# A Distributed Supervisor Architecture for a General Wafer Production System

^{1}

^{2}

^{*}

*Sensors*in 2023)

## Abstract

**:**

## 1. Introduction

## 2. Modelling of the Manufacturing Process

#### 2.1. Parametric Notation and Configuration of the Process

- (i)
- ${R}_{1}$ to transfer products from ${L}_{in}$ to ${C}_{1,1}$, from ${C}_{1,1}$ and ${C}_{1,2}$ to ${B}_{1}$, and from ${C}_{1,2}$ to ${L}_{out}$,
- (ii)
- ${R}_{i}$, where $i\in \{2,\dots ,n\}$, to transfer products from ${C}_{i,1}$ and ${C}_{i,2}$ to ${B}_{i}$ and ${B}_{i-1}$, and vice versa, i.e., from ${B}_{i}$ and ${B}_{i-1}$ to ${C}_{i,1}$ and ${C}_{i,2}$,
- (iii)
- ${R}_{n+1}$ to transfer products from ${C}_{n+1,1}$,…,${C}_{n+1,m}$ to ${B}_{n}$ and vice versa.

#### 2.2. The Models of the Production Stations

#### 2.3. The Models of the Buffers

#### 2.4. The Models of the Loading Docks

#### 2.5. The Models of the Robotic Manipulators

## 3. Desired Behavior

#### 3.1. The Rules of the Desired Behavior

- Only if a station (or a buffer) is empty will the respective serving robotic manipulator drop a product to the station (or the buffer).
- Only if a station has a manufactured product (or a buffer has a product) will the respective serving robotic manipulator pick a product from the station (or the buffer).
- The desired production sequence, namely the sequence of the placements of a wafer, is presented in the transportation diagram in Figure 9.
- Once a wafer is picked up by ${R}_{i}$, where $i\in \{1,\dots ,n\}$, if ${C}_{i+1,1}$ is full, then the transportation of the wafer to ${B}_{i}$ may cause blocking. Hence, a reasonable requirement to avoid blocking is to guarantee an empty slot in ${C}_{i+1,1}$ before ${R}_{i}$ initiates the wafer transportation to ${B}_{i}$.
- Once a wafer is picked up by ${R}_{i}$, where $i\in \{2,\dots ,n+1\}$, if ${C}_{i-1,2}$ is full, then the transportation of the wafer to ${B}_{i-1}$ may cause blocking. Hence, a reasonable requirement to avoid blocking is to guarantee an empty slot in ${C}_{i-1,2}$ before ${R}_{i}$ initiates the wafer transportation to ${B}_{i-1}$.

#### 3.2. The Desired Behavior in the Form of Regular Languages

**Proposition**

**1.**

**Proof of Proposition**

**1.**

## 4. Distributed Supervisory Control Architecture

- (i)
- The events of ${R}_{i}$;
- (ii)
- the single uncontrollable event of ${C}_{i,j}$, namely the event ${}^{i,j}e{}_{u}$, where $j\in \{1,2\}$;
- (iii)
- The events ${}^{i+1,i}e{}_{BP}$, ${}^{i+1,i}e{}_{BD}$, ${}^{i+1,1}e{}_{P}$, and ${}^{i+1,2}e{}_{P}$, of ${R}_{i+1}$;
- (iv)
- the events ${}^{i-1,i-1}e{}_{BP}$, ${}^{i-1,i-1}e{}_{BD}$, ${}^{i-1,1}e{}_{P}$, and ${}^{i-1,2}e{}_{P}$ of ${R}_{i-1}$, where $i\in \{2,\dots ,n\}$.

- (i)
- The events of ${R}_{n+1}$;
- (ii)
- The uncontrollable events of ${C}_{n+1,j}$, namely the events ${}^{n+1,j}e{}_{u}$, where $j\in \{1,\dots ,m\}$;
- (iii)
- The events ${}^{n,n}e{}_{BP}$, ${}^{n,n}e{}_{BD}$, ${}^{n,1}e{}_{P}$, and ${}^{n,2}e{}_{P}$ of ${R}_{n}$.

## 5. Supervisor Realization

#### 5.1. Supervisors of the Production Stations

#### 5.2. Supervisors of the Buffers

#### 5.3. Supervisors of the Robotic Manipulators

## 6. Establishment of the Satisfactory Performance of the Controlled Automaton

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

**Lemma**

**1.**

**Proof of Lemma**

**1.**

**Corollary**

**1.**

**Lemma**

**2.**

**Proof of Lemma**

**2.**

**Theorem**

**3.**

**Remark**

**2.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof of Theorem**

**5.**