Symmetric Bipartite Containment Tracking of High-Order Networked Agents via Predefined-Time Backstepping Control

Abstract

Signed networks, which incorporate both cooperative and antagonistic interactions, naturally give rise to symmetric behaviors in multi-agent systems. One such behavior is bipartite containment tracking, where follower agents converge to a symmetric configuration determined by multiple groups of leaders with opposing influence. Moreover, a timely response is critical to ensuring high performance in containment tracking tasks, particularly for high-order multi-agent systems operating in dynamic and uncertain environments. To this end, this paper investigates the predefined-time bipartite containment tracking problem for high-order multi-agent systems affected by external disturbances. A robust tracking control scheme is developed based on the backstepping method to ensure that the tracking errors converge to a predefined residual set within a user-specified time. The convergence time is explicitly adjustable through a design parameter, and the proposed scheme effectively avoids the singularities often encountered in conventional predefined-time control approaches. The stability and robustness of the proposed scheme are rigorously established through Lyapunov-based analysis, and extensive simulation results are provided to validate our theoretical findings.

1. Introduction

With the rapid development of information technologies, multi-agent systems have become a central research topic in control theory due to their wide applications in unmanned aerial vehicle (UAV) coordination, intelligent transportation, and distributed sensor networks [1,2,3,4,5,6,7]. In practical engineering scenarios, cooperative control of multi-agent systems is often subject to various types of disturbances, and containment control has attracted considerable attention [8,9]. The primary objective of containment control is to regulate the states of follower agents via a subset of designated leaders, such that the followers asymptotically converge into the convex hull spanned by the leaders, thereby achieving effective group coordination [10,11,12].

In recent years, extensive research efforts have been devoted to the problem of tracking the containment of multi-agent systems [13,14,15,16]. For instance, the authors of [13] established an explicit upper bound on communication delays to ensure containment tracking convergence in networks with nonuniform time delays. In [15], a distributed formation-containment control scheme was developed for underactuated hovercrafts subjected to compound disturbances, wherein an adaptive linear extended state observer and a radial basis function neural network (RBFNN) were employed to estimate and compensate for external disturbances. The authors of [16] proposed a collision-free formation-containment control strategy for multiple unmanned surface vehicles under input constraints by integrating a dual-layer Lyapunov-stable control architecture with zeroing control barrier functions, thereby ensuring both inter-agent safety and obstacle avoidance. These works primarily focus on the design of distributed control laws that guarantee that the follower agents converge into the convex hull formed by the leaders.

However, conventional asymptotic convergence approaches cannot provide explicit guarantees on convergence time, and are therefore often inadequate for time-critical engineering applications. To address this limitation, recent studies have shifted attention toward finite-time containment strategies [17], which can accelerate the convergence speed of the system. The upper bound of the convergence time of the finite-time control strategy depends on the initial state of the system. On this basis, the fixed-time consensus-tracking control strategy decouples the initial state and convergence time of the system [18]. Despite their improved convergence performance, fixed-time control methods still suffer from a drawback whereby the upper bound of the convergence time depends implicitly on the control parameters and thus cannot be explicitly prescribed. To overcome this limitation, the predefined-time control strategy has recently gained attention, as it enables system trajectories to converge to a desired residual set within a user-specified time, regardless of the initial conditions or control parameters [19,20,21,22]. In this context, Yang and Dong, in [19], developed an adaptive containment control strategy for uncertain nonlinear heterogeneous multi-agent systems with multiple leaders, which guarantees predefined-time convergence and high funnel performance despite the presence of unknown sensor and actuator faults. In [20], a two-layer hierarchical control framework was proposed for the predefined-time time-varying formation-containment tracking problem in disturbed multiple Euler–Lagrange systems, where nonsingular terminal sliding mode controllers were employed, and Lyapunov-based analysis was conducted to ensure the stability of both leader formation and follower containment. Furthermore, the work in [21] presented a prescribed-time fault-tolerant containment scheme for fully actuated heterogeneous multi-agent systems, in which distributed observers were designed to estimate the convex hull of the leaders based on partial access to leader information.

It is worth noting that most of the aforementioned studies were conducted under conventional cooperative interaction frameworks [13,14,15,16,17,18,19,20,21]. However, in many practical scenarios, agent interactions may involve both cooperative and antagonistic behaviors [23,24,25,26,27]. For instance, in UAV or robotic formations, bipartite formation behaviors emerge through coexisting cooperative and adversarial interactions, thereby achieving expanded coverage or search capabilities [28,29]. Similarly, such cooperative–competitive relationships are commonly observed in social network dynamics [30]. In such cases, agents are typically partitioned into two antagonistic groups, leading to more complex and challenging bipartite containment problems. In this direction, the authors of [24] studied finite-time bipartite containment control for heterogeneous linear multi-agent systems over directed graphs by designing distributed bipartite compensators to achieve cooperative output regulation. Time-varying adaptive robust protocols were developed to ensure that follower agents affected by various uncertainties converge to the convex hull spanned by the leaders, with system stability guaranteed via Lyapunov-based analysis. In [25], an adaptive neural-network-based control scheme was proposed for nonaffine fractional-order multi-agent systems affected by unknown dynamics and external disturbances. A novel finite-time convergence lemma was introduced, and the mean-value theorem was employed to address the challenges posed by nonaffine nonlinearities. Furthermore, the work in [26] proposed a periodic event-triggered bipartite containment strategy for multi-agent systems with unmeasurable states and input delays. The design integrates Padé approximation for delay compensation, fuzzy state observers for state reconstruction, and a novel periodic event-triggering mechanism to reduce communication load. Additionally, machine learning and artificial intelligence approaches can also effectively handle uncertainties within the system while enhancing its robustness [31,32,33,34]. It is also important to note that existing studies have predominantly focused on system models of the first or second order, whereas real-world engineering systems often exhibit high-order dynamic characteristics [35,36]. Moreover, external disturbances, which are ubiquitous in practical applications, can significantly degrade control performance [37,38,39,40].

