Adaptive containment control algorithm design of second-order multi-agent systems under disturbances

Abstract

This study addresses the robustness challenges in hierarchical coordination of second-order networked multi-agent systems operating under exogenous perturbations. A decentralized adaptive control framework is developed to dynamically regulate follower agents toward a time-varying convex domain spanned by leaders, with explicit compensation mechanisms for bounded yet unstructured disturbances. Unlike conventional containment strategies relying on fixed-gain feedback, the proposed algorithm integrates a nonlinear sigmoidal adaptation law (Equation 6) that autonomously scales control efforts based on real-time error metrics, ensuring asymptotic convergence without prior knowledge of disturbance bounds. Global stability of the closed-loop system is rigorously proven via a composite Lyapunov function that jointly accounts for state errors and adaptive gain dynamics. Parametric studies further reveal a tunable trade-off between convergence speed and control energy consumption. Experimental validations under heterogeneous disturbance profiles—including non-smooth square waves and stochastic noise—demonstrate superior containment accuracy and rapid settling time.

Keywords

Adaptive control multi-agent systems containment control external disturbances

Introduction

Recent advancements in multi-agent systems (MASs) have revolutionized applications ranging from autonomous robotics to smart grids, with a growing emphasis on distributed control strategies that enhance adaptability in dynamic environments. The central objective of MAS research is to design efficient distributed control algorithms that enable individual agents to achieve desired global objectives based on local information. Among the foundational problems in MAS cooperative control, the consensus problem serves as a cornerstone, underpinning advancements in formation control and flocking behavior (Olfati-Saber et al., 2007; Su et al., 2009; Zhang et al., 2023). Consensus in MASs emphasizes decentralized coordination to align agent states through local interactions (Liang et al., 2021). Extending this paradigm, containment control introduces a hierarchical structure in which followers are driven to remain within a time-varying convex region spanned by leaders, a critical requirement for scenarios such as UAV swarm navigation under dynamic threats (Jin et al., 2025; Pang et al., 2022; Yu et al., 2023). For the problem of stochastic environmental disturbances in autonomous underwater vehicles (AUVs), a terminal sliding mode observer combined with a dual-power reaching law is designed to estimate unmeasurable velocities. An adaptive controller is proposed to compensate for stochastic disturbances, and its effectiveness in complex marine environments is validated through theoretical analysis and experiments, providing a significant reference for underwater vehicle formation control (Yan et al., 2023). Also, the processing of random disturbance of AUV further promotes the practical application of MAS in uncertain physical systems.

However, the widespread presence of external disturbances and system uncertainties in real-world environments significantly complicates MAS control problems. External disturbances not only degrade control precision but may also destabilize the system (Xiao et al., 2022). Therefore, designing robust distributed control algorithms capable of mitigating the impact of such disturbances is critical. In recent years, researchers have proposed various effective methods to address the issues of MAS consensus and containment control under disturbances, including robust and adaptive control strategies to counteract external perturbations (Li et al., 2020a; Qian et al., 2020a). For instance, addressing disturbance scenarios in single-integrator systems, Jiang et al. (2024) proposed a fully distributed preset-time containment control method. By employing a preset-time disturbance observer to regulate the convergence time of observation errors, their approach eliminates reliance on global topological information. Sui et al. (2025) extended this research to nonlinear non-strict feedback systems, integrating barrier Lyapunov functions with preset performance functions. While handling full-state constraints, their adaptive strategy ensures follower errors converge to specified boundaries, offering a new framework for robust containment control in complex dynamical systems.

To address diverse task requirements, MAS control strategies have been extensively studied, encompassing consensus control, containment control, and event-triggered control. In the realm of consensus control, a distributed event-triggered adaptive control algorithm was proposed in Qian et al. (2020b) to achieve robust consensus for linear MASs under disturbances while reducing communication and computational burdens. For specific system types, Gong et al. (2019) explored observer-based control methods for discrete-time positive systems, offering solutions for consensus in specialized networked systems. For inherent nonlinear MASs, Li et al. (2024) developed a distributed adaptive finite-time control algorithm. Utilizing a dual-power reaching law to address high- and low-order nonlinear terms, their method eliminates the need for Lipschitz continuity assumptions and significantly enhances anti-disturbance capabilities in complex dynamic systems. Regarding heterogeneous MASs, a fixed-time control approach was proposed in Duan et al. (2023) to tackle formation and containment control challenges involving time-varying outputs, while Yang et al. (2023) addressed bidirectional containment control of fractional-order MASs with input delays, introducing a delay-compensated protocol to overcome difficulties arising from fractional calculus, time delays, and dynamic switching topologies.

