Observer-based fixed-time consensus tracking for second-order multi-agent systems with disturbances

Abstract

This paper researches into a fixed-time consensus tracking problem for a class of second-order multi-agent systems (MASs) subject to external disturbances based on a fixed-time observer. First, we adopt a state observer to change the states of each agent from unknown to known. The observer is designed so that observed states and the leader’s state reach consensus. Second, according to the designed controller, states of each agent and observed states achieve consensus, namely leader-following consensus is realized and the setting time of which is obtained without depending on initial states. By employing Lyapunov stability theory, some sufficient criteria are obtained to achieve fixed-time consensus tracking for second-order MASs with disturbances. Finally, the validity of the design is verified by numerical arithmetic examples.

Introduction

Enlightened by biological behavior, the topic of cooperative control of multi-agent systems (MASs) has drown lots of interest, which has higher reliability, greater maneuverability, and efficiency than individual agent. As these problems are applied in various domains such as mobile robot (Ren et al., 2022; Sharifi, 2022), consensus (Han and Zheng, 2021; Zhao et al., 2020), and formation (Liu et al., 2019, 2020), it has drawn a tremendous amount of attention. Consensus problems play a critical role in collaborative control. The basic challenge in considering consensus is to conceive suitable protocols that make agents realize common states through local information interactions between neighboring agents. Some productive and significant results have been accomplished through researches in this direction (Ni and Cheng, 2010; Qin et al., 2017).

According to the presence or absence of a leader in the considered MASs, consensus protocols can be divided into two kinds. One is leader-following consensus protocol and the other is leaderless consensus protocol. Leader-following consensus can as well be referred to as consensus tracking. So far, there are plenty of fruitful results in this area. In the work by Zhao et al. (2018), an event-triggered protocol without the need for constant communication among followers was conceived to tackle consensus tracking problems for second-order nonlinear MASs. Based on the observer, problems of consensus tracking for nonlinear MASs were looked into in the work by Zhang et al. (2018), where MASs have state constraints and unknown disturbances. In the work by Gu et al. (2020), a sliding mode controller was constructed by introducing the statistical information of channel fading to reach consensus tracking for second-order MASs. In the work by Yoo (2018), recursive design approaches were proposed to devise local general adaptive controllers to tackle distributed consensus tracking problems for switched nonlinear MASs with directed graph.

Convergence rate is the key to determine whether consensus protocols are good or not. Asymptotic consensus with infinite convergence time was first introduced, and then finite-time consensus was raised. In contrast to asymptotic consensus, it is not difficult to derive that finite-time consensus can deliver better anti-interference performances as well as faster convergence rates. In the work by Li et al. (2011), the finite-time consensus problem in the absence of a leader and the finite-time observer-based consensus problem with external disturbances were examined for second-order MASs, respectively. The robust nonlinear finite-time consensus problem under undirected topology was addressed for MASs with external disturbances in the work by Zuo and Tie (2016). Using backstepping design method to come up with the event-based finite-time controller to achieve consensus for second-order MASs with input delay was addressed in the work by Ran et al. (2020). In the work by Tian et al. (2018), for second-order MASs that possess unknown velocities and external disturbances, robust finite-time consensus tracking problems were tackled using a high-gain observer.

In recent years, the terminal sliding mode control method has been widely used to study the fixed-time consensus tracking problems of MASs (Gao et al., 2020; Wang et al., 2019; Zhao et al., 2017). What is particularly noteworthy is that despite favorable performances of finite-time consensus, convergence times hinge heavily on initial states of the agents. The inability to acquire information about initial conditions of agents in advance leads to limit practical applications, which is considered to be the biggest drawback. Therefore, the notion of fixed-time consensus was born and was first cited in the work by Polyakov (2011) to be used for the purpose of surmounting the above drawbacks. In the work by Zou et al. (2019), fixed-time consensus problems for heterogeneous nonlinear MASs in the presence and absence of a leader was considered, respectively. Problems of fixed-time consensus for MASs under directed graphs with linear and nonlinear state measurements were addressed in the work by Zhang and Jia (2015). In the work by Ning et al. (2017), fixed-time consensus problems for first-order MASs with discontinuous nonlinear intrinsic dynamics were treated. In the work by Han et al. (2021b), two distributed bipartite average tracking protocols were constructed to resolve problems of finite-time and fixed-time consensus with bounded disturbance. The problem of fixed-time consensus tracking for second-order MASs based on terminal sliding mode control was first addressed in the work by Zuo (2015). The sliding mode surface is designed as a terminal sliding mode form, which can ensure that the state converges in finite time after reaching the sliding mode surface.

