A Prescribed-Time Consensus Algorithm for Distributed Time-Varying Optimization Based on Multiagent Systems

Abstract

This paper presents a distributed optimization algorithm for time-varying objective functions utilizing a prescribed-time convergent multi-agent system within undirected communication networks. Departing from conventional time-invariant optimization paradigms with static optimal solutions, our approach specifically addresses the challenge of tracking dynamic optimal trajectories in evolving environments. A novel continuous-time distributed optimization algorithm is developed based on prescribed-time consensus, guaranteeing the consensus attainment among agents within a user-defined timeframe while asymptotically converging to the time-dependent optimal solution. The proposed methodology enables explicit predetermination of convergence duration, representing a significant advancement over existing asymptotic convergence methods. Moreover, two simulation examples on the rendezvous problem and multi-robots control are presented to validate the theoretical results, exhibiting precise time-controlled convergence characteristics and effective tracking performance for time-varying optimization targets.

1. Introduction

Distributed optimization is widely encountered in various areas, such as optimal control [1], Nash equilibrium [2] and smart grids [3]. To deal with these optimization problems, a variety of distributed optimization methods grounded on multi-agent systems have been designed. For example, the classical optimization methods include the alternating direction multiplier method [4], push-sum method [5], Lagrangian multiplier method [6], and so on. In addition, there are distributed optimization methods based on control theory [7,8,9]. Note that the aforementioned works are concentrated on distributed optimization problems with time-independent cost functions and searching for some constant optimal solutions, i.e., the cost functions and optimal solutions are independent of time.

Generally, time-varying optimization problems appear in a large amount of practical applications because the local performance functions or the constraints of engineering technology might evolve over time, such as in time-varying environment allocation [10], network traffic management [11], and optimal matching and formation of multi-robot systems [12]. In such cases, for time-varying distributed optimization, it is required to track an optimal trajectory instead of some fixed points, which is more challenge in comparison with time-invariant optimization problems. Recently, time-varying distributed optimization has attracted many researchers’ increasing attention. For example, ref. [13] proposes a prediction-correction method to deal with a matter of unconstrained optimization subject to time-varying objective functions. With the addition of a term of Newton correction, ref. [14] extends the prediction-correction method [13] into a continuous-time multi-agent system for addressing time-varying distributed optimization problems with inequality constraints. Furthermore, ref. [15] proposes an interior-point method on account of a log-barrier penalty function for general time-varying constraints with an exponential convergent rate to optimal solutions.

To cope with time-varying optimization problems with time-dependent quadratic cost functions, ref. [16] develops a distributed gradient searching approach to track optimal trajectories. To solve a bounded constraint, ref. [17] presents a distributed projected-gradient algorithm, which is convergent to a neighborhood of optimal solutions asymptotically. Furthermore, in order to track and follow the exact time-varying optimal solutions, ref. [18] proposes a distributed protocol based on sliding control. In practical applications, there are expectations that the convergence can be arrived within a finite time for some requirements. For example, a group of robots are required to achieve an expected formation, and preserve signal connectivity in finite time or fixed time in the meantime, minimizing the cost/energy functions [19,20,21,22,23]. However, the aforementioned distributed optimization methods are only capable of reaching agreement over infinite time. In particular, ref. [20] investigated a finite-time consensus tracking control problem in uncertain nonlinear multi-agent systems, showing that the all signals remain semi-global practical finite-time stable, and the tracking errors converge to a neighborhood of the origin in finite time. However, the time for reaching consensus is not given explicitly. Ref. [19] develops a distributed law to achieve a finite-time consensus, where the consensus time heavily counts on agents’ original states. To overcome these drawbacks, on the basis of fixed-time stability [24], fixed-time convergent algorithms are presented in [23,25] for time-dependent distributed optimization. In these algorithms, the time for achieving fixed-time consensus is independent of initial conditions.

However, the upper bound of settling time functions for both distributed fixed-time and finite-time algorithms is dominated by the controller parameters, communication networks, and number of agents. For these limitations, some topics on prescribed-time consensus for nonlinear systems have been developed recently. The notion of prescribed-time stability can be found in [26,27,28,29], where the dynamic system can reach at the state agreement in a preset time. For example, ref. [26] proposes a consensus-based algorithm to enable all agents to reach the Nash equilibrium within a prescribed time. Inspired by these observations, a prescribed-time convergent algorithm is designed for distributed optimization with time-varying cost functions in this paper.

The main contributions of this paper are summarized as follows. First, the time-varying distributed optimization approach proposed herein generalizes prior works [4,6,7,30,31,32] on time-invariant distributed optimization. Moreover, the time-varying objective functions in [10] can be regarded as a special case of this framework. Second, the proposed algorithm allows arbitrary initial conditions and relaxes bound constraints on the optimal state, in contrast to the distributed optimization algorithms in [17,22,33]. Specifically, ref. [33] requires initial states to be bounded, while [17,22] necessitate bounded derivatives of optimal solutions. Third, the presented algorithm achieves prescribed-time consensus with a preset settling time, in contrast to the infinite-time consensus approaches in [13,14,15,16,17,18,33]; the finite-time consensus methods in [19,22,34] with the settling time function depending on original conditions; and the fixed-time consensus methods in [23,35] with the settling time governed by control parameters, number of agents, and communication network.