Despite these advancements, traditional control strategies often suffer from high communication costs. Event-triggered control mechanisms have emerged as a promising alternative (Liu et al., 2024), significantly alleviating communication burdens while improving control efficiency. For example, an event-triggered adaptive control strategy was developed in Xing et al. (2017) for uncertain nonlinear systems, which combines adaptive gain and event-triggered mechanisms to ensure stability while reducing communication overhead. Similarly, Zhou et al. (2021) introduced an observer-based event-triggered fuzzy adaptive control strategy to address bidirectional consensus under input quantization, enhancing the robustness and stability of MASs under constrained communication and computation conditions. For heterogeneous systems, Feng et al. (2024) proposed an adaptive event-triggered control strategy targeting time-varying output formation containment. By designing dual-trigger conditions to decouple triggering sequences, they effectively reduced communication costs in large-scale heterogeneous networks.

In recent years, adaptive control for stochastic MASs has also witnessed significant progress. To address the effects of stochastic disturbances and nonlinear dynamics, distributed adaptive neural control strategies have been developed, enabling each agent to rely solely on local and neighbor information to achieve control objectives (Li et al., 2020b, 2021a, 2021b; Wang et al., 2017). These studies have not only extended the applicability of MAS control methods but also provided theoretical insights into tackling dynamic challenges in complex environments.

Given the above progress, despite notable achievements in theoretical analysis and algorithm design, challenges remain in realizing adaptive containment control under disturbances while ensuring rapid convergence and high-precision control. To address these issues, the main contributions of this study are as follows:

This paper proposes a novel nonlinear adaptive control protocol for the containment control problem of second-order MASs. Unlike traditional fixed-gain strategies, the proposed dynamic gain mechanism evolves based on instantaneous tracking errors, enabling adaptive regulation of the control input. The design integrates both state error and adaptive gain dynamics, and rigorously guarantees the global asymptotic stability of the closed-loop system.

In contrast to conventional simulations under a single disturbance type, this study conducts comprehensive numerical experiments under various disturbance scenarios, including sinusoidal signals, square waves, random noise, and mixed noise. The simulation results clearly demonstrate that the proposed algorithm exhibits superior performance in terms of convergence speed and robustness against disturbances.

The subsequent sections of this paper are structured to systematically address the challenges of adaptive containment control under disturbances. Section “Preliminaries and model establishment” establishes foundational graph-theoretic concepts and notations critical for modeling multi-agent interactions. Building on this framework, section “Analysis of adaptive containment control” formulates the adaptive containment control problem and develops a distributed control law with dynamic parameter adaptation. To validate the theoretical analysis, section “Simulation example” conducts comprehensive numerical simulations under heterogeneous disturbance scenarios, including sinusoidal and discontinuous signals. Finally, section “Conclusion” summarizes the key contributions and discusses potential extensions for adaptive control strategies in networked systems with switching topologies.

Notations: ℝ stands for all real numbers, and $ℝ^{n}$ denotes Euclidean space with dimension n. For real numbers a, b, $sign (a)$ denotes the sign function, $si g^{b} (a) ≜ | a |^{b} sign (a)$ . For a vector A, $A^{T}$ denotes the transposition of A, and its Euclidean norm is expressed as $∥ A ∥$ . $dist (x, Co (X))$ denotes Euclidean distance of scalar x to the convex hull of X, where $Co (X)$ means convex hull of set $X \subset ℝ$ . $λ_{\min} (\cdot)$ , $λ_{\max} (\cdot)$ denote the maximum and minimum eigenvalues of a matrix, respectively.

Preliminaries and model establishment

Preliminary knowledge

Without loss of generality, let networks consisting of n agents, which share identical state space ℝ. To facilitate the research, the following definitions are provided: graph $G = (V, E, A)$ represent the structure of the entire system, where node set $V = {v_{1}, \dots, v_{n}}$ , edge set $E \subseteq V \times V$ , and adjacency matrix $A = {[a_{ij}]}_{n \times n}$ . Furthermore, $(v_{j}, v_{i}) \in E$ means $a_{ji} \neq 0$ . It also implies that node $v_{j}$ can obtain information from node $v_{i}$ . Graph $G$ has no self-loops means $a_{ii} = 0$ and is undirected implies $a_{ij} = a_{ji}$ .

Assumption 1. (Olfati-Saber et al., 2007). The communication topology $G$ is designed such that every follower agent maintains at least one uninterrupted path from a leader node, that is

Path (j \to i) \subseteq E

(1)

Lemma 1. (Cao et al., 2013). Consider an MAS with a directed spanning tree (exists at least one root node (e.g. a leader), and all other nodes (followers) have directed paths originating from the root node) in its communication topology $G$ .

The following properties hold for the submatrix $L_{F}$ and interaction matrix $L_{FL}$ : (1) Stability: All eigenvalues of $L_{F}$ lie strictly in the right-half complex plane ( $Re (λ > 0)$ ); (2) non-negativity: every entry of the composite matrix $L_{F}^{- 1} L_{FL}$ is non-negative; (3) row stochasticity: the row-sum of $L_{F}^{- 1} L_{FL}$ satisfies 1 for all i.

Definition 1. (Li et al., 2018). Let sets $X = {x_{1}, x_{2}, \dots, x_{n}}$ and $V = {v_{1}, v_{2}, \dots, v_{n}}$ be subsets of the real vector space ℝ. The positions X and velocities V of follower $i (i = 1, \dots, m)$ are said to converge to the convex hull within a predetermined time if and only if they satisfy the convergence condition

C_{O} (X) = {\sum_{i = 1}^{n} λ_{i} x_{i} | x_{i} \in X, λ_{i} > 0, \sum_{i = 1}^{n} λ_{i} = 1}

(2)

C_{O} (V) = {\sum_{i = 1}^{n} λ_{i} v_{i} | x_{i} \in V, λ_{i} > 0, \sum_{i = 1}^{n} λ_{i} = 1}

(3)

Problem description

Consider a second-order MAS consisting of both followers and leaders. In practical scenarios, where external disturbances are present in the control inputs, the dynamics of the follower agents can be expressed as follows

{\begin{matrix} ẋ_{i} (t) = v_{i} (t), \\ {\overset{\cdot}{v}}_{i} (t) = u_{i} (t) + w_{i} (t), \end{matrix} i \in ℱ

(4)

In this paper, we consider the case of static leaders, such that ${\overset{\cdot}{v}}_{i} (t) = 0, i \in ℛ$ where the dynamics of the leader agents are governed by the following equation:

{\begin{matrix} ẋ_{i} (t) = v_{i} (t), \\ {\overset{\cdot}{v}}_{i} (t) = 0, \end{matrix} i \in ℛ,

(5)

where $x_{i} (t)$ , $v_{i} (t)$ , $u_{i} (t)$ are known as the position, velocity, and control input, respectively. $w_{i} (t)$ denotes the external disturbances.

Denote $ℱ \equiv {m, m + 1, \dots, n}$ and $ℛ \equiv {1, 2, \dots, m}$ as the follower set and the leader set, respectively. For clarity, the stacks of positions and velocities of the leader and follower agents are expressed as follows

{\begin{matrix} x_{L} (t) = {x_{1} (t) \cdot \cdot \cdot x_{m} (t)}^{T} \\ v_{L} (t) = {v_{1} (t) \cdot \cdot \cdot v_{m} (t)}^{T} \end{matrix}

(6)

{\begin{matrix} x_{F} (t) = {x_{m + 1} (t) \cdot \cdot \cdot x_{n} (t)}^{T} \\ v_{F} (t) = {v_{m + 1} (t) \cdot \cdot \cdot v_{n} (t)}^{T} \end{matrix}

(7)

The objective of the design is to develop an adaptive controller that converges more rapidly to the convex hull of the leader within a finite time, capable of efficiently handling uncertainties, external disturbances, and dynamic changes. This controller is expressed as

\begin{matrix} u_{i} (t) = c (t) {si g^{α_{1}} (3 \sum_{j \in N_{i}} a_{ij} (x_{j} (t) - x_{i} (t))) \\ + si g^{α_{2}} (3 \sum_{j \in N_{i}} a_{ij} (v_{j} (t) - v_{i} (t))) \\ + β_{1} \sum_{j \in N_{i}} a_{ij} (x_{j} (t) - x_{i} (t)) \\ + β_{2} \sum_{j \in N_{i}} a_{ij} (v_{j} (t) - v_{i} (t))} \end{matrix}

(8)

where $α_{1}$ , $α_{2}$ , $β_{1}$ , $β_{2}$ are real numbers, $0 < α_{1} < 1$ , $α_{2} = \frac{2 α_{1}}{α_{1} + 1}$ . And $c (t)$ denotes the adaptive coupling strength.

The adaptive law for $c (t)$ is defined as

\overset{\cdot}{c} (t) = e^{a_{1} t} ρ e^{T} (t) e (t)

(9)

where ρ and $a_{1}$ are positive constants, $0 < a_{1} < λ_{\min} (L_{F})$ , and $e (t) = e_{x} (t) + e_{v} (t)$ represents the combined error.

Definition 2. (Chen et al., 2009). The containment control objective for the networked agents in (4)–(5) is achieved when the follower states, governed by a distributed protocol, asymptotically approach a time-varying convex combination of leader states. Formally, for any initial condition $x_{i} (0)$ , there exists $T > 0$ such that

\lim_{t \to T} dist (x_{F} (t), Co (x_{L} (t))) = 0

(10)

The Laplacian matrix L is partitioned to isolate leader–follower interactions, a configuration critical for decentralized containment in networked systems

L = [\begin{matrix} 0_{m \times m} & 0_{m \times (n - m)} \\ L_{FL (n - m) \times m} & L_{F (n - m) \times (n - m)} \end{matrix}]

(11)

where $L_{F} \in ℝ^{(n - m) \times m}$ is symmetric and $L_{FL} \in ℝ^{(n - m) \times (n - m)}$ . $L_{F}$ characterizes the interactions among followers. The diagonal entries denote the total input weights of followers, whereas the off-diagonal entries are defined as negative adjacency weights. While $L_{FL}$ characterizes leader-to-follower couplings, where each element measures the influence exerted by a leader on a follower.

The control input integrates both nonlinear and linear feedback terms

\begin{matrix} u_{i} (t) = c (t) { & {sig}^{α_{1}} (- 3 Lx (t)) + {sig}^{α_{2}} (- 3 Lv (t)) \\ - β_{1} Lx (t) - β_{2} Lv (t)} \end{matrix}

(12)

where the nonlinear terms dominate during transient phases to rapidly mitigate disturbances, while the linear terms ensure steady-state precision.

Assumption 2. (Xiao et al., 2022). The external disturbance $w_{i} (t)$ is assumed to be bounded, that is, there exists a positive constant $W_{i} > 0$ such that

∥ w_{i} (t) ∥ \leq W_{i}, i \in ℱ

(13)

Remark 1. The proposed control law (12) integrates a nonlinear feedback term, a linear feedback term, and an adaptive gain, whose synergistic action achieves robust containment control in the presence of disturbances. The nonlinear component dominates during the transient phase when the system error is large, rapidly suppressing initial disturbances and accelerating convergence; the linear component provides smooth and precise regulation in the steady state when the error is small. Moreover, the adaptive gain dynamically scales the overall control effort, autonomously enhancing robustness under perturbations and thus attaining a balance between speed and stability. Compared to a fixed-gain strategy alone, this approach offers significant advantages in attenuating nonsmooth disturbances and reducing control energy consumption.

Remark 2. The developed adaptive control algorithm employs a dynamic adjustment gain $c (t)$ in combination with a nonlinear feedback term $si g^{α} (\cdot)$ , effectively suppressing various types of bounded disturbances, including sinusoidal signals, random noise, and non-smooth bounded excitations. The key advantage lies in the adaptive law relying solely on the real-time tracking error $e (t)$ , thereby eliminating the need for precise boundary information or prior knowledge of the disturbances. Meanwhile, the use of a composite Lyapunov function (as shown in equation (20)) rigorously ensures the global stability of the closed-loop system, even in the presence of unknown bounded disturbances. Furthermore, the nonlinear feedback term $si g^{α} (\cdot)$ acts instantaneously to guide the control input for rapid disturbance suppression, while the linear feedback component enhances the control precision during the steady-state phase, achieving a balanced performance between dynamic regulation capability and steady-state stability.

Remark 3. The simulation results demonstrate that the proposed method exhibits robust disturbance rejection performance across various disturbance types. For instance, under sinusoidal, stochastic noise, and square wave disturbances, the steady-state error converges to a small neighborhood (approximately 2.85), validating its effectiveness. Although the convergence time under square wave disturbance is slightly longer (0.98 s), this is attributed to the non-smooth nature of the signal, which increases the challenge for the $c (t)$ gain to adapt rapidly. In contrast, random noise and sinusoidal disturbances are comparatively easier to suppress due to the statistical characteristics of the nonlinear term.

