ARTICLE TYPE

Sampled-Data Based Consensus in the Presence of Asymmetric
Heterogeneous Actuator Saturation

Juan Qian<fnref n="1" />,<fnref n="2" /> | Xiaoling Wang<fnref n="1" />,<fnref n="3" /> | Daniele Astolfi<fnref n="2" /> | Housheng Su<fnref n="4" /> | Guo-Ping Jiang<fnref n="1" />

Correspondence

*Housheng Su, School of Artificial
Intelligence and Automation, Image
Processing and Intelligent Control Key
Laboratory of Education Ministry of China,
Huazhong University of Science and
Technology, Luoyu Road 1037, Wuhan
430074, PR China. Email:
houshengsu@gmail.com

Funding Information

This research was supported by the National
Natural Science Foundation of China under
Grants 62473207, 62425602, 62373197,
Natural Science Fund for Excellent Young
Scholars of Jiangsu Province under Grant
No. BK20220104, the Fundamental
Research Funds for the Central Universities
under Grant No. 2024SMECP03, the French
National Agency of Research via the grant
ANR Alligator (ANR-22-CE48-0009-01).

Summary

In this article we investigate the problem of consensus of multi-agent systems in the presence of input saturation constraints and sampled-data control. An improved low-gain feedback strategy is proposed for achieving semi-global consensus while mitigating the adverse effects of saturation. Furthermore, a distributed dynamic saturation reconstruction protocol is designed to tackle the global consensus problem in systems with asymmetric heterogeneous actuator saturation. The results in this two approaches all reveal a novel aspect by providing an explicit upper bound for the sampling periods of heterogeneous saturated agents, posing difficulties to consensus analysis schemes. These findings, as shown in simulations, contribute to advancing the understanding and implementation of consensus in multi-agent systems.

KEYWORDS:

Asymmetric heterogeneous actuator saturation, sampled-data control, dynamic saturation reconstruction, global consensus

<footnote n="1">¹College of Automation & AI, Nanjing
University of Posts and
Telecommunications, and Jiangsu
Engineering Lab for IOT Intelligent
Robots(IOTRobot), Nanjing 210023, PR
China</footnote>

<footnote n="2">²Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43 boulevard du 11 novembre 1918, F-69100, Villeurbanne, France</footnote>

<footnote n="3">³Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China</footnote>

<footnote n="4">⁴School of Artificial Intelligence and
Automation, Image Processing and
Intelligent Control Key Laboratory of
Education Ministry of China, Huazhong
University of Science and Technology,
Luoyu Road 1037, Wuhan 430074, PR
China</footnote>
# 1 | INTRODUCTION

The consensus problem stands as a fundamental issue in the context of distributed coordinated control research ¹. Recent theoretical developments have increasingly focused on actuator saturation ² (also referred to as input saturation, imperfect actuators, or bounded control)—due to its critical relevance in practical industrial applications. Notably, the investigation of stabilization problems for saturated single systems serves as a precursor to the research on consensus in saturated multi-agent systems. Typically, a single continuous-time system is described as $\dot{x} = Ax + Bsat_{\Delta}(u)$. For a detailed exploration, readers are referred to ³,⁴,⁵, while this paper introduces only the essential concepts.

Based on the initial states, stabilization control can be categorized into two types: semi-global and global. In semi-global stabilization, the initial states of $x$ are confined to a bounded region, while in global stabilization, this restriction is removed ⁶. For high-order integrator systems with an integral chain length greater than 2, asymptotically null controllable with bounded controls (ANCBC), i.e., ($A, B$) is stabilizable and all eigenvalues of $A$ lie in the closed left-half plane, is a key prerequisite for achieving both semi-global and global stabilization ⁷.

When the saturated system dynamics are governed by general linear differential equations, Sussmann et al., Wang et al. ⁸,⁹ have highlighted that nonlinear distributed controllers are necessary to address the global consensus under the ANCBC condition, and several approaches ¹⁰,¹¹,¹²,¹³ have been proposed. In the work of Su et al. ¹⁰, a scheduled low-gain feedback method, inspired by Lin's work ³, was introduced to mitigate actuator saturation by dynamically adjusting the low-gain parameter based on real-time inputs. However, the dependence of this parameter on real-time agents' states contradicts the distributed nature of coordinated systems. In the work of Zhang ¹², a fully distributed control protocol was designed, demonstrating excellent robust performance against communication topology changes. However, this method is only applicable to neutrally stable systems. For further insights into this area, please refer to our recent paper ¹³.

Current research on saturated multi-agent systems, as mentioned above, primarily addresses actuator saturation with homogeneous levels. However, in practical engineering, asymmetric ¹⁴ and heterogeneous saturation actuators are more common. Some efforts have been made to address the first aspect. For instance, Qian and Zhang et al. ¹⁵ proposed a distributed tracking strategy for unmanned aerial vehicle formations under asymmetric input constraints and actuator faults. He et al. ¹⁶ emphasized the role of asymmetric saturation in mobile train lifting jacks for controlling speed and minimizing vibration during EMU maintenance. Qian and Wang et al. ¹⁷ addressed robust tracking control in leader-follower systems under asymmetric saturation and external disturbances. However, while the second scenario was partially considered in our recent work ¹³, the combined effect of asymmetric and heterogeneous saturation, which we term as “asymmetric heterogeneous actuator saturation”, has not yet been systematically studied, posing a significant challenge. Notably, asymmetric heterogeneous saturation encompasses asymmetric, heterogeneous, and homogeneous saturation as special cases. Thus, the proposed method is also applicable to these three types of actuator saturation. This prompts a critical question: how can consensus be achieved in such systems under the ANCBC condition while maintaining distributed feature?

Building on this challenge, the sampled-data paradigm enhances both the efficiency and feasibility of networked systems. In this approach, a digital signal processor generates discrete signals for each agent, which are then converted into continuous signals using a zero-order-holder ¹⁸. These continuous signals are subsequently transmitted to the actuators for further act on the agents. The integration of sampled-data paradigm with existing control algorithms is a step-by-step process. In the context of linear plants with asymmetric input saturation, Mariano et al. ¹⁹ proposed a shifted stabilizer to guarantee the region of attraction. Braun et al. ²⁰,²¹ extended and optimized this result, with one work ²⁰ focusing on single-input systems, while another work ²¹ not only considered multi-input systems but also applied a sampled-data framework to update parameters. For general linear multi-agent systems, there exist a wide range of applications, Zhang and Tang, Zhao and Hua ²²,²³ conducted research on the sampled-data leaderless mean square consensus problem. Furthermore, Liu et al. ²⁴ introduced a sampled-data consensus control law, and provided explicit upper bounds on sampling periods for all agents based on a technical lemma. Additionally, Zhao et al. ²⁵ developed a hybrid systems approach that allows for larger upper bounds on the sampling periods.

Despite the growing interest in sampled-data paradigm, literature exploring their application in asymmetric heterogeneous saturated multi-agent systems remains limited. For example, the sampled-data based protocol proposed in Zou's work ²⁶ is restricted to first-order, single-input systems, making it inadequate for higher-order, multi-input scenarios. As such, there remains a significant gap in the consensus of multi-input multi-output (MIMO) agents with asymmetric heterogeneous input saturation constraints.

Motivated by the statements above, the key novelties of this paper are outlined as follows:
(i) This paper tackles the consensus problem for MIMO systems with asymmetric heterogeneous saturation, where asymmetry refers to different upper and lower saturation bounds, while heterogeneity involves varying saturation across agents and dimensions. Compared to Zou's work²⁶, our research is applied to general linear multi-agent systems within a sampled-data framework, covering single- and double-integrator systems. Moreover, since the considered saturation setting generalizes several existing special cases, the developed approach naturally extends to multi-agent systems with purely homogeneous, heterogeneous, or asymmetric actuator saturation.

(ii) Building on several work³,²⁷,²⁸, this paper addresses the challenges of asymmetric and heterogeneous saturation by improving the traditional low-gain method to achieve semi-global consensus. Within the unified sampled-data framework, particular attention is paid to the influence of the low-gain parameter on the allowable sampling intervals. A fundamental trade-off is revealed: selecting a smaller low-gain parameter helps suppress the influence of input saturation, but also leads to a reduction in the maximum admissible sampling period.

(iii) Furthermore, to refine the results, we delve deeply into the principle of dynamic regulation for reaching global consensus, which is influenced by the concept of dynamic saturation level in our recent paper¹³, and this concept is derived from pioneering works²⁹,³⁰,³¹. The redesigned saturation functions are designed in the form of first order differential equation for each agent, comparing against the asymmetric and heterogeneous saturation levels in the dynamics. Due to the unnecessary energy consumption caused by real-time updating in the dynamic regulation process, we adopt a sampled-data framework to minimize the frequency of updates as much as possible.

(iv) Our findings contribute a breakthrough feature of providing an explicit upper bound for the sampling periods of all agents in a MIMO system, essentially a case in which sampling is synchronous. This bound not only guarantees the stability of the closed-loop system but also presents challenges to consensus analysis schemes.