2. Preliminaries

2.1. Notations

Let $\mathbb{R}$ and ${\mathbb{R}}_{+}$ be the real number set and nonnegative real number set. Let ${\mathbf{1}}_N$, $0_N$ be the column vectors, where all elements in the vectors are 1 and 0, respectively. $I_N\in$${\mathbb{R}}^{N\times N}$ represents the identity matrix. The superscripts ${(·)}^{\top }$ and ${\parallel ·\parallel }_2$ stand for the transpose and Euclidean norm, respectively. For $ϵ\in {\mathbb{R}}^N$, we denote $\left|ϵ\right|=\left(\right|{ϵ}_1|,|{ϵ}_2|,\dots ,|{ϵ}_N{\left|\right)}^{\top }$ and $\mathrm{sgn}\left(ϵ\right)={(\mathrm{sgn}\left({ϵ}_1\right),\mathrm{sgn}\left({ϵ}_2\right),\dots ,\mathrm{sgn}\left({ϵ}_N\right))}^{\top }$, in which the symbol of $\mathrm{sgn}(·)$ is the abbreviation for the signum function. The symbol of ${\lambda }_2\left(L\right)$ represents the second smallest eigenvalue of the matrix L. Let $\mathcal{M}\otimes \mathcal{N}$ be the Kronecker product of matrices $\mathcal{M}$ and $\mathcal{N}$.

2.2. Graph Theory

Consider a multi-agent system that contains N agents. The graph $\mathcal{G}=(\mathcal{V},\mathit{E},\mathcal{A})$ is modeled to describe the interaction among agents, where $\mathcal{V}=\{{\mathbf{l}}_1,{\mathbf{l}}_2,\dots ,{\mathbf{l}}_N\}$ is the set of vertexes, $\mathit{E}\subseteq \left\{({\mathbf{l}}_i,{\mathbf{l}}_j)\right|{\mathbf{l}}_i,{\mathbf{l}}_j\in \mathcal{V},{\mathbf{l}}_i\ne {\mathbf{l}}_j\}$ with $\left|\mathit{E}\right|=p$ is the set of edges, and $\mathcal{A}={\left[a_{ij}\right]}_{q\times q}$ is the adjacent matrix. In $\mathcal{A}$, its entry $a_{ij}=1$ when $({\mathbf{l}}_j,{\mathbf{l}}_i)\in \mathit{E}$, otherwise $a_{ij}=0$. Let node $l_i^{\prime }s$ neighbors be ${\mathcal{N}}_i=\{{\mathbf{l}}_j\mid ({\mathbf{l}}_j,{\mathbf{l}}_i)\in \mathit{E}\}$. The edge $({\mathbf{l}}_j,{\mathbf{l}}_i)$$(({\mathbf{l}}_j,{\mathbf{l}}_i)\in \mathit{E})$ indicates that the information can be transmitted between node ${\mathbf{l}}_i$ and node ${\mathbf{l}}_j$. In addition, a path is defined when there is an edge sequence between node ${\mathbf{l}}_{k1}$ and node ${\mathbf{l}}_{kj}$ in the manner of $({\mathbf{l}}_{k1},{\mathbf{l}}_{k2})$, $({\mathbf{l}}_{k2},{\mathbf{l}}_{k3}),\dots ,({\mathbf{l}}_{kj-1},{\mathbf{l}}_{kj})$. An undirected graph is deemed connected if there exists a path between any pairs of vertexes. Let $\mathrm{D}={\left[{\mathcal{D}}_{ij}\right]}_{p\times q}$ be the incidence matrix, and it is represented as $h_{ik}=-1$ when the node i is the beginning of an edge $(i,k)$, and $h_{ik}=1$ when the node i is the ending of an edge $(i,k)$, otherwise $h_{ik}=0$. $L={\left[{\mathcal{L}}_{ij}\right]}_{q\times q}$ is the Laplacian matrix associated with graph $\mathcal{G}$, where its elements are represented as ${\mathcal{L}}_{ii}={\sum }_{j=1,j\ne i}^N{a}_{ij}$, and ${\mathcal{L}}_{ij}=-a_{ij}$ with $j\ne i$. It is noted that there are $L=L^{\top }$, $L={\mathrm{D}}^T\mathrm{D}$, and $L{1}_N=0_N$ for an undirected graph.

2.3. Problem Formulation

Consider a dynamic equation

$$\dot{\xi }=g(\mathbf{t},\mathcal{X}),\mathcal{X}\left({\mathbf{t}}_0\right)={\xi }_0,$$

(1)

where $\mathcal{X}(\mathcal{X}\in {\mathbb{R}}^n)$ is the state of system (1), ${\mathbf{t}}_0$ is the initial time, and $g:{\mathbb{R}}_{+}\times {\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ is a nonlinear function. If at some points g is discontinuous, then the solution $\xi \left(\mathbf{t}\right)$ to (1) is described as the Filippov solution [36].

Lemma 1

([27]). Define $\mathcal{I}\subseteq {\mathbb{R}}^n$ as a domain that contains equilibrium points of system (1). Suppose ${\Gamma }_1\left(\mathcal{X}\right)$ and ${\Gamma }_2\left(\mathcal{X}\right)$ to be continuous and positive definite functions on $\mathcal{I}$. For a real-valued continuously differentiable function $\mathbb{V}:\tau \times \mathcal{I}\mapsto {\mathbb{R}}_{+}$, if there exist a constant $\sigma \ge 1$ and a time interval $\tau =[{\mathbf{t}}_0,{\mathbf{t}}_f]$ such that

(i) 

${\Gamma }_1\left(\mathcal{X}\right)\le \mathbb{V}(\mathbf{t},\mathcal{X})\le {\Gamma }_2\left(\mathcal{X}\right),$$\forall \mathbf{t}\in \tau ,$$\forall \mathcal{X}\in \mathcal{I}\setminus \left\{0\right\},$

(ii) 

$\mathbb{V}(\mathbf{t},0)=0,$$\forall \mathbf{t}\in \tau$,

(iii) 

$\dot{\mathbb{V}}\le -{\displaystyle \frac{\sigma (1-e^{-\mathbb{V}})}{{\mathbf{t}}_f-\mathbf{t}}}$, $\mathbb{V}\ne 0,$$\forall \mathbf{t}\in \tau$,

then an equilibrium point $\mathcal{X}=0$ is prescribed-time stable, where τ is a time interval, ${\mathbf{t}}_f={\mathbf{t}}_0+T$, and T is set in advance.