Under the influence of above discussions, based on a fixed-time observer, this paper inquires into the fixed-time consensus tracking problem for second-order MASs subject to external disturbances using the terminal sliding mode control and fixed-time stability theory. The contributions of this paper are threefold. First, unlike the works by Ran et al. (2020), Zou et al. (2019), Zhang and Jia (2015), Ning et al. (2017), this paper takes into account inevitable external disturbances. In real life and applications, external disturbances exist everywhere, which makes it indispensable to consider fixed-time consensus tracking problems under the influence of disturbances. Second, although there are many results on the finite-time consensus and fixed-time consensus, as far as we know, there is few result on the fixed-time consensus of MASs based on the distributed observer. Compared with the works by Zuo and Tie (2016), Zou et al. (2019), Zhang and Jia (2015), Ning et al. (2017), this paper designs distributed fixed-time observers to change the states of each agent from unknown to known. It can observe states of leader to ensure that MASs can reach fixed-time consensus. And it only relies on information between neighboring agents. Even if the agent doesn’t link to the leader directly, the state information of the agent can be observed.

The remainder of this paper is organized by five parts. Section “Preliminaries” proposes preliminaries and system description. Section “Main results” discusses the problem of fixed-time consensus tracking problem for second-order MASs. Simulations are performed in section “Simulations” and it summarizes in section “Conclusion.”

Notations: Let ${\mathbb{R}}^n$ denote a set of $n$ -dimensional real vectors. |·| denote the absolute value of a real number. $1_n=[1,\dots ,1{]}^T\in {\mathbb{R}}^n$ . $\mathrm{sgn}(·)$ represents a sign function and $\mathrm{sig}(·{)}^{\alpha }=\mathrm{sign}(·)|·{|}^{\alpha }$ .

Preliminaries

Graph theory

A directed network can be expressed as $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ , including a vertex set $\mathcal{V}=\{v_1,v_2,\dots ,v_n\}$ , an edge set $\mathcal{E}\subseteq \{(v_i,v_j):v_i,v_j\in \mathcal{V}\}$ , and an adjacency weight matrix $\mathcal{A}=[a_{\mathrm{ij}}]\in {\mathbb{R}}^{n\times n}$ such that $a_{\mathrm{ij}}\ne 0,\mathrm{if}(v_i,v_j)\in \mathcal{E}$ . Assume that $\mathcal{G}$ has no selfloops, that is, $a_{\mathrm{ii}}=0,\forall i\in I_n$ and is undirected, that is, $(v_i,v_j)\in \mathcal{E}$ if and only if $(v_i,v_j)\in \mathcal{E}$ . For a directed edge $(v_i,v_j)\in \mathcal{E}$ , we call $v_j$ a neighbor of $v_i$ as ${\mathcal{N}}_i=\{j:(v_j,v_i)\in \mathcal{E}\}$ . Denote the leader’s adjacency matrix as $B=\mathrm{diag}\{b_1,b_2,\dots ,b_n\}$ where $b_i\ge 0$ denotes that the $i\mathrm{th}$ follower can accept commands from the leader, and $b_i=0$ otherwise. Laplacian is expressed as $\mathcal{L}$ and is described as $\mathcal{L}=\mathcal{D}-\mathcal{A}$ where $\mathcal{D}=\mathrm{diag}\{d_1,d_2,\dots ,d_n\}$ .