# 2 | PROBLEM FORMULATION & PRELIMINARIES

## 2.1 | Notation

The notations in this paper are as follows. $\mathbb{R}$, resp. $\mathbb{R}^{m \times n}$, resp. $\mathbb{R}^N$, denotes the set of real number, resp. $m \times n$ matrices, resp. $N$-dimensional real vector space. $\mathbb{N}$ equals $\{0, \mathbb{Z}^+\}$ with $\mathbb{Z}^+$ being the set of all positive integers. Given two vectors $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^p$, we compactly write $(x, y) := (x^\top, y^\top)^\top \in \mathbb{R}^{n+p}$. $\Delta_i = \Delta_j$ with $\Delta_i = [\Delta_{i1}, \dots, \Delta_{im}] \in \mathbb{R}^m$ and $\Delta_j = [\Delta_{j1}, \dots, \Delta_{jm}] \in \mathbb{R}^m$ means $\Delta_{ik} = \Delta_{jk}$, $k = 1, \dots, m$. $\mathbb{S}_1 \times \mathbb{S}_2$ denotes the Cartesian product of two sets $\mathbb{S}_1$ and $\mathbb{S}_2$, and $e_k$ represents a unit column vector with compatible dimension in the $k$-th entry being 1 and all the others being 0. For $\zeta \in \mathbb{R}^m$, $|\zeta|_{\text{cw}}^2 := (|\zeta_1|^2, \dots, |\zeta_m|^2)$ as in the work of Wang et al.¹³, and $|z|$ is the absolute value of constant $z$. Besides, for a matrix $Q$, $\lambda(Q)$ and $\bar{\lambda}(Q)$ are the minimum and maximum eigenvalues of $Q$, and $Q > 0$ means that $Q$ is a positive definite matrix.

## 2.2 | Graph theory

For an undirected graph $G = (\mathcal{V}, \mathcal{E}, \mathcal{A})$, $\mathcal{V} = \{1, \dots, N\}$ is the node set of all $N$ agents, while $\mathcal{E} = \{(i, j) \mid (i, j) \in \mathcal{V}, i \neq j\}$ is the edge set, representing the connections between all agents. The graph $G$ is assumed to be connected, meaning there exists a path between any two nodes. The adjacency matrix $\mathcal{A}$ describes the undirected information flow between each two nodes with values 0 and 1. If there is an edge directly connected between the $i$th and the $j$th agents, it is represented by 1, namely $a_{ij} = 1$, $i, j = 1, \dots, N$, or otherwise $a_{ij} = 0$. For $\forall i \in \mathcal{V}, \mathbb{N}_i = \{j \mid (i, j) \in \mathcal{E}\}$ is the neighbor set of $i$. Based on $\mathcal{A}$, the Laplacian matrix $L$ of $G$ is defined as $L = (l_{ij}) \in \mathbb{R}^{N \times N}$. Specifically, $l_{ij} = -a_{ij}$ with $i \neq j$, and $l_{ij} = \sum_{j=1, j \neq i}^N a_{ij}$ with $i = j$. For connected undirected graph $G$, since $L$ is real symmetric, there exists an orthogonal matrix $U \in \mathbb{R}^{N \times N}$ such that $L = U^\top \Lambda U$ with $\Lambda = \text{diag}\{\lambda_1(L), \dots, \lambda_N(L)\} = \text{diag}\{\lambda_1, \dots, \lambda_N\}$, where $0 = \lambda_1 < \lambda_2 \le \dots \le \lambda_N$. For simplicity, we denote $\lambda_i(L) = \lambda_i$ for $i = 1, \dots, N$.

The symmetric property of $L$ plays a fundamental role in the distributed controller design and the Lyapunov-based consensus analysis. More importantly, such undirected graph structures frequently arise in practical systems with symmetric communication or mutual sensing, such as wireless sensor networks and multi-agent vehicle systems with bidirectional links.
## 2.3 | Model description

In this paper, a multi-agent system with asymmetric heterogeneous actuator saturation is considered, where the N agents move on an n-dimensional Euclidean space and regulate themselves according to the following dynamics:

$$ \dot{x}_i(t) = Ax_i(t) + B \mathbf{sat}_{\Delta_i}(u_i(t)), \quad i = 1, \dots, N, \qquad (1) $$

where for each agent, $x_i(t) \in \mathbb{R}^n$ is the state, $u_i(t) \in \mathbb{R}^m$ is the control input which will be designed later, $\mathbf{sat}_{\Delta_i}(\cdot)$ is the saturation function of the physical devices of the systems with $\Delta_i = (\Delta_{i1}, \dots, \Delta_{im}) \in \mathbb{R}^m$ being the constant saturation level, A and B are constant matrices with compatible dimension, and the pair (A, B) satisfy the following Assumption.

**Assumption 1.** The pair (A, B) satisfy ANCBC, that is, (A, B) is stabilizable and all eigenvalues of A are in the closed left-half plane.

Back to $\mathbf{sat}_{\Delta_i}(\cdot)$ in (1), it is an asymmetric saturation function with $\Delta_i^+ \neq \Delta_i^-$, where $\Delta_i^+ > 0$ and $\Delta_i^- > 0$ are the known upper bound and lower bound of the saturation level $\Delta_i$, respectively. Besides, the specific definition of $\Delta_i = [\Delta_{i1}, \dots, \Delta_{im}]^T$ will be further divided into the following three cases based on whether $\Delta_i = \Delta_j$ or $\Delta_{ik} = \Delta_{i\ell}$ with $i, j = 1, \dots, N, k, \ell = 1, \dots, m$ and $i \neq j, k \neq \ell$ is true:

* Asymmetry and heterogeneity in agent: i.e., for $i, j = 1, \dots, N$ and $i \neq j$, there is $\Delta_i \neq \Delta_j$ and $\Delta_{ik} = \Delta_{i\ell}$ with $k, \ell = 1, \dots, m$ and $k \neq \ell$ but $\Delta_i^+ \neq \Delta_i^-$. Symbolically,

$$ \mathbf{sat}_{\Delta_i}(u_i) = \begin{cases} \Delta_i^+, & \text{if } u_i > \Delta_i^+, \\ u_i, & \text{if } -\Delta_i^- \le u_i \le \Delta_i^+, \\ -\Delta_i^-, & \text{if } u_i < -\Delta_i^-. \end{cases} \qquad (2) $$

* Asymmetry and heterogeneity in component: i.e., for $i, j = 1, \dots, N$ and $i \neq j$, there is $\Delta_i = \Delta_j$ and $\Delta_{ik} \neq \Delta_{i\ell}$ with $k, \ell = 1, \dots, m$ and $k \neq \ell$ but $\Delta_{ik}^+ \neq \Delta_{i\ell}^-$. Symbolically,

$$ \mathbf{sat}_{\Delta_{ik}}(u_{ik}) = \begin{cases} \Delta_{ik}^+, & \text{if } u_{ik} > \Delta_{ik}^+, \\ u_{ik}, & \text{if } -\Delta_{ik}^- \le u_{ik} \le \Delta_{ik}^+, \\ -\Delta_{ik}^-, & \text{if } u_{ik} < -\Delta_{ik}^- \end{cases} \qquad (3) $$

* Asymmetry and heterogeneity both in agent and component: i.e., for $i, j = 1, \dots, N$ and $i \neq j$, there is $\Delta_i \neq \Delta_j$ and $\Delta_{ik} \neq \Delta_{i\ell}$ with $k, \ell = 1, \dots, m$ and $k \neq \ell$ but $\Delta_{ik}^+ \neq \Delta_{i\ell}^-$. Symbolically,

## 2.4 | Problem statement

The objective of this paper is to work out the consensus problem of system (1) under a sampled-data framework. First, we suppose that the information exchange actions in the controller are not continuous. We reorder the sampling time instants of all agents in an ascending sequence as $0 = t_0 < t_1 < \dots < t_p < t_{p+1}$. The distributed controller satisfies

$$ u_i(t) = f \left( \sum_j l_{ij} x_j(t_p) \right), \quad t_p \le t < t_{p+1}, \qquad (4) $$

where $t_{p+1} = t_p + T_p, p \in \mathbb{N}, \underline{T} \le T_p \le \overline{T}$ with $\underline{T} \le \overline{T}$ being two positive real numbers. Note that the desired constraint implies that the aggregated information exchanged to each agent is sampled, that is $\sum_j l_{ij} x_j(t)$ is not available continuously, but we can use only $\sum_j l_{ij} x_j(t_p)$ at each sampling time $t_p$. This control structure captures a physically meaningful hybrid setting: while agents update their control signals continuously, they only rely on sampled neighbor information, which reduces communication burden and models realistic digital implementations of distributed systems. The goal of this article is to design a sampled-data based distributed control algorithm $u_i$, based on previous aggregate information, to achieve state synchronization, that is,

$$ \lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \quad \forall i, j \in \{1, \dots, N\}. $$

We conclude this section by recalling the following two technical lemmas which will be used throughout the article.
**Lemma 1.** 32 For Laplacian matrix L associated with an un-directed connected graph G, zero is a simple eigenvalue of L if and only if the graph is connected. Then one has $\lambda_2 \ge 2 \left[1 - \cos\left(\frac{\pi}{N}\right)\right]$ and $\lambda_N \le N$ for $N \ge 2$, where the strict equality holds if G is a chain or a complete graph, respectively.