Lemma 2

([27]). For ${\theta }_1,{\theta }_2\in \mathbb{R}$, if ${\theta }_1\ge {\theta }_2>0$, there is

$$-{\theta }_1(1-e^{-{\theta }_1})\le -{\theta }_2(1-e^{-{\theta }_2}).$$

For $\theta \in {\mathbb{R}}^n,$ there is

$$-{\theta }^{\top }(I_n-e^{-\mathrm{diag}\left(\theta \right)}){\mathbf{1}}_n\le -{\Vert \theta \Vert }_2(1-e^{-{\Vert \theta \Vert }_2}).$$

Lemma 3

([37]). For $y\in {\mathbb{R}}^n,$ if $\mu >\nu >0$, the following inequality holds:

$${\Vert \mathit{y}\Vert }_{\mu }\le {\Vert y\Vert }_{\nu }\le n^{{\displaystyle \frac{1}{\nu }}-{\displaystyle \frac{1}{\mu }}}{\Vert y\Vert }_{\mu }.$$

(2)

Lemma 4

([38]). When $h\left(\mathcal{X}\right):{\mathbb{R}}^n\mapsto \mathbb{R}$ is a convex and continuously differentiable function with respect to $\mathcal{X}$, $h\left(\mathcal{X}\right)$ will be minimized at ${\xi }^{*}$ if and only if the gradient $\nabla h\left({\xi }^{*}\right)=0$.

3. Main Results

Here, a multi-agent system consisting of n agents is considered based on an undirected communication graph. Assume that the agents in the multi-agent system meet the dynamics defined below:

$${\dot{\xi }}_i\left(\mathbf{t}\right)={\mathbf{g}}_i\left(\mathbf{t}\right),$$

(3)

where ${\xi }_i({\xi }_i\left(\mathbf{t}\right)\in {\mathbb{R}}^d)$ is the state of the ith agent, and ${\mathbf{g}}_i({\mathbf{g}}_i\left(\mathbf{t}\right)\in {\mathbb{R}}^d)$ is the control input.

Assumption 1.

The graph $\mathcal{G}$ associated with the communication topology of the multi-agent system is connected and undirected.

Remark 1.

Unlike static optimization problems with some fixed points, time-varying ones result in time-dependent optimal solutions, increasing the challenge of algorithm design. To the best of our knowledge, there are limited works on time-varying distributed optimization problems with prescribed-time convergence. These novel results have been attained under strict assumptions. Interesting future research directions include relevant extensions by the relaxation of Assumption 1 on undirected graphs to directed topologies.

Definition 1.

The multi-agent system (3) is said to achieve prescribed-time consensus if and only if

$$\underset{\mathbf{t}\to T}{lim}\left|\right|{\xi }_i\left(\mathbf{t}\right)-{\xi }_j\left(\mathbf{t}\right)\left|\right|=0,$$

and ${\xi }_i\left(\mathbf{t}\right)={\xi }_j\left(\mathbf{t}\right)$ for $\mathbf{t}\ge T$ with any initial conditions, where the value of T is set offline in advance.

For a multi-agent system, every agent i is assigned a local state ${\xi }_i({\xi }_i\in {\mathbb{R}}^d)$, $i=1,2,\dots ,n$, and privately owns a local objective function ${\mathrm{F}}_i$, with ${\mathrm{F}}_i({\xi }_i,\mathbf{t}):$${\mathbb{R}}^d\times {\mathbb{R}}^{+}\to \mathbb{R}$. Note that the local state and cost function are only known to agent i itself. A distributed protocol ${\mathbf{g}}_i\left(\mathbf{t}\right)$ is designed to guarantee that all agents collaboratively trace the optimal trajectory ${\xi }^{*}\left(\mathbf{t}\right)$ for the following time-varying convex optimization:

$$\underset{\xi \left(\mathbf{t}\right)\in {\mathbb{R}}^d}{min}\sum _{i=1}^n{\mathrm{F}}_i(\xi \left(\mathbf{t}\right),\mathbf{t}).$$

(4)

Assumption 2.

The time-varying cost function ${\mathrm{F}}_i(\xi ,\mathbf{t})$ is at least twice continuously differentiable with respect to x and t. In addition, for any x and t, the Hessian matrix $H_i(\xi \left(\mathbf{t}\right),\mathbf{t})$ is invertible.

It is worth noting that the time-varying optimization problem (4) can be regarded as an equivalence of the consensus problem in multi-agent systems, where all agents arrive at a consensus and at the same time minimize the whole total objective function, i.e.,

$$\begin{array}{cc}\hfill \underset{\xi \left(\mathbf{t}\right)\in {\mathbb{R}}^{nd}}{min}& {\displaystyle \sum _{i=1}^n}{\mathrm{F}}_i({\xi }_i\left(\mathbf{t}\right),\mathbf{t}),\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\xi }_i\left(\mathbf{t}\right)={\xi }_j\left(\mathbf{t}\right),i,j=1,2,\dots ,n,\hfill \end{array}$$

(5)

where $\xi \left(\mathbf{t}\right)={({\xi }_1^{\top }\left(\mathbf{t}\right),{\xi }_2^{\top }\left(\mathbf{t}\right),\dots ,{\xi }_n^{\top }\left(\mathbf{t}\right))}^{\top }$, and ${\xi }_i\left(\mathbf{t}\right)\in {\mathbb{R}}^d$. For simplicity, we remove the time dependence associated with the variables when appropriate.