System description

Let a set of $n+1$ agents composed of a leader and $n$ followers under an undirected interaction graph. The dynamics of the $i$ th follower is represented as follows

$$\begin{array}{c}{\stackrel{·}{q}}_i(t)=p_i(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{p}}_i(t)=u_i(t)+d_i(t)}\end{array}$$

(1)

where $Q_i=[q_i,p_i{]}^T\in {\mathbb{R}}^2$ is the state of agent $i$ , $u_i\in \mathbb{R}$ is the control input of agent $i$ , and $d_i\in \mathbb{R}$ is the uncertain term which satisfies Assumption 1. The dynamics of the leader is described as follows:

$$\begin{array}{c}{\stackrel{·}{q}}_0(t)=p_0(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{p}}_0(t)=u_0(t)}\end{array}$$

(2)

where $Q_0=[q_0,p_0{]}^T\in {\mathbb{R}}^2$ is the state of the leader, $u_0\in \mathbb{R}$ is the control input of the leader which satisfies Assumption 2.

Definition 1. The control objective is to design a controller $u_i$ to reach consensus within a fixed time $\mathcal{T}$ , that is

$$\{\begin{array}{c}\underset{t\to \mathcal{T}}{lim}||Q_i(t)-Q_0(t)||=0\hfill \\ \multicolumn{1}{c}{Q_i(t)=Q_0(t),\forall t>\mathcal{T}}\end{array}$$

Assumption 1. The uncertain term $d_i(t)$ is bounded, that is, there exists ${\rho }_1>0$ such that $|d(t)|\le {\rho }_1$ .

Assumption 2. Suppose $|u_0|\le {\rho }_2$ , $\forall t\ge 0$ with ${\rho }_2$ a known positive constant.

Lemmas

Lemma 1 (Polyakov, 2011). For a system

$$\stackrel{·}{\phi }=-{\iota }_1{\phi }^{\omega }-{\iota }_2{\phi }^{\upsilon },\phi (0)={\phi }_0$$

(3)

where ${\iota }_1,{\iota }_2>0,0<\omega <1,\upsilon >1$ . Then, the origin of the consider system (3) is globally fixed-time stable with setting time satisfies

$$\mathcal{T}\le \frac{1}{{\iota }_1(1-\omega )}+\frac{1}{{\iota }_2(\upsilon -1)}$$

Lemma 2. Let ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_n\ge 0$ , and $0<\omega <1$ . Then

$$\sum _{i=1}^n{\kappa }_i^{\omega }\ge {\left(\sum _{i=1}^n{\kappa }_i\right)}^{\omega }$$

Lemma 3 (Zuo et al., 2016). Let ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_n\ge 0$ , and $\omega >1$ . Then

$$\sum _{i=1}^n{\kappa }_i^{\omega }\ge n^{1-\omega }{\left(\sum _{i=1}^n{\kappa }_i\right)}^{\omega }$$

Main results

A distributed fixed-time observer

Due to the fact that not all agents have a direct link to the leader, states of the leader is required to be estimated, so the fixed-time observer is designed as follows

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{\hat{q}}}_i={\hat{p}}_i-c_{q1}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\hat{q}}_i-{\hat{q}}_j)+b_i({\hat{q}}_i-q_0))}^{\frac{{\eta }_1}{{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{q2}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\hat{q}}_i-{\hat{q}}_j)+b_i({\hat{q}}_i-q_0))}^{\frac{{\eta }_2}{{\varsigma }_2}}}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{p}}}_i=-c_{p1}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\hat{p}}_i-{\hat{p}}_j)+b_i({\hat{p}}_i-p_0))}^{\frac{{\eta }_1}{{\varsigma }_1}}}\\ \multicolumn{1}{c}{-c_{p2}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\hat{p}}_i-{\hat{p}}_j)+b_i({\hat{p}}_i-p_0))}^{\frac{{\eta }_2}{{\varsigma }_2}}}\\ \multicolumn{1}{c}{-{\rho }_2\mathrm{sign}(\sum _{j=1}^n{a}_{\mathrm{ij}}({\hat{p}}_i-{\hat{p}}_j)+b_i({\hat{p}}_i-p_0))}\\ \multicolumn{1}{c}{i=1,2,\dots ,n}\end{array}\hfill \end{array}$$