In the subsequent sections, the primary results are presented and discussed in detail based on the above framework.

Analysis of adaptive containment control

Theorem 1. For any given initial condition $x_{i} (0)$ , $i \in ℱ$ , if there exists a control protocol $u_{i}$ that ensures the states of the followers stably converge to the convex hull formed by the states of the leaders, then the system described by (3) and (4) achieves leader–follower consensus. This result effectively addresses the containment control problem, ensuring that the agents $i (i = m, m + 1, \dots, n)$ not only satisfy the specified constraints but also guarantee the stability and convergence of the followers’ states to the target region defined by the leaders.

{\begin{matrix} \lim_{t \to \infty} | | x_{F} (t) + L_{F}^{- 1} L_{FL} x_{L} (t) | | = 0 \\ \lim_{t \to \infty} | | v_{F} (t) + L_{F}^{- 1} L_{FL} v_{L} (t) | | = 0 \end{matrix}

(14)

The containment error functions are defined as follows:

Step 1: Relative position state information:

\begin{matrix} e_{x} (t) = \sum_{j = 1}^{n - m} a_{ij} (x_{i} (t) - x_{j} (t)) + \sum_{l = 1}^{m} a_{il} (x_{i} (t) - x_{l} (t)) \end{matrix}

(15)

Step 2: Relative velocity state information:

\begin{matrix} e_{v} (t) = \sum_{j = 1}^{n - m} a_{ij} (v_{i} (t) - v_{j} (t)) + \sum_{l = 1}^{m} a_{il} (v_{i} (t) - v_{l} (t)) \end{matrix}

(16)

Furthermore

{\begin{matrix} e_{x} (t) = L_{F} x_{F} (t) + L_{FL} x_{L} (t) \\ e_{v} (t) = L_{F} v_{F} (t) + L_{FL} v_{L} (t) \end{matrix}

(17)

According to (5), we can get that

\begin{matrix} \begin{matrix} u (t) = c (t) {si g^{α_{1}} (- 3 L_{FL} x_{L} (t) - 3 L_{F} x_{F} (t)) \\ + si g^{α_{2}} (- 3 L_{FL} v_{L} (t) - 3 L_{F} v_{F} (t)) \\ - β_{1} (L_{FL} x_{L} (t) + L_{F} x_{F} (t)) \\ - β_{2} (L_{FL} v_{L} (t) + L_{F} v_{F} (t))} \end{matrix} \end{matrix}

(18)

By combining equations (14) and (15), the control input is derived as follows

\begin{matrix} u (t) = c (t) { & si g^{α_{1}} (- 3 e_{x} (t)) + si g^{α_{2}} (- 3 e_{v} (t)) \\ - β_{1} (e_{x} (t)) - β_{2} (e_{v} (t))} \end{matrix}

(19)

Proof. To analyze the closed-loop stability, a composite Lyapunov function is designed as

V (t) = V_{1} (t) + V_{2} (t)

(20)

where the specific forms of $V_{1} (t)$ and $V_{2} (t)$ will be determined as follows

V_{1} (t) = \frac{1}{2} {e_{v}}^{T} (t) {L_{F}}^{- 1} e_{v} (t)

(21)

V_{2} (t) = \sum_{i = 1}^{n} V_{2 i} (t) + \int_{0}^{χ^{T} (t)} W_{i} ds

(22)

Then

V_{2 i} (t) = \int_{0}^{χ^{T} (t)} \tilde{c} (si g^{α_{1}} (3 s) + β_{1} s) ds

(23)

χ (t) = \sum_{j = 1}^{n - m} a_{ij} (x_{j} (t) - x_{i} (t)) + \sum_{l = 1}^{m} a_{il} (x_{l} (t) - x_{i} (t))

(24)

It is easy to see that $V_{2}$ is nonnegative, the time derivative of $V_{1}$ can be obtained as

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = {e_{v}}^{T} (t) L_{F}^{- 1} ė_{v} (t) \\ = {e_{v}}^{T} (t) L_{F}^{- 1} L_{F} (u_{i} (t) + w_{i} (t)) \end{matrix}

(25)