To deal with problem (5), the distributed protocol is proposed for the ith agent as below:

$$\begin{array}{c}{\mathbf{g}}_i=\left\{\begin{array}{c}{\mathcal{S}}_i^1+{\mathcal{S}}_i^2+{\mathcal{K}}_i,\mathrm{if}{\mathbf{t}}_0\le \mathbf{t}<{\mathbf{t}}_{\mathbf{f}},\hfill \\ {\mathcal{S}}_i^2+{\mathcal{K}}_i,\mathrm{if}\mathbf{t}\ge {\mathbf{t}}_{\mathbf{f}},\hfill \end{array}\right.\hfill \end{array}$$

(6)

where ${\xi }_i={({\xi }_{i1},{\xi }_{i2},\dots ,{\xi }_{id})}^{\top }\in {\mathbb{R}}^d$, ${\mathbf{t}}_{\mathbf{f}}={\mathbf{t}}_0+T$,

$$\begin{array}{ccc}\hfill {\mathcal{K}}_i& =& -H_i^{-1}({\xi }_i,\mathbf{t})({\displaystyle \frac{\partial }{\partial t}}\nabla {\mathrm{F}}_i({\xi }_i,\mathbf{t})+\nabla {\mathrm{F}}_i({\xi }_i,\mathbf{t})),\hfill \\ \hfill {\mathcal{S}}_i^1& =& -{\displaystyle \frac{\sigma }{T-t}}\mathrm{diag}\left(\sum _{j\in {\mathcal{N}}_i}\mathrm{sgn}({\xi }_i-{\xi }_j)\right){\theta }_i,\hfill \\ \hfill {\mathcal{S}}_i^2& =& -{\gamma }_i\sum _{j\in {\mathcal{N}}_i}\mathrm{sgn}({\xi }_i-{\xi }_j),\hfill \end{array}$$

${\theta }_i:=[I_d-e^{-\mathrm{diag}\left(\right|{\sum }_{j\in {\mathcal{N}}_i}({\xi }_i-{\xi }_j)\left|\right)}]{\mathbf{1}}_d$, $\sigma \ge {\displaystyle \frac{1}{{\lambda }_2\left(L\right)}}$, and ${\gamma }_i>{\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{2}}$. The terms $\nabla {\mathrm{F}}_i({\xi }_i,\mathbf{t})$ and $H_i({\xi }_i,\mathbf{t})$ are the first derivative and the Hessian matrix of the cost function ${\mathrm{F}}_i({\xi }_i,\mathbf{t})$ with respect to ${\xi }_i$, respectively. Moreover, ${\displaystyle \frac{\partial }{\partial t}}\nabla {\mathrm{F}}_i({\xi }_i,\mathbf{t})$ is the partial derivative of the gradient $\nabla {\mathrm{F}}_i({\xi }_i,\mathbf{t})$ with respect to $\mathbf{t}$. In the distributed protocol (6), the purpose of ${\mathcal{S}}_i^1$ is to reach a prescribed-time consensus, while the terms of ${\mathcal{S}}_i^2$ and ${\mathcal{K}}_i$ aim at tracing the evolution in line with the local cost function ${\mathrm{F}}_i({\xi }_i,\mathbf{t})$ and urging the state $\xi \left(\mathbf{t}\right)$ toward the time-dependent optimal solutions.

Assumption 3.

There exists a constant $\overline{l}(\overline{l}>0)$ such that $\left|\right|{\mathcal{K}}_i-{\mathcal{K}}_j{\left|\right|}_2\le \overline{l}$ for $i,j=1,2,\dots ,n$, where ${\mathcal{K}}_i$ is defined in (6).

Furthermore, system (3) combined with the distributed protocol (6) is capable of being compactly notated as in the form below:

$$\begin{array}{ccc}\hfill \dot{\xi }(\mathbf{t})& =& \left\{\begin{array}{c}{\mathbb{S}}_1+{\mathbb{S}}_2+\mathbb{K},\mathrm{if}{\mathbf{t}}_0\le \mathbf{t}<{\mathbf{t}}_f,\hfill \\ {\mathbb{S}}_2+\mathbb{K},\mathrm{if}\mathbf{t}\ge {\mathbf{t}}_f,\hfill \end{array}\right.\hfill \\ \hfill {\mathbb{S}}_1& =& -{\displaystyle \frac{\sigma }{T-t}}{\mathcal{D}}^{\top }\mathrm{diag}(\mathrm{sgn}(\mathcal{D}\xi (\mathbf{t})))\Pi ,\hfill \\ \hfill {\mathbb{S}}_2& =& -{\mathcal{D}}^{\top }\mathrm{sgn}(\mathcal{D}\xi ),\hfill \\ \hfill \mathbb{K}& =& -{\mathbf{H}}^{-1}(\xi ,\mathbf{t})(\nabla \mathrm{F}(\xi ,\mathbf{t})+{\displaystyle \frac{\partial }{\partial t}}\nabla \mathrm{F}(\xi ,\mathbf{t}))),\hfill \\ \hfill {\mathbf{H}}^{-1}(\xi ,\mathbf{t})& =& \mathrm{diag}\{H_1^{-1}({\xi }_1,\mathbf{t}),\dots ,H_n^{-1}({\xi }_n,\mathbf{t})\},\hfill \\ \hfill \nabla \mathrm{F}(\xi ,\mathbf{t})& =& {({\nabla }^{\top }{\mathrm{F}}_1({\xi }_1,\mathbf{t}),\dots ,{\nabla }^{\top }{\mathrm{F}}_n({\xi }_n,\mathbf{t}))}^{\top },\hfill \\ \hfill \Pi & =& [I_{nd}-e^{-\mathrm{diag}(|\mathcal{D}\mathit{\xi }(\mathbf{t})|)}]{\mathbf{1}}_{nd},\hfill \end{array}$$

