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

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## 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.**

## References

- Shi, X.; Cao, J.; Wen, G.; Perc, M. Finite-time consensus of opinion dynamics and its applications to distributed optimization over digraph. IEEE Trans. Cybern.
**2018**, 49, 3767–3779. [Google Scholar] [CrossRef] [PubMed] - Dai, H.; Fang, X.; Jia, J. Consensus-based distributed fixed-time optimization for a class of resource allocation problems. J. Frankl. Inst.
**2022**, 359, 11135–11154. [Google Scholar] [CrossRef] - Mao, S.; Tang, Y.; Dong, Z.; Meng, K.; Dong, Z.Y.; Qian, F. A privacy preserving distributed optimization algorithm for economic dispatch over time-varying directed networks. IEEE Trans. Ind. Inform.
**2020**, 17, 1689–1701. [Google Scholar] [CrossRef] - Dougherty, S.; Guay, M. An extremum-seeking controller for distributed optimization over sensor networks. IEEE Trans. Autom. Control
**2016**, 62, 928–933. [Google Scholar] [CrossRef] - Nedic, A. Distributed gradient methods for convex machine learning problems in networks: Distributed optimization. IEEE Signal Process. Mag.
**2020**, 37, 92–101. [Google Scholar] [CrossRef] - Yang, T.; Yi, X.; Wu, J.; Yuan, Y.; Wu, D.; Meng, Z.; Hong, Y.; Wang, H.; Lin, Z.; Johansson, K.H. A survey of distributed optimization. Annu. Rev. Control
**2019**, 47, 278–305. [Google Scholar] [CrossRef] - Zak, M. Terminal attractors in neural networks. Neural Netw.
**1989**, 2, 259–274. [Google Scholar] [CrossRef] - Polyakov, A. Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems. IEEE Trans. Autom. Control
**2012**, 57, 2106–2110. [Google Scholar] [CrossRef][Green Version] - Sánchez-Torres, J.D.; Gómez-Gutiérrez, D.; López, E.; Loukianov, A.G. A class of predefined-time stable dynamical systems. IMA J. Math. Control. Inf.
**2018**, 35, i1–i29. [Google Scholar] [CrossRef][Green Version] - Jiménez-Rodríguez, E.; Muñoz-Vázquez, A.J.; Sánchez-Torres, J.D.; Defoort, M.; Loukianov, A.G. A Lyapunov-like characterization of predefined-time stability. IEEE Trans. Autom. Control
**2020**, 65, 4922–4927. [Google Scholar] [CrossRef][Green Version] - Aldana-López, R.; Gómez-Gutiérrez, D.; Jiménez-Rodríguez, E.; Sánchez-Torres, J.D.; Defoort, M. Generating new classes of fixed-time stable systems with predefined upper bound for the settling time. Int. J. Control
**2022**, 95, 2802–2814. [Google Scholar] [CrossRef] - Sánchez-Torres, J.D.; Defoort, M.; Muñoz-Vázquez, A.J. Predefined-time stabilisation of a class of nonholonomic systems. Int. J. Control
**2020**, 93, 2941–2948. [Google Scholar] [CrossRef] - Aldana-López, R.; Gómez-Gutiérrez, D.; Jiménez-Rodríguez, E.; Sánchez-Torres, J.D.; Loukianov, A.G. On predefined-time consensus protocols for dynamic networks. J. Frankl. Inst.
**2020**, 357, 11880–11899. [Google Scholar] [CrossRef] - Jiménez-Rodríguez, E.; Aldana-López, R.; Sánchez-Torres, J.D.; Gómez-Gutiérrez, D.; Loukianov, A.G. Consistent Discretization of a Class of Predefined-Time Stable Systems. In Proceedings of the 21st IFAC World Congress 2020—1st Virtual IFAC World Congress (IFAC-V 2020), Berlin, Germany, 11–17 July 2020. [Google Scholar]
- Ning, B.; Han, Q.L.; Zuo, Z. Distributed optimization of multiagent systems with preserved network connectivity. IEEE Trans. Cybern.
