Finite-Time Partial Component Consensus for Nonlinear Leader-Following Multi-Agent Systems

Abstract

The problem of finite-time partial component consensus (FTPCC) for first-order nonlinear multi-agent systems (MASs) is investigated in this paper for the first time. By incorporating the permutation matrix approach, we derive a novel error system for identical components, which facilitates stability analysis. Leveraging partial variable stability theory and related foundational knowledge, we devise two adaptable protocols. These protocols are tailored to achieve FTPCC in nonlinear MASs, one for systems without disturbances and another for those with bounded disturbances. To validate our findings, numerical examples are provided, demonstrating the effectiveness of the proposed results.

1. Introduction

In recent years, because of the rapid development of artificial intelligence and computer communication, MASs have received extensive attention from scholars in the fields of mathematics, physics, biology, and control theory. In real life, the application of MASs can be seen in the cooperative control of unmanned aircraft, distributed sensor networks, formation control, and other problems, as discussed in [1,2]. At present, the main issues of multi-agent systems include cluster, tracking and consensus [3,4,5].

Multi-agent systems originate from the exploration and research of biological behavior in nature, and are defined as coupled structures composed of multiple agents that interact with each other and solve complex problems. Consensus, which is a hot topic in MASs, has attracted extensive attention from scholars and has also achieved a great number of research results, such as group consensus in [6], finite-time consensus in [7], projective group consensus in [8], lag consensus in [9], input-output consensus in [10], and so on. Consensus is mainly used to design appropriate control protocols to make systems’ states eventually converge to a target state over time. Common control protocols include impulsive control, intermittent control, event-triggered control, and pinning control. In addition, some scholars also considered the influence of other factors on the consensus of MASs, including external noise, input saturation, dynamic model, and so on. In [6,11], cluster consensus was studied based on two different pinning controls, namely, piecewise pinning and adaptive intermittent pinning, respectively. By designing a new protocol associated with neighbours’ states and standard white noises, the containment control problem with random multiplicative noise was solved in [12], including first-order and second-order MASs. To solve the problem of input saturation, a novel distributed protocol which was based on event-triggered control was put forward in [13] to investigate group consensus for leaderless heterogeneous MASs. Furthermore, the consensus problem for leader-following MASs with uncertainty and disturbance under undirected graph and directed graph was discussed in [14].

Furthermore, since convergence in finite time can make multi-agent systems have higher accuracy, anti-interference, and robustness, finite-time consensus (FTC) has substantial industrial application value. Therefore, FTC is more attractive, and there are many results about this issue. Under Euler digraph, FTC for nonlinear MASs via pinning control was investigated in [15]. For second-order MASs with input saturation, a new protocol was designed, which could cope with finite-time group consensus in [16], which contained appropriate saturation functions. In addition, by using integral sliding mode control and event-triggered control, the problem of FTC of nonlinear heterogeneous MASs was solved in [17]. To solve the distributed adaptive tracking problem in finite time, the corresponding solution, which was dependent on the recursion algorithm and neural networks, was provided in [18]. Tong designed a distributed consensus protocol to solve FTC for MASs under continuous time-varying interaction topology in [19]. To deal with the problem of finite-time stability for discrete-time stochastic MASs, Hu presented a novel control criterion which made full use of the piecewise-like delay method in [20]. Furthermore, a novel distributed finite-time adaptive control framework was proposed based on the barrier function in [21].

However, agents are usually disturbed by some factors in the information transmission. Therefore, it is inevitable to investigate the influence of disturbance on consensus. To study the fixed-time consensus problems for MASs, two protocols without/with disturbances were designed over signed directed graphs in [22]. For the second-order system with input saturation and disturbance, a continuous integral sliding mode method was utilized to solve the FTC in [23]. Furthermore, the group consensus problem for heterogeneous MASs with bounded disturbances was addressed in [24].

It is noted that the research objects of the above literature are all state components in MASs. However, not all MASs require all components to achieve consensus in real life. In this case, some scholars have begun to study partial component consensus (PCC) [25]. For example, when a flying fleet performs in a special attitude in the air, the displacement components of all aircraft in the fleet are exactly the same in the forward direction. However, the displacement in other directions does not need to be consistent [26]. That is to say, PCC is more general than identical consensus. For the past few years, some scholars have studied partial component consensus, and a few results have been obtained. The problem of PCC for MASs in discrete time was considered in [27] for the first time. Furthermore, not only intermittent control but also pinning control were considered to reach PCC for MASs in [28]. Furthermore, for the high-order MASs, Schur complement was fully taken advantage of when dealing with the PCC for the second-order MASs and the third-order MASs in [25]. The study of oscillatory behavior of differential equations was inspired by references [29,30,31,32], and [21] introduced a novel and developed a numerical technique that solved nonlinear time fractional differential equations.

Motivated by above discussions, we found two problems still waiting to be solved. On the one hand, in the study of FTC problems, almost all of them are about all components; on the other hand, all results of PCC are infinite time. Therefore, this paper intends to consider the FTPCC for leader-following MASs. Compared to existing research results, the contributions mainly include the following three points:

• The FTPCC for the first-order nonlinear MASs is studied for the first time and the relevant definition of FTPCC is given in this paper.
• For nonlinear MASs without external disturbances and nonlinear MASs with external disturbances, control protocols that can achieve FTPCC are proposed. Furthermore, the designed consensus protocols are more flexible because parameters can be different, which can be used more widely in practical applications.
• The agents in the MASs considered in this paper are all n-dimensional rather than one-dimensional, which makes the theoretical results given by this paper can be applied to many fields.

The rest of the structure of the article is as follows. Section 2 mainly introduces the related preliminaries. Section 3 explains the new concept of FTPCC, and then gives main results and theories. Section 4 gives two numerical examples to demonstrate the validity of the theoretical results. Finally, Section 5 summarizes this paper and offers further perspectives.

The study of the oscillatory behavior of differential equations in this paper is inspired by [29,30,31,32,33,34,35,36,37,38], which studied the controllability and stability of differential equations. In this paper, we study first-order nonlinear differential equations. The resolution of first-order differential equations enables the comprehension of the governing principles of dynamic processes and furnishes a theoretical foundation for the resolution of practical problems. In the control systems, first-order differential equations can be employed to assess the stability and dynamic performance of the system.