(7)

where $\xi \left(\mathbf{t}\right)={({\xi }_1^{\top }\left(\mathbf{t}\right),\dots ,{\xi }_n^{\top }\left(\mathbf{t}\right))}^{\top },\mathcal{D}=D^{\prime }\otimes I_d$, and $D^{\prime }={\left[h_{ij}^{\prime }\right]}_{m\times N}$ with $h_{ij}^{\prime }=-{\gamma }_i{a}_{ij}$ if the node i is an endpoint that an edge leaves, and $h_{ij}^{\prime }={\gamma }_i{a}_{ij}$ if the node i is an endpoint that an edge enters, otherwise $h_{ij}^{\prime }=0$.

Theorem 1.

Suppose that Assumptions 1–3 hold. The multi-agent system (3) combined with the distributed law (6) reaches prescribed-time consensus T, i.e., ${\xi }_i\left(\mathbf{t}\right)={\xi }_j\left(\mathbf{t}\right)$ as $\mathbf{t}\ge {\mathbf{t}}_0+T$ for $i,j=1,2,...,n,$ where the time interval T can be set in advance.

Proof. 

Define ${\mathcal{E}}_i\left(\mathbf{t}\right)={\xi }_i\left(\mathbf{t}\right)-{\displaystyle \frac{1}{n}}{\sum }_{\iota =1}^n{\xi }_{\iota }\left(\mathbf{t}\right)$, $i=1,2,\dots ,n$, and

$$\mathcal{E}\left(\mathbf{t}\right)={({\mathcal{E}}_1^{\top }\left(\mathbf{t}\right),{\mathcal{E}}_2^{\top }\left(\mathbf{t}\right),\dots ,{\mathcal{E}}_n^{\top }\left(\mathbf{t}\right))}^{\top }.$$

Then, there is

$$\mathcal{E}=((I_n-{\displaystyle \frac{1}{n}}{\mathbf{11}}^{\top })\otimes I_d)\mathit{\xi }.$$

(8)

It is easy to see that ${\xi }_i\left(\mathbf{t}\right)={\xi }_j\left(\mathbf{t}\right)$ if ${\mathcal{E}}_i\left(\mathbf{t}\right)=0$.

Furthermore, substituting (8) into (7), we can obtain

$$\dot{\mathcal{E}}\left(\mathbf{t}\right)=\left\{\begin{array}{c}{\widehat{\mathbb{S}}}_1+{\widehat{\mathbb{S}}}_2+\widehat{\mathbb{K}},\mathrm{if}{\mathbf{t}}_0\le \mathbf{t}<{\mathbf{t}}_f,\hfill \\ {\widehat{\mathbb{S}}}_2+\widehat{\mathbb{K}},\mathrm{if}\mathbf{t}\ge {\mathbf{t}}_f,\hfill \end{array}\right.$$

(9)

where

$$\begin{array}{ccc}\hfill {\widehat{\mathbb{S}}}_1& =& -{\displaystyle \frac{\sigma }{T-t}}{\mathcal{D}}^{\top }\mathrm{diag}\left(\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\right)\right)\widehat{\Pi },\hfill \\ \hfill \widehat{\Pi }& =& [I_{nd}-e^{-\mathrm{diag}\left(\right|\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\left|\right)}]{\mathbf{1}}_{nd},\hfill \\ \hfill {\widehat{\mathbb{S}}}_2& =& -{\mathcal{D}}^{\top }\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\right),\hfill \\ \hfill \widehat{\mathbb{K}}& =& -(\Xi \otimes I_d)\mathbb{K},\hfill \\ \hfill \Xi & =& I_n-{\displaystyle \frac{1}{n}}\mathbf{1}{\mathbf{1}}^{\top }.\hfill \end{array}$$

The fact that $\Xi D^{\prime }=D^{\prime }$ is used follows. Choose a candidate Lyapunov function $\mathcal{V}\left(\mathbf{t}\right)$, where $\mathcal{V}\left(\mathbf{t}\right)={\displaystyle \frac{1}{2}}{\mathcal{E}}^{\top }\left(\mathbf{t}\right)\mathcal{E}\left(\mathbf{t}\right).$ The time derivative of $\mathcal{V}\left(\mathbf{t}\right)$ along (9) is described as

$$\begin{array}{cc}\dot{\mathcal{V}}\left(\mathbf{t}\right)& ={\mathcal{E}}^{\top }\left(\mathbf{t}\right)\dot{\mathcal{E}}\left(\mathbf{t}\right).\hfill \end{array}$$

(10)

According to Formulas (9) and (10), for ${\mathbf{t}}_0\le \mathbf{t}<{\mathbf{t}}_f$, one can obtain

$$\begin{array}{ccc}\hfill & & {\mathcal{E}}^{\top }\left(\mathbf{t}\right){\widehat{\mathbb{S}}}_1\hfill \\ & =& {\mathcal{E}}^{\top }\left(\mathbf{t}\right)(-{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}{\mathcal{D}}^{\top }\mathrm{diag}\left(\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\right)\right)[I_{nd}-e^{-\mathrm{diag}\left(\right|\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\left|\right)}]{\mathbf{1}}_{nd})\hfill \\ & =& -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}{\left(\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\right)}^{\top }\mathrm{diag}\left(\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\right)\right)[I_{nd}-e^{-\mathrm{diag}\left(\right|\mathcal{D}e\left(\mathbf{t}\right)\left|\right)}]{\mathbf{1}}_{nd}\hfill \\ & =& -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}{\left|\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\right|}_1^{\top }[I_{nd}-e^{-\mathrm{diag}\left(\right|\mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\left|\right)}]{\mathbf{1}}_{nd}\hfill \\ & \le & -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}{\parallel \mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{-{\parallel \mathcal{D}\mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right)\hfill \\ & \le & -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right).\hfill \end{array}$$