Substituting (14) and (16) into (22) yields

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = {e_{v}}^{T} (t) {c (t) [si g^{α_{1}} (- 3 e_{x} (t)) + si g^{α_{2}} (- 3 e_{v} (t)) \\ - β_{1} (e_{x} (t)) - β_{2} (e_{v} (t))] + w (t)} \\ \leq {e_{v}}^{T} (t) c (t) [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)] \\ + {e_{v}}^{T} (t) c (t) [| 3 e_{v} (t) |^{α_{2}} - β_{2} e_{v} (t)] \\ + {e_{v}}^{T} (t) w (t) \end{matrix}

(26)

Without loss of generality, let $\tilde{c} \geq c (t)$ , we get from (23) that

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) \leq {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)] \\ + {e_{v}}^{T} (t) \tilde{c} [| 3 e_{v} (t) |^{α_{2}} - β_{2} e_{v} (t)] + {e_{v}}^{T} (t) w (t) \\ \leq {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)] \\ - \tilde{c} 3^{α_{2}} | e_{v} |^{α_{2} + 1} - \tilde{c} β_{2} | e_{v} |^{2} + {e_{v}}^{T} (t) w (t) \end{matrix}

(27)

Considering that $β_{2} > 3^{α_{2}}$ and $α_{2} < 1$

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) & \leq {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)] \\ + {e_{v}}^{T} (t) w (t) \end{matrix}

(28)

Then, the time derivative of $V_{2}$ can be obtained as

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) & = \overset{\cdot}{χ} (t) \tilde{c} (si g^{α_{1}} (3 χ (t)) + β_{1} χ (t)) + χ (t) W_{i} \\ = - {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)] \\ - {e_{v}}^{T} (t) W_{i} \end{matrix}

(29)

Above all, by (25) and (26), it follows from (17) that

\overset{\cdot}{V} = V_{1} + {\overset{\cdot}{V}}_{2} \leq 0

(30)

Then, it follows that V is negative semi-definite, implying that V is nonincreasing. Since $\overset{\cdot}{V} (t) \leq 0$ and $V (t)$ is bounded, $c (t)$ converges and is monotonically increasing, eventually approaching a constant steady-state value. Therefore, $\overset{\cdot}{V} (t) = 0$ occurs only at the equilibrium point, which implies $e_{v} (t) = 0$ . According to LaSalle’s invariance principle, it follows from $\overset{\cdot}{V} (t) \leq 0$ that $\lim_{t \to \infty} ∥ e_{v} (t) ∥ = 0$ and $\lim_{t \to \infty} ∥ e_{x} (t) ∥ = 0$ .

Thus, $\lim_{t \to \infty} ∥ x_{F} (t) + L_{F}^{- 1} L_{FL} x_{L} (t) ∥ = 0$ and $\lim_{t \to \infty}$ $∥ v_{F} (t) + L_{F}^{- 1} L_{FL} v_{L} (t) ∥ = 0$ , and it follows from Theorem 1 that the MAS achieves containment control.

Corollary 1. In the absence of external disturbances $w (t)$ , that is, $w (t) = 0$ , the convergence proof follows a similar approach to that of Theorem 1. By applying Theorem 1, it can be concluded that the second-order MAS achieves leader–follower containment consensus within a finite time.

Proof. Consider the Lyapunov function defined as

V (t) = V_{1} (t) + V_{2} (t)

(31)

where $V_{1} (t)$ and $V_{2} (t)$ are given by equations (12) and (13), respectively.

{\overset{\cdot}{V}}_{1} (t) \leq {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)]

(32)

{\overset{\cdot}{V}}_{2} (t) = - {e_{v}}^{T} (t) \tilde{c} [si g^{α_{1}} (- 3 e_{x} (t)) - β_{1} e_{x} (t)]

(33)

The time derivative of $V_{2}$ can be obtained as

\overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2} \leq 0

(34)

Remark 4. In contrast to existing controllers designed for disturbance-free or simple disturbance models, the proposed algorithm is theoretically capable of handling second-order MASs with bounded disturbances. The adaptive mechanism of the algorithm relies on the estimation of the disturbance boundary rather than precise modeling, allowing the controller to achieve stable containment control even under unknown disturbances.

Remark 5. Although the robustness of the proposed algorithm has been validated in this section through simulations under five representative types of disturbances (i.e. sinusoidal, square wave, random noise, exponential decay, and mixed disturbances), along with detailed evaluations of convergence time, steady-state error, and control input magnitude, experimental validation on a physical platform remains a crucial step in assessing the algorithm’s performance in real-world environments. To this end, we have initiated a system requirements analysis and hardware selection process for a UAV formation flight platform. Based on the parameter ranges proposed in this paper, we are conducting phased debugging of the UAV flight control loops. In the next phase, we plan to refine the control parameters using experimental data and explore the integration of event-triggered or intermittent communication strategies to further optimize the overall performance in practical applications.

Simulation example

This section presents numerical simulations to evaluate the proposed adaptive containment framework under various disturbance scenarios. The results demonstrate the algorithm’s superior performance in containment precision and transient response compared to existing methods.

The communication topology among the agents is illustrated in Figure 1, including three leaders (nodes 1, 2, and 3) and five followers (nodes 4–8). And all edge weights are set to 1.

Figure 1.

The communication graph.

Consider an MAS consisting of three leaders and five followers. Assume that the initial positions and velocities of the three leaders are $x_{L} (0) = [8; 3; 2]$ and $v_{L} (0) = [5; 2; 3]$ , respectively. The initial positions and velocities of the five followers are $x_{F} (0) = [2; 4; 5; 1; 3]$ and $v_{F} (0) = [2; 3; 6; 8; 10]$ , respectively.

The Laplacian matrices $L_{FL}$ and $L_{F}$ are given as

L_{FL} = [\begin{matrix} - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix}] L_{F} = [\begin{matrix} 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 3 & - 1 & 0 \\ 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & - 1 & 2 \end{matrix}]

The controller parameters are chosen as $α_{1} = 0.8$ , $α_{2} = 0.8889$ , $β_{1} = 9$ , $β_{2} = 4$ . Select $α_{1} = 0.1, 0.3, 0.6, 0.8$ , respectively, for comparison. When $α_{1} = 0.1, 0.3$ , the control inputs will all experience toothed fluctuations and it will take a long time to reach stability. When $α_{1} = 0.6, 0.8$ , chattering is virtually eliminated, and the latter case achieves faster stabilization and shorter convergence time compared to the former. And the parameters of the adaptive law are $a_{1} = 0.04$ , $ρ = 0.01$ .

First, when the disturbance is a sinusoidal signal, which is also one of the most common types of disturbances.

Case 1: $w_{i} (t) = A \sin (ωt)$ , $i \in {1, 2, \dots, 5}$

When it comes to this case, the parameters are chosen with an amplitude of $A = 1.5$ and a frequency of $ω = 3$ .

As shown in Figure 2, the followers (thin lines) initially exhibit oscillatory behaviors due to sinusoidal disturbances but progressively converge to the convex hull spanned by leaders (bold lines) within 10 s. Notably, the adaptive gain $c (t)$ in Figure 5 undergoes rapid adjustment during the initial 2 s, reflecting the algorithm’s sensitivity to abrupt disturbance changes.

Figure 2.

Agent trajectories under sinusoidal disturbances.

Figure 3.

Agent velocities under sinusoidal disturbances.

Figure 4.

State trajectory control inputs.

Figure 5.

Adaptive response curves under sinusoidal disturbances.

In practical scenarios, various types of disturbances may arise, such as exponential decay disturbances, random signal disturbances, square-wave disturbances, and mixed signal disturbances.

In the following analysis, we consider the performance of the dynamic system under these different types of disturbances to demonstrate the superior capabilities of the proposed method.

The velocity trajectories and adaptive response curves for individual agents (specifically, followers 1 and 2) under different disturbance types are shown in Figures 6 and 7.

Figure 6.

Velocities of followers $(F 4, F 5)$ under various disturbances: (a) shows the velocity trajectories of F4 under five types of disturbances and (b) shows the velocity trajectories of F5 under five types of disturbances.

Figure 7.

Adaptive response curves of followers $(F 4, F 5)$ under various disturbances: (a) shows the adaptive response curves of F4 under five types of disturbances and (b) shows the adaptive response curves of F5 under five types of disturbances.

As shown in Figures 8 and 9, under different types of disturbances, the states of the followers are effectively contained within the convex hull formed by the leader. Additionally, the adaptive response curves of the followers rapidly converge to a fixed constant value.

Figure 8.

Velocity of agents under various disturbances (a) shows the velocity trajectories of the agents under sinusoidal disturbance; (b) shows the velocity trajectories of the agents under exponential disturbance; (c) shows the velocity trajectories of the agents under random disturbance; (d) shows the velocity trajectories of the agents under square wave disturbance; (e) shows the velocity trajectories of the agents under mixed disturbance.

Figure 9.

Adaptive response curves under various disturbances: (a) shows the adaptive response curves of the agents under sinusoidal disturbance; (b) shows the adaptive response curves of the agents under exponential disturbance; (c) shows the adaptive response curves of the agents under random disturbance; (d) shows the adaptive response curves of the agents under square wave disturbance; (e) shows the adaptive response curves of the agents under mixed disturbance.

To facilitate a better comparison of the containment performance of the agents under different disturbances, three groups of parameters were selected for comparison.

Case 1: $a_{1} = 0.04$ , $ρ = 0.01$

The performance results are shown in Table 1. The system exhibits varying behavior under different disturbance types. Under random disturbances, the system achieves the best performance, with the shortest convergence time, smallest steady-state error (2.83), and control input remaining within a reasonable range (1.17). In contrast, under square-wave disturbances, the steady-state error reaches its maximum, indicating relatively weaker adaptability to such discontinuous signals.

Table 1.

Performance under different disturbance types.

Disturbance type	Containment time (s)	Steady-state error mean	Control input mean
Sin	0.96	2.8327	1.1768
Exp	1.00	2.8544	1.1765
Square	0.98	2.8625	1.1763
Random	0.91	2.8308	1.1769
Mixed	0.94	2.8318	1.1767

Case 2: $a_{1} = 0.1$ , $ρ = 0.05$

No significant improvement is observed in convergence time across the five disturbance types. However, the control input increases substantially to 1.87—approximately a 60% increase compared to Case 1—resulting in considerably higher energy consumption. The steady-state error under random disturbance remains minimal (2.83), while it increases to 2.86 under square-wave disturbance.

Case 3: $a_{1} = 0.35$ , $ρ = 0.1$

In this case, the convergence time under sinusoidal disturbance is prolonged. The steady-state error is smallest (2.845) under both random and hybrid disturbances. However, this improvement comes at the cost of the highest energy consumption under random disturbance, with the control input reaching 2.86.

Based on the comprehensive comparison, it is observed that increasing the gain adjustment rate tends to amplify the cumulative error effect. Meanwhile, higher values of both ρ and $a_{1}$ inevitably lead to an increase in control input energy, indicating that over-optimization of these parameters may compromise energy efficiency. Among the three parameter sets, the most pronounced performance degradation is observed under square-wave disturbances, which could be attributed to the exponent-based nonlinearity (e.g. $si g^{a} (\cdot)$ ) being sensitive to discontinuous signals. In contrast, the best overall performance is observed under random disturbances.

Nevertheless, the system demonstrates relatively stable performance across all five disturbance types. The average control input values under different disturbances remain comparable, reflecting the controller’s consistency in energy usage and robustness in the presence of various perturbations.

Conclusion

This paper studies second-order MASs subject to external disturbances by designing an adaptive control law with dynamic gain. Numerical simulations indicate that under various disturbances (sinusoidal, square-wave, and stochastic noise), the proposed method achieves faster convergence and lower steady-state error compared to conventional fixed-gain and linear-gain approaches, demonstrating strong robustness and disturbance rejection capabilities; thus, it provides a high-performance, reliable solution for containment control of second-order networked systems.

Although the proposed algorithm performs well under bounded disturbances, several issues warrant further investigation in future work: extending the algorithm to time-varying topologies to verify convergence; redesigning the control law and conducting time-delay stability analysis when delays are introduced; incorporating sliding-mode disturbance observers to handle more complex or unbounded noise; and generalizing the approach to higher-order or more complex dynamics to enhance applicability and reliability under broader dynamic network and disturbance conditions.

Footnotes

Author contributions

Conceptualization: Ronghua Chi, Ruitian Yang

Methodology: Ruitian Yang, Lei Wang

Formal Analysis: Dagao Tian

Original Draft: Ruitian Yang, Dagao Tian

Review & Editing: All authors

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Jiangsu Province Industry University Research Cooperation Project (BY20230745); Wuxi Young Scientific and Technological Talent Support Initiative, Project Number: TJXD-2024-203, the Wuxi University research start-up fund for introduced talents (Grant No. 550223012), the Wuxi University research start-up fund for introduced talents (Grant No. 550222025).

ORCID iDs

Ruitian Yang

Ronghua Chi

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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