# Enhancing Strong Neighbor-Based Optimization for Distributed Model Predictive Control Systems

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**Definition**

**1.**

- to achieve a good optimization performance of the entire closed-loop system.
- to guarantee the feasibility of target tracking.
- to simplify the information connectivity among controllers to guarantee good structural flexibility and error-tolerance of the distributed control framework.

## 3. DMPC Design

#### 3.1. Strong-Coupling Neighbor-Based Optimization for Tracking

**Definition**

**2.**

**Definition**

**3.**

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

Algorithm 1 Enhancing Strong Neighbor-Based Optimization DMPC |

#### 3.2. Warm Start

Algorithm 2 Warm Start for Iterative Algorithm |

#### 3.3. RPI Control Law and RPI Set

#### 3.4. Determination of Strong Coupling

## 4. Stability and Convergence

**Theorem**

**1.**

**Proof.**

## 5. Simulation

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

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**Figure 1.**An illustration of the structure of a distributed system and its distributed control framework. MPC: model predictive control; DMPC: distributed MPC.

**Figure 2.**States of each subsystem under the control of strong-coupling neighbor-based DMPC (SCN-DMPC) and cooperative DMPC.

Item | ${\mathcal{S}}_{1}$ | ${\mathcal{S}}_{2}$ | ${\mathcal{S}}_{3}$ | ${\mathcal{S}}_{4}$ | ${\mathcal{S}}_{5}$ |
---|---|---|---|---|---|

MSE | 0.5771 | 1.1512 | 0.7111 | 0.1375 | 0.9162 |

System | SCN-DMPC | Cooperative DMPC |
---|---|---|

${\mathcal{S}}_{1}$ | 1 | 2 |

${\mathcal{S}}_{2}$ | 2 | 3 |

${\mathcal{S}}_{3}$ | 2 | 4 |

${\mathcal{S}}_{4}$ | 2 | 3 |

${\mathcal{S}}_{5}$ | 1 | 1 |

$\mathcal{S}$ | 8 | 13 |

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**MDPI and ACS Style**

Gao, S.; Zheng, Y.; Li, S.
Enhancing Strong Neighbor-Based Optimization for Distributed Model Predictive Control Systems. *Mathematics* **2018**, *6*, 86.
https://doi.org/10.3390/math6050086

**AMA Style**

Gao S, Zheng Y, Li S.
Enhancing Strong Neighbor-Based Optimization for Distributed Model Predictive Control Systems. *Mathematics*. 2018; 6(5):86.
https://doi.org/10.3390/math6050086

**Chicago/Turabian Style**

Gao, Shan, Yi Zheng, and Shaoyuan Li.
2018. "Enhancing Strong Neighbor-Based Optimization for Distributed Model Predictive Control Systems" *Mathematics* 6, no. 5: 86.
https://doi.org/10.3390/math6050086