(11)

Lemmas 2 and 3 are utilized to obtain the first and the second inequalities, respectively, and the last inequality holds due to ${\parallel \mathcal{D}\mathcal{E}\parallel }_2=\sqrt{{\mathcal{E}}^{\top }{\mathcal{D}}^{\top }\mathcal{D}\mathcal{E}}=\sqrt{{\mathcal{E}}^{\top }(L\otimes I_d)\mathcal{E}}\ge \sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\parallel }_2$.

Furthermore, on one hand, we have

$$\begin{array}{ccc}\hfill {\mathcal{E}}^{\top }{\widehat{\mathbb{S}}}_2& =& -{\mathcal{E}}^{\top }{\mathcal{D}}^{\top }\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\right)\hfill \\ & =& -{\left(\mathcal{D}\mathcal{E}\right)}^{\top }\mathrm{sgn}\left(\mathcal{D}\mathcal{E}\right)\hfill \\ & =& -\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\displaystyle \frac{{\gamma }_i}{2}}{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1.\hfill \end{array}$$

(12)

On the other hand, one can obtain

$$\begin{array}{ccc}\hfill {\mathcal{E}}^{\top }\widehat{\mathbb{K}}& =& {\mathcal{E}}^{\top }(\Xi \otimes I_d)\mathbb{K}\hfill \\ & =& {\displaystyle \frac{1}{2n}}\sum _{i=1}^n\sum _{j=1}^n({\mathcal{E}}_i-{\mathcal{E}}_j)({\mathcal{K}}_i-{\mathcal{K}}_j)\hfill \\ & \le & {\displaystyle \frac{1}{2n}}\sum _{i=1}^n\sum _{j=1}^n\parallel {\mathcal{E}}_i-{\mathcal{E}}_j{\parallel }_1{\parallel {\mathcal{K}}_i-{\mathcal{K}}_j\parallel }_2\hfill \\ & \le & {\displaystyle \frac{\overline{\mathcal{K}}}{2n}}\sum _{i=1}^n\sum _{j=1}^n{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1\hfill \\ & \le & {\displaystyle \frac{\overline{\mathcal{K}}}{2}}{\max }_i\{\sum _{j=1,j\ne i}^n\parallel {\mathcal{E}}_i-{\mathcal{E}}_j{\parallel }_1\}\hfill \\ & \le & {\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{4}}\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1.\hfill \end{array}$$

(13)

Combining this with the Formulas (11)–(13), for ${\mathbf{t}}_0\le \mathbf{t}<{\mathbf{t}}_f$, Formula (10) is able to be rewritten as the following representation:

$$\begin{array}{ccc}\hfill \dot{\mathcal{V}}& \le & -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right)-\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\displaystyle \frac{{\gamma }_i}{2}}{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1\hfill \\ & & +{\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{4}}\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1\hfill \\ & \le & -{\displaystyle \frac{\sigma }{{\mathbf{t}}_f-t}}\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right)\hfill \\ & & -\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\displaystyle \frac{{\gamma }^{*}}{2}}\parallel {\mathcal{E}}_i-{\mathcal{E}}_j{\parallel }_1+{\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{4}}\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\parallel {\mathcal{E}}_i-{\mathcal{E}}_j\parallel }_1\hfill \\ & =& -{\displaystyle \frac{\sigma \sqrt{{\lambda }_2\left(L\right)}}{{\mathbf{t}}_f-t}}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right)\hfill \\ & & +\left({\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{4}}-{\displaystyle \frac{{\gamma }^{*}}{2}}\right)\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\parallel {\mathcal{E}}_i\left(\mathbf{t}\right)-{\mathcal{E}}_j\left(\mathbf{t}\right)\parallel }_1\hfill \\ & \le & -{\displaystyle \frac{\sigma \sqrt{{\lambda }_2\left(L\right)}}{{\mathbf{t}}_f-t}}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2\left(1-e^{\sqrt{{\lambda }_2\left(L\right)}{\parallel \mathcal{E}\left(\mathbf{t}\right)\parallel }_2}\right),\hfill \end{array}$$

(14)

in which ${\gamma }^{*}={\inf }_i\left\{{\gamma }_i\right\}$, and ${\gamma }^{*}={\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{2}}$.

According to the definition of $\mathcal{V}$, Formula (14) can be rewritten as

$$\dot{\mathcal{V}}\le -{\displaystyle \frac{\sigma \sqrt{{\lambda }_2\left(L\right)}\sqrt{2\mathcal{V}}}{T-t}}\left(1-e^{-\sqrt{{\lambda }_2\left(L\right)}\sqrt{2\mathcal{V}}}\right).$$

Let $\varrho =\sqrt{2{\lambda }_2\left(L\right)\mathcal{V}}$; one obtains

$$\begin{array}{ccc}\hfill \dot{\varrho }& =& {\displaystyle \frac{\sqrt{{\lambda }_2\left(L\right)}}{\sqrt{2\mathcal{V}}}}\dot{\mathcal{V}}\hfill \\ & \le & -{\displaystyle \frac{\sigma \sqrt{{\lambda }_2\left(L\right)}}{T-t}}\left(1-e^{-\varrho }\right).\hfill \end{array}$$

(15)