According to the above discussion, this paper investigates the bipartite predefined-time containment tracking problem for high-order multi-agent systems with disturbances, and designs a predefined-time control scheme based on the backstepping method. The main contributions of this paper are as follows:

(1)

A novel predefined-time robust containment tracking control scheme is developed for high-order multi-agent systems over signed networks. By integrating the backstepping technique with dynamic surface control, the proposed scheme systematically constructs virtual control laws and actual control inputs while avoiding repeated differentiation of fractional power terms and effectively circumventing the singularities typically encountered in conventional predefined-time [19,20,21].

(2)

The proposed controller guarantees bipartite containment tracking within a user-defined predefined time, which can be user-defined. Compared with traditional finite-time [17] or fixed-time approaches [18], the proposed scheme offers enhanced design flexibility and stronger timing guarantees, and mitigates limitations such as asymptotic convergence delays or time-overestimation.

(3)

The control design explicitly accounts for external disturbances by incorporating a robust compensation term into the control input. A rigorous Lyapunov-based stability analysis demonstrates that, under bounded disturbances, the tracking errors converge to a small residual set within the predefined time, thereby significantly enhancing the robustness and practical applicability of the control framework in uncertain and dynamic environments.

The rest of this paper is organized as follows. Section 2 introduces the necessary preliminaries and formulates the bipartite containment tracking problem, including basic concepts from graph theory and system modeling over signed networks. Section 3 presents the main theoretical results, where a predefined-time robust containment tracking controller is developed using backstepping and dynamic surface control techniques. Section 4 validates the effectiveness of the proposed method through two simulation examples. Section 5 concludes the paper.

Notations: ${\mathbf{I}}_m$ denotes an identity matrix with dimensions of $m\times m$. ${\mathbf{1}}_n$ represents a column vector where all elements are 1. ${\mathbf{0}}_n$ represents a column vector where all elements are 0. The Kronecker product is represented by ⊗. ${\mathbf{diag}}_i^n\left[k_i\right]\stackrel{\Delta }{=}\mathbf{diag}(k_1,···,k_n)$ represents a diagonal matrix, where $k_i$ presents the $i_{th}$ diagonal element. ${\mathbf{col}}_i^n\left[p_i\right]\stackrel{\Delta }{=}\mathbf{col}(p_1,···,p_n)\stackrel{\Delta }{=}{[p_1^T,···,p_n^T]}^T$ denotes a column vector. $\mathrm{sign}\left(·\right)$ represents the sign function.

2. Preliminaries and Problem Formulation

2.1. Graph Theory

Consider a multi-agent system consisting of m leaders and n followers. Let ${\mathcal{V}}_L=\{1,2,\dots ,m\}$ and ${\mathcal{V}}_F=\{m+1,m+2,\dots ,m+n\}$ denote the sets of leader and follower agents, respectively. The overall agent set is defined as $\mathcal{V}={\mathcal{V}}_L\cup {\mathcal{V}}_F$. To characterize the interaction topology among agents, a signed directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ is employed, where $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ represents the edge set and $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{(m+n)\times (m+n)}$ is the weighted adjacency matrix. Specifically, $a_{ij}\ne 0$ if there exists a directed edge from agent j to agent i (i.e., $(i,j)\in \mathcal{E}$), and $a_{ij}=0$ otherwise. The sign of $a_{ij}$ encodes the type of interaction: $a_{ij}=1$ indicates a cooperative interaction, while $a_{ij}=-1$ represents an antagonistic (competitive) interaction.

The degree matrix is defined as $D=\mathbf{diag}(d_1,\dots ,d_{m+n})$, where each diagonal entry is given by $d_i={\sum }_{j=1}^{m+n}\left|a_{ij}\right|$, representing the absolute degree of agent i. The (signed) Laplacian matrix of the graph is then defined as $L=D-\mathcal{A}$.

2.2. Structurally Balanced Graph

A structurally balanced graph is a signed graph that satisfies specific structural properties related to the partitioning of its vertices and the consistency of its edge signs. The graph $\mathcal{G}$ is structurally balanced if and only if one of the following equivalent conditions holds: (1) The overall agent set $\mathcal{V}$ can be partitioned into two disjoint subsets, $V_1$ and $V_2$ (where $V_1\cup V_2=V$ and $V_1\cap V_2=\varnothing$), such that every edge between vertices within the same subset is positive and every edge between vertices from different subsets is negative. (2) Every cycle in $\mathcal{G}$ contains an even number of negative edges.

2.3. Problem Description

In this paper, we consider a multi-agent system composed of N follower agents, where the dynamics of each follower are modeled by a class of high-order nonlinear systems described as follows:

$$\left\{\begin{array}{c}{\dot{p}}_{mi}\left(t\right)=p_{(m+1)i}\left(t\right)+q_{mi}\left({\overline{p}}_{mi}\left(t\right)\right),\hfill \\ {\dot{p}}_{ni}\left(t\right)={\overline{\tau }}_i\left(t\right)+q_{ni}\left({\overline{p}}_{ni}\left(t\right)\right),\hfill \\ {\overline{\tau }}_i\left(t\right)={\tau }_i\left(t\right)+f_i\left(t\right),\hfill \\ y_i\left(t\right)=p_{1i}\left(t\right),m=1,\dots ,n-1;i=1,\dots ,N,\hfill \end{array}\right.$$

(1)

where ${\overline{p}}_{mi}\left(t\right)={[p_{1i}\left(t\right),p_{2i}\left(t\right),\dots ,p_{mi}\left(t\right)]}^T\in {\mathbb{R}}^m$ denotes the state vector of agent i, and $y_i\left(t\right)$ is its measurable output. The function $q_{ni}\left({\overline{p}}_{ni}\left(t\right)\right)$ is a known nonlinear function that characterizes the internal dynamics of agent i. The term $f_i\left(t\right)$ represents unknown external disturbances acting on agent i, while ${\overline{\tau }}_i\left(t\right)$ denotes the actual (disturbed) control input.

Assumption 1.

The interaction topology among agents is modeled as a connected and structurally balanced graph. In addition, it is assumed that at least one follower has access to the information of every leader, thereby ensuring the feasibility of information propagation across the entire network.