Notations: $\mathbb{N}=\left\{1,2,\cdots \right\}$ represent the set of all positive integers. ${\mathbb{R}}^{N\times N}$ is the set that is made up of all $N\times N$ real matrices and $1_N={(1,1,\cdots ,1)}^{\mathrm{T}}\in {\mathbb{R}}^N$. The matrix $diag(d_1,d_2,\cdots ,d_N)$ denotes a $N\times N$ diagonal matrix. $E\in {\mathbb{R}}^{n\times n}$ is the n-order identity matrix. The symbol ⊗ is the Kronecker product of matrices. $sig{\left(p\right)}^{\alpha }=sgn\left(p\right){\left|p\right|}^{\alpha }$, $\alpha \in {\mathbb{R}}^{+}$, ${\mathbb{R}}^{+}$ is the set of positive real numbers. We use $p_i={(p_{i1},p_{i2},\cdots ,p_{in})}^{\mathrm{T}}\in {\mathbb{R}}^n$, $p={(p_1^{\mathrm{T}},p_2^{\mathrm{T}},\cdots ,p_N^{\mathrm{T}})}^{\mathrm{T}}$ to denote the vector in ${\mathbb{R}}^{nN}$. $sig{\left(p\right)}^{\alpha }={(sig{\left(p_1\right)}^{\alpha },sig{\left(p_2\right)}^{\alpha },\cdots ,sig{\left(p_N\right)}^{\alpha })}^{\mathrm{T}}$ with $sig{\left(p_i\right)}^{\alpha }={(sig{\left(p_{i1}\right)}^{\alpha },sig{\left(p_{i2}\right)}^{\alpha },\cdots ,sig{\left(p_{in}\right)}^{\alpha })}^{\mathrm{T}}$. $\parallel ·\parallel$ and ${\parallel ·\parallel }_1$ indicate the Euclidean norm and 1 norm, respectively. For $A\in {\mathbb{R}}^{N\times N}$, $\lambda$ stands for its the smallest non-zero eigenvalue.

2. Preliminaries and Model Description

Before studying the problem of FTPCC, we give some important preliminaries.

2.1. Graph Theory

A directed graph can be represented by $\mathcal{G}=(\mathcal{V},\mathcal{E},A)$, where $\mathcal{V}=\left\{v_1,\cdots ,v_N\right\}$ is the set of edges and $A=\left(a_{pq}\right)\in {\mathbb{R}}^{N\times N}$ is the adjacency matrix of $\mathcal{G}$. An edge of $\mathcal{G}$ is denoted by $e_{pq}=(v_p,v_q)\in \mathcal{E}$; if $e_{pq}\in \mathcal{E}$, then $a_{pq}>0$, which can indicate that the pth node can receive the qth node’s information, i.e., $e_{pq}\in \mathcal{E}\iff a_{pq}>0$; otherwise, $e_{pq}\notin \mathcal{E}\iff a_{pq}=0$. $N_p=\left\{v_q\in \mathcal{V}:(v_p,v_q)\in \mathcal{E}\right\}$ represents all neighbours of the pth node. The Laplacian matrix $L=\left(l_{pq}\right)\in {\mathbb{R}}^{N\times N}$ is defined as $l_{pq}=-a_{pq}(p\ne q)$, $l_{pq}={\sum }_{q=1}^N{a}_{pq}$ and ${\sum }_{q=1}^N{l}_{pq}=0$, which is related to graph $\mathcal{G}$.

2.2. Partial Variable Stability Theory

Consider a classical n-dimensional ordinary differential nonautonomous equation as follows

$${\displaystyle \frac{dx}{dt}}=\mathsf{\Phi }(t,x)$$

(1)

where $\mathsf{\Phi }(t,x)\in C\left[{\mathbb{R}}^{+}\times {\mathbb{R}}^n,{\mathbb{R}}^n\right]$, ${\mathbb{R}}^{+}=\left[0,+\infty \right)$, $t\in {\mathbb{R}}^{+}$, $\mathsf{\Phi }(t,0)\equiv 0$, $x={(y^{\mathrm{T}},z^{\mathrm{T}})}^{\mathrm{T}}\in {\mathbb{R}}^n$, $y={(x_1,x_2,\cdots ,x_m)}^{\mathrm{T}}\in {\mathbb{R}}^m$, $z={(x_{m+1},x_{m+2},\cdots ,x_n)}^T\in {\mathbb{R}}^p$, $m+p=n$, $\parallel x\parallel ={\left({\sum }_{i=1}^n{x}_i^2\right)}^{1/2}$, $\parallel y\parallel ={\left({\sum }_{i=1}^m{x}_i^2\right)}^{1/2}$, $\parallel z\parallel ={\left({\sum }_{i=m+1}^n{x}_i^2\right)}^{1/2}$.

Definition 1

([39]). The trivial solution of nonautonomous system (1) with respects to partial variables y can reach y-stable if for any positive real numbers ϵ and $t_0$, there exists a $\delta >0$ which is dependent on ϵ and $t_0$ such that $\parallel x_0\parallel < \delta$. That is to say, $\parallel y(t,t_0,x_0)\parallel < ϵ$ holds for all $t\ge t_0$, and then MASs can achieve finite-time consensus.

2.3. Problem Statement

Without the generality, we consider the first-order MASs. Furthermore, the number of leaders is 1, and the number of followers is N. The leader $x_0$ is an isolated agent, and the specific behaviour ia as follows

$${\stackrel{.}{x}}_0\left(t\right)=\varphi \left(x_0\left(t\right)\right),$$

(2)

where $x_0={(x_{01},x_{02},\cdots ,x_{0n})}^T\in {\mathbb{R}}^n$, $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a continuous mapping, and $\varphi \left(x_0\right)={({\varphi }_1\left(x_0\right),{\varphi }_2\left(x_0\right),\cdots ,{\varphi }_n\left(x_0\right))}^T\in {\mathbb{R}}^n$.

The ith follower’s dynamic behaviour can be explained by

$${\stackrel{.}{x}}_i\left(t\right)=\varphi \left(x_i\left(t\right)\right)+u_i\left(t\right),$$

(3)

where $x_i={(x_{i1},x_{i2},\cdots ,x_{in})}^{\mathrm{T}}\in {\mathbb{R}}^n$, $i=1,2,\cdots ,N$, and $\varphi \left(x_i\right)={({\varphi }_1\left(x_i\right),{\varphi }_2\left(x_i\right),\cdots ,{\varphi }_n\left(x_i\right))}^T\in {\mathbb{R}}^n$. $u_i\left(t\right)$ represents the consensus protocol of the ith follower.