If $\widehat{\sigma }=\sigma \sqrt{{\lambda }_2\left(L\right)}\ge 1$, then (15) can be transformed as $\dot{\varrho }\le -(\widehat{\sigma }/{\mathbf{t}}_f-t)(1-e^{-\varrho })$. It can be obtained that $\varrho \le \ln (\rho {({\mathbf{t}}_f-t)}^{\widehat{\sigma }}+1)$, where $\rho =\left(e^{\varrho \left({\mathbf{t}}_0\right)-1}\right)/{({\mathbf{t}}_f-t)}^{\widehat{\sigma }}$ and $\varrho \left(0\right)=\varrho \left({\mathbf{t}}_0\right)$. Then, the time derivative of $\varrho$ is able to be translated into $\dot{\varrho }\le -\widehat{\sigma }\rho {({\mathbf{t}}_f-t)}^{\widehat{\sigma }-1}/(\rho {({\mathbf{t}}_f-t)}^{\widehat{\sigma }}+1)$. Therefore, it can be observed that there is $\varrho =0$ and $\dot{\varrho }=0$ when $\mathbf{t}={\mathbf{t}}_f$. As a result of $\varrho =\sqrt{2{\lambda }_2\left(L\right)\mathcal{V}},$ it can be noted that $\mathcal{V}(\mathbf{t},\mathcal{E})=0$ for $\mathbf{t}={\mathbf{t}}_f$ and $\mathcal{V}(\mathbf{t},0)=0$ for ${\mathbf{t}}_0\le t\le {\mathbf{t}}_f$. Based on Lemma 1, system (3) is prescribed-time stable, and ${\xi }_i={\xi }_j$ for $\mathbf{t}={\mathbf{t}}_f$.

Moreover, for $\mathbf{t}\ge {\mathbf{t}}_f$, based on (9) and (10), combining with (12) and (13), the following is obtained:

$$\begin{array}{ccc}\hfill \dot{\mathcal{V}}& \le & \left({\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{4}}-{\displaystyle \frac{{\gamma }^{*}}{2}}\right)\sum _{i=1}^n\sum _{j\in {\mathcal{N}}_i}{\parallel {\mathcal{E}}_i\left(\mathbf{t}\right)-{\mathcal{E}}_j\left(\mathbf{t}\right)\parallel }_1\hfill \\ & <& 0.\hfill \end{array}$$

Note that $\mathcal{V}\left(\mathbf{t}\right)=0$ is sustained for $\mathbf{t}\ge {\mathbf{t}}_f$. On the basis of Definition 1, it could be concluded that the prescribed-time consensus is able to be reached if $\sigma \ge 1/\sqrt{{\lambda }_2\left(L\right)}$ and ${\gamma }_i>{\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{2}}$. Hence, the distributed protocol (6) drives the agents of system (3) to reach a consensus within prescribed time and keep moving in consensus for $\mathbf{t}\ge {\mathbf{t}}_f$. □

Theorem 2.

Suppose that Assumptions 1–3 hold. If $\sigma \ge 1/\sqrt{{\lambda }_2\left(L\right)}$ and ${\gamma }_i>{\displaystyle \frac{(n-1)\overline{\mathcal{K}}}{2}}$, the multi-agent system (3) with the distributed protocol (6) minimizes the time-varying problem (4) as $t\to +\infty$.

Proof. 

Based on Theorem 1, the multi-agent system (3) with the distributed law (6) achieves a consensus in prescribed time, i.e., ${\xi }_i\left(\mathbf{t}\right)=1/n{\sum }_{j=1}^n{\xi }_j\left(\mathbf{t}\right)={\xi }_j\left(\mathbf{t}\right)$ for $\mathbf{t}\ge {\mathbf{t}}_0+T$.

Define ${\xi }^{*}\left(\mathbf{t}\right)\triangleq {\sum }_{j=1}^n{\xi }_j\left(\mathbf{t}\right)$. According to the above analysis, one can obtain ${\dot{\xi }}^{*}\left(\mathbf{t}\right)={\displaystyle \frac{1}{n}}{\sum }_{j=1}^n{\dot{\xi }}_j\left(\mathbf{t}\right)=-H_j^{-1}(\nabla {\mathrm{F}}_j({\xi }_j,\mathbf{t})+{\displaystyle \frac{\partial }{\partial t}}\nabla {\mathrm{F}}_j({\xi }_j,\mathbf{t}))$ for $\mathbf{t}\ge {\mathbf{t}}_0+T$. Construct a Lyapunov function

$$W\left(\mathbf{t}\right)={\displaystyle \frac{1}{2}}{\left(\sum _{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*}\left(\mathbf{t}\right),\mathbf{t})\right)}^{\top }\left(\sum _{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*}\left(\mathbf{t}\right),\mathbf{t})\right),$$

where $W\left(\mathbf{t}\right)$ is positive definite with respect to ${\sum }_{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})$. Then, according to Assumptions 1 and 3, for $\mathbf{t}\ge {\mathbf{t}}_0+T$, the time derivative of $W\left(\mathbf{t}\right)$ can be described as follows:

$$\begin{array}{ccc}\hfill \dot{W}\left(\mathbf{t}\right)& =& {\left(\sum _{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})\right)}^{\top }\left(\sum _{j=1}^n{H}_j({\xi }^{*},\mathbf{t}){\dot{\xi }}^{*}\left(\mathbf{t}\right)+\sum _{j=1}^n{\displaystyle \frac{\partial }{\partial t}}\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})\right)\hfill \\ & =& -{\left(\sum _{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})\right)}^{\top }\left(\sum _{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})\right).\hfill \end{array}$$

Therefore, there is $\dot{W}\left(\mathbf{t}\right)<0$ for ${\sum }_{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})\ne 0$. This indicates that ${\sum }_{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})$ will be asymptotic convergent to zero, i.e., ${lim}_{t\to +\infty }{\sum }_{j=1}^n\nabla {\mathrm{F}}_j({\xi }^{*},\mathbf{t})=0$. Based on Lemma 4, the time-varying cost function ${\sum }_{i=1}^n{\mathrm{F}}_i({\xi }_i,\mathbf{t})$ will be minimized as $t\to +\infty$. □

4. Examples and Simulations

4.1. Prescribed-Time Rendezvous

The rendezvous problem can be modeled as a consensus problem [39], and it is depicted as the following strongly convex optimization:

$$\underset{{\xi }_i\left(\mathbf{t}\right)}{min}f({\xi }_i,\mathbf{t})={\displaystyle \frac{1}{2}}\sum _{i=1}^n{\parallel {\xi }_i\left(\mathbf{t}\right)-p_i\left(\mathbf{t}\right)\parallel }_2^2,$$

(16)

For the shortest distance rendezvous problem in two-dimensional space, we set $n=10$. A communication network with 10 agents and circular connectivity is used for (16). Let $p_i\left(\mathbf{t}\right)=0.01i*{(sin\left(\mathbf{t}\right)+2e^t,-cos\left(\mathbf{t}\right))}^{\top }$. The parameter $\sigma$ is set as $\sigma =20$, and the control gain ${\gamma }_i=120$.

The simulation results are displayed in Figure 1 with the prescribed time ${\mathbf{t}}_f=0.06$ s and Figure 2 with the prescribed time ${\mathbf{t}}_f=0.1$ s. In Figure 1a and Figure 2a, it is shown that the states based on algorithm (6) can reach consensus within the respective prescribed times, where the convergent time can be preset by the designer and is independent of any initial conditions. The errors between the state and optimal solution are displayed in Figure 1b and Figure 2b, which show that the errors eventually converge to zero. It should be noted that the times at which the errors converge are almost close to the prescribed times.

4.2. Multi-Robots Control

In this subsection, the target for a multi-agent system with n agents is to cope with the multi-robot control problem. In the system, an amount of smart agents are required to trace a moving point and sustain a certain formation. The example is inspired by a formation control problem in [33,40], where a distributed controller based on multi-agent systems is desired to address the following modeled optimization problem:

$$\begin{array}{c}\underset{\xi \left(\mathbf{t}\right)\in {\mathbb{R}}^{2n}}{min}\sum _{i=1}^n\left(\theta \left|\right|{\xi }_i\left(\mathbf{t}\right)-{\xi }_i^{\mathrm{ref}}{\left|\right|}_2^2+\sum _{j\in {\mathcal{N}}_i}\left|\right|{\xi }_i\left(\mathbf{t}\right)-{\xi }_i^{\mathrm{ref}}-({\xi }_j\left(\mathbf{t}\right)-{\xi }_j^{\mathrm{ref}}){\left|\right|}_2^2\right),\end{array}$$

(17)

where $\theta >0$ is a scaling factor, and ${\xi }_i\left(\mathbf{t}\right)\in {\mathbb{R}}^2$ and ${\xi }_i^{\mathrm{ref}}\in {\mathbb{R}}^2$ are the location position and reference point of agent i, respectively.

We set $n=4$ and $\theta =0.5$. The expected formation of the multi-robots team is reformulated as ${\xi }_i^{\mathrm{ref}}={(2\mathbf{t}+cos(2(i-1)\pi /4),2\mathbf{t}+sin(2(i-1)\pi /4))}^{\top }(i=1,2,3,4)$, which moves a rotating rectangle. The parameters in (6) are selected as $\sigma =3$ and ${\gamma }_i=100$ for $i=1,2,3,4$.

The communication topological graph associated with the introduced multi-agent system is illustrated in Figure 3. The relative errors $\parallel {\xi }_i-{\xi }_i^{*}{\parallel }_2/{\parallel {\xi }_i^{*}\parallel }_2$ with the prescribed time ${\mathbf{t}}_f=1$ s are described in Figure 4. It can be seen that the errors between the optimums and states converge to zero as the time approaches $1.5$ s, which is almost equal to the preset time. To make a rectangle formation, let $x_1^{\mathrm{ref}}={(0.3,0.4)}^{\top }$, $x_2^{\mathrm{ref}}={(-0.3,0.4)}^{\top }$, $x_3^{\mathrm{ref}}={(-0.3,-0.4)}^{\top }$, $x_4^{\mathrm{ref}}={(0.3,-0.4)}^{\top }$. Figure 5 illustrates the transient positions of four robots at several sampling times: each corresponding to a specific time $t=0$, $t=0.3$ s, $t=0.6$ s, $t=0.8$ s, $t=1$ s, and $t=2$ s. The agents are represented by different colored circles: Agent 1 in red, Agent 2 in green, Agent 3 in blue, and Agent 4 in magenta. Additionally, there are two special points marked: the target point in black and the optimal point in cyan. It can be seen that the robots are moving towards an optimal point over time and can constitute the rectangle formation moving around the target point.

5. Conclusions

This paper investigates a type of time-varying optimization problem on the basis of multi-agent systems with continuous single-integrator dynamics in a distributed manner. Based on the predefined-time consensus controller, a new distributed law is presented to guarantee that all agents reach a consensus in a preset time while minimizing the sum of time-dependent local cost functions. Unlike the existent distributed consensus algorithms for time-varying optimization on account of multi-agent systems, the time for reaching consensus herein can be prescribed in advance. On the basis of Lyapunov stability theory and convex optimization theory, the prescribed-time convergence and asymptotic optimality of the presented approach are proven. Lastly, the effectiveness of the designed algorithm is examined through two simulations.