(4)

where ${\hat{q}}_i,{\hat{p}}_i$ are estimate values of the leader state variables for followers. For the $i\mathrm{th}$ agent, define observer errors as follows

$$\begin{array}{c}{\epsilon }_{q,i}={\hat{q}}_i-q_0\hfill \\ \multicolumn{1}{c}{{\epsilon }_{p,i}={\hat{p}}_i-p_0}\end{array}$$

(5)

Then, the derivative of equation (5) can be written as follows

$$\begin{array}{c}{\stackrel{·}{\epsilon }}_{q,i}={\epsilon }_{p,i}-c_{q1}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{q,i}-{\epsilon }_{q,j})+b_i{\epsilon }_{q,i})}^{\frac{{\eta }_1}{{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{q2}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{q,i}-{\epsilon }_{q,j})+b_i{\epsilon }_{q,i})}^{\frac{{\eta }_2}{{\varsigma }_2}}}\end{array}$$

$$\begin{array}{l}{\stackrel{.}{\epsilon }}_{p,i}=-c_{p1}sig{\left(\sum _{j=1}^n{a}_{ij}({\epsilon }_{p,i}-{\epsilon }_{p,j})+b_i{\epsilon }_{p,i}\right)}^{\frac{{\eta }_1}{{\varsigma }_1}}\\ -c_{p2}sig{\left(\sum _{j=1}^n{a}_{ij}({\epsilon }_{p,i}-{\epsilon }_{p,j})+b_i{\epsilon }_{p,i}\right)}^{\frac{{\eta }_2}{{\varsigma }_2}}\\ -{\rho }_{2}sign\left(\sum _{j=1}^n{a}_{ij}({\epsilon }_{p,i}-{\epsilon }_{p,j})+b_i{\epsilon }_{p,i}\right)-u_0\end{array}$$

(6)

Let

$$\begin{array}{c}{\sigma }_{q,i}=\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{q,i}-{\epsilon }_{q,j})+b_i{\epsilon }_{q,i}\hfill \\ \multicolumn{1}{c}{{\sigma }_{p,i}=\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{p,i}-{\epsilon }_{p,j})+b_i{\epsilon }_{p,i}}\end{array}$$

(7)

Define ${\epsilon }_q=[{\epsilon }_{q,1},\dots ,{\epsilon }_{q,n}{]}^T,{\epsilon }_p=[{\epsilon }_{p,1},\dots ,{\epsilon }_{p,n}{]}^T,{\sigma }_q=$ $[{\sigma }_{q,1},\dots ,{\sigma }_{q,n}{]}^T,{\sigma }_p=[{\sigma }_{p,1},\dots ,{\sigma }_{p,n}{]}^T$ . One has

$$\begin{array}{c}{\sigma }_q=\overline{L}{\epsilon }_q\hfill \\ \multicolumn{1}{c}{{\sigma }_p=\overline{L}{\epsilon }_p}\end{array}$$

(8)

where $\overline{L}=\mathcal{L}+\mathcal{B}$ . Moreover, the control framework of the proposed fixed-time algorithm is shown in Figure 1.

Figure 1.

Control framework of the proposed fixed-time algorithm.

Theorem 1. Suppose Assumption 2 holds, the observer error dynamics (5) reach stability in fixed time ${\mathcal{T}}_O$ , if the observer is chosen as (4), where ${\eta }_1,{\varsigma }_1,{\eta }_2,{\varsigma }_2$ are positive constants and ${\eta }_1>{\varsigma }_1,{\eta }_2<{\varsigma }_2$ .