Based on Assumption 1, the communication topology among the agents can be described by a connected and structurally balanced graph, where each follower has at least an indirect or direct information path to the leaders. Accordingly, the Laplacian matrix L associated with such a topology can be partitioned as

$$L=\left[\begin{array}{cc}{\mathbf{0}}_{m\times m}& {\mathbf{0}}_{m\times n}\\ L_1\in {\mathbb{R}}^{n\times m}& L_2\in {\mathbb{R}}^{n\times n}\end{array}\right],$$

(2)

where the upper-left and upper-right blocks correspond to the absence of incoming edges for the leaders, $L_1$ characterizes the interconnection from leaders to followers, and $L_2$ represents the internal topology among followers.

Lemma 1

([41]). Under Assumption 1, the follower–follower submatrix $L_2$ is nonsingular. Moreover, all entries of the matrix product $-L_2^{-1}{L}_1$ are non-negative, and the sum of elements in each row of $-L_2^{-1}{L}_1$ equals one. That is, $\left(-L_2^{-1}{L}_1\right){\mathbf{1}}_m={\mathbf{1}}_n$, implying that $-L_2^{-1}{L}_1$ is a row-stochastic matrix.

Assumption 2.

The external disturbance $f_i\left(t\right)$ in system (1) is bounded by an unknown positive constant ${\overline{f}}_i$, i.e.,

$$|f_i\left(t\right)|\le {\overline{f}}_i,\forall t\ge 0.$$

The assumption that the external disturbance $f_i\left(t\right)$ is bounded by an unknown constant is reasonable in practical scenarios. This assumption reflects the fact that many real-world disturbances are energy-limited due to physical or environmental constraints. It does not require prior knowledge of the exact bound, since the proposed control design ensures stability and robustness without relying on the precise value of ${\overline{f}}_i$.

Lemma 2

([42]). Let G be a connected structurally balanced graph. Then the associated Laplacian matrix L has a simple eigenvalue at zero, and all remaining eigenvalues are strictly positive.

Lemma 3.

Consider the nonlinear system

$$\dot{\eta }=\varphi \left(\eta \right),$$

(3)

and suppose there exists a continuously differentiable Lyapunov function $V\left(\eta \right)$ such that

$$\dot{V}\left(\eta \right)\le -\alpha V^a\left(\eta \right)-\beta V^b\left(\eta \right)+c,$$

(4)

where $\alpha >0$, $\beta >0$, $a>1$, $0<b<1$, and $c>0$. If the parameters

$$\alpha ={\displaystyle \frac{1}{(a-1)d{T}_p}},\beta ={\displaystyle \frac{1}{(1-b)(1-d)T_p}},$$

are selected, where $T_p>0$ is the user-specified predefined time and $0<d<1$, then the system (3) is practically predefined-time stable. That is, the state $\eta \left(t\right)$ satisfies ${lim}_{t\to T}\parallel \eta \left(t\right)\parallel \le \delta$, for some residual error bound $\delta >0$, with convergence time $T\le T_p$.

Proof. 

According to Lemma 2 in [43], system (3) is practically fixed-time stable under inequality (4), with the convergence time upper bounded by

$$\overline{T}={\displaystyle \frac{1}{\alpha (a-1)}}+{\displaystyle \frac{1}{\beta (1-b)}}.$$

Substituting the parameter values yields

$$\overline{T}={\displaystyle \frac{1}{\frac{1}{(a-1)d{T}_p}(a-1)}}+{\displaystyle \frac{1}{\frac{1}{(1-b)(1-d)T_p}(1-b)}}=d{T}_p+(1-d)T_p=T_p.$$

Hence, the system state $\eta \left(t\right)$ reaches a neighborhood of the origin within the predefined time $T_p$, completing the proof. □

Lemma 4

([44]). For any non-negative real numbers $r_i\ge 0$ with $i=1,2,\dots ,n$, the following inequality holds:

$${\left(\sum _{i=1}^n{r}_i\right)}^2\le n\sum _{i=1}^n{r}_i^2.$$

(5)

Lemma 5

([45]). For any $s\in [0,1]$ and $r_i\in \mathbb{R}$ with $i=1,2,\dots ,n$, the following inequality holds:

$${\left(\sum _{i=1}^n\left|r_i\right|\right)}^s\le \sum _{i=1}^n{\left|r_i\right|}^s.$$

(6)

Definition 1

([46]). Given a set of leader states ${\eta }_{01},\dots ,{\eta }_{0m}$, their convex hull is defined as

$$\mathrm{con}\left\{{\eta }_{01},\dots ,{\eta }_{0m}\right\}=\left\{\sum _{i=1}^m{\rho }_i{\eta }_{0i}|{\rho }_i\in \mathbb{R},\sum _{i=1}^m{\rho }_i=1\right\}.$$

(7)

Definition 2

([41]). The bipartite containment tracking objective for the multi-agent system described in (1) is said to be achieved if a subset of followers asymptotically converges to the convex hull $\mathrm{con}\left\{{\eta }_{0j}\mid j\in {\mathcal{V}}_L\right\}$ while the remaining followers converge to its symmetric counterpart $-\mathrm{con}\left\{{\eta }_{0j}\mid j\in {\mathcal{V}}_L\right\}$.

3. Main Results

In this section, a predefined-time robust tracking controller is developed for the multi-agent system described in (1) to achieve predefined-time practical containment tracking. The proposed control framework is constructed based on the backstepping methodology, augmented with dynamic surface filtering. This design effectively addresses the singularity issues commonly encountered in conventional predefined-time control schemes that rely on time-scaling functions. Moreover, the incorporation of dynamic surface filtering eliminates the need for repeated differentiation of fractional power terms, which is typically required in finite-time and fixed-time control designs, thereby simplifying the implementation and enhancing the robustness of the overall scheme.

Furthermore, a block diagram of the proposed containment tracking control scheme is shown in Figure 1. The control scheme proposed in this paper integrates a dynamic surface filter, virtual variables, a tracking controller, and a predefined-time parameter. By designing a robust predefined-time tracking controller based on the backstepping method, the high-order networked system is capable of achieving bipartite containment tracking within a user-specified finite time while effectively suppressing the adverse effects of external disturbances.

