference to the traditional expert-knowledge based design, a two-level framework of transfer learning methodology is constructed to design the 3D fuzzy multivariable controller (3D FMC) with the help of a multi-output support vector regression (M-SVR). The 3D FMC defines a qualitative spatial fuzzy structure that will be transferred to the M-SVR. On the other hand, the structure parameters of the M-SVR will be learned from data and transferred to design quantitative parameters of the 3D FMC. Under the framework of transfer learning, the control laws (e.g. human control experience) hidden in spatio-temporal data can be extracted and formulated back into multi-output 3D fuzzy rules. The proposed method provides an effective integration of the spatial fuzzy inference and the transfer learning for 3D FLC design. The newly developed method is applied to the temperature uniformity control of a rapid thermal chemical vapor deposition (RTCVD) system at the set temperature 1000K, and the maximum non-uniformity along the wafer radius is close to 1K.

Appendices

Appendix 1

1.1 3D fuzzy multivariable controller: operations and mathematical derivation

A 3D FMC consists of three parts: 3D fuzzifier, multiple-output 3D fuzzy rule inference, and defuzzifier. Different to 3D FLC in [13], the 3D fuzzy rules in 3D FMC are designed with multiple outputs in the consequent parts, and then 3D fuzzy rule inference has to be carried out for each output. Therefore, defuzzifier will be executed on each output.

• (1) 3D fuzzifier

The 3D fuzzifier transforms the crisp spatial input $\boldsymbol {x}(\bar {Z})$ into a 3D fuzzy input $\bar A_{X}$ as in (19). It is an extension of traditional fuzzifier with a space dimension and has two types: singleton type and non-singleton type.

$$ \begin{array}{l} {{\bar A}_{X}} = \sum\limits_{z \in \bar Z} {\sum\limits_{{{x}_{1}}(z) \in {X_{1}}} {...\sum\limits_{{{x}_{s}}(z) \in {X_{s}}} {{\mu_{{{\bar A}_{X}}}}({{x}_{1}}(z),...,} } } {\text{ }}{{x}_{s}}(z),z)/({{x}_{1}}(z),...,{{x}_{s}}(z),z)\\ {\text{ }} = \sum\limits_{z \in \bar Z} {\sum\limits_{{{x}_{1}}(z) \in {X_{1}}} {...\sum\limits_{{{x}_{s}}(z) \in {X_{s}}} {{\mu_{{X_{1}}}}({{x}_{1}}(z),z) * ...} } } * {\mu_{{X_{s}}}}({{x}_{s}}(z),z)/({{x}_{1}}(z),...,{{x}_{s}}(z),z) \end{array} $$

(19)

where ∗ denotes the t-norm operation.

In this study, singleton fuzzifier is used for brevity.

• (2) Multi-output 3D fuzzy rule inference

The multi-output 3D fuzzy rule inference consists of three operations: spatial information fusion, dimension reduction, and traditional inference [13]. It can cope with space information with two functions. The first one is capturing spatial information and the second one is inferring in a traditional way.

Assumed that multi-output 3D fuzzy rules are designed as in (20).

$$ \begin{array}{l} \bar R_{l}^{~}:{\text{ if }}{x}_{1}^{~}(\bar Z){\text{ is }}\bar C_{1l}^{~}{\text{ and }} {\cdots} {\text{ }}x_{m}^{~}(\bar Z){\text{ is }}\bar C_{ml}^{~}{\text{ }} {\cdots} {\text{ and }}x_{s}^{~}(\bar Z){\text{ is }}\bar C_{sl}^{~}\\ {\text{ then }}{u^{1}}{\text{ is }}B_{1l}^{~}{\text{, }} {\cdots} {\text{ }},{u^{i}}{\text{ is }}B_{il}^{~}, {\cdots} {\text{ }},{u^{L}}{\text{ is }}B_{Ll}^{~}{\text{ }} \end{array} $$

(20)

Then, for each fired rule, L fuzzy relations are formulated as below:

$${\bar R_{l}}:\left\{ {\begin{array}{*{20}{c}} {\bar C_{1l}^{~} \times {\cdots} \times \bar C_{sl}^{~} \to {B_{1l}}}\\ {\vdots} \\ {\bar C_{1l}^{~} \times {\cdots} \times \bar C_{sl}^{~} \to {B_{Ll}}} \end{array}} \right. $$