Proof. Consider a Lyapunov function

$$V_1=\frac{1}{2}{{\epsilon }_p}^T\overline{L}{\epsilon }_p$$

Taking the derivative of $V_1$ yields that

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}{\stackrel{·}{V}}_1=[{\sigma }_{p,1},\dots ,{\sigma }_{p,n}]\hfill \\ \multicolumn{1}{c}{\left(\begin{array}{c}-c_{p1}{[\mathrm{sig}{({\sigma }_{p,1})}^{\frac{{\eta }_1}{{\varsigma }_1}},\dots ,\mathrm{sig}{({\sigma }_{p,n})}^{\frac{{\eta }_1}{{\varsigma }_1}}]}^T\hfill \\ \multicolumn{1}{c}{-c_{p2}{[\mathrm{sig}{({\sigma }_{p,1})}^{\frac{{\eta }_2}{{\varsigma }_2}},\dots ,\mathrm{sig}{({\sigma }_{p,n})}^{\frac{{\eta }_2}{{\varsigma }_2}}]}^T}\\ \multicolumn{1}{c}{-{\rho }_2{[\mathrm{sig}({\sigma }_{p,1}),\dots ,\mathrm{sig}({\sigma }_{p,n})]}^T-1_n{u}_0}\end{array}\right)}\\ \multicolumn{1}{c}{\le -c_{p1}\sum _{i=1}^n|{\sigma }_{p,i}{|}^{\frac{{\eta }_1}{{\varsigma }_1}+1}-c_{p2}\sum _{i=1}^n|{\sigma }_{p,i}{|}^{\frac{{\eta }_2}{{\varsigma }_2}+1}}\\ \multicolumn{1}{c}{-{\rho }_2\sum _{i=1}^n|{\sigma }_{p,i}|-{{\sigma }_p}^T{1}_n{u}_0}\\ \multicolumn{1}{c}{\le -c_{p1}\sum _{i=1}^n|{\sigma }_{p,i}{|}^{\frac{{\eta }_1}{{\varsigma }_1}+1}-c_{p2}\sum _{i=1}^n|{\sigma }_{p,i}{|}^{\frac{{\eta }_2}{{\varsigma }_2}+1}}\\ \multicolumn{1}{c}{-({\rho }_2-|u_0|)\sum _{i=1}^n|{\sigma }_{p,i}|.}\end{array}}\end{array}$$

(9)

By Lemma 2, Lemma 3 and Assumption 2, we can further get

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{V}}_1\le -c_{p1}{n}^{\frac{{\varsigma }_1-{\eta }_1}{2{\varsigma }_1}}{(\sum _{i=1}^n|{\sigma }_{p,i}{|}^2)}^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{p2}{(\sum _{i=1}^n|{\sigma }_{p,i}{|}^2)}^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}}\end{array}\hfill \end{array}$$

Since ${\sum }_{i=1}^n|{\sigma }_{p,i}{|}^2={\epsilon }_p^T\overline{L}\overline{L}{\epsilon }_p\ge {\lambda }_{min}(\overline{L}){\epsilon }_p^T\overline{L}{\epsilon }_p=2{\lambda }_{min}(\overline{L})V_1$ , one obtains

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{V}}_1\le -c_{p1}{n}^{\frac{{\varsigma }_1-{\eta }_1}{2{\varsigma }_1}}{(2{\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}{V}_1^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{p2}{(2{\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}{V}_1^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}}\end{array}\hfill \end{array}$$

In the issue, by Lemma 1, we can obtain $V_1(t)\equiv 0,\forall t\ge {\mathcal{T}}_{O1}$ , where ${\mathcal{T}}_{O1}=\frac{n^{\frac{{\eta }_1-{\varsigma }_1}{2{\varsigma }_1}}}{c_{p1}{2}^{\frac{{\eta }_1-{\varsigma }_1}{2{\varsigma }_1}}{({\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}}\frac{{\varsigma }_1}{{\eta }_1-{\varsigma }_1}+\frac{1}{c_{p2}{2}^{\frac{{\eta }_2-{\varsigma }_2}{2{\varsigma }_2}}{({\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}}\frac{{\varsigma }_2}{{\varsigma }_2-{\eta }_2}$ , which means after ${\mathcal{T}}_{O1}$ , ${\hat{p}}_i=p_0\equiv 0$ . As a consequence, the first equation in equation (6) can be reduced to

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{\epsilon }}_{q,i}=-c_{q1}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{q,i}-{\epsilon }_{q,j})+b_i{\epsilon }_{q,i})}^{\frac{{\eta }_1}{{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{q2}\mathrm{sig}{(\sum _{j=1}^n{a}_{\mathrm{ij}}({\epsilon }_{q,i}-{\epsilon }_{q,j})+b_i{\epsilon }_{q,i})}^{\frac{{\eta }_2}{{\varsigma }_2}}}\end{array}\hfill \end{array}$$

(10)