**Lemma 2.** 33 For any given positive definite matrix $G \in \mathbb{R}^{n \times n}$, and vector function $M : [a, b] \to \mathbb{R}^n$, there holds

$$ \left( \int_a^b M(s)ds \right)^\top G \left( \int_a^b M(s)ds \right) \le (b-a) \int_a^b M(s)^\top G M(s)ds. $$

# 3 | IMPROVED LOW-GAIN FEEDBACK BASED SEMI-GLOBAL CONSENSUS

The traditional low-gain approach in Lin's work 3 is quite common to deal with the homogeneous case, i.e., when the saturation level of N different agents is uniform, and the upper and lower bounds of saturation levels in each agent are symmetric. In this section, we show how to adapt this methodology in order to solve a consensus problem in the presence of distributed controllers with sampled data and asymmetric heterogeneous saturation.

## 3.1 | Feedback design

The main idea we want to follow is to guarantee that for all times $t \ge 0$ the feedback laws satisfy $|u_i(t)| \le \min\{\Delta_{ik}^+, \Delta_{ik}^-\}$, for $i = 1, \dots, N$, with $\Delta_{ik}^+ \ne 0$ and $\Delta_{ik}^- \ne 0$. To this end, let us first introduce the following compact notation that represents state coupling term and will be used throughout this section

$$ \xi_i = \sum_{j=1}^{N} l_{ij} x_j, \quad \xi = (\xi_1, \dots, \xi_N) \in \mathbb{R}^{nN}, \qquad (5) $$

and let the feedback law $u_i$ selected for any $t \in [t_p, t_{p+1})$ as

$$ \begin{aligned} u_i(t) &= -B^\top P_i(\varepsilon_i) \sum_{j \in \mathcal{N}_i} a_{ij} (x_i(t_p) - x_j(t_p)) \\ &= -B^\top P_i(\varepsilon_i) \xi_i(t_p), \end{aligned} \qquad (6) $$

where $\varepsilon_i \in (0, 1]$ is the low gain parameter for the i-th agent, and $P_i(\varepsilon_i) \in \mathbb{R}^{n \times n} > 0$ is the solution of the parameterized algebraic Riccati equation (PARE)

$$ A^\top P_i(\varepsilon_i) + P_i(\varepsilon_i)A - 2 \left(1 - \cos\left(\frac{\pi}{N}\right)\right) P_i(\varepsilon_i)BB^\top P_i(\varepsilon_i) + \varepsilon_i I_n = 0. \qquad (7) $$

Notice that the ANCBC assumption guarantees the unique existence of the solution of PARE (7). Moreover, the fact that $\lim_{\varepsilon_i \to 0} P_i(\varepsilon_i) = 0$ motivates the term of low-gain feedback.

Define $\mathbf{P} \in \mathbb{R}^{nN \times nN}$ and $\mathbf{E} \in \mathbb{R}^{nN \times nN}$ as

$$ \begin{aligned} \mathbf{P} &= \operatorname{diag}\{P_1(\varepsilon_1), \dots, P_N(\varepsilon_N)\}, \\ \mathbf{E} &= \operatorname{diag}\{\varepsilon_1 I_n, \dots, \varepsilon_N I_n\}, \end{aligned} $$

then (6) and (7) can be written in the compact form as

$$ u(t) = -(I_N \otimes B^\top) \mathbf{P} \xi(t_p), \qquad (8) $$

and

$$ (I_N \otimes A^\top) \mathbf{P} + \mathbf{P} (I_N \otimes A) - 2 \left(1 - \cos\left(\frac{\pi}{N}\right)\right) \mathbf{P} (I_N \otimes B B^\top) \mathbf{P} + \mathbf{E} = 0, \qquad (9) $$

respectively, where $u = (u_1, \dots, u_N)$.

Throughout the section, if the low-gain parameter $\varepsilon_i$ for each agent is chosen properly, one can get $|u_{ik}(t)| \le \min\{\Delta_{ik}^+, \Delta_{ik}^-\}$ with the limitation that $\Delta_{ik}^+ \ne 0$ as well as $\Delta_{ik}^- \ne 0$, which further leads to $-\Delta_{ik}^- \le u_{ik}(t) \le \Delta_{ik}^+$, so that the nonlinear dynamics in (1) turns to be a linear form as

$$ \dot{x}_i(t) = A x_i(t) + B u_i(t), \quad i = 1, \dots, N, \qquad (10) $$

with specific explanations being given in the next subsection.
## 3.2 | Theoretical analysis

In this subsection, theoretical analysis and main results are provided for semi-global consensus of system steered by (1) with linear distributed control law $u_i(t)$ in (6).

**Theorem 1.** Consider the network of N agents (1) over an undirected and connected graph $G$, and the linear control protocol $u_i(t)$ in (6) with distributed sampled-data $\xi_i(t_p)$ and suppose that Assumption 1 holds. There exists a set of values $\bar{\epsilon}_i > 0$ such that, for any $\epsilon_i \in (0, \bar{\epsilon}_i]$ and any sampling period $T_p$ satisfying $\underline{T} \le T_p \le \bar{T}$ with

$$0 < \underline{T} \le \bar{T} \le \sqrt{\frac{\lambda(\mathbf{P})}{\bar{\lambda}(\mathbf{P})} \frac{\|\mathbf{E}\|^2}{\alpha^2 (\|\mathbf{A}\| + \kappa)^2}}, \quad (11)$$

where $\alpha = 2N (\|\mathbf{B}\|\|\mathbf{P}\|)^2$, and $\kappa = N \|\mathbf{B}\mathbf{B}^\top\|\|\mathbf{P}\|$, then, the semi-global consensus problem of the multi-agent system in (1) is solved, namely $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0$ for $x_i(0) \in X, x_j(0) \in X, \forall i, j = 1, \dots, N$, where $X \in \mathbb{R}^n$ is any prior given bounded set.

*Proof.* With the notation introduced in (5), it is readily seen that $\xi \to 0$ implies $x_i - x_j \to 0$. As a consequence, consider the Lyapunov function $V: \mathbb{R}^{nN} \to \mathbb{R}_{\ge 0}$ for $\xi$ as

$$V(\xi) = \sum_{i=1}^{N} V_i = \xi^{\top} \mathbf{P} \xi, \quad V_i = \xi_i^{\top} P_i(\epsilon_i) \xi_i. \quad (12)$$

Note that

$$\underline{\lambda}(\mathbf{P}) \|\xi\|^2 \le V(\xi) \le \bar{\lambda}(\mathbf{P}) \|\xi\|^2. \quad (13)$$

To begin, we will give a brief explanation of the transformation of the dynamics into a linear form in (10). Let us define the Lyapunov level set $\Omega(c) = \{V(\xi) \le c\}$. Since the initial states $x_i(0)$ are taken from a bounded set $X$ and the fact that $\lim_{\epsilon_i \to 0} P_i(\epsilon_i) = 0$, then for any given constant $c > 0$, there exist $\epsilon_i^* \in (0, 1]$ such that

$$\sup_{\epsilon_i \in (0, \epsilon_i^*], x_i(0) \in X, i=1, \dots, N} V(\xi) \le c.$$

As a consequence, with $u_i(t)$ in (6), one has $u_i^\top u_i \le \text{tr}(\mathbf{B}^\top P_i(\epsilon_i) \mathbf{B})c$. Hence, there exists $\bar{\epsilon}_i \in (0, \epsilon_i^*]$ such that for $\forall \epsilon_i \in (0, \bar{\epsilon}_i]$, the following holds $\|u_i\| \le \min\{\Delta_{ik}^+, \Delta_{ik}^-\}$. The above explanation indicates that as long as the low gain parameters $\epsilon_i$ are chosen small enough, the input saturation function $\text{sat}_{\Delta_{ik}}(\cdot)$ will no longer have an impact on the system if the Lyapunov level set $\Omega(c)$ is invariant, thus recovering the linear dynamics as in (10). In such a case, the overall $\xi$-linear dynamics read, in compact form, as

$$\dot{\xi}(t) = (I_N \otimes A)\xi(t) - (L \otimes BB^\top)\mathbf{P}\xi(t_p) \quad (14)$$

for all $t \in [t_p, t_{p+1})$. According to the previous equation, the origin is an equilibrium, that is $\xi(t_p) = 0$ implies $\xi(t) = 0$ for all $t \ge t_p$. Taking the dynamics equation of the closed-loop system (14) into the derivative of the Lyapunov function $V$, adding and subtracting the term $(L \otimes BB^\top)\mathbf{P}\xi(t)$ one obtains