Firstly, the bipartite containment tracking error variables are defined as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\delta }_{i1}={\displaystyle \sum _{j\in V_F}^N}\left|a_{ij}\right|(p_{1i}-\mathrm{sign}\left(a_{ij}\right)p_{1j})+{\displaystyle \sum _{j\in V_L}^N}\left|a_{ij}\right|(p_{1i}-\mathrm{sign}\left(a_{ij}\right){\eta }_{0j}),i\in V_F\hfill \\ {\delta }_{i2}={\dot{\delta }}_{i1},\hfill \\ ⋮\hfill \\ {\delta }_{in}={\dot{\delta }}_{i(n-1)},\hfill \end{array}\right.\end{array}$$

(8)

where ${\eta }_{0j}$ is the leader’s state.

The dynamic surface filter error variables are defined as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\phi }_{i2}=p_{2c}-p_{2d},\hfill \\ {\phi }_{i3}=p_{3c}-p_{3d},\hfill \\ ⋮\hfill \\ {\phi }_{in}=p_{nc}-p_{nd},\hfill \end{array}\right.\end{array}$$

(9)

where $p_{2d},p_{3d},\dots ,p_{nd}$ are the virtual variables to be designed later.

Step 1. When $m=1$, the following Lyapunov function is chosen:

$$\begin{array}{c}\hfill V_{1i}={\displaystyle \frac{1}{2}}{\delta }_{i1}^2.\end{array}$$

(10)

Then, it follows that

$$\begin{array}{c}\hfill {\dot{V}}_{1i}={\delta }_{i1}({\delta }_{i2}+{\phi }_{i2}+{\phi }_{2d}+q_{1i}-{\dot{p}}_{1c}).\end{array}$$

(11)

The virtual variable is presented as

$$\begin{array}{c}\hfill p_{2d}=-q_{1i}-k_{11}{\displaystyle \frac{{\delta }_{i1}}{\sqrt{{\delta }_{i1}^2+{\left(\frac{{\varpi }_1}{k_{11}}\right)}^2}}}-k_{12}{\delta }_{i1}^3-k_{13}{\delta }_{i1}+{\dot{p}}_{1c},\end{array}$$

(12)

where $k_{11}$, $k_{12}$, $k_{13}$, and ${\varpi }_1$ are the parameters to be designed. Substituting (12) into (11) yields

$$\begin{array}{c}\hfill {\dot{V}}_{1i}\le -\left(k_{13}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2k_{c23}}}\right){\delta }_{i1}^2+{\delta }_{i1}{\delta }_{i2}+{\displaystyle \frac{k_{c23}}{2}}{\phi }_{i2}^2-k_{11}\left|{\delta }_{i1}\right|+{\varpi }_1-k_{12}{\delta }_{i1}^4.\end{array}$$

(13)

Step 2.

When $m=2$, the following Lyapunov function is chosen:

$$\begin{array}{c}\hfill V_{2i}={\displaystyle \frac{1}{2}}{\delta }_{i2}^2+{\displaystyle \frac{1}{2}}{\phi }_{i2}^2.\end{array}$$

(14)

Then, one determines that

$$\begin{array}{c}\hfill {\dot{V}}_{2i}={\delta }_{i2}({\delta }_{i3}+{\phi }_{i3}+p_{3d}+q_{2i}-{\dot{p}}_{1c})+{\phi }_{i2}({\dot{p}}_{2c}-{\dot{p}}_{2d}).\end{array}$$

(15)

Furthermore, the dynamic surface filter is defined as

$$\begin{array}{c}\hfill {\dot{p}}_{2c}=-k_{c21}{\displaystyle \frac{{\phi }_{i2}}{\sqrt{{\phi }_{i2}^2+{\left(\frac{s_2}{k_{c21}}\right)}^2}}}-k_{c22}{\phi }_{i2}^3-k_{c23}{\phi }_{i2},\end{array}$$

(16)

where $k_{c21}$, $k_{c22}$, $k_{c23}$, and $s_2$ are the parameters to be designed.

The virtual variable is designed as follows:

$$\begin{array}{c}\hfill p_{3d}=-q_{2i}-k_{21}{\displaystyle \frac{{\delta }_{i2}}{\sqrt{{\delta }_{i2}^2+{\left(\frac{{\varpi }_2}{k_{21}}\right)}^2}}}-k_{22}{\delta }_{i2}^3-k_{23}{\delta }_{i2}+{\dot{p}}_{2c}-{\delta }_{i1},\end{array}$$

(17)

where $k_{21}$, $k_{22}$, $k_{23}$, and ${\varpi }_2$ are the parameters to be designed.

Substituting (16) and (17) into (15) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{2i}\le & -\left(k_{23}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2k_{c33}}}\right){\delta }_{i2}^2-{\delta }_{i1}{\delta }_{i2}+{\delta }_{i2}{\delta }_{i3}+{\displaystyle \frac{k_{c33}}{2}}{\phi }_{i3}^2-k_{21}\left|{\delta }_{i2}\right|+{\varpi }_2-k_{22}{\delta }_{i2}^4\hfill \\ & -k_{c21}\left|{\phi }_{i2}\right|+s_2-k_{c22}{\phi }_{i2}^4-k_{c23}{\phi }_{i2}^2-{\dot{p}}_{2d}{\phi }_{i2}.\hfill \end{array}$$

(18)

Step m. When $2<m\le n-1$, the following Lyapunov function is chosen:

$$\begin{array}{c}\hfill V_{mi}={\displaystyle \frac{1}{2}}{\delta }_{im}^2+{\displaystyle \frac{1}{2}}{\phi }_{im}^2.\end{array}$$

(19)

Then, it can be determined that

$$\begin{array}{c}\hfill {\dot{V}}_{mi}={\delta }_{im}({\delta }_{i(m+1)}+{\phi }_{i(m+1)}+p_{(m+1)d}+q_{mi}-{\dot{p}}_{mc})+{\phi }_{im}({\dot{p}}_{mc}-{\dot{p}}_{md}).\end{array}$$

(20)

Furthermore, the dynamic surface filter is defined as

$$\begin{array}{c}\hfill {\dot{p}}_{mc}=-k_{cm1}{\displaystyle \frac{{\phi }_{im}}{\sqrt{{\phi }_{im}^2+{\left(\frac{s_m}{k_{cm1}}\right)}^2}}}-k_{cm2}{\phi }_{im}^3-k_{cm3}{\phi }_{im},\end{array}$$

(21)

where $k_{cm1}$, $k_{cm2}$, $k_{cm3}$, and ${\theta }_m$ are parameters to be designed.