The flexible consensus protocol of this paper is designed as follows

$$u_i\left(t\right)=-k_1\left[sig{\left({\sum }_{j=1}^N{a}_{ij}\mathsf{\Gamma }(x_i-x_j)\right)}^{\alpha }\right]-k_2\left[sig{\left(b_i\mathsf{\Gamma }(x_i-x_0)\right)}^{\beta }\right],$$

(4)

where $k_1>0$, $k_2>0$ are coupling strengths. The matrix $\mathsf{\Gamma }=diag(r_1,r_2,\cdots ,r_n)$ denotes the inner coupling matrix of the MASs, and $r_i\ge 0$. $b_i>0$, if the ith follower can receive the information from the leader; otherwise, $b_i=0$.

Remark 1.

The protocol (4) is called flexible because the selection of α and β is free. If $\alpha =\beta$ and $\mathsf{\Gamma }=E$, (4) turns into the special protocol in [40], which can achieve fully distributed adaptive FTC for nonlinear MASs with uncertainty.

Remark 2.

$b_i=0$ for any $i=1,2,\cdots ,N$, if the MAS is a leaderless system. Consequently, the protocol given in (4) becomes the consensus protocol in [41]. Considering the second-order leaderless system, if the velocity state information is added into (4), similarly with [42], the resulting protocol can solve the FTPCC problem for the second-order MASs.

Assumption 1.

The nonlinear function $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies

$${(y-y^{\prime })}^T\mathsf{\Lambda }(\varphi \left(y\right)-\varphi \left(y^{\prime }\right))\le {ϵ}_1{(y-y^{\prime })}^T\mathsf{\Lambda }(y-y^{\prime }),$$

(5)

in which ${ϵ}_1>0$, $\mathsf{\Lambda }=diag(\stackrel{l}{\overbrace{1,1,\cdots ,1}},0,0,\cdots ,0)$ and $y\in {\mathbb{R}}^n$, $y^{\prime }\in {\mathbb{R}}^n$.

Remark 3.

Compared with the Lipschitz condition in [43], Assumption 1 is a weaker condition, which is used to analyze PCC problems by some scholars in [25,26,27,28].

Definition 2.

The Kronecker product of $A(p\times q)$ and $B(m\times n)$ is denoted $A\otimes B$ and is a $pm\times qn$ matrix defined by

$$A\otimes B\triangleq \left[\begin{array}{cccc}{a}_{11}B& a_{12}B& \dots & a_{1q}B\\ a_{21}B& \dots & & \\ \dots & & & \\ a_{p1}B& \dots & & a_{pq}B\end{array}\right].$$

Next, we need some other lemmas to analyze FTPCC better.

Lemma 1

([44]). For matrices $Q_1$, $Q_2$, $Q_3$, and $Q_4$ with appropriate dimensions, the Kronecker product ⊗ which satisfies:

(i) 

$(Q_1+Q_2)\otimes Q_3=Q_1\otimes Q_3+Q_2\otimes Q_3;$

(ii) 

$(Q_1\otimes Q_2)(Q_3\otimes Q_4)=(Q_1{Q}_3\otimes Q_2{Q}_4);$

(iii) 

${(Q_1\otimes Q_2)}^{\mathrm{T}}=Q_1^{\mathrm{T}}\otimes Q_2^{\mathrm{T}};$

(iv) 

$\left(k{Q}_1\right)\otimes Q_2=Q_1\otimes \left(k{Q}_2\right),$ where k is a constant.

Lemma 2

([26]). Suppose that $M_1=\left(a_{ij}\right)\in {\mathbb{R}}^{N\times N}$, $M_2=\left(b_{ij}\right)\in {\mathbb{R}}^{n\times n}$. Then, there is a $nN$ order matrix $Q=Q_1{Q}_2\cdots Q_s$ which satisfies

$$Q(M_1\otimes M_2)Q^{-1}=M_2\otimes M_1,$$

(6)

where the matrix $Q_i$ is the first class of the elementary row transformation matrix, $i=1,2,\cdots ,s$.

Lemma 3

([45]). Supposing that the Lyapunov function $V\left(x\right)$ is defined on an origin’s neighbourhood U, $V\left(x\right)$ satisfies $\stackrel{.}{V}\left(x\right)\le -c{V}^{\alpha }\left(x\right)+kV\left(x\right)$, where $0<\alpha <1$, $c>0$, $k>0$. As a result, the origin is finite-time stable and its upper bound of settling time is ${\displaystyle \frac{ln(1-\frac{k}{c}V{\left(0\right)}^{1-\alpha })}{k(\alpha -1)}}$.

Lemma 4

([46]). Let ${\eta }_1,{\eta }_2,\cdots ,{\eta }_s\ge 0$ and $0<q\le 1$. Then, ${\sum }_{i=1}^s{\eta }_i^q\ge {\left({\sum }_{i=1}^s{\eta }_i\right)}^q$.

3. Main Result

3.1. FTPCC Without Bounded Disturbances

Under the flexible consensus protocol (4), the dynamic behaviour of the ith follower of MASs (3) is

$$\begin{array}{cc}\hfill {\dot{x}}_i\left(t\right)=& \varphi \left(x_i\left(t\right)\right)-k_1\left[sig{(\sum _{j=1}^N{a}_{ij}\mathsf{\Gamma }(x_i-x_j))}^{\alpha }\right]\hfill \\ & -k_2\left[sig{\left(b_i\mathsf{\Gamma }(x_i-x_0)\right)}^{\beta }\right],\hfill \end{array}$$

(7)

where $i=1,2,\cdots ,N$.

Taking the ith state error as $e_i\left(t\right)=x_i\left(t\right)-x_0\left(t\right)$, the error system can be written as

$$\begin{array}{cc}\hfill {\dot{e}}_i\left(t\right)=& \varphi \left(x_i\left(t\right)\right)-\varphi \left(x_0\left(t\right)\right)-k_1\left[sig{(\sum _{j=1}^N{l}_{ij}\mathsf{\Gamma }{e}_j(t))}^{\alpha }\right]\hfill \\ & -k_2\left[sig{\left(b_i\mathsf{\Gamma }{e}_i\left(t\right)\right)}^{\beta }\right],\hfill \end{array}$$

(8)

where $i=1,2,\cdots ,N$.

According to (8), the compact form of the error system is

$$\stackrel{.}{e}\left(t\right)=\mathsf{\Phi }\left(e\left(t\right)\right)-k_{1}sig{(L\otimes \mathsf{\Gamma }e\left(t\right))}^{\alpha }-k_{2}sig{(B\otimes \mathsf{\Gamma }e\left(t\right))}^{\beta },$$

(9)

where

$$\begin{array}{cc}& e\left(t\right)={(e_1{\left(t\right)}^{\mathrm{T}},e_2{\left(t\right)}^{\mathrm{T}},\cdots ,e_N{\left(t\right)}^{\mathrm{T}})}^{\mathrm{T}}\in {\mathbb{R}}^{Nn},\hfill \\ & e_i\left(t\right)={(e_{i1}\left(t\right),e_{i2}\left(t\right),\cdots ,e_{in}\left(t\right))}^{\mathrm{T}}\in {\mathbb{R}}^n,\hfill \end{array}$$

(10)

and

$$\mathsf{\Phi }\left(e\left(t\right)\right)={({\mathsf{\Phi }}_1{\left(t\right)}^{\mathrm{T}},{\mathsf{\Phi }}_2{\left(t\right)}^{\mathrm{T}},\cdots ,{\mathsf{\Phi }}_N{\left(t\right)}^{\mathrm{T}})}^{\mathrm{T}},$$

(11)

where ${\mathsf{\Phi }}_i\left(t\right)=\varphi \left(x_i\left(t\right)\right)-\varphi \left(x_0\left(t\right)\right)$.

Then, by making use of the method of the permutation matrix, we obtain a new error system as

$$\stackrel{.}{\tilde{e}}\left(t\right)=\tilde{\mathsf{\Phi }}\left(\tilde{e}\left(t\right)\right)-k_{1}sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_{2}sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta },$$

(12)

in which

$$\begin{array}{cc}& \tilde{e}\left(t\right)={({\tilde{e}}_1{\left(t\right)}^{\mathrm{T}},{\tilde{e}}_2{\left(t\right)}^{\mathrm{T}},\cdots ,{\tilde{e}}_n{\left(t\right)}^{\mathrm{T}})}^{\mathrm{T}}\in {\mathbb{R}}^{nN},\hfill \\ & {\tilde{e}}_j\left(t\right)={({\tilde{e}}_{1j}\left(t\right),{\tilde{e}}_{2j}\left(t\right),\cdots ,{\tilde{e}}_{Nj}\left(t\right))}^{\mathrm{T}},\hfill \\ & {\tilde{e}}_{ij}\left(t\right)=x_{ij}\left(t\right)-x_{0j}\left(t\right),j=1,2,\cdots ,N,\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}& \tilde{\mathsf{\Phi }}\left(\tilde{e}\left(t\right)\right)={({\tilde{\varphi }}_1{\left(t\right)}^{\mathrm{T}},{\tilde{\varphi }}_2{\left(t\right)}^{\mathrm{T}},\cdots ,{\tilde{\varphi }}_n{\left(t\right)}^{\mathrm{T}})}^{\mathrm{T}},\hfill \\ & {\tilde{\varphi }}_j{\left(t\right)}^{\mathrm{T}}={({\varphi }_j\left(x_1\left(t\right)\right),{\varphi }_j\left(x_2\left(t\right)\right),\cdots ,{\varphi }_j\left(x_N\left(t\right)\right))}^{\mathrm{T}}-{\varphi }_j\left(x_0\left(t\right)\right)1_N.\hfill \end{array}$$

(14)

Remark 4.

In the proof of the general FTC problems in [14,15], the corresponding Lyapunov function always relies on the same agent error vector $e_i\left(t\right)$. However, to analyze the FTPCC problem, we need to construct the Lyapunov function which merely depends on the same component error system in this paper.

Definition 3

([47]). If the solution of (2) and (3) with any initial states satisfies

$$\underset{t\to T_{\ast }}{lim}{\sum }_{q=1}^l\parallel {\tilde{e}}_q\left(t\right)\parallel =0,$$

(15)

then the FTPCC with respect to the first l components is said to be achieved, in which ${\tilde{e}}_q\left(t\right)=({\tilde{e}}_{1q}\left(t\right)$, ${\tilde{e}}_{2q}\left(t\right),\cdots ,{\tilde{e}}_{Nq}{\left(t\right))}^{\mathrm{T}}={(x_{1q}\left(t\right)-x_{0q}\left(t\right),x_{2q}\left(t\right)-x_{0q}\left(t\right),\cdots ,x_{Nq}\left(t\right)-x_{0q}\left(t\right))}^{\mathrm{T}}\in {\mathbb{R}}^N$ and $T_{\ast }$ is defined in Lemma 3.

Remark 5.

From the definition of the FTPCC, if $l=n$, it means that all components reach FTC. Then, FTPCC is turned into FTC. Consequently, it is easy to obtain that FTPCC is more general than FTC.

Now, a main theorem about FTPCC is given.

Theorem 1.

Supposing that Assumption 1 holds. Let $\alpha \in (0,1)$, $\beta \in (0,1)$, $\gamma =min\left\{\alpha ,\beta \right\}$, $R=\underset{i\in \left\{1,2,\cdots ,n\right\}}{min}\left\{r_i^{\gamma }|r_i>0\right\}$. Then, FTPCC in systems (2) and (3) with protocol (4) is achieved.

Proof. 

Take the Lyapunov function candidate as follows

$$V\left(t\right)={\displaystyle \frac{1}{2}}{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\tilde{e}\left(t\right),$$

(16)

where $\mathsf{\Lambda }=diag(\stackrel{l}{\overbrace{1,1,\cdots ,1}},0,0,\cdots ,0)$. Combining with vector form, one has

$$\begin{array}{cc}\hfill V\left(t\right)& ={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}{\tilde{e}}_1^{\mathrm{T}}\left(t\right)& {\tilde{e}}_2^{\mathrm{T}}\left(t\right)& \cdots & {\tilde{e}}_n^{\mathrm{T}}\left(t\right)\end{array}\right)\mathsf{\Lambda }{\left(\begin{array}{cccc}{\tilde{e}}_1^{\mathrm{T}}\left(t\right)& {\tilde{e}}_2^{\mathrm{T}}\left(t\right)& \cdots & {\tilde{e}}_n^{\mathrm{T}}\left(t\right)\end{array}\right)}^{\mathrm{T}}\hfill \\ & ={\displaystyle \frac{1}{2}}{\sum }_{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right){\tilde{e}}_j\left(t\right)\hfill \\ & ={\displaystyle \frac{1}{2}}{\sum }_{j=1}^l\left(\begin{array}{cccc}{\tilde{e}}_{j1}\left(t\right)& {\tilde{e}}_{j2}\left(t\right)& \cdots & {\tilde{e}}_{jN}\left(t\right)\end{array}\right){\left(\begin{array}{cccc}{\tilde{e}}_{j1}\left(t\right)& {\tilde{e}}_{j2}\left(t\right)& \cdots & {\tilde{e}}_{jN}\left(t\right)\end{array}\right)}^{\mathrm{T}}\hfill \\ & ={\displaystyle \frac{1}{2}}{\sum }_{j=1}^l{\sum }_{i=1}^N{\tilde{e}}_{ji}\left(t\right){\tilde{e}}_{ji}\left(t\right)\hfill \end{array}$$

(17)

By Assumption 1, we have

$$\begin{array}{cc}\hfill \stackrel{.}{V}\left(t\right)& ={\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\stackrel{.}{\tilde{e}}\left(t\right)\hfill \\ & ={\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }(\tilde{\mathsf{\Phi }}(\tilde{e}(t))-k_{1}sig{(\mathsf{\Gamma }\otimes L\tilde{e}(t))}^{\alpha }-k_{2}sig{(\mathsf{\Gamma }\otimes B\tilde{e}(t))}^{\beta })\hfill \\ & ={\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\tilde{\mathsf{\Phi }}\left(\tilde{e}\left(t\right)\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & \le {ϵ}_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\tilde{e}\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & =2{ϵ}_{1}V\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }.\hfill \end{array}$$

(18)

For the last term of (18), we can obtain

$$\begin{array}{cc}& \begin{array}{cc}& k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \end{array}\hfill \\ & \begin{array}{cc}& =k_2\left(\begin{array}{cccccccc}{\tilde{e}}_1^{\mathrm{T}}\left(t\right)\hfill & {\tilde{e}}_2^{\mathrm{T}}\left(t\right)\hfill & \dots \hfill & {\tilde{e}}_l^{\mathrm{T}}\left(t\right)\hfill & 0\hfill & 0\hfill & \dots \hfill & 0\hfill \end{array}\right)sig{\left(\left(\begin{array}{c}{r}_{1}B{\tilde{e}}_1^{\mathrm{T}}\left(t\right)\\ r_{2}B{\tilde{e}}_2^{\mathrm{T}}\left(t\right)\\ ⋮\\ r_{n}B{\tilde{e}}_n^{\mathrm{T}}\left(t\right)\end{array}\right)\right)}^{\beta }\hfill \\ & =k_2\left(\begin{array}{cccccccc}{\tilde{e}}_1^{\mathrm{T}}\left(t\right)\hfill & {\tilde{e}}_2^{\mathrm{T}}\left(t\right)\hfill & \cdots \hfill & {\tilde{e}}_l^{\mathrm{T}}\left(t\right)\hfill & 0\hfill & 0\hfill & \cdots \hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}sig{\left(r_{1}B{\tilde{e}}_1^{\mathrm{T}}\left(t\right)\right)}^{\beta }\\ sig{\left(r_{2}B{\tilde{e}}_2^{\mathrm{T}}\left(t\right)\right)}^{\beta }\\ ⋮\\ sig\left(r_{n}B{\tilde{e}}_n^{\mathrm{T}}{\left(t\right)}^{\beta }\right.\end{array}\right)\hfill \\ & =k_2\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)sig\left(r_{j}B{\tilde{e}}_j^{\mathrm{T}}{\left(t\right)}^{\beta }\right)\hfill \\ & =k_2\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)sig{\left(r_j\left(\begin{array}{cccc}{b}_1& & & \\ & b_2& & \\ & & \ddots & \\ & & & b_n\end{array}\right)\left(\begin{array}{c}{\tilde{e}}_{j1}\left(t\right)\\ {\tilde{e}}_{j2}\left(t\right)\\ ⋮\\ {\tilde{e}}_{jN}\left(t\right)\end{array}\right)\right)}^{\beta }\hfill \end{array}\hfill \\ & \begin{array}{cc}& =k_2\sum _{j=1}^l{r}_j^{\beta }\left(\begin{array}{cccc}{\tilde{e}}_{j1}\left(t\right)\hfill & {\tilde{e}}_{j2}\left(t\right)\hfill & \cdots \hfill & {\tilde{e}}_{jN}\left(t\right)\hfill \end{array}\right)\left(\begin{array}{c}sig{\left(b_1{\tilde{e}}_{j1}\left(t\right)\right)}^{\beta }\\ sig{\left(b_2{\tilde{e}}_{j2}\left(t\right)\right)}^{\beta }\\ ⋮\\ sig{\left(b_N{\tilde{e}}_{jN}\left(t\right)\right)}^{\beta }\end{array}\right)\hfill \\ & =k_2\sum _{j=1}^l{r}_j^{\beta }\sum _{i=1}^N{\tilde{e}}_{ji}\left(t\right)sig{\left(b_i{\tilde{e}}_{ji}\left(t\right)\right)}^{\beta }.\hfill \end{array}\hfill \end{array}$$

(19)

For the second term of (18), we also can obtain

$$\begin{array}{cc}& k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes \lambda E\tilde{e}\left(t\right))}^{\alpha }\hfill \\ \hfill =& k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)sig{\left(r_{j}L{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }-k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)sig{\left(r_j\lambda E{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }\hfill \\ \hfill =& k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\left[sig{\left(r_{j}L{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }-sig{\left(r_j\lambda E{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }\right]\hfill \\ \hfill =& k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\left[sig{\left(r_{j}L\right)}^{\alpha }sig{\left({\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }-sig{\left(r_j\lambda E\right)}^{\alpha }sig{\left({\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }\right]\hfill \\ \hfill =& k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)\left[sig{\left(r_{j}L\right)}^{\alpha }-sig{\left(r_j\lambda E\right)}^{\alpha }\right]sig{\left({\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }\hfill \\ \hfill =& k_1\sum _{j=1}^l{\tilde{e}}_j^{\mathrm{T}}\left(t\right)Hsig{\left({\tilde{e}}_j^{\mathrm{T}}\left(t\right)\right)}^{\alpha }\hfill \end{array}$$

(20)

where $H=\left(\begin{array}{ccc}sig{\left(r_j{l}_{11}\right)}^{\alpha }-sig{\left(r_j\lambda \right)}^{\alpha }& \cdots & sig{\left(r_j{l}_{1N}\right)}^{\alpha }\\ ⋮& & ⋮\\ sig{\left(r_j{l}_{N1}\right)}^{\alpha }& \cdots & sig{\left(r_j{l}_{NN}\right)}^{\alpha }-sig{\left(r_j\lambda \right)}^{\alpha }\end{array}\right)$; it is not difficult to find that all nondiagonal elements of H are non-positive.

If ${\tilde{e}}_{qp}\left(t\right)\ge 0$, then we can have

$${\tilde{e}}_{qp}\left(t\right)sig{\left({\tilde{e}}_{qp}\left(t\right)\right)}^{\alpha }={\tilde{e}}_{qp}\left(t\right)sgn\left({\tilde{e}}_{qp}\left(t\right)\right)|{\tilde{e}}_{qp}{\left(t\right)|}^{\alpha }={|{\tilde{e}}_{qp}\left(t\right)|}^{\alpha +1}.$$

(21)

If ${\tilde{e}}_{qp}\left(t\right)<0$, then we also have

$${\tilde{e}}_{qp}\left(t\right)sig{\left({\tilde{e}}_{qp}\left(t\right)\right)}^{\alpha }={\tilde{e}}_{qp}\left(t\right)sgn\left({\tilde{e}}_{qp}\left(t\right)\right)|{\tilde{e}}_{qp}{\left(t\right)|}^{\alpha }=-{\tilde{e}}_{qp}\left(t\right)|{\tilde{e}}_{qp}{\left(t\right)|}^{\alpha }={|{\tilde{e}}_{qp}\left(t\right)|}^{\alpha +1}.$$

(22)

Therefore, we can obtain

$${\tilde{e}}_{ji}\left(t\right)sig{\left({\tilde{e}}_{ji}\left(t\right)\right)}^{\alpha }\ge 0.$$

(23)

Combining (20) and (23), we have

$$k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }\ge k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes \lambda E\tilde{e}\left(t\right))}^{\alpha }.$$

(24)

Combining with (19) and (24), one obtains

$$\begin{array}{cc}\hfill \stackrel{.}{V}\left(t\right)& \le 2{ϵ}_{1}V\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & \le 2{ϵ}_{1}V\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes \lambda E\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & =2{ϵ}_{1}V\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes \lambda E\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & =2{ϵ}_{1}V\left(t\right)-k_1{\sum }_{j=1}^l{r}_j^{\alpha }{\sum }_{i=1}^N{\tilde{e}}_{ij}\left(t\right)sig{\left(\lambda {\tilde{e}}_{ij}\left(t\right)\right)}^{\alpha }\hfill \\ & \hspace{1em}\hspace{1em}-k_2{\sum }_{j=1}^l{r}_j^{\beta }{\sum }_{i=1}^N{\tilde{e}}_{ji}\left(t\right)sig{\left(b_i{\tilde{e}}_{ji}\left(t\right)\right)}^{\beta }\hfill \\ & \le 2{ϵ}_{1}V\left(t\right)-{\sum }_{j=1}^l{r}_j^{\gamma }{\sum }_{i=1}^N{\tilde{e}}_{ij}\left(t\right)(k_{1}sig{\left(\lambda \right)}^{\alpha }+k_{2}sig{\left(b_i\right)}^{\beta })sig{\left({\tilde{e}}_{ij}\left(t\right)\right)}^{\gamma }\hfill \\ & \le 2{ϵ}_{1}V\left(t\right)-R{\sum }_{j=1}^l{\sum }_{i=1}^N{\tilde{e}}_{ij}\left(t\right)(k_{1}sig{\left(\lambda \right)}^{\gamma }+k_{2}sig{\left(b_i\right)}^{\gamma })sig{\left({\tilde{e}}_{ij}\left(t\right)\right)}^{\gamma }\hfill \\ & \le 2{ϵ}_{1}V\left(t\right)-RM{\sum }_{j=1}^l{\sum }_{i=1}^N{\tilde{e}}_{ij}\left(t\right)sig{\left({\tilde{e}}_{ij}\left(t\right)\right)}^{\gamma }\hfill \\ & =2{ϵ}_{1}V\left(t\right)-RM{\sum }_{j=1}^l{\sum }_{i=1}^N{\left({\tilde{e}}_{ij}{\left(t\right)}^2\right)}^{{\displaystyle \frac{\gamma +1}{2}}}\hfill \\ & \le 2{ϵ}_{1}V\left(t\right)-2^{{\displaystyle \frac{\gamma +1}{2}}}RMV{\left(t\right)}^{{\displaystyle \frac{\gamma +1}{2}}}.\hfill \end{array}$$

(25)

where $M=min\left\{k_{1}sig{\left(\lambda \right)}^{\gamma }+k_{2}sig{\left(b_i\right)}^{\gamma }\right\}>0$, ${ϵ}_1>0$, $2^{{\displaystyle \frac{\gamma +1}{2}}}RM>0$, $k_1$ and $k_2$ are defined in (4).

Let $k=2{ϵ}_1$, $c=RM{2}^{{\displaystyle \frac{\gamma +1}{2}}}>0$; based on Lemma 3, $V\left(t\right)$ will be zero in finite time ${\displaystyle \frac{2ln(1-\frac{k}{c}V{(0)}^{\frac{1-\gamma }{2}})}{k(\gamma -1)}}$, which means that ${\tilde{e}}_j\left(t\right)(j=1,2,\cdots ,l)$ will be zero in finite time. Therefore, the multi-agent systems (2) and (3) can achieve FTPCC. □

Remark 6.

Note that [47] comprehensively investigates the fixed-time partial component consensus problem for nonlinear MASs and the control protocols contain seven items. However, compared with the control protocol in [47], the protocol (4) designed in this paper is simpler and achieving FTPCC is easier.

3.2. FTPCC with Bounded Disturbances

In this subsection, the FTPCC problem of nonlinear MASs with bounded disturbances is investigated.

The leader’s dynamical behaviour is the same as in (2).

The ith follower agent’s dynamic behaviour is

$${\stackrel{.}{x}}_i\left(t\right)=\varphi \left(x_i\left(t\right)\right)+{\widehat{u}}_i\left(t\right)+d\left(x_i\left(t\right)\right).$$

(26)

where $i=1,2,\cdots ,N$. Except for ${\widehat{u}}_i\left(t\right)$ and $d\left(x_i\left(t\right)\right)$, the rest in (26) are the same as in (4). $d\left(x_i\right)={(d_1\left(x_i\right),d_2\left(x_i\right),\cdots ,d_n\left(x_i\right))}^{\mathrm{T}}$ is the disturbance of the ith follower. ${\widehat{u}}_i\left(t\right)$ is the consensus protocol which will be given later.

Assumption 2.

Assuming that the disturbance $d\left(x_i\left(t\right)\right)$ is continuous with respect to t and satisfies

$$\parallel d\left(x_i\left(t\right)\right)\parallel \le d_{max},$$

(27)

where $i=1,2,\cdots ,N$, $d_{max}>0$ is a positive constant.

In order to solve FTPCC problem with bounded disturbances, the consensus protocol ${\widehat{u}}_i\left(t\right)$ is proposed:

$$\begin{array}{cc}\hfill {\widehat{u}}_i\left(t\right)=& -k_1\left[sig{\left({\sum }_{j=1}^N{a}_{ij}\mathsf{\Gamma }(x_i-x_j)\right)}^{\alpha }\right]-k_2\left[sig{\left(b_i\mathsf{\Gamma }(x_i-x_0)\right)}^{\beta }\right]\hfill \\ & -dsign\left(\mathsf{\Gamma }(x_i-x_0)\right),\hfill \end{array}$$

(28)

where $k_1>0$, $k_2>0$, $d_{max}\le d$. The matrix $\mathsf{\Gamma }$ is defined in (4).

Similarly, one has

$$\begin{array}{cc}\hfill {\stackrel{.}{e}}_i\left(t\right)=& \varphi \left(x_i\left(t\right)\right)-\varphi \left(x_0\left(t\right)\right)-k_1\left[sig{\left({\sum }_{j=1}^N{l}_{ij}\mathsf{\Gamma }{e}_j\left(t\right)\right)}^{\alpha }\right]-k_2\left[sig{\left(b_i\mathsf{\Gamma }{e}_i\left(t\right)\right)}^{\beta }\right]\hfill \\ & -dsign\left(\mathsf{\Gamma }\left(e_i\left(t\right)\right)\right)+d\left(x_i\left(t\right)\right),\hfill \end{array}$$

(29)

where $i=1,2,\cdots ,N$.

Next, we can have

$$\begin{array}{cc}\hfill \stackrel{.}{e}\left(t\right)=\mathsf{\Phi }\left(e\left(t\right)\right)& -k_{1}sig{(L\otimes \mathsf{\Gamma }e\left(t\right))}^{\alpha }-k_{2}sig{(B\otimes \mathsf{\Gamma }e\left(t\right))}^{\beta }\hfill \\ & -dsign\left(\mathsf{\Gamma }\right(e\left(t\right)\left)\right)+D\left(x\right(t\left)\right),\hfill \end{array}$$

(30)

where

$$D\left(x\left(t\right)\right)={(d{\left(x_1\left(t\right)\right)}^{\mathrm{T}},d{\left(x_2\left(t\right)\right)}^{\mathrm{T}},\cdots ,d{\left(x_N\left(t\right)\right)}^{\mathrm{T}})}^{\mathrm{T}}\in {\mathbb{R}}^{Nn}.$$

(31)

Based on the permutation matrix method, we can obtain the new $nN$ order error system

$$\begin{array}{cc}\hfill \stackrel{.}{\tilde{e}}\left(t\right)=\tilde{\mathsf{\Phi }}\left(\tilde{e}\left(t\right)\right)& -k_{1}sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_{2}sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & -dsign\left(\mathsf{\Gamma }\tilde{e}\left(t\right)\right)+\tilde{D}\left(\tilde{x}\left(t\right)\right)\hfill \end{array}$$

(32)

in which

$$\begin{array}{cc}& \tilde{D}\left(\tilde{x}\left(t\right)\right)={({\tilde{d}}_1{\left(t\right)}^{\mathrm{T}},{\tilde{d}}_2{\left(t\right)}^{\mathrm{T}},\cdots ,{\tilde{d}}_n{\left(t\right)}^{\mathrm{T}})}^{\mathrm{T}},\hfill \\ & {\tilde{d}}_j{\left(t\right)}^{\mathrm{T}}={(d_j\left(x_1\left(t\right)\right),d_j\left(x_2\left(t\right)\right),\cdots ,d_j\left(x_N\left(t\right)\right))}^{\mathrm{T}},\hfill \end{array}$$

(33)

$\stackrel{.}{e}\left(t\right)$, $\mathsf{\Phi }\left(e\right(t\left)\right)$, $\stackrel{.}{\tilde{e}}\left(t\right)$ and $\tilde{\mathsf{\Phi }}\left(\tilde{e}\left(t\right)\right)$ are the same as in the previous subsection.

Theorem 2.

Supposing that Assumptions 1 and 2 hold. Let $\alpha \in (0,1)$, $\beta \in (0,1)$, $\gamma =min\left\{\alpha ,\beta \right\}$, $R=\underset{i\in \left\{1,2,\cdots ,n\right\}}{min}\left\{r_i^{\gamma }|r_i>0\right\}$. Then, FTPCC with bounded disturbances in systems (2) and (26) with protocol (28) is achieved.

Proof. 

Choose the following Lyapunov function

$$V_1\left(t\right)={\displaystyle \frac{1}{2}}{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\tilde{e}\left(t\right),$$

(34)

Substituting (32) into (34), we have

$$\begin{array}{cc}\hfill {\stackrel{.}{V}}_1\left(t\right)& ={\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }\stackrel{.}{\tilde{e}}\left(t\right)\hfill \\ & \le 2{ϵ}_1{V}_1\left(t\right)-k_1{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes L\tilde{e}\left(t\right))}^{\alpha }-k_2{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }sig{(\mathsf{\Gamma }\otimes B\tilde{e}\left(t\right))}^{\beta }\hfill \\ & \hspace{1em}\hspace{1em}+{\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }(-dsign\left(\mathsf{\Gamma }\tilde{e}\left(t\right)\right)+\tilde{D}\left(\tilde{x}\left(t\right)\right)).\hfill \end{array}$$

(35)

For the last term of (35), one obtains

$${\tilde{e}}^{\mathrm{T}}\left(t\right)\mathsf{\Lambda }(-dsign\left(\mathsf{\Gamma }\tilde{e}\left(t\right)\right)+\tilde{D}\left(\tilde{x}\left(t\right)\right))\le (-d+d_{max}){\sum }_{j=1}^l{\parallel {\tilde{e}}_j\left(t\right)\parallel }_1.$$

(36)

Because $d\ge d_{max}$, one obtains $-d+d_{max}\le 0$. That is to say, the last term of (35) is nonpositive.