Similar to the above steps, we construct Lyapunov function as

$$V_2=\frac{1}{2}{\epsilon }_q^T\overline{L}{\epsilon }_q$$

one has

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{V}}_2\le -c_{q1}{n}^{\frac{{\varsigma }_1-{\eta }_1}{2{\varsigma }_1}}{(2{\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}{V}_2^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}\hfill \\ \multicolumn{1}{c}{-c_{q2}{(2{\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}{V}_2^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}}\end{array}\hfill \end{array}$$

Hence, by Lemma 1, we can easily obtain $V_2(t)\equiv 0,\forall t\ge {\mathcal{T}}_{O2}$ , where ${\mathcal{T}}_{O2}=\frac{n^{\frac{{\eta }_1-{\varsigma }_1}{2{\varsigma }_1}}}{c_{p1}{2}^{\frac{{\eta }_1-{\varsigma }_1}{2{\varsigma }_1}}{({\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_1+{\eta }_1}{2{\varsigma }_1}}}\frac{{\varsigma }_1}{{\eta }_1-{\varsigma }_1}+\frac{1}{c_{p2}{2}^{\frac{{\eta }_2-{\varsigma }_2}{2{\varsigma }_2}}{({\lambda }_{min}(\overline{L}))}^{\frac{{\varsigma }_2+{\eta }_2}{2{\varsigma }_2}}}\frac{{\varsigma }_2}{{\varsigma }_2-{\eta }_2}$ , which means that after ${\mathcal{T}}_O={\mathcal{T}}_{O1}+{\mathcal{T}}_{O2}$ , ${\hat{p}}_i=p_0\equiv 0$ , that is to say that the observer error achieves stable in fixed time ${\mathcal{T}}_O$ .

A fixed-time tracking controller

On the basis of the above fixed-time observer, follwers can make an accurate estimate of leader states. Define

$$\begin{array}{c}{e}_{p,i}=q_i-{\hat{q}}_i\hfill \\ \multicolumn{1}{c}{e_{q,i}=p_i-{\hat{p}}_i}\end{array}$$

(11)

as the tracking error. After ${\mathcal{T}}_O$ , equation (11) can be written as follows

$$\begin{array}{c}{e}_{p,i}=q_i-q_0\hfill \\ e_{q,i}=p_i-p_0\hfill \end{array}$$

(12)

The fixed-time sliding mode surface is constructed as follows

$$s_i={\stackrel{·}{e}}_{q,i}+{\gamma }_1\mathrm{sig}(e_{q,i}{)}^{\frac{{\eta }_3}{{\varsigma }_3}}+{\gamma }_2\mathrm{sig}(e_{q,i}{)}^{\frac{{\eta }_4}{{\varsigma }_4}}$$

(13)

and the controller is chosen as follows

$$\begin{array}{c}\begin{array}{c}{u}_i=-{\gamma }_1\frac{{\eta }_3}{{\varsigma }_3}|e_{q,i}{|}^{\frac{{\eta }_3}{{\varsigma }_3}-1}-{\gamma }_2\frac{{\eta }_4}{{\varsigma }_4}|e_{q,i}{|}^{\frac{{\eta }_4}{{\varsigma }_4}-1}\hfill \\ \multicolumn{1}{c}{-({\kappa }_1|s_i{|}^{\frac{{\eta }_5}{{\varsigma }_5}}+{\kappa }_2|s_i{|}^{\frac{{\eta }_6}{{\varsigma }_6}})\mathrm{sign}(s_i)}\\ \multicolumn{1}{c}{+u_0-{\rho }_1\mathrm{sign}(s_i)}\end{array}\hfill \end{array}$$

(14)

where ${\eta }_3,{\varsigma }_3,{\eta }_4,{\varsigma }_4,{\eta }_5,{\varsigma }_5,{\eta }_6,{\varsigma }_6$ are positive odd integers satisfying ${\eta }_3<{\varsigma }_3,{\eta }_4>{\varsigma }_4,{\eta }_5<{\varsigma }_5,{\eta }_6>{\varsigma }_6$ .