The virtual variable is designed as follows:

$$\begin{array}{c}\hfill p_{(m+1)d}=-q_{mi}-k_{m1}{\displaystyle \frac{{\delta }_{im}}{\sqrt{{\delta }_{im}^2+{\left(\frac{{\varpi }_m}{k_{m1}}\right)}^2}}}-k_{m2}{\delta }_{m2}^3-k_{m3}{\delta }_{m2}+{\dot{p}}_{mc}-{\delta }_{i(m-1)},\end{array}$$

(22)

where $k_{m1}$, $k_{m2}$, $k_{m3}$, and ${\varpi }_m$ are the parameters to be designed.

Substituting (21) and (22) into (20) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{mci}\le & -\left(k_{m3}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2k_{c(m+1)3}}}\right){\delta }_{im}^2-{\delta }_{i(m-1)}{\delta }_{im}+{\delta }_{im}{\delta }_{i(m+1)}+{\displaystyle \frac{k_{c(m+1)3}}{2}}{\phi }_{i(m+1)}^2\hfill \\ & -k_{m1}\left|{\delta }_{im}\right|+{\varpi }_m-k_{m2}{\delta }_{im}^4-k_{cm1}\left|{\phi }_{im}\right|+s_m-k_{cm2}{\phi }_{im}^4-k_{cm3}{\phi }_{im}^2-{\dot{p}}_{md}{\phi }_{im}.\hfill \end{array}$$

(23)

Step n. The following Lyapunov function is chosen:

$$\begin{array}{c}\hfill V_{ni}={\displaystyle \frac{1}{2}}{\delta }_{in}^2+{\displaystyle \frac{1}{2}}{\phi }_{in}^2.\end{array}$$

(24)

Then we have

$$\begin{array}{c}\hfill {\dot{V}}_{ni}={\delta }_{in}({\tau }_i+f_i+q_{ni}-{\dot{p}}_{nc})+{\phi }_{in}({\dot{p}}_{nc}-{\dot{p}}_{nd}).\end{array}$$

(25)

The dynamic surface filter is defined as

$$\begin{array}{c}\hfill {\dot{p}}_{nc}=-k_{cn1}{\displaystyle \frac{{\phi }_{in}}{\sqrt{{\phi }_{in}^2+{\left(\frac{s_n}{k_{cn1}}\right)}^2}}}-k_{cn2}{\phi }_{in}^3-k_{cn3}{\phi }_{in},\end{array}$$

(26)

where $k_{cn1}$, $k_{cn2}$, $k_{cn3}$, and $s_n$ are the parameters to be designed.

The predefined-time robust containment tracking controller is designed as follows:

$$\begin{array}{c}\hfill {\tau }_i=-q_{ni}-\mathrm{sign}\left({\delta }_{in}\right){\overline{f}}_i-k_{n1}{\displaystyle \frac{{\delta }_{in}}{\sqrt{{\delta }_{in}^2+{\left(\frac{{\varpi }_n}{k_{n1}}\right)}^2}}}-k_{n2}{\delta }_{n2}^3-k_{n3}{\delta }_{n2}+{\dot{p}}_{nc}-{\delta }_{i(n-1)},\end{array}$$

(27)

where $k_{n1}$, $k_{n2}$, $k_{n3}$, and ${\varpi }_n$ are the parameters to be designed. Substituting (26) and (27) into (25) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{ni}\le & -\left(k_{n3}-{\displaystyle \frac{1}{2}}\right){\delta }_{in}^2-{\delta }_{i(n-1)}{\delta }_{in}-k_{n1}\left|{\delta }_{in}\right|+{\varpi }_n-k_{n2}{\delta }_{in}^4-k_{cn1}\left|{\phi }_{in}\right|+s_n\hfill \\ & -k_{cn2}{\phi }_{in}^4-k_{cn3}{\phi }_{in}^2-{\dot{p}}_{nd}{\phi }_{in}.\hfill \end{array}$$

(28)

Remark 1.

In the design of controller (27), a compensation term $\mathrm{sign}\left({\delta }_{in}\right){\overline{f}}_i$ is introduced, which incorporates the known upper bound of the disturbance ${\overline{f}}_i$. By substituting Equations (26) and (27) into (25), and utilizing the inequality ${\delta }_{in}·\mathrm{sign}\left({\delta }_{in}\right){\overline{f}}_i=\left|{\delta }_{in}\right|{\overline{f}}_i\ge {\delta }_{in}{f}_i$, we obtain Equation (28). This confirms that, through the designed compensation term $\mathrm{sign}\left({\delta }_{in}\right){\overline{f}}_i$ and the adoption of appropriate scaling techniques, the disturbance $f_i$ in the system model can be effectively canceled and suppressed, thereby enhancing the robustness of the control scheme.

Theorem 1.

Consider the multi-agent system (1) under Assumptions 1 and 2. Let the control protocol be designed as in (27), and the associated dynamic surface filters be constructed according to (16), (21), and (26). Then, there exist positive constants, $k_{m1}$, $k_{m2}$, $k_{m3}$, ${\varpi }_m$, $k_{cm1}$, $k_{cm2}$, $k_{cm3}$, and $s_m$, for all $m=1,\dots ,n$, such that the containment tracking error ${\delta }_{i1}$ converges to a bounded neighborhood of the origin within the predefined time T.

Proof. 

The following Lyapunov function is selected:

$$\begin{array}{c}\hfill V_i=\sum _{m=1}^n{\displaystyle \frac{1}{2}}{\delta }_{im}^2+\sum _{m=2}^n{\displaystyle \frac{1}{2}}{\phi }_{im}^2.\end{array}$$

(29)

Then, we have

$$\begin{array}{cc}\hfill {\dot{V}}_{ti}\le & \sum _{m=1}^n\left(-\sqrt{2}{k}_{m1}{\left(\frac{1}{2}{\delta }_{im}^2\right)}^{\frac{1}{2}}-4k_{m2}{\left(\frac{1}{2}{\delta }_{im}^2\right)}^2\right)\hfill \\ \hfill & +\sum _{m=2}^n\left(-\sqrt{2}{k}_{cm1}{\left(\frac{1}{2}{\phi }_{im}^2\right)}^{\frac{1}{2}}-4k_{cm2}{\left(\frac{1}{2}{\phi }_{im}^2\right)}^2\right)+Q,\hfill \end{array}$$

(30)

where

$$Q=\sum _{m=1}^n{\varpi }_m+\sum _{m=2}^n\left(s_m+{\displaystyle \frac{{\overline{P}}_m^2}{2k_{cm3}}}\right).$$

From Lemmas 4 and 5, by selecting control gains such that $k_{11}<k_{m1}<k_{cm1}$ and $k_{12}<k_{m2}<k_{cm2}$ for all $m=2,\dots ,n$, it follows that when $V_{ti}\ge \sqrt{{\displaystyle \frac{Q}{{\alpha }_4}}}$, we obtain

$${\dot{V}}_{ti}\le -c_1{V}_{ti}^{1/2}-c_2{V}_{ti}^2+Q,$$

(31)

where $c_1=\sqrt{2}{k}_{11}$ and $c_2=4k_{12}$.

We choose the following control parameters:

$$k_{11}={\displaystyle \frac{2\sqrt{2}}{T}},k_{12}={\displaystyle \frac{1}{2T}}.$$

Then, according to Lemma 3, the Lyapunov function $V_{ti}$ will converge to the residual set

$${\Omega }_{V_i}=\left\{V_{ti}\le \sqrt{{\displaystyle \frac{Q}{c_2}}}\right\}$$

within the predefined time T. Consequently, the tracking error ${\delta }_{im}$ converges to the residual set

$${\Omega }_{{\delta }_{im}}=\left\{{\delta }_{im}\le \sqrt{2\sqrt{{\displaystyle \frac{Q}{c_2}}}}\right\}.$$

Furthermore, under Assumption 1 and Lemma 2, it follows that the tracking error vectors ${\delta }_1,\dots ,{\delta }_n$ are uniformly bounded. Therefore, the overall tracking error $e_i$ converges to the residual set ${\Omega }_{{\delta }_{ti}}$ within the predefined time T.

Finally, by invoking Lemma 1 and Theorem 8 in [41], the networked agent system achieves predefined-time bipartite containment tracking. □

In this section, a predefined-time robust containment tracking controller was developed for high-order multi-agent systems affected by external disturbances. The proposed control scheme adopts a backstepping design integrated with dynamic surface control to systematically construct virtual control laws and actual control inputs. By incorporating nonlinear damping terms and employing dynamic surface filters to avoid repeated differentiation, the controller guarantees that the bipartite containment tracking errors converge to a predefined residual set within a user-specified time. A rigorous Lyapunov-based analysis confirms that all tracking errors remain bounded and reach the desired set within the predefined time, thereby achieving robust practical containment tracking under signed network topologies.

Table 1. Comparison of control strategies in terms of convergence time and stability characteristics.

Control Strategy	Convergence Time and Stability Characteristics
Finite-time control strategy [17]	The upper bound of the convergence time depends on the system’s initial conditions.
Fixed-time control strategy [18]	The upper bound of the convergence time is independent of the initial conditions and determined solely by the control parameters.
Conventional prescribed-time control strategy [19,20,21]	The convergence time can be predefined by the user, but the control design often suffers from singularity issues.
Proposed predefined-time control strategy	The convergence time is user-defined and the design is free from singularities, ensuring robust performance.

illustrates the limitations of finite-time, fixed-time, and conventional prescribed-time control strategies in terms of control performance, as well as the superiority of the proposed predefined-time control strategy presented in this paper.

Remark 2.

According to Table 1, we can determine that unlike the finite-time control strategy [17] and the fixed-time control strategy [18], the proposed control strategy decouples the system’s convergence time from both the initial conditions and control parameters while enabling user-defined specification, thereby enhancing the system’s overall control performance. Furthermore, compared with existing predefined-time control approaches [19,20,21], which often suffer from singularity issues and require higher-order derivatives involving fractional power terms, the proposed predefined-time robust control scheme overcomes these limitations by incorporating dynamic surface filtering and utilizing Lemma 3. Moreover, in contrast to conventional containment strategies that primarily focus on cooperative networks, the proposed design addresses the bipartite containment tracking problem over signed graphs with antagonistic interactions while ensuring convergence within a user-specified finite time.

Remark 3.

The proposed control scheme is inherently scalable with respect to the number of agents, making it readily applicable to both sparse and dense network topologies without modification. Furthermore, while directed graphs indeed introduce challenges due to the asymmetry of their Laplacian matrices (which prevents similarity diagonalization), our theoretical framework does not rely on the symmetry or diagonalizability of the Laplacian. This allows the controller to be directly applied to directed topologies, as evidenced in our stability analysis.

4. Simulation Results

In this section, numerical simulations are presented to validate the effectiveness and predefined-time performance of the proposed control scheme.

4.1. Simulation Example 1

The communication topology of the multi-agent system is illustrated in Figure 2, where agents 1–8 denote the followers, and agents A and B denote the leaders (${\eta }_{01}$ and ${\eta }_{02}$).