The proof for the second term and the third term in (35) is similar to the derivation process of (24) and (19) in Theorem 1. For the second term of (35), one can obtain similar inference to (19), and likewise for the third term of (35), one can obtain similar inference to (24), so that one can obtain

$${\stackrel{.}{V}}_1\left(t\right)\le 2{ϵ}_1{V}_1\left(t\right)-2^{{\displaystyle \frac{\gamma +1}{2}}}RM{V}_1{\left(t\right)}^{{\displaystyle \frac{\gamma +1}{2}}},$$

(37)

where M is defined in (25).

Based on (37) and Lemma 3, one can obtain that MASs (2) and (26) can reach FTPCC and the upper bound of settling time is ${\displaystyle \frac{2ln(1-\frac{k}{c}{V}_1{(0)}^{\frac{1-\gamma }{2}})}{k(\gamma -1)}}$, where $k=2{ϵ}_1$ and $c=RM{2}^{{\displaystyle \frac{\gamma +1}{2}}}>0$. □

Remark 7.

To compensate the effect of bounded disturbances, we add the term $-dsign(x_i-x_0)$ to consensus protocol (4) so that we obtain a new flexible protocol (28) which can be used to achieve FTPCC for MASs with bounded disturbances. This method was also used in [22].

Remark 8.

Theorems 1 and 2 are the results obtained under the directed topology. Furthermore, it is easy to prove that two theorems are also correct for FTPCC under undirected topology.

4. Numerical Examples

In the section, we give some numerical examples to display the effectiveness of Theorem 1 and Theorem 2, respectively. We consider the MASs which is made up of a leader and four followers. Furthermore, the nonlinear function $h(·)$ can be described by Chua’s circuit in [28].

$$\varphi \left(x_i\left(t\right)\right)=\left(\begin{array}{c}{x}_{i2}-3x_{i1}+\left[|x_{i1}+1|-|x_{i1}-1|\right]\\ -x_{i1}+x_{i2}-x_{i2}{x}_{i3}^2\\ 2x_{i2}\end{array}\right).$$

(38)

The specific network communication topology is given by Figure 1. The practical application can be seen as a fleet of five aircraft flying in the air. The leader can be regarded as the host of the flying fleet, and the four followers can be regarded as the secondary aircraft of the flying fleet. When the flying fleet presents various formation performances, the five aircraft need to be consistent in the forward direction of the displacement, but do not need to be consistent in other directions.

4.1. Example 1

For simplicity, the matrix $\mathsf{\Gamma }=diag(1,1,0)$. $b_2=1$, $b_i=0$, $i=1,3,4$. The corresponding Laplacian matrix is $L=\left(\begin{array}{cccc}1& 0& -1& 0\\ 0& 0& 0& 0\\ 0& -1& 1& 0\\ 0& -1& 0& 1\end{array}\right).$ The initial values are taken as $\alpha =0.7$, $\beta =0.75$, $k_1=0.2$, $k_2=0.4$, ${ϵ}_1=0.3$, $x_0\left(0\right)={\left[0,0,2\right]}^{\mathrm{T}}$, $x_1\left(0\right)={\left[-1,1.2,-3.5\right]}^{\mathrm{T}}$, $x_2\left(0\right)={\left[1.3,1,-3\right]}^{\mathrm{T}}$, $x_3\left(0\right)={\left[-1.8,1.4,-0.3\right]}^{\mathrm{T}}$, and $x_4\left(0\right)={\left[0.5,0.8,1.5\right]}^{\mathrm{T}}$.

From Figure 2 and Figure 3, it is easy to find that the first two components of the whole MAS can reach finite-time consensus. However, Figure 4 indicates that the third component does not achieve consensus within 3 s and the small figure in Figure 4 shows that the third component also cannot achieve consensus in a longer time. Consequently, the whole multi-agent system achieves FTPCC. After the first two components achieve partial component consensus, the system achieves partial stability. At this point, since the state error is zero, there is no longer a need for control input, which can save the resource utilization of the controller, reducing certain costs and resource losses to a certain extent. Furthermore, we calculate that the settling time is $22.46$ s and observe that the first component and the second component can achieve FTPCC in $T_{\ast }=0.6$ s. As a result, the example indicates that Theorem 1 is right.

4.2. Example 2

This example is based on Example 1. We choose the bounded disturbances $d\left(x_i\left(t\right)\right)={\left[0.01cos\left(x_{i1}\left(t\right)\right),0.01sin\left(x_{i2}\left(t\right)\right),0.01cos\left(x_{i3}\left(t\right)\right)\right]}^{\mathrm{T}}$, where $i=1,2,3,4$. $d=0.01$ is taken in this example. The initial states, the communication topology, the parameters, and the nonlinear function are the same as in Example 1.

From Figure 5 and Figure 6, we can also find that the first two components can reach consensus. After reaching FTPCC for the system, the control protocol does not work, reducing costs and resources to a certain extent. However, Figure 7 indicates that the third component cannot reach FTPCC within 3 s and the small figure in Figure 7 shows that the third component cannot reach consensus even after a longer time. Furthermore, we calculate that the settling time is $22.46$ s and observe that the first component and the second component can achieve FTPCC in $T_{\ast }=0.8$ s. Therefore, this example illustrates that Theorem 2 is right.

Based on the above examples, we can compare their results in

Table 1. Comparison.

	Example 1	Example 2
Disturbance	with disturbance	without disturbance
PCC	the first two components	the first two components
Time of PCC	0.6 s	0.8 s

. We can find that under the corresponding control protocol, MASs both with or without external interference can achieve PCC within a finite time. Additionally, providing external interference to the system will prolong the time required to achieve FTPCC. Meanwhile, the discussions above demonstrate the validity of the theorem.

5. Conclusions

This paper delves into the FTPCC problem for nonlinear first-order multi-agent systems (MASs), both with and without bounded disturbances. By employing permutation matrices and finite-time stability theory, FTPCC is transformed into the stability analysis of a new error system for identical components. Two adaptable protocols are then introduced to accomplish FTPCC in nonlinear MASs. Numerical examples are presented to demonstrate the accuracy of the theoretical findings. In real-world scenarios, not all multi-agent systems necessitate full consensus among all components. For instance, in an airplane maintaining a specific flight attitude, the forward displacement components remain consistent, while displacements in other directions may vary. Hence, this paper offers a broader perspective on consensus phenomena.

It is noteworthy that several intriguing challenges remain unresolved, such as the partial component consensus (PCC) for fractional-order MASs and MASs subject to stochastic disturbances. Additionally, while this paper focuses on finite-time partial component consensus, practical applications may encounter interference factors like random noise. Therefore, future research could build upon these results to explore fixed-time partial component consensus. These complex issues will be addressed in subsequent work.