**2018**, 49, 3980–3990. [Google Scholar] [CrossRef] [PubMed] - Chen, G.; Li, Z. A fixed-time convergent algorithm for distributed convex optimization in multi-agent systems. Automatica
**2018**, 95, 539–543. [Google Scholar] [CrossRef] - Wang, X.; Wang, G.; Li, S. A distributed fixed-time optimization algorithm for multi-agent systems. Automatica
**2020**, 122, 109289. [Google Scholar] [CrossRef] - Gong, X.; Cui, Y.; Shen, J.; Xiong, J.; Huang, T. Distributed Optimization in Prescribed-Time: Theory and Experiment. IEEE Trans. Netw. Sci. Eng.
**2021**, 9, 564–576. [Google Scholar] [CrossRef] - Deng, Z.; Chen, T. Distributed algorithm design for constrained resource allocation problems with high-order multi-agent systems. Automatica
**2022**, 144, 110492. [Google Scholar] [CrossRef] - Ma, L.; Hu, C.; Yu, J.; Wang, L.; Jiang, H. Distributed Fixed/Preassigned-Time Optimization Based on Piecewise Power-Law Design. IEEE Trans. Cybern.
**2022**. [Google Scholar] [CrossRef] - Tang, Y. Distributed optimization for a class of high-order nonlinear multiagent systems with unknown dynamics. Int. J. Robust Nonlinear Control
**2018**, 28, 5545–5556. [Google Scholar] [CrossRef][Green Version] - Adibzadeh, A.; Suratgar, A.A.; Menhaj, M.B.; Zamani, M. Distributed optimization in heterogeneous dynamical networks. Iran. J. Sci. Technol. Trans. Electr. Eng.
**2020**, 44, 473–483. [Google Scholar] [CrossRef] - Wang, X.; Wang, G.; Li, S. Distributed finite-time optimization for integrator chain multiagent systems with disturbances. IEEE Trans. Autom. Control
**2020**, 65, 5296–5311. [Google Scholar] [CrossRef] - Tran, N.T.; Wang, Y.W.; Yang, W. Distributed optimization problem for double-integrator systems with the presence of the exogenous disturbance. Neurocomputing
**2018**, 272, 386–395. [Google Scholar] [CrossRef] - Li, S.; Nian, X.; Deng, Z.; Chen, Z. Predefined-time distributed optimization of general linear multi-agent systems. Inf. Sci.
**2022**, 584, 111–125. [Google Scholar] [CrossRef] - Aldana-López, R.; Seeber, R.; Haimovich, H.; Gómez-Gutiérrez, D. On inherent robustness and performance limitations of a class of prescribed-time algorithms. arXiv
**2022**, arXiv:2205.02528. [Google Scholar] - Yu, Z.; Yu, S.; Jiang, H.; Mei, X. Distributed fixed-time optimization for multi-agent systems over a directed network. Nonlinear Dyn.
**2021**, 103, 775–789. [Google Scholar] [CrossRef] - Boyd, S.; Boyd, S.P.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Hiriart-Urruty, J.B. Optimisation et Analyse Convexe: Exercices Corrigés; In the Series Enseignement SUP-Maths; EDP Sciences: Les Ulis, France, 2009. [Google Scholar]
- Lin, W.T.; Wang, Y.W.; Li, C.; Yu, X. Predefined-time optimization for distributed resource allocation. J. Frankl. Inst.
**2020**, 357, 11323–11348. [Google Scholar] [CrossRef] - Lu, J.; Tang, C.Y. Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case. IEEE Trans. Autom. Control
**2012**, 57, 2348–2354. [Google Scholar] [CrossRef][Green Version] - Ren, W.; Beard, R.W. Distributed Consensus in Multi-Vehicle Cooperative Control; Springer: London, UK, 2008. [Google Scholar]

**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.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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