In this paper, the nonlinear multi-agent system (1) is formulated as a third-order nonlinear dynamical system, expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{p}}_{1i}\left(t\right)=p_{2i}\left(t\right)+q_{1i}\left({\overline{p}}_{1i}\left(t\right)\right),\hfill \\ {\dot{p}}_{2i}\left(t\right)=p_{3i}\left(t\right)+q_{2i}\left({\overline{p}}_{2i}\left(t\right)\right),\hfill \\ {\dot{p}}_{3i}\left(t\right)={\overline{\tau }}_i\left(t\right)+q_{3i}\left({\overline{p}}_{3i}\left(t\right)\right),\hfill \\ {\overline{\tau }}_i\left(t\right)={\tau }_i\left(t\right)+f_i\left(t\right),\hfill \\ y_i\left(t\right)=p_{1i}\left(t\right),i=1,···,N\hfill \end{array}\right.\end{array}$$

where the components are defined as:

$$\begin{array}{cc}\hfill & q_{11}\left({\overline{p}}_{11}\left(t\right)\right)=p_{11}+sin\left(p_{11}\right),q_{12}\left({\overline{p}}_{12}\left(t\right)\right)=p_{12}+sin\left(p_{12}\right),q_{13}\left({\overline{p}}_{13}\left(t\right)\right)=p_{13}+sin\left(p_{13}\right),\hfill \\ \hfill & q_{14}\left({\overline{p}}_{14}\left(t\right)\right)=p_{14}+sin\left(p_{14}\right),q_{15}\left({\overline{p}}_{15}\left(t\right)\right)=p_{15}+sin\left(p_{15}\right),q_{16}\left({\overline{p}}_{16}\left(t\right)\right)=p_{16}+sin\left(p_{16}\right),\hfill \\ \hfill & q_{17}\left({\overline{p}}_{17}\left(t\right)\right)=p_{17}+sin\left(p_{17}\right),q_{18}\left({\overline{p}}_{18}\left(t\right)\right)=p_{18}+sin\left(p_{18}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & q_{21}\left({\overline{p}}_{21}\left(t\right)\right)=p_{11}+sin\left(p_{21}\right),q_{22}\left({\overline{p}}_{22}\left(t\right)\right)=p_{12}+sin\left(p_{22}\right),q_{23}\left({\overline{p}}_{23}\left(t\right)\right)=p_{13}+sin\left(p_{23}\right),\hfill \\ \hfill & q_{24}\left({\overline{p}}_{24}\left(t\right)\right)=p_{14}+sin\left(p_{24}\right),q_{25}\left({\overline{p}}_{25}\left(t\right)\right)=p_{15}+sin\left(p_{25}\right),q_{26}\left({\overline{p}}_{26}\left(t\right)\right)=p_{16}+sin\left(p_{26}\right),\hfill \\ \hfill & q_{27}\left({\overline{p}}_{27}\left(t\right)\right)=p_{17}+sin\left(p_{27}\right),q_{28}\left({\overline{p}}_{28}\left(t\right)\right)=p_{18}+sin\left(p_{28}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & q_{31}\left({\overline{p}}_{31}\left(t\right)\right)=cos\left(p_{31}\right)+p_{21}+log(1+p_{11}^2),q_{32}\left({\overline{p}}_{32}\left(t\right)\right)=cos\left(p_{32}\right)+p_{22}+log(1+p_{12}^2),\hfill \\ \hfill & q_{33}\left({\overline{p}}_{33}\left(t\right)\right)=cos\left(p_{33}\right)+p_{23}+log(1+p_{13}^2),q_{34}\left({\overline{p}}_{34}\left(t\right)\right)=cos\left(p_{34}\right)+p_{24}+log(1+p_{14}^2),\hfill \\ \hfill & q_{35}\left({\overline{p}}_{35}\left(t\right)\right)=cos\left(p_{35}\right)+p_{25}+log(1+p_{15}^2),q_{36}\left({\overline{p}}_{36}\left(t\right)\right)=cos\left(p_{36}\right)+p_{26}+log(1+p_{16}^2),\hfill \\ \hfill & q_{37}\left({\overline{p}}_{37}\left(t\right)\right)=cos\left(p_{37}\right)+p_{27}+log(1+p_{17}^2),q_{38}\left({\overline{p}}_{38}\left(t\right)\right)=cos\left(p_{38}\right)+p_{28}+log(1+p_{18}^2),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & f_1=1.2cos\left(0.5t\right)+1,f_2=0.8sin\left(1.2t\right)+0.5,f_3=1.5e^{-2t}+0.3,f_4=0.5sin\left(t\right),\hfill \\ \hfill & f_5=0.7cos\left(0.5t\right)+0.6,f_6=0.5sin\left(0.5t\right)+0.3,\hfill \\ \hfill & f_7=0.4sin\left(0.2t\right)+0.6,f_8=0.7sin\left(0.8t\right)+0.2.\hfill \end{array}$$

The dynamical characteristics of the leader system are set as follows:

$$\begin{array}{c}\hfill {\eta }_{01}\left(t\right)=sin\left(0.5t\right)+5,{\eta }_{02}\left(t\right)=sin\left(0.5t\right)+3,\end{array}$$

The following initial conditions are chosen: $p_{11}\left(0\right)=2, p_{12}\left(0\right)=1.5, p_{13}\left(0\right)=2.1, p_{14}\left(0\right)=1.8,$$p_{15}\left(0\right)=1.3, p_{16}\left(0\right)=2.2, p_{17}\left(0\right)=-1.2, p_{18}\left(0\right)=-1.5$, $p_{21}\left(0\right)=1, p_{22}\left(0\right)=0.5,$$p_{23}\left(0\right)=-1, p_{24}\left(0\right)=1.2, p_{25}\left(0\right)=1.7, p_{26}\left(0\right)=1.3, p_{27}\left(0\right)=1.6, p_{28}\left(0\right)=1.4$, $p_{31}\left(0\right)=5, p_{32}\left(0\right)=-5, p_{33}\left(0\right)=-7, p_{34}\left(0\right)=8, p_{35}\left(0\right)=4, p_{36}\left(0\right)=3, p_{37}\left(0\right)=2,$$p_{38}\left(0\right)=-1$.

The parameters are set as follows: $k_{11}=2/3\sqrt{2}, k_{12}={\displaystyle \frac{1}{6}}, k_{13}=4, k_{21}=4, k_{22}=20,$$k_{23}=4, {\varpi }_1=0.001, {\varpi }_2=0.001,$$k_{c21}=5, k_{c22}=20, k_{c23}=10, k_{c31}=5, k_{c32}=10,$$k_{c33}=15, s_2=0.001, s_3=0.001, T=3$.

Remark 4.