• Theorem 2. Suppose Assumption 1 holds. Using the above sliding mode controller (14) for systems (1) and (2), then equation (11) will achieve stability in fixed time $\mathcal{T}={\mathcal{T}}_O+{\mathcal{T}}_E$ .

Proof. Substituting equations (1), (2), (12), and (14) into equation (13) and differentiating s yield that

$${\stackrel{·}{s}}_i=-({\kappa }_1|s_i{|}^{\frac{{\eta }_5}{{\varsigma }_5}}+{\kappa }_2|s_i{|}^{\frac{{\eta }_6}{{\varsigma }_6}})\mathrm{sign}(s_i)-{\rho }_1\mathrm{sign}(s_i)+d_i(t)$$

Construct the Lyapunov function as follows

$$V_3=\mathrm{sign}(s)s$$

(15)

If $s\ne 0$ , the derivative of equation (15) is

$$\begin{array}{c}\begin{array}{cc}{\stackrel{·}{V}}_3\hfill & \le -{\kappa }_1|s_i{|}^{\frac{{\eta }_5}{{\varsigma }_5}}-{\kappa }_2|s_i{|}^{\frac{{\eta }_6}{{\varsigma }_6}}-({\rho }_1-|d_i(t)|).\hfill \end{array}\hfill \end{array}$$

Due to Assumption 1, we can further obtain

$${\stackrel{·}{V}}_3\le -{\kappa }_1|s_i{|}^{\frac{{\eta }_5}{{\varsigma }_5}}-{\kappa }_2|s_i{|}^{\frac{{\eta }_6}{{\varsigma }_6}}$$

Hence, ${\mathcal{T}}_{E1}=({\varsigma }_5/{\kappa }_1({\varsigma }_5-{\eta }_5))+({\varsigma }_6/{\kappa }_2({\eta }_6-{\varsigma }_6))$ is the time of reaching sliding mode surfaces. Once reaching sliding mode surfaces, that is, $s=0$ , from equation (13), one has

$${\stackrel{·}{e}}_{q,i}=-{\gamma }_1\mathrm{sig}(e_{q,i}{)}^{\frac{{\eta }_3}{{\varsigma }_3}}-{\gamma }_2\mathrm{sig}(e_{q,i}{)}^{\frac{{\eta }_4}{{\varsigma }_4}}$$

Consider a Lyapunov candidate

$$V_4=\mathrm{sign}(e_q)e_q$$

Its derivative is

$$\begin{array}{c}\begin{array}{c}{\stackrel{·}{V}}_4=-{\gamma }_1|e_{p,i}{|}^{\frac{{\eta }_3}{{\varsigma }_3}}-{\gamma }_2|e_{p,i}{|}^{\frac{{\eta }_4}{{\varsigma }_4}}\hfill \\ \multicolumn{1}{c}{=-{\gamma }_1{(V_4)}^{\frac{{\eta }_3}{{\varsigma }_3}}-{\gamma }_2{(V_4)}^{\frac{{\eta }_4}{{\varsigma }_4}}}\end{array}\hfill \end{array}$$

Based on Lemma 1, $e_q$ will converge to 0 in fixed time ${\mathcal{T}}_E={\mathcal{T}}_{E1}+{\mathcal{T}}_{E2}$ , where ${\mathcal{T}}_{E2}={\varsigma }_3/{\gamma }_1({\varsigma }_3-{\eta }_3)+{\varsigma }_4/{\gamma }_2({\eta }_4-{\varsigma }_4)$ , which means $e_{p,i}$ converge to $0$ within $\mathcal{T}={\mathcal{T}}_O+{\mathcal{T}}_E$ .

• Remark 1. The states of each agent discussed in this paper may be unknown, which will bring great difficulties to the design and effectiveness of our controller. To solve this problem, we adopt a state observer to change the states of each agent from unknown to known. What is more, in the actual control process, the initial states of each agent often does not conform to the preset situation, so it is very necessary to study the fixed-time consensus, which will greatly reduce the dependence of our control on the initial value. Also, there is often no way to avoid the generation of disturbances in reality. Therefore, we discuss this situation. To sum up, the controller designed in this paper is reasonable, reliable, and practical.

Simulations

In this section, the validity of theorems is verified by numerical arithmetic examples. Consider a system of $n=5$ agents, including four followers and one leader with a directed interaction graph by Figure 2. The adjacency matrix $\mathcal{A}$ and related matrix $\mathcal{B}$ are chosen as follows

$$A=\left(\begin{array}{cccc}0\hfill & 1\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 1\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \end{array}\right)$$

and

$$\mathcal{B}=\left(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \end{array}\right)$$

Figure 2.

Interaction topology $\mathcal{G}$ .

The dynamics of agents is given by equation (1) with disturbance term $d_i(t)=-0.1\sin (\mathrm{t})$ . The control input of the leader is $u_0=-1.4*0.2*0.2\cos (0.2t)$ . Initial conditions are provided as follows

$$\begin{array}{c}q(0)=[0.71,1.03,0.97,-0.43]\hfill \\ \multicolumn{1}{c}{p(0)=[0.62,-1.32,0.82,-1.87]}\end{array}$$

and

$$\begin{array}{c}\hat{q}(0)=[-0.31,1.66,1.17,1.84]\hfill \\ \multicolumn{1}{c}{\hat{p}(0)=[0.62,-1.86,1.4,1.74]}\end{array}$$

Parameters of the observer are set as ${\rho }_2=0.1$ , $c_{q1}=8$ , $c_{q2}=8$ , $c_{p1}=8$ , $c_{p2}=8$ , ${\eta }_1=3$ , ${\eta }_2=5$ , ${\varsigma }_1=5$ , ${\varsigma }_2=3$ . Parameters of the controller are set as ${\rho }_1=0.5$ , ${\eta }_3=1$ , ${\eta }_4=3$ , ${\eta }_5=1$ , ${\eta }_6=3$ , ${\varsigma }_3=3$ , ${\varsigma }_4=1$ , ${\varsigma }_5=3$ , ${\varsigma }_6=1$ , ${\gamma }_1=10$ , ${\gamma }_2=7$ , ${\kappa }_1=8$ , ${\kappa }_2=11$ .

From the above parameters, we can obtain the following simulation results (see Figures 3–8). Figures 3 and 4 show trajectories of ${\hat{q}}_i$ , $q_0$ , ${\hat{p}}_i$ , and $p_0$ . Figures 5 and 6 draw evolutions of $q_i-{\hat{q}}_i$ and $p_i-{\hat{p}}_i$ . States of agents are revealed in Figures 7 and 8, which confirm that fixed-time consensus is successfully achieved. Figure 9 shows our reproduction of simulation results for Theorem 1 in the work by Li et al. (2022). For the simulation of Li et al. (2022), we have selected the same initial states of multi-agent systems and similar parameters. By comparing Figures 7 and 8 with Figure 9, it is easy to see that this paper has a faster convergence rate, which indicates that this paper has better performance. Moreover, compared with Theorem 1 in the work by Li et al. (2022), this paper extends from first-order system to second-order system, which is closer to the actual model. And the observer is adopted to estimate the state of each agent more accurately, so that the results of the consensus problems are more widely applicable, accurate, and reliable.

Figure 3.

Trajectories of ${\hat{q}}_i$ and $q_0$ .

Figure 4.

Trajectories of ${\hat{p}}_i$ and $p_0$ .

Figure 5.

Evolutions of $q_i-{\hat{q}}_i$ .

Figure 6.

Evolutions of $p_i-{\hat{p}}_i$ .

Figure 7.

Position state trajectories of each agent.

Figure 8.

Velocity state trajectories of each agent.

Figure 9.

Fixed-time bipartite consensus of Theorem 1 in the work by Li et al. (2022).

Conclusion

This paper centers around the problem of fixed-time consensus tracking for second-order MASs with external disturbances under undirected graphs. To implement the fixed-time tracking task, the design thinking is divided into two steps. First, a distributed fixed-time observer is engineered to make each agent estimate leader’s states at a fixed time. Second, based on the estimation of the leader state, each follower tracks the leader at a fixed time using a fixed-time sliding surface. Simulation results showed that the proposed control laws could guarantee the closed-loop system had fixed-time convergence rate and desired tracking performance. Future research can be extended toward the study of the directed graph and predefined time.