The 3D fuzzy input ${\bar A}_{X}$ goes through the multi-output 3D fuzzy inference engine. In (20), it is assumed that Gaussian type 3D membership function (MF) is chosen to describe a 3D fuzzy set. Then, 3D MF of ${{x}_{i}}(\bar Z)$ is expressed as

$$\mu_{Gi}^{l} = \exp \left( { - {{\left( {{{({{x}_{i}}(\bar Z) - {c_{i}^{l}}(\bar Z))} {\left/{\vphantom {{({\boldsymbol{x}_{i}}(\bar Z) - {c_{i}^{l}}(\bar Z))} {{\sigma_{i}^{l}}(\bar Z)}}} \right.} {{\sigma_{i}^{l}}(\bar Z)}}} \right)}^{2}}} \right)$$

where ${c_{i}^{l}}(\bar Z)=({c_{i}^{l}}({z_{1}}), \cdots , {c_{i}^{l}}({z_{p}}))'$ is the center of the i th Gaussian type 3-D MF $\bar C_{il}$ in the l th rule, ${\sigma _{i}^{l}}(\bar Z)=({\sigma _{i}^{l}}({z_{1}}), \cdots , {\sigma _{i}^{l}}({z_{p}}))'$ is the width of the i th Gaussian type 3-D MF $\bar C_{il}$ in the l th rule. And 2D MF of ${{x}_{i}}(\bar Z)$ at z = z_i is given as

$$\mu_{Gij}^{l} = \exp \left( { - {{\left( {{{({x_{i}}({z_{j}}) - c_{ij}^{l})} {\left/{\vphantom {{({x_{i}}({z_{j}}) - c_{ij}^{l})} {\sigma_{ij}^{l}}}} \right.} {\sigma_{ij}^{l}}}} \right)}^{2}}} \right)$$

where $c_{ij}^{l}={c_{i}^{l}}({z_{j}})$ and $\sigma _{ij}^{l}={\sigma _{i}^{l}}({z_{j}})$.

Firstly, as for each fuzzy relation, spatial information fusion operation is executed. A spatially distributed set W^l is formulated and its grade of the MF is given as

$$ \begin{array}{l} {\mu_{{W^{l}}}}(z) = {\mu_{{{\bar A}_{X}} \circ (\bar C_{1l}^{~} \times {\cdots} \times \bar C_{sl}^{~})}}(x(\bar Z),z)\\ = {\sup_{{x_{1}}(z) \in {X_{1}},...,{x_{s}}(z) \in {X_{s}}}}[{\mu_{{{\bar A}_{X}}}}(x(\bar Z),z) * {\mu_{\bar C_{1l}^{~} \times {\cdots} \times \bar C_{sl}^{~}}}(x(\bar Z),z)]\\ = \left\{ {{{\sup }_{{x_{1}}(z) \in {X_{1}}}}[{\mu_{{X_{1}}}}({x_{1}}(z),z) * {\mu_{\bar C_{1l}^{~}}}({x_{1}}(z),z)]} \right\} * {\cdots} * \\\quad\left\{ {{{\sup }_{{x_{s}}(z) \in {X_{s}}}}[{\mu_{{X_{s}}}}({x_{s}}(z),z) * {\mu_{\bar C_{sl}^{~}}}({x_{s}}(z),z)]} \right\}\\ = \prod\limits_{i = 1}^{s} {\exp \left( { - {{\left( {{{({x_{i}}(z) - {c_{i}^{l}}(z))} {\left/{\vphantom {{({x_{i}}(z) - {c_{i}^{l}}(z))} {{\sigma_{i}^{l}}(z)}}} \right.} {{\sigma_{i}^{l}}(z)}}} \right)}^{2}}} \right)} \quad \quad \quad \quad \quad z \in \bar Z \end{array} $$

(21)

where ∗ denotes the t-norm operation.

Then, dimension reduction operation is carried out. For instance, if a weighted aggregation dimension reduction [14] is chosen, we will have a 2D set χ^l

$$ \begin{array}{l} {\mu_{{\chi^{l}}}} = {\kappa_{1}}{\mu_{{W^{l}}}}({z_{1}}) + {\kappa_{2}}{\mu_{{W^{l}}}}({z_{2}}) + {\cdots} + {\kappa_{p}}{\mu_{{W^{l}}}}({z_{p}})\\ = \sum\limits_{j = 1}^{p} {{\kappa_{j}}\prod\limits_{i = 1}^{s} {\exp \left( { - {{\left( {{{({x_{i}}({z_{j}}) - c_{ij}^{l})} {\left/ {\vphantom {{({x_{i}}({z_{j}}) - c_{ij}^{l})} {\sigma_{ij}^{l}}}} \right.} {\sigma_{ij}^{l}}}} \right)}^{2}}} \right)} } \end{array} $$

(22)

In the next place, traditional inference operation is carried out. To begin with, implication operation is executed. For brevity, Mamdani implication is used. As for each fuzzy relation of $\bar R_{l}$, a fuzzy output set is derived as follows.

$$ {\mu_{{V^{il}}}}({u^{i}}) = {\mu_{{\chi^{l}}}} * {\mu_{{B_{il}}}}({u^{i}}){\text{ }}\quad \quad \quad {u^{i}} \in U, i = 1,2, {\cdots} ,L $$

(23)

where ∗ is a t-norm; $\mu _{{B_{il}}}({u^{i}})$ and V^il are the membership grade of B_il and the output fuzzy set of $\bar R_{l}$ for uⁱ, respectively.

Finally, all the fired rules are combined by the inference engine. We have a composite output fuzzy set for each fuzzy relation of $\bar R_{l}$

$$ {V^{i}} = \cup_{l = 1}^{N}{V^{il}}{\text{ \quad (}}i = 1, \cdots, L) $$

(24)

where N is the number of fired rules.

• (3) Defuzzifier

Defuzzifier is used to transform an output fuzzy set into a crisp output. If a linear defuzzifier [36] is chosen and B_il is a singleton fuzzy set, we have L crisp outputs, whose nonlinear mathematical expressions are given as the following.

$$ \begin{array}{l} {u^{i}}(x(\bar Z)) = \sum\limits_{l = 1}^{N} {{\zeta_{l}^{i}}} \sum\limits_{j = 1}^{p} {\kappa_{j}^{~}\prod\nolimits_{m = 1}^{s} {{\mu_{\overline C_{ml}^{~}}}(x_{m}^{~}(z_{j}^{~}))} } \\ = \sum\limits_{l = 1}^{N} {{\zeta_{l}^{i}}} \sum\limits_{j = 1}^{p} {\kappa_{j}^{~}\prod\nolimits_{m = 1}^{s} {\exp \left( { - {{\left( {\frac{{x_{m}^{~}(z_{j}^{~}) - c_{mj}^{l}}}{{\sigma_{mj}^{l}}}} \right)}^{2}}} \right)} } {\text{ \quad (}}i = 1,2, {\cdots} ,L) \end{array} $$

(25)

where ${\zeta _{l}^{i}}$ is the nonzero value in the singleton fuzzy set B_il of the output variable uⁱ in the l th rule.

Appendix 2

2.1 Multi-output SVR

In this study, we introduce an M-SVR with ε-insensitive cost function to settle the problem of regression with multiple variables, which is based on a previous contribution [37]. For simplicity, M-SVR with ε-insensitive cost function is abbreviated as M-SVR.

Let D = {[x_i,y_i] ∈ R^s × R^m,i = 1,⋯,r} be a training set with r pairs (x₁,y₁), (x₂,y₂), ⋯, (x_r,y_r), where ${{\text {x}}_{i}}{\text { = [}}{x_{i1}}{\text {,}} \cdots , {x_{is}}{]^{\prime }}$ is of s-dimension, ${{\text {y}}_{i}} = [{y_{1}^{i}}, \cdots , {y_{m}^{i}}]'$ is of m-dimension, and both of them are continuous. The goal is to find m functions f_j(x,w_j) (j = 1,⋯ ,m) so that all training patterns in D has a maximum deviation ε from the target values.

In terms of the structural risk minimization [38], by introducing the slack variables ξ_i and $\xi _{i}^{*}$, the multi-output regression problem can be transformed into a convex optimization problem in (26).

$$ \min\limits_{{w_{k}},{b_{k}},{\xi_{k}^{j}},\xi_{k}^{{j^{*}}}} \left\{ {\frac{1}{2}\sum\limits_{k = 1}^{m} {||{w_{k}}|{|^{2}}} + C\sum\limits_{k = 1}^{m} {\sum\limits_{j = 1}^{r} {({\xi_{k}^{j}} + \xi_{k}^{{j^{*}}})} } } \right\} $$

(26)

subject to

$$ \left\{ \begin{array}{l} \begin{array}{*{20}{c}} {{y_{k}^{j}} - < {w_{k}},{\text{x}} > - {b_{k}} \le {\varepsilon_{k}} + {\xi_{k}^{j}}}\\ { < {w_{k}},{\text{x}} > + {b_{k}} - {y_{k}^{j}} \le {\varepsilon_{k}} + \xi_{k}^{j*}}\\ {{\xi_{k}^{j}},\xi_{k}^{j*} \ge 0} \end{array}\\ k = 1, \cdots, m;j = 1, \cdots, r \end{array} \right. $$

(27)

where C is a constant that will be selected by the user.

By introducing the Lagrange multipliers ${\alpha _{k}^{j}}$, $\alpha _{k}^{j*}$, γ_k and applying the saddle point condition, a dual optimization problem of (26) is yielded as follows.

$$ \max\limits_{{\alpha_{k}^{j}},\alpha_{k}^{j*}} \left\{ \begin{array}{l} - \frac{1}{2}\sum\limits_{k = 1}^{m} {\sum\limits_{j,l = 1}^{r} {({\alpha_{k}^{j}} - \alpha_{k}^{j*})({\alpha_{k}^{l}} - \alpha_{k}^{l*}) < {{\text{x}}_{l}},{{\text{x}}_{j}} > } } \\ + \sum\limits_{k = 1}^{m} {\sum\limits_{j = 1}^{r} {({\alpha_{k}^{j}} - \alpha_{k}^{j*}){y_{k}^{j}}} } - \sum\limits_{k = 1}^{m} {\sum\limits_{j = 1}^{r} {({\alpha_{k}^{j}} + \alpha_{k}^{j*}){\varepsilon_{k}}} } \end{array} \right\} $$

(28)

subject to

$$ \begin{array}{l} \sum\limits_{j = 1}^{r} {({\alpha_{k}^{j}} - \alpha_{k}^{j*})} = 0\\ 0 \le {\alpha_{k}^{j}} \le C,{\text{ }}0 \le \alpha_{k}^{j*} \le C\\ k = 1,2,...,m \end{array} $$

(29)

Solving (28)-(29), we obtain the best regression hyper-surface f_k(x,w_k) (k = 1, 2,...,m ) with optimal weight vector w_k and optimal bias b_k as given in (30).

$$ \begin{array}{l} {f_{k}}({\text{x,}}{w_{k}}) = < {w_{k}},{\text{x}} > + {b_{k}}\\ = \sum\limits_{j = 1}^{r} {({\alpha_{k}^{j}} - \alpha_{k}^{j*}) < {\text{x}},{{\text{x}}_{j}} > } + {b_{k}}\\ = \sum\limits_{j \in SV}^{~} {({\alpha_{k}^{j}} - \alpha_{k}^{j*}) < {\text{x}},{{\text{x}}_{j}} > + {b_{k}}} \end{array} $$

(30)

where ${w_{k}} = \sum \limits _{j = 1}^{r} {({\alpha _{k}^{j}} - \alpha _{k}^{j*}){{\text {x}}_{j}}}$, ${b_{k}} = \frac {1}{r}\left ({\sum \limits _{j = 1}^{r} {({y_{k}^{j}} - \left \langle {{w_{k}},{{\text {x}}_{j}})} \right \rangle } } \right )$. The training pattern x_j with nonzero ${\alpha _{k}^{j}} - \alpha _{k}^{j*}$ is called Support Vector (SV).

Fig. 18

Multi-output three-layer network structure of an M-SVR

To make the M-SVR nonlinear, and avoid a direct mapping, the kernel trick is used. Then, the best regression hyper-surface f_k(x,w_k) (k = 1,⋯ ,m) can be written as

$$ {f_{k}}({\text{x,}}{w_{k}}) = {b_{k}} + \sum\limits_{j \in SV}^{~} {({\alpha_{k}^{j}} - \alpha_{k}^{j*})K({\text{x, }}{{\text{x}}_{j}})} $$

(31)

where K(x, x_j) is a Kernel Function (KF), which satisfy the Mercer’s theorem.

Let x₁, x₂, ⋯, x_N represent support vectors. The solution of the M-SVR is described by a multi-output three-layer network structure as shown in Fig. 18.