In practical applications, the computational complexity of the proposed control scheme is influenced by both the system order and the number of agents in the networked system. Specifically, higher system orders and larger network sizes tend to result in increased computational demand. Without loss of generality, the simulation experiments in this study were carried out using a third-order system comprising eight agents to validate the effectiveness of the proposed method.

As illustrated in Figure 3, the multi-agent system successfully achieves bipartite containment tracking under the proposed control scheme. The follower agents are observed to converge either to the convex hull formed by the leader agents or to its symmetric counterpart, thereby validating the effectiveness of the controller in handling cooperative–competitive interactions.

Furthermore, Figure 4 depicts the time evolution of the tracking error $e_i$. It can be seen that all tracking errors converge to a predefined residual set within approximately 2.6 s, which is strictly less than the predefined convergence time $T=3$ s. This result confirms the predefined-time performance of the proposed scheme and demonstrates its capability to guarantee convergence within a user-specified time horizon, regardless of the initial conditions and external disturbances.

4.2. Simulation Example 2

To further verify the effectiveness and flexibility of the proposed containment tracking control scheme, an additional simulation is conducted with a smaller predefined convergence time, specifically set as $T=2$ s. Accordingly, the controller parameters are updated to $k_{11}=\sqrt{2}$ and $k_{12}=\frac{1}{4}$, in accordance with the design guidelines established in Lemma 3. All other parameters remain identical to those used in Simulation Example 1 to ensure a consistent and fair comparison.

As shown in Figure 5, the multi-agent system successfully achieves bipartite containment tracking under the proposed predefined-time control scheme. Specifically, follower agents are observed to converge either to the convex hull formed by the leaders or to its symmetric counterpart, according to the definition of bipartite containment. This validates the effectiveness of the proposed controller in handling antagonistic interactions. In addition, Figure 6 illustrates the evolution of the tracking error $e_i$ over time. It can be observed that all tracking errors converge to a predefined residual set within approximately 1.5 s, which is strictly less than the user-specified predefined time $T=2$ s. This confirms the predefined-time convergence property of the proposed scheme and further demonstrates its capability to guarantee high performance with explicit time bounds, despite the presence of nonlinearity and external disturbances.

In order to highlight the contributions of this paper, we consider the work in [47], which represents a relatively advanced approach to consensus tracking control based on the backstepping method for networked multi-agent systems. A comparative analysis is conducted between [47] and the proposed method by examining the relevant simulation results, including Figure 6 and Figure 7, and

Table 2. Tracking errors of each agent at $T=1.5\mathrm{s}$.

Method	Agent (1–4)
Our study	−0.002478	0.000331	−0.001143	0.002313
Ref. [47]	−0.078926	−0.164816	−0.212413	−0.163083
Method	Agent (5–8)
Our study	0.004537	0.001531	0.004458	0.000304
Ref. [47]	0.549299	0.582721	0.558298	0.442251

.

From Figure 6 and Figure 7, and Table 2 and

Table 3. Comparison of tracking error performance after stabilization.

Agent	Mean		Variance
	Ours	Ref. [47]	Ours	Ref. [47]
Agent 1	−0.000973	−0.005092	3.443 × ${10}^{-6}$	1.051 × ${10}^{-5}$
Agent 2	−0.000746	−0.005462	2.410 × ${10}^{-6}$	1.021 × ${10}^{-5}$
Agent 3	0.000723	−0.004643	1.126 × ${10}^{-6}$	1.860 × ${10}^{-5}$
Agent 4	−0.000737	−0.006240	2.615 × ${10}^{-6}$	1.613 × ${10}^{-5}$
Agent 5	0.000719	0.126300	1.741 × ${10}^{-6}$	1.752 × ${10}^{-5}$
Agent 6	−0.000730	0.172900	1.135 × ${10}^{-6}$	1.203 × ${10}^{-5}$
Agent 7	0.000715	0.172900	1.039 × ${10}^{-6}$	1.201 × ${10}^{-5}$
Agent 8	−0.000723	0.124700	1.771 × ${10}^{-6}$	1.842 × ${10}^{-5}$

, it can be observed that, compared with the method presented in [47], the prescribed-time backstepping control scheme proposed in this paper achieves smaller tracking errors within the prescribed time frame while demonstrating superior steady-state performance with both a reduced mean and variance of tracking errors after system convergence, thereby improving the control performance of the system. Moreover, the upper bound of the convergence time can be explicitly specified and flexibly adjusted within the admissible range of actuator capabilities, thereby offering enhanced adaptability to task-specific temporal requirements.

Remark 5.

Regarding the selection of the key control parameters $k_{11}$ and $k_{12}$, it is observed that increasing their values reduces the system’s convergence time. However, this improvement comes at the expense of larger control inputs, as evidenced by Simulation Examples 1 and 2. Therefore, in practical implementations, the parameters should be judiciously tuned according to task-specific requirements, striking a balance between ensuring convergence within the desired time frame and minimizing control effort to reduce energy consumption. Consequently, a trade-off must be considered between convergence performance and energy efficiency in parameter tuning.

5. Conclusions

In this paper, a predefined-time robust control scheme was proposed to achieve bipartite containment tracking for high-order multi-agent systems affected by external disturbances. By leveraging the backstepping method, a distributed controller was systematically designed to guarantee that the bipartite containment tracking errors converge to a predefined residual set within a user-specified time, irrespective of the initial conditions. This predefined-time property not only enhances the timeliness and responsiveness of the control strategy, but also significantly improves its robustness against modeling uncertainties and external perturbations. Simulation results were obtained to demonstrate the effectiveness and practical applicability of the proposed scheme under various scenarios.

It is worth noting that the proposed predefined-time control scheme currently relies on continuous communication among agents, which may impose a heavy burden on network resources in large-scale or resource-constrained systems. Therefore, future research will focus on integrating event-triggered communication strategies into the proposed scheme and implementing practical application of our proposed control scheme in real-world scenarios. By designing appropriate triggering conditions, the communication load can be significantly reduced while preserving system performance and ensuring predefined-time convergence. This research direction is expected to further improve the practicality and scalability of the containment tracking control scheme.