# Distributed Predefined-Time Optimization for Second-Order Systems under Detail-Balanced Graphs

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## Abstract

**:**

## 1. Introduction

- The algorithm can be applied to undirected graphs and detail-balanced graphs. None of the discussed papers considers this extension.
- The initial optimization step of the algorithm requires fewer adjustable parameters than many other algorithms found in the literature.
- Compared to many existing works, the gradients and Hessians are not shared among agents.
- Contrary to [25], the proposed algorithm is robust in the presence of matched disturbances and does not use a TBG.

## 2. Preliminaries

#### 2.1. Notation

#### 2.2. Graph Theory

#### 2.3. Convex Analysis

#### 2.4. Predefined-Time Stability

**Definition**

**1**

- Lyapunov is stable if for any ${x}_{0}\in {\mathbb{R}}^{m}$, the solution $\Phi (t,{x}_{0})$ is defined for all $t\ge 0$, and for any $\u03f5>0$, there is $\delta >0$ such that for any ${x}_{0}\in {\mathbb{R}}^{m}$, if ${x}_{0}\in {B}_{\delta}\left(0\right)$ then $\Phi (t,{x}_{0})\in {B}_{\u03f5}\left(0\right)$ for all $t\ge 0$;
- It is finite-time stable if it is Lyapunov stable and for any ${x}_{0}\in {\mathbb{R}}^{m}$, there exists $0\le \tau <\infty $ such that $\Phi (t,{x}_{0})=0$ for all $t\ge \tau $. The function $T\left({x}_{0}\right)=inf\left\{\tau \ge 0:\Phi (t,{x}_{0})=0,\phantom{\rule{0.166667em}{0ex}}\forall t\ge \tau \right\}$ is said the settling-time function of system (2);
- It is fixed-time stable if it is finite-time stable, and the settling-time function of system (2), $T\left({x}_{0}\right)$, is bounded on ${\mathbb{R}}^{m}$, i.e., there exists ${T}_{max}$ such that ${sup}_{{x}_{0}\in {\mathbb{R}}^{m}}T\left({x}_{0}\right)\le {T}_{max}$;
- It is predefined-time stable if it is fixed-time stable and for any ${T}_{c}\in {\mathbb{R}}_{+}$ there exists some $\rho \in {\mathbb{R}}^{b}$ such that the settling-time function of system (2) satisfies$$\underset{{x}_{0}\in {\mathbb{R}}^{m}}{sup}T\left({x}_{0}\right)\le {T}_{c}.$$

**Proposition**

**1**

**Proposition**

**2**

## 3. Problem Statement

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Remark**

**1.**

## 4. Main Results

#### 4.1. Distributed Predefined-Time Optimal Signal Generator (DPTOSG)

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 4.2. Predefined-Time Reference Tracking—PTRT

## 5. Numerical Example

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DPTOSG | Distributed Predefined-Time Optimal Signal Generator |

MAS | Multi-Agent System |

PTRT | Predefined-Time Reference Tracking |

TBG | Time-Base Generator |

UBST | Upper Bound of the Settling Time |

ZGS | Zero Gradient Sum |

## Appendix A. Useful lemmas

**Lemma**

**A1**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

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**Figure 3.**Response curves for the outputs of distributed optimal signal generator. (

**a**) ${z}_{i1}$ (

**b**) ${z}_{i2}$.

**Figure 4.**Evolution of the position of the agents in the presence of matched disturbances with respect to time. (

**a**) ${x}_{i1}$ (

**b**) ${x}_{i2}$.

**Figure 6.**System evolution with respect to time in the presence of matched disturbances. (

**a**) Norm of the tracking error in position. (

**b**) Norm of the tracking error in velocity.

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**MDPI and ACS Style**

De Villeros, P.; Sánchez-Torres, J.D.; Muñoz-Vázquez, A.J.; Defoort, M.; Fernández-Anaya, G.; Loukianov, A.
Distributed Predefined-Time Optimization for Second-Order Systems under Detail-Balanced Graphs. *Machines* **2023**, *11*, 299.
https://doi.org/10.3390/machines11020299

**AMA Style**

De Villeros P, Sánchez-Torres JD, Muñoz-Vázquez AJ, Defoort M, Fernández-Anaya G, Loukianov A.
Distributed Predefined-Time Optimization for Second-Order Systems under Detail-Balanced Graphs. *Machines*. 2023; 11(2):299.
https://doi.org/10.3390/machines11020299

**Chicago/Turabian Style**

De Villeros, Pablo, Juan Diego Sánchez-Torres, Aldo Jonathan Muñoz-Vázquez, Michael Defoort, Guillermo Fernández-Anaya, and Alexander Loukianov.
2023. "Distributed Predefined-Time Optimization for Second-Order Systems under Detail-Balanced Graphs" *Machines* 11, no. 2: 299.
https://doi.org/10.3390/machines11020299