$$\begin{aligned} \dot{V}(t) & \le -\xi^\top(t)\mathbf{E}\xi(t) + \alpha\|\xi(t)\|\|\xi(t_p) - \xi(t)\| \\ & \le -\frac{\|\mathbf{E}\|}{2}\|\xi(t)\|^2 + \frac{\alpha^2}{2\|\mathbf{E}\|}\|\xi(t_p) - \xi(t)\|^2 \\ & \le -\frac{\|\mathbf{E}\|}{2}\|\xi(t)\|^2 + \frac{\alpha^2}{2\|\mathbf{E}\|} \left( \int_{t_p}^t \|\dot{\xi}(\tau)\| d\tau \right)^2, \end{aligned} \quad (15)$$

for all $t \in [t_p, t_{p+1})$, where $\alpha$ is defined as in the statement of the theorem, the first inequality holds due to PARE in (9) and the fact that $|L| \le N$ (see Lemma 1), and the second inequality holds due to Yang inequality, and in the third step we used the fact that $\xi$ is differentiable for all $t \in [t_p, t_{p+1})$. Note that, based on inequality (15), it is possible to show, using arguments by contradiction similarly to the argument in Appendix A of Wang's work$^{34}$ that

$$\max_{\tau \in [t_p, t_{p+1})} V(\xi(\tau)) = V(\xi(t_p)).$$
Then, by using (14), and the bound in (13), we obtain

$$
\begin{aligned}
\|\dot{\xi}(t)\| &= \|(I_N \otimes A)\xi(t) - (L \otimes BB^T)\mathbf{P}\xi(t_p)\| \\
&\leq (\|A\| + \kappa) \sup_{\tau \in [t_p, t_{p+1})} \|\xi(t)\|^2 \leq \frac{\|A\| + \kappa}{\sqrt{\underline{\lambda}(\mathbf{P})}} V(t_p),
\end{aligned}
$$

for any $t \in [t_p, t_{p+1})$, where we used previous definition of $\kappa$. Recall that $t_{p+1} - t_p = T_p \le \bar{T}$. Moreover, for any $t \in [t_p, t_{p+1})$, let

$$
\beta_1 = \frac{\|\mathbf{E}\|}{2\bar{\lambda}(\mathbf{P})}, \quad \beta_2 = \frac{\alpha^2}{2\|\mathbf{E}\|} \frac{(\|A\| + \kappa)^2}{\underline{\lambda}(\mathbf{P})} \frac{\bar{T}^2}{T},
$$

Then, one has

$$
\dot{V}(t) \leq -\beta_1 V(t) + \beta_2 V(t_p). \tag{16}
$$

Integrating both sides of the inequality from $t_p$ to $t$ to get

$$
V(t) \le \left[ \frac{\beta_2}{\beta_1} + \left( 1 - \frac{\beta_2}{\beta_1} \right) e^{-\beta_1(t-t_p)} \right] V(t_p). \quad (17)
$$

By the sampling period $T_p$ in (11), we have $\beta_2 < \beta_1$ and $\rho = \frac{\beta_2}{\beta_1} + (1 - \frac{\beta_2}{\beta_1})e^{-\beta_1(t-t_p)} < 1$ for $t \in (t_p, t_{p+1})$. In addition, since $V(t)$ is continuous for all $t \ge 0$, and therefore

$$
V(t_{p+1}) = \lim_{t \to t_{p+1}^+} V(t) < \rho V(t_p). \quad (18)
$$

As a consequence, with such a choice, any Lyapunov level set $\Omega(c)$ is invariant and the $\xi$-dynamics are always linear for $\varepsilon$ small enough. This further shows that for $p \to \infty$, $V(t_p)$ converges to zero leading to $\lim_{t \to \infty} V(t) = 0$ and moreover $\lim_{t \to \infty} \xi(t) = 0$. By the properties of the Laplacian matrix $L$ in the connected graph $G$, this implies $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \forall (i, j) \in \mathbb{E}$, concluding the proof. Thus the proof is complete. $\square$

3.3 | Technical comments

In general, a linear distributed controller in sampled-data framework to achieve semi-global consensus for asymmetric hetero-
geneous saturated multi-agent systems is designed. Although the control law imposes a conservative constraint based on the
symmetric bound min {Δ̄<sub>ik</sub>, Δ<sup>+</sup><sub>ik</sub>}, it constitutes a nontrivial extension of the classical low-gain method<sup>3</sup> to more realistic sce-
narios involving asymmetric and heterogeneous saturation. This subsection aims to summarize the performance limitations
of the preceding method and highlight several technical insights, which naturally motivate the introduction of a more flexible
alternative in the next section.

**Remark 1.** Compared to traditional low-gain approach, this section has resolved the consensus problem for asymmetric hetero-
geneous saturation that it could not achieve in$^3$. However, there is an obvious fact to be improved that the solution above is
relatively restrictive in terms of $\varepsilon_i$. For one thing, we need to take the minimum between all levels $\Delta_{ik}$ to define the bounds $\bar{\varepsilon}_i$ for
further copying with heterogeneous case. For another thing, the dynamics considered here is essentially in a symmetric form,
i.e., $\min \{\bar{\Delta}_{ik}^{-}, \bar{\Delta}_{ik}^{+}\}$, which must satisfy $\bar{\Delta}_{ik}^{-} \neq 0$ and $\bar{\Delta}_{ik}^{+} \neq 0$.

**Remark 2.** We would like to delve deeper into the influence of the low-gain parameter $\varepsilon_i$ on the admissible sampling period $\bar{T}$ in (11), which directly relates to the trade-off discussed in the contribution section, and we refer to the explicit solution given in^35 Example 2, which shows that in some cases $\lim_{\varepsilon_i \to 0} \bar{T} = 0$. This indicates a performance limitation: while a smaller $\varepsilon_i$ can effectively reduce the influence of input saturation, it necessitates a higher sampling rate, thereby increasing implementation cost and complexity in practical systems.

**Remark 3.** The improved low-gain methodology works out the semi-global consensus via sampled-data control, where the domain of $x_i(0)$ and $x_j(0)$ is any prior given bounded set.

Therefore, a new method is urgently needed to design the distributed controller in a sampled framework to make the objective of this paper better to achieve.
# 4 | DYNAMIC SATURATION RECONSTRUCTION TECHNIQUE BASED GLOBAL CONSENSUS

Following the recent ideas¹³, we investigate here a new sampled-data based distributed control algorithm which can solve the global consensus of multi-agent systems in the presence of asymmetric heterogeneous actuator saturation.

## 4.1 | Feedback design

In order to introduce our new feedback law, let us first define the following algebraic Riccati equation (ARE)

$$A^T Q + QA - 2 \left[ 1 - \cos\left(\frac{\pi}{N}\right) \right] Q B B^T Q + I_n = 0. \quad (19)$$

Based on the matrix Q so defined, we define the following variables $\zeta_i = B^T Q \xi_i$ and $\zeta_i \in \mathbb{R}^m$ as

$$\begin{align}
\zeta_i &= B^T Q \sum_{j=1}^{N} l_{ij} x_j = (e_i^T L \otimes B^T Q) x, \tag{20} \\
\zeta &= (\zeta_1, \dots, \zeta_N), \nonumber
\end{align}$$

where we recall that $e_i$ denotes the column with all zeros but one in the i-th element, as defined in the notation section. It can be seen that the k-th element of $\zeta_i$ is computed as

$$\zeta_{ik} = -(e_i^T L \otimes e_k^T B^T Q) x, \quad k = 1, \dots, m. \quad (21)$$

Next, in order to deal with the saturation level of N agents in asymmetry and heterogeneity characteristics, we introduce the following notation. In particular, given a vector $\omega = (\bar{\omega}, \underline{\omega}) \in \mathbb{R}^2$ with non-negative entries $\bar{\omega} \ge 0$ and $\underline{\omega} \ge 0$, we define the function $\sigma_{\sqrt{\bar{\omega}}} : \mathbb{R} \to \mathbb{R}$ as the asymmetric saturation function

$$\sigma_{\sqrt{\bar{\omega}}}(s) = \max\{-\sqrt{\bar{\omega}}, \min\{\sqrt{\bar{\omega}}, s\}\}.$$

Similarly, given a vector $\omega = (\omega_1, \dots, \omega_m) \in \mathbb{R}^{2m}$ where $\omega_k = (\bar{\omega}_k, \underline{\omega}_k) \in \mathbb{R}^2$ is a vector with non-negative entries $\bar{\omega}_k \ge 0$ and $\underline{\omega}_k \ge 0$ for any $k \in \{1, \dots, m\}$, we define the distributed asymmetric saturation function $\sigma_{\sqrt{\bar{\omega}}} : \mathbb{R}^m \to \mathbb{R}^m$ as

$$\sigma_{\sqrt{\bar{\omega}}}(s) = (\sigma_{\sqrt{\bar{\omega}_1}}(s_1), \dots, \sigma_{\sqrt{\bar{\omega}_m}}(s_m)).$$

