# Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Agent Consensus System Preliminaries

#### 2.1. Graph Theory

#### 2.2. Multi-Agent Forced Consensus Control

#### 2.3. Information Flow Topology

## 3. DC Motor Speed Consensus Control

#### 3.1. Three Agents Cyclic Topology

#### 3.2. Fixed Reference

**Remark**

**1.**

#### 3.3. Time Varying Reference

## 4. Robust Consensus Analysis

#### 4.1. Robust Consensus for the Multi-Agent System

#### 4.2. A Practical Example of Computing the Time Delay Margin in Communication between Agents

## 5. Experimental Results

^{®}. Before the tests, the WiFi communication was used to configure the controllers parameters, but, during the tests, it was only used to send the reference value from the computer to the leader controller. The gains ${k}_{p}=0.07$ and ${k}_{cm}=1$ were chosen to have an underdamped response. Note that while controllers 2 and 3 need one speed state information via Bluetooth, the controller 1, the leader, needs to have all speeds to calculate the center of mass. Approximate motor parameter values are shown in Table 1. It is important to note that, in this system, there may be several sources of uncertainty, namely motor parameters, measurement noise, and transmission delays. A schematic diagram of the multi-agent system is shown in Figure 2, and the experimental implementation is presented in Figure 3.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Information flow configuration: (

**a**) cyclic topology; (

**b**) chain topology. The gray circle represents the leader agent and white circles are followers.

${R}_{m}$ | 3.3 $\Omega $ | ${B}_{l}$ | $4.41\times {10}^{-6}$ Nm/(rad/s) |

${k}_{m}$ | $7.68\times {10}^{-3}$ V/(rad/s) | ${k}_{t}$ | $6.9\times {10}^{-3}$ Nm/A |

${k}_{g}$ | 53 | ${\eta}_{m}$ | 0.69 |

${\eta}_{g}$ | 0.90 | ${J}_{m}$ | $3.90\times {10}^{-7}$${\mathrm{kgm}}^{2}$ |

${J}_{l}$ | $1.03\times {10}^{-4}$${\mathrm{kgm}}^{2}$ | ${B}_{m}$ | $1.07\times {10}^{-9}$ Nm/(rad/s) |

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**MDPI and ACS Style**

Olivares, D.; Romero, G.; Guerrero, J.A.; Lozano, R.
Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization. *Appl. Sci.* **2020**, *10*, 6521.
https://doi.org/10.3390/app10186521

**AMA Style**

Olivares D, Romero G, Guerrero JA, Lozano R.
Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization. *Applied Sciences*. 2020; 10(18):6521.
https://doi.org/10.3390/app10186521

**Chicago/Turabian Style**

Olivares, Daniel, Gerardo Romero, Jose A. Guerrero, and Rogelio Lozano.
2020. "Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization" *Applied Sciences* 10, no. 18: 6521.
https://doi.org/10.3390/app10186521