Based on the function $\sigma$ so defined, we propose the following distributed nonlinear algorithm for the i-th agent in (1)

$$u_i(t) = \sigma_{\sqrt{\omega_i(t)}}(\zeta_i(t_p)), \quad t \in [t_p, t_{p+1}), \quad i \in \{1, \dots, N\}, \quad (22)$$

with $\omega_i \in \mathbb{R}^{2m}$, so that the k-th component of $u_i$ reads as

$$u_{ik}(t) = \sigma_{\sqrt{\omega_{ik}(t)}}(\zeta_{ik}(t_p)), \quad t \in [t_p, t_{p+1}), \quad k \in \{1, \dots, m\}, \quad (23)$$

where $\omega_{ik} = (\bar{\omega}_{ik}, \underline{\omega}_{ik})$ are defined by the dynamics

$$\begin{align}
\dot{\bar{\omega}}_{ik}(t) &= -\bar{\theta}_{ik} \bar{\omega}_{ik}(t) + \bar{r}_{ik}(t) \zeta_{ik}^2(t_p), \quad \bar{\omega}_{ik}(0) \ge 0, \nonumber \\
\dot{\underline{\omega}}_{ik}(t) &= -\underline{\theta}_{ik} \underline{\omega}_{ik}(t) + \underline{r}_{ik}(t) \zeta_{ik}^2(t_p), \quad \underline{\omega}_{ik}(0) \ge 0. \tag{24}
\end{align}$$

In the whole algorithm, $\underline{\theta}_{ik} > 0$ and $\bar{\theta}_{ik} > 0$ are constant saturation parameters, while $\underline{r}_{ik}(t) > 0$ and $\bar{r}_{ik}(t) > 0$ are time-varying saturation parameters to be defined. We highlight that $\sqrt{\bar{\omega}_{ik}}$ and $\sqrt{\underline{\omega}_{ik}}$ are the time-varying upper and lower bounds of dynamic saturation level $\sqrt{\omega_{ik}}$, which is adjusted adaptively in close relation to the state coupling term $\zeta_{ik}(t_p)$ in (21) to obtain the desirable properties. Furthermore, due to the choice of initial condition, $\bar{\omega}_{ik}(0) \ge 0$, $\underline{\omega}_{ik}(0) \ge 0$, $\omega_{ik}(0) \ge 0$ is constrained to evolve in the forward invariant set $\mathbb{R}_{\ge 0}$, that is, $\bar{\omega}_{ik}(t) \ge 0$ and $\underline{\omega}_{ik}(t) \ge 0$ for all $t \ge 0$.

## 4.2 | A technical lemma for the dynamic saturation level

In this subsection, we will establish a technical lemma about the choice of dynamic saturation level. By further analyzing the dynamics (24), for ease of explanation here, we will not distinguish the two dynamics $\bar{\omega}_{ik}$ and $\underline{\omega}_{ik}$ and in the following we simply consider a scalar dynamics

$$\dot{\omega}(t) = -\theta\omega(t) + r(t)\zeta^2(t_p), \quad \omega(0) \ge 0, \quad t \in [t_p, t_{p+1}), \quad (25)$$
where we further dropped the index *ik* and where $\theta > 0$ and $r(t) > 0$ are the constant saturation parameter and the time-varying saturation parameter, respectively. First, we remark that for any function $r(t)$ which is almost $C^0$ on $[t_p, t_{p+1})$ and sequence of constant signals $\zeta(t_p)$ on any interval of time $[t_p, t_{p+1})$, the corresponding solution $\omega(t)$ is $C^0$ for all $t \ge 0$ and $C^1$ almost everywhere (i.e. it is differentiable on each interval $[t_p, t_{p+1})$). Then, we have the following result.

**Lemma 3.** Let $r(t) > 0$ be a time-varying parameter satisfying $r(t)\zeta^2(t_p) \le \theta\Delta^2$ for all $t \in [t_p, t_{p+1})$. Then for any initial condition $\omega(0)$ satisfying $0 \le \omega(0) \le \Delta^2$, the following holds

$$0 \le \omega(t) \le \Delta^2, \quad \forall t \in [t_p, t_{p+1}].$$

*Proof.* By integrating equation (25) and applying the condition of $r(t)\zeta^2(t_p) \le \theta\Delta^2$ on $t \in [t_p, t_{p+1})$, one obtains

$$\begin{aligned}
\omega(t) &\le e^{-\theta(t-t_p)}\omega(t_p) + \theta\Delta^2 \int_{t_p}^t e^{-\theta(t-\tau)} d\tau \tag{26} \\
&= \Delta^2 + e^{-\theta(t-t_p)}(\omega(t_p) - \Delta^2).
\end{aligned}$$

Therefore, for $t \in [0, t_1)$, we obtain $\omega(t) \le \Delta^2$ since $\omega(0) \le \Delta^2$, and $\omega(t_1) = \lim_{t \to t_1^-} \omega(t) \le \Delta^2$. Moreover, for $t \in [t_1, t_2)$, there is $\omega(t) \le \Delta^2$ since $\omega(t_1) \le \Delta^2$, and $\omega(t_2) = \lim_{t \to t_2^-} \omega(t) \le \Delta^2$. Hence, for any $t \in [t_p, t_{p+1}]$, there is $0 \le \omega(t) \le \Delta^2$. $\square$

According to this Lemma, we have the following summary about the dynamic saturation function constructed in (24).

**Summary:**

* Given an upper bound $\Delta_{ik}^+$ of saturation level, as long as $\bar{r}_{ik}(t)\zeta_{ik}^2(t_p) \le \bar{\theta}_{ik}\Delta_{ik}^{+2}$, then for any $0 \le \bar{\omega}_{ik}(0) \le \Delta_{ik}^{+2}$, we have $0 \le \bar{\omega}_{ik}(t) \le \Delta_{ik}^{+2}$, $\forall t \ge 0$.

* Given a lower bound $\Delta_{ik}^-$ of saturation level, as long as $\underline{r}_{ik}(t)\zeta_{ik}^2(t_p) \le \underline{\theta}_{ik}\Delta_{ik}^{-2}$, then for any $0 \le \underline{\omega}_{ik}(0) \le \Delta_{ik}^{-2}$, there is $0 \le \underline{\omega}_{ik}(t) \le \Delta_{ik}^{-2}$, $\forall t \ge 0$.

Therefore, for any selection of $\bar{r}_{ik}(t)$ and $\underline{r}_{ik}(t)$ satisfying the previous summary, one obtains $-\Delta_{ik}^- \le u_{ik}(t) \le \Delta_{ik}^+$, and system (1) for $i = 1, \dots, N$ is equivalent to

$$\begin{aligned}
\dot{x}_i(t) &= Ax_i(t) + B\mathbf{sat}_{\Delta_i}(\boldsymbol{\sigma}_{\sqrt{\omega_i(t)}}(\zeta_i(t_p))) \\
&= Ax_i(t) + B\boldsymbol{\sigma}_{\sqrt{\omega_i(t)}}(\zeta_i(t_p)), \quad t \in [t_p, t_{p+1}).
\end{aligned} \tag{27}$$

In particular, this can be achieved if one selects the time-varying parameters $\bar{r}_{ik}(t)$ and $\underline{r}_{ik}(t)$ as follows:

$$\begin{aligned}
\bar{r}_{ik}(t) &= \max \left\{ \psi(t) \in \left(0, \frac{\bar{\theta}_{ik}}{2}\right] : \psi(t)\zeta_{ik}^2(t_p) \le \bar{\theta}_{ik}\Delta_{ik}^{+2} \right\}, \\
\underline{r}_{ik}(t) &= \max \left\{ \psi(t) \in \left(0, \frac{\underline{\theta}_{ik}}{2}\right] : \psi(t)\zeta_{ik}^2(t_p) \le \underline{\theta}_{ik}\Delta_{ik}^{-2} \right\},
\end{aligned} \tag{28}$$

Note that with such a choice, $\bar{r}_{ik}(t)$ and $\underline{r}_{ik}(t)$ are guaranteed to be $\bar{r}_{ik}(t_p)$ and $\underline{r}_{ik}(t_p)$ on any interval of time $[t_p, t_{p+1})$.

## 4.3 | Theoretical analysis

This subsection will analyse the consensus of multi-agent systems steered by (1) with $u_{ik}(t)$ in (23) and $\sqrt{\omega_{ik}(t)}$ in (24). We have the following main result showing that the global consensus of continuous-signal-driven system affected by asymmetric heterogeneous input saturation can be achieved by sampled-data based distributed feedback control law.

**Theorem 2.** Consider the network of N agents (1) over an undirected and connected graph $G$, and the dynamic input saturation reconstruction protocol $u_i(t)$ in (23) with distributed sampled-data $\zeta_{ik}(t_p)$ in (21) as well as the dynamic saturation level $\sqrt{\omega_{ik}(t)}$ in (24) with $\bar{r}_{ik}(t)$ and $\underline{r}_{ik}(t)$ selected as in (28). Suppose that Assumption 1 holds and that the sampling period $T_p$ satisfies

$$\left\{
\begin{array}{l}
\underline{T} \le T_p \le \bar{T} \quad \text{with } 0 < \underline{T} \le \bar{T} \le \sqrt{\frac{1}{32\|A\|^2}}, \quad \text{if } \|A\| \ne 0, \\
T_p > 0, \quad \text{if } \|A\| = 0.
\end{array}
\right.
\tag{29}$$
Then, for any

$$ \bar{\theta}_{ik} > 2, \quad \underline{\theta}_{ik} > 2, \qquad (30) $$

and $\sqrt{\omega_{ik}(0)} \in [0, \Delta_{ik}]$, the consensus problem of the multi-agent system in (1) is solved, namely $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0$ for $x_i(0) \in \mathbb{R}^n, x_j(0) \in \mathbb{R}^n, \forall i, j = 1, \dots, N$.

*Proof.* For $i = 1, \dots, N$, let $\tilde{x}_i(t) = x_i(t) - x_i(t_p), t \in [t_p, t_{p+1})$. Then, there is $\tilde{\zeta}_i(t) = \zeta_i(t) - \zeta_i(t_p)$ and $\tilde{\zeta}_i(t) \in \mathbb{R}^m$ satisfying

$$ \tilde{\zeta}_i(t) = -B^\top Q \sum_{j \in \mathbb{N}_i} a_{ij}(\tilde{x}_i(t) - \tilde{x}_j(t)), \quad t \in [t_p, t_{p+1}). \qquad (31) $$

Furthermore, Define $\tilde{x} = (\tilde{x}_1, \dots, \tilde{x}_N)$, and for $k = 1, \dots, m$, there is $\tilde{\zeta}_{ik}(t) = - (e_i^\top L \otimes e_k^\top B^\top Q) \tilde{x}(t)$. The dynamics of $x(t) = (x_1(t), \dots, x_N(t))$ on $t \in [t_p, t_{p+1})$ meets the following compact form

$$ \dot{x}(t) = (I_N \otimes A)x(t) + (I_N \otimes B)\sigma_{\sqrt{\omega}(t)}(\zeta(t_p)), \qquad (32) $$

where $\zeta(t_p) = -(L \otimes B^\top Q)x(t_p)$.

Consider now the Lyapunov function for the $x(t)$ and $\omega_{ik}(t)$ dynamics as follows

$$ \begin{aligned} W(t) &= \epsilon_1 V_x(t) + V_\omega(t) + 4\epsilon_1 V_T(x, \zeta) \\ V_x(t) &= x^\top(t) (L \otimes Q) x(t) \\ V_\omega(t) &= \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2} \omega_{ik}(t) + \max \left\{ \frac{2r_{ik}(t)}{\theta_{ik}} (\zeta_{ik}(t) - \tilde{\zeta}_{ik}(t))^2 - \omega_{ik}(t), 0 \right\} \right) \end{aligned} \qquad (33) $$

$$ V_T(x, \zeta) = \bar{T} \int_{-\bar{T}}^{0} \int_{t+r}^{t} \dot{x}(s)^\top (L^2 \otimes QBB^\top Q) \dot{x}(s) ds dr, $$

where $\epsilon_1 > 0$ is a sufficient small constant. It can be verified that $W(t)$ is positive definite.

Next, in order to compute the time-derivative of $W$ along solutions, we remark that such a Lyapunov function is locally Lipschitz and thus treatable by using Clarkes generalized gradient$^{36}$, that, among other things, enjoys useful and well-established properties. See, e.g.$^{36}$ Theorem 2.5.1.p. 63 and corollary, p. 64 and also$^{37}$ p. 99. As a consequence, we will use the standard symbol $\dot{W}$ for time derivative in order to simplify the notation. Note indeed, that $W$ is non-differentiable in the zero measure set where $\frac{2r_{ik}(t)}{\theta_{ik}}(\zeta_{ik}(t) - \tilde{\zeta}_{ik}(t))^2 - \omega_{ik}(t) = 0$.

Then, we split $N$ agent elements into two sets, that is, let $\mathcal{I} \subset \{1, \dots, N\} \times \{1, \dots, m\}$ be the subset of indexes $(i, k)$ for which $\zeta_{ik}(t_p) \ge 0$ and $\mathcal{J} = (\{1, \dots, N\} \times \{1, \dots, m\}) \setminus \mathcal{I}$ be the set of indexes $(i, k)$ for which $\zeta_{ik}(t_p) < 0$. Moreover, for set $\mathcal{I}$, there are $\bar{\omega}_{ik}(t) \ge \frac{2\bar{r}_{ik}(t)}{\bar{\theta}_{ik}}\zeta_{ik}^2(t_p)$ and $\bar{\omega}_{ik}(t) < \frac{2\bar{r}_{ik}(t)}{\bar{\theta}_{ik}}\zeta_{ik}^2(t_p)$ hold for $(i, k) \in \mathcal{I}_1$ and $(i, k) \in \mathcal{I}_2$. Similarly, for set $\mathcal{J}$, there are $\underline{\omega}_{ik}(t) \ge \frac{2\underline{r}_{ik}(t)}{\underline{\theta}_{ik}}\zeta_{ik}^2(t_p)$ and $\underline{\omega}_{ik}(t) < \frac{2\underline{r}_{ik}(t)}{\underline{\theta}_{ik}}\zeta_{ik}^2(t_p)$ hold for $(i, k) \in \mathcal{J}_1$ and $(i, k) \in \mathcal{J}_2$.

Therefore, one obtains

$$ V_{\omega}(t) = \underbrace{\sum_{\langle i,k \rangle \in \mathcal{I}_1} \frac{1}{2}\bar{\omega}_{ik}(t) + \sum_{\langle i,k \rangle \in \mathcal{I}_2} (-\frac{1}{2})\bar{\omega}_{ik}(t) + \sum_{\langle i,k \rangle \in \mathcal{I}_1} \frac{1}{2}\underline{\omega}_{ik}(t) + \sum_{\langle i,k \rangle \in \mathcal{I}_2} (-\frac{1}{2})\underline{\omega}_{ik}(t)}_{V_{\omega_1}} + \underbrace{\sum_{\langle i,k \rangle \in \mathcal{I}_2} \frac{2\bar{r}_{ik}(t)}{\bar{\theta}_{ik}}\zeta_{ik}^2(t_p) + \sum_{\langle i,k \rangle \in \mathcal{I}_2} \frac{2r_{-ik}(t)}{\underline{\theta}_{-ik}}\zeta_{ik}^2(t_p)}_{V_{\omega_2}}. \qquad (34) $$

In the rest of the proof, the argument (t) will be dropped for space reasons, when clear from the context.

First, for the derivative of $V_x(t)$, define

$$ \dot{x} = (\dot{x}_1, \dots, \dot{x}_N) = (U \otimes I_n)x, $$

where $U$ has been defined in the Graph Theory section, that is such that $L = U^\top \Lambda U$. Moreover, given $\omega, s$ denote

$$ \mathrm{d}z_{\sqrt{\omega}}(s) = s - \sigma_{\sqrt{\omega}}(s). $$
Hence, by definition of $\zeta$ we also get

$$ 
\begin{aligned}
\boldsymbol{\sigma}_{\sqrt{\omega}}(\zeta) &= -(L \otimes B^\mathrm{T} Q)x - \mathrm{d}\mathbf{z}_{\sqrt{\omega}}(\zeta) \\
&= -(U^\mathrm{T} \Lambda \otimes B^\mathrm{T} Q)\check{x} - \mathrm{d}\mathbf{z}_{\sqrt{\omega}}(\zeta).
\end{aligned}
$$

Using these relations, we evaluate the derivative of the function $W$ for any $t \in [t_p, t_{p+1})$. The first term $V_x$ reads

$$ 
\begin{aligned}
\dot{V}_x \le \sum_{i=2}^{N} \lambda_i \check{x}_i^\mathrm{T} (QA + A^\mathrm{T}Q - 2\lambda_i QBB^\mathrm{T}Q) \check{x}_i \\
& \quad - 2x^\mathrm{T}(L \otimes QB)\zeta(t_p) - 2x^\mathrm{T}(L \otimes QB)\tilde{\zeta} + 2x^\mathrm{T}(L \otimes QB)\boldsymbol{\sigma}_{\sqrt{\omega}}(\zeta(t_p)).
\end{aligned}
\quad (35) $$

Then adopting Young inequality, we obtain

$$ 
\begin{aligned}
-2x^\mathrm{T}(L \otimes QB)\zeta(t_p) &\le \frac{1}{4}x^\mathrm{T}(L^2 \otimes QBB^\mathrm{T}Q)x + 4\zeta(t_p)^\mathrm{T}\zeta(t_p), \\
2x^\mathrm{T}(L \otimes QB)\boldsymbol{\sigma}_{\sqrt{\omega}}(\zeta(t_p)) &\le \frac{1}{4}x^\mathrm{T}(L^2 \otimes QBB^\mathrm{T}Q)x + 4\boldsymbol{\sigma}_{\sqrt{\omega}}^\mathrm{T}(\zeta(t_p))\boldsymbol{\sigma}_{\sqrt{\omega}}(\zeta(t_p)), \\
-2x^\mathrm{T}(L \otimes QB)\tilde{\zeta} &\le \frac{1}{4}x^\mathrm{T}(L^2 \otimes QBB^\mathrm{T}Q)x + 4\tilde{\zeta}^\mathrm{T}\tilde{\zeta}.
\end{aligned}
$$

Recalling the definition of $\mathbb{I}$ and $\mathbb{J}$ given at the beginning of the proof, it obtains that

$$ 
\begin{aligned}
\dot{V}_x \le \sum_{i=2}^{N} \lambda_i \check{x}_i^\mathrm{T} (QA + A^\mathrm{T}Q - \frac{5}{4}\lambda_i QBB^\mathrm{T}Q) \check{x}_i \\
& \quad + \sum_{\langle i,k \rangle \in \mathbb{I}_1} \left( \frac{2\bar{\theta}_{ik}}{\bar{r}_{ik}} + 4 \right) \bar{\omega}_{ik} + \sum_{\langle i,k \rangle \in \mathbb{I}_2} \left( 4 + \frac{8\bar{r}_{ik}}{\bar{\theta}_{ik}} \right) \zeta_{ik}^2(t_p) \\
& \quad + \sum_{\langle i,k \rangle \in \mathbb{J}_1} \left( \frac{2\theta_{ik}}{r_{ik}} + 4 \right) \omega_{ik} + \sum_{\langle i,k \rangle \in \mathbb{J}_2} \left( 4 + \frac{8r_{ik}}{\theta_{ik}} \right) \zeta_{ik}^2(t_p) + 4\tilde{\zeta}^\mathrm{T}\tilde{\zeta}.
\end{aligned}
\quad (36) $$

Next, we evaluate the term $V_{\omega_1}$ in $V_\omega$. Substituting (24), its derivative satisfies

$$ 
\dot{V}_{\omega_1} \le -\frac{1}{2} \sum_{\langle i,k \rangle \in \mathbb{I}_1} \bar{\theta}_{ik} \bar{\omega}_{ik} + \sum_{\langle i,k \rangle \in \mathbb{I}_1} \frac{\bar{\theta}_{ik}}{4\bar{r}_{ik}} \bar{r}_{ik} \bar{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} \left(-\frac{1}{2}\right) \bar{\theta}_{ik} \bar{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} \frac{1}{2} \bar{r}_{ik} \zeta_{ik}^2(t_p). \quad (37) $$

Similarly, the derivative of $V_{\omega_2}$ satisfies

$$ 
\dot{V}_{\omega_2} \le -\frac{1}{2} \sum_{\langle i,k \rangle \in \mathbb{I}_1} \underline{\theta}_{ik} \underline{\omega}_{ik} + \sum_{\langle i,k \rangle \in \mathbb{I}_1} \frac{\underline{\theta}_{ik}}{4r_{ik}} r_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} \left(-\frac{1}{2}\right) \underline{\theta}_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} \frac{1}{2} r_{ik} \zeta_{ik}^2(t_p). \quad (38) $$

Concerning the term $V_{\omega_3}$ in $V_\omega(t)$, since $\zeta_{ik}(t_p)$ is constant between any $t \in [t_p, t_{p+1})$, we obtain $\dot{V}_{\omega_3} = 0$. Finally, for the derivative of $V_T$, we compute

$$ 
\begin{aligned}
& \frac{d}{dt} \left( \bar{T} \int_{-\bar{T}}^{0} \int_{t+r}^{t} \dot{x}^{\mathrm{T}}(s) (L^2 \otimes QBB^{\mathrm{T}}Q) \dot{x}(s) ds dr \right) \\
& = \bar{T}^2 \dot{x}^{\mathrm{T}}(t) (L^2 \otimes QBB^{\mathrm{T}}Q) \dot{x}(t) - \bar{T} \int_{t-\bar{T}}^{t} \dot{x}^{\mathrm{T}}(s) (L^2 \otimes QBB^{\mathrm{T}}Q) \dot{x}(s) ds.
\end{aligned}
\quad (39) $$

Then using (32) and Young inequality, we can get

$$ 
\begin{aligned}
\bar{T}^2 \dot{x}^\mathrm{T}(t) (L^2 \otimes QBB^\mathrm{T}Q) \dot{x}(t) \le 2\bar{T}^2 \left[ x^\mathrm{T}(t) (L^2 \otimes A^\mathrm{T}QBB^\mathrm{T}QA)x(t) \right. \\
\left. \qquad + \boldsymbol{\sigma}_{\sqrt{\omega(t)}}(\zeta(t_p))^\mathrm{T} (L^2 \otimes B^\mathrm{T}QBB^\mathrm{T}QB) \boldsymbol{\sigma}_{\sqrt{\omega(t)}}(\zeta(t_p)) \right],
\end{aligned}
\quad (40) $$
and recalling that $t - t_p \le \bar{T}$, and Lemma 2, we obtain

$$ 
\begin{aligned}
-\bar{T} \int_{t-\bar{T}}^{t} \dot{x}^{\mathrm{T}}(s)(L^2 \otimes QBB^{\mathrm{T}}Q)\dot{x}(s)ds &\le -\bar{T} \int_{t_p}^{t} \dot{x}^{\mathrm{T}}(s)(L^2 \otimes QBB^{\mathrm{T}}Q)\dot{x}(s)ds \\
&\le -(t-t_p) \int_{t_p}^{t} \dot{x}^{\mathrm{T}}(s)(L^2 \otimes QBB^{\mathrm{T}}Q)\dot{x}(s)ds \\
&\le -(x(t)-x(t_p))^{\mathrm{T}}(L^2 \otimes QBB^{\mathrm{T}}Q)(x(t)-x(t_p)) \\
&= -\tilde{\zeta}^{\mathrm{T}}(t)\tilde{\zeta}(t).
\end{aligned}
\quad (41) $$

Therefore, with the definition of $\sigma_{\sqrt{\omega}}(s)$ and the definition of $\mathbb{I}$ and $\mathbb{J}$, we can get

$$ 
\begin{aligned}
\dot{V}_T &\le 2\bar{T}^2 \|A\|^2 x^{\mathrm{T}}(L^2 \otimes QBB^{\mathrm{T}}Q)x + 2\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \sigma_{\sqrt{\omega}}(\zeta(t_p))^{\mathrm{T}} \sigma_{\sqrt{\omega}}(\zeta(t_p)) - \tilde{\zeta}^{\mathrm{T}}\tilde{\zeta} \\
&\le 2\bar{T}^2 \|A\|^2 x^{\mathrm{T}}(L^2 \otimes QBB^{\mathrm{T}}Q)x - \tilde{\zeta}^{\mathrm{T}}\tilde{\zeta} \\
&\quad + 2\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \sum_{\langle i,k \rangle \in \mathbb{I}_1} \bar{\omega}_{ik} + 2\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \sum_{\langle i,k \rangle \in \mathbb{I}_2} \frac{2\bar{r}_{ik}}{\bar{\theta}_{ik}} \zeta_{ik}^2(t_p) \\
&\quad + 2\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \sum_{\langle i,k \rangle \in \mathbb{J}_1} \underline{\omega}_{ik} + 2\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \sum_{\langle i,k \rangle \in \mathbb{J}_2} \frac{2\underline{r}_{ik}}{\underline{\theta}_{ik}} \zeta_{ik}^2(t_p).
\end{aligned}
\quad (42) $$

Combined with (36), (37), (38), and (42), it obtains that

$$ 
\begin{aligned}
\dot{W} \le \epsilon_1 \sum_{i=2}^{N} \lambda_i \tilde{x}_i^{\mathrm{T}} S \tilde{x}_i - \sum_{\langle i,k \rangle \in \mathbb{I}_1} \bar{\phi}_{ik} \bar{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} (-\frac{1}{2}) \bar{\theta}_{ik} \bar{\omega}_{ik} \\
- \sum_{\langle i,k \rangle \in \mathbb{J}_1} \bar{\rho}_{ik} \zeta_{ik}^2(t_p) - \sum_{\langle i,k \rangle \in \mathbb{J}_1} \underline{\varphi}_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{J}_2} (-\frac{1}{2}) \underline{\theta}_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{J}_2} \underline{\rho}_{ik} \zeta_{ik}^2(t_p),
\end{aligned}
\quad (43) $$

where

$$ 
\begin{aligned}
S &= QA + A^{\mathrm{T}}Q - \left(\frac{5}{4} - 8\bar{T}^2 \|A\|^2\right) \lambda_i QBB^{\mathrm{T}}Q \\
\bar{\phi}_{ik} &= \frac{\bar{\theta}_{ik}}{4} - \left(\frac{2\bar{\theta}_{ik}}{\bar{r}_{ik}} + 4\right) \epsilon_1 - 8\epsilon_1 \bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2, \\
\bar{\rho}_{ik} &= \frac{1}{2}\bar{r}_{ik} - \left(4 + \frac{8\bar{r}_{ik}}{\bar{\theta}_{ik}}\right) \epsilon_1 - 16\epsilon_1 \bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \frac{\bar{r}_{ik}}{\bar{\theta}_{ik}}, \\
\underline{\varphi}_{ik} &= \frac{\underline{\theta}_{ik}}{4} - \left(\frac{2\underline{\theta}_{ik}}{\underline{r}_{ik}} + 4\right) \epsilon_1 - 8\epsilon_1 \underline{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2, \\
\underline{\rho}_{ik} &= \frac{1}{2}\underline{r}_{ik} - \left(4 + \frac{8\underline{r}_{ik}}{\underline{\theta}_{ik}}\right) \epsilon_1 - 16\epsilon_1 \underline{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \frac{\underline{r}_{ik}}{\underline{\theta}_{ik}}.
\end{aligned}
$$

Substitute (28), $\bar{T}$ in (29) and the saturation parameters $\bar{\theta}_{ik}$, $\underline{\theta}_{ik}$ in (30) into the above equation, we can select $\epsilon_1 \le \min\{\epsilon_1^a, \epsilon_1^b\}$ with

$$ 
\begin{aligned}
\epsilon_1^a &= \frac{\bar{r}_{ik}}{4\bar{\theta}_{ik} + 8\bar{r}_{ik} + 16\bar{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \bar{r}_{ik}}, \\
\epsilon_1^b &= \frac{\underline{r}_{ik}}{4\underline{\theta}_{ik} + 8\underline{r}_{ik} + 16\underline{T}^2 N^2 \|B^{\mathrm{T}}QB\|^2 \underline{r}_{ik}}.
\end{aligned}
$$

Note that $\bar{r}_{ik}$ and $\underline{r}_{ik}$ in (28) will not be zero, which implies there always exists $\epsilon_1$ sufficiently small so that

$$ 
\begin{aligned}
\dot{W} \le -2\epsilon_1 \left(1 - \cos\left(\frac{\pi}{N}\right)\right) \|x\|^2 &- \sum_{\langle i,k \rangle \in \mathbb{I}_1} \bar{\phi}_{ik} \bar{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{I}_2} \left(-\frac{1}{2}\right) \bar{\theta}_{ik} \bar{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{J}_2} \bar{\rho}_{ik} \zeta_{ik}^2(t_p) \\
&- \sum_{\langle i,k \rangle \in \mathbb{J}_1} \underline{\varphi}_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{J}_2} \left(-\frac{1}{2}\right) \underline{\theta}_{ik} \underline{\omega}_{ik} - \sum_{\langle i,k \rangle \in \mathbb{J}_2} \underline{\rho}_{ik} \zeta_{ik}^2(t_p),
\end{aligned}
\quad (44) $$
where $\bar{\varphi}_{ik} > 0$, $\bar{\rho}_{ik} > 0$, $\underline{\varphi}_{ik} > 0$, $\underline{\rho}_{ik} > 0$. In view of $W(t)$ in (33) and $\dot{W}(t)$ in (44), one can get

$$ \dot{W}(t) \leq -\eta(\epsilon_1 V_x(t) + V_\omega(t)), \quad (45) $$

which indicates that $W(t)$ is asymptotically stable with

$$ \eta = \min \left\{ \frac{2 \left( 1 - \cos \left( \frac{\pi}{N} \right) \right)}{N \|Q\|}, 2\bar{\varphi}_{ik}, \bar{\theta}_{ik}, \frac{\bar{\rho}_{ik}}{2\frac{\bar{r}_{ik}}{\bar{\theta}_{ik}}}, 2\underline{\varphi}_{ik}, \underline{\theta}_{ik}, \frac{\underline{\rho}_{ik}}{2\frac{\underline{r}_{ik}}{\underline{\theta}_{ik}}} \right\} $$

a positive constant. Therefore, using Barbalat's lemma, we obtain that when $x \neq 0$, $\omega_{ik} \neq 0$, there holds $\dot{W}(t) < 0$, $t \in [t_p, t_{p+1})$. Below we provide the proof of $W(t_{p+1}^+) \leq W(t_{p+1}^-)$.

From $\dot{W}(t) < 0$, $t \in [t_p, t_{p+1})$, one has $W(t_{p+1}^-) < W(t_p)$,

$$ W(t_p) = \epsilon_1 V_x(t_p) + 4\epsilon_1 V_T(t_p) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2}\omega_{ik}(t_p) + \max \left\{ \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p), 0 \right\} \right), \quad (46) $$

$$ W(t_{p+1}^-) = \epsilon_1 V_x(t_{p+1}) + 4\epsilon_1 V_T(t_{p+1}) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2}\omega_{ik}(t_{p+1}) + \max \left\{ \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}), 0 \right\} \right), \quad (47) $$

$$ \begin{aligned} W(t_{p+1}^+) &= \epsilon_1 V_x(t_{p+1}) + 4\epsilon_1 V_T(t_{p+1}) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2}\omega_{ik}(t_{p+1}) + \max \left\{ \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) - \omega_{ik}(t_{p+1}), 0 \right\} \right) \\ &= \epsilon_1 V_x(t_{p+1}) + 4\epsilon_1 V_T(t_{p+1}) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2}\omega_{ik}(t_{p+1}) \right. \\ & \qquad \left. + \max \left\{ \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) + \left( \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) \right), 0 \right\} \right). \end{aligned} \quad (48) $$

We will discuss the above equation in four situations.

* **Case 1:** $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) \leq 0$ and $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p) \leq 0$. In this case, we can obtain $W(t_{p+1}^+) = W(t_{p+1}^-)$ directly.

* **Case 2:** $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) > 0$ and $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p) \leq 0$. Since $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p) \leq 0$ leading to $\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) < W(t_{p+1}^-)$ easily.

* **Case 3:** $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) > 0$ and $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p) > 0$. In this case, we use the method of proof by contradiction to argue. Suppose that $\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) > 0$ holds, i.e. $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}}\zeta_{ik}^2(t_{p+2}) > \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) > \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p)$, there must be an interval $[t_p, t_{p+1}]$ that makes $\left\{ \frac{\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) < \Delta_{ik}^2}{\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) = \Delta_{ik}^2} \right. $ leading to $\frac{r_{ik}(t_p)}{\theta_{ik}} = \frac{1}{2}$, $\frac{r_{ik}(t_{p+1})}{\theta_{ik}} \leq \frac{1}{2}$, and $\zeta_{ik}^2(t_p) < \zeta_{ik}^2(t_{p+1}) < \zeta_{ik}^2(t_{p+2})$. According to the relationship of parameters in (28), then one has $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}}\zeta_{ik}^2(t_{p+2}) = \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) = \Delta_{ik}^2$. That's the opposite of $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}}\zeta_{ik}^2(t_{p+2}) > \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1})$. Thus, the assumption of $\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) > 0$ cannot hold. Instead, we have $\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\zeta_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) \leq 0$, then we can obtain $W(t_{p+1}^+) \leq W(t_{p+1}^-)$.

* **Case 4:** $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) \leq 0$ and $\frac{2r_{ik}(t_p)}{\theta_{ik}}\zeta_{ik}^2(t_p) - \omega_{ik}(t_p) > 0$. This situation is clearly not valid according to the proof of **Case 3**.

Taking all the above proofs into account, we can conclude from the perspective of energy decay that $\lim_{t \to \infty} W(t) = 0$, which implies $\lim_{t \to \infty} x^\top(L \otimes Q)x = 0$. Since $Q > 0$, one has

$$ \lim_{t \to \infty} x^\top(L \otimes I_n)x = \frac{1}{2} \lim_{t \to \infty} \sum_{i=1}^{N} \sum_{j \in \mathcal{N}_i} a_{ij} \|x_i - x_j\|^2 = 0, $$

which further leads to

$$ \lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \forall i, j = 1, \dots, N. $$

□
**Remark 4.** If the matrix A is a zero matrix, i.e., all elements of A are zero$^{38}$, and the matrix B contains n linearly independent columns, then Assumption 1 holds.

**Remark 5.** The relationship in (29) reflects the fact that for a system with known system matrices A and B, the bounds of the sampling period are uniform and the bound is independent of the number of agents. Moreover, compared to the results in Theorem 1, it is not necessary to reduce the sampling time depending on the selection of the control gain parameters.

5 | SIMULATION

To explain the effectiveness of the sampled-data based input saturation reconstruction algorithm in Section 4, system (1) of a network with N = 6 agents is adopted for simulation with the matrices satisfying Assumption 1 given by

$$A = \begin{bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}. \qquad (49)$$

The connected communication topology associated with N = 6 agents is described by the Laplacian matrix L, and the solution to the algebraic Riccati equation (ARE) (19) determines the matrix Q

$$L = \begin{bmatrix} 2 & -1 & 0 & 0 & 0 & -1 \\ -1 & 3 & -1 & 0 & 0 & -1 \\ 0 & -1 & 3 & -1 & -1 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & -1 & -1 & 3 & -1 \\ -1 & -1 & 0 & 0 & -1 & 3 \end{bmatrix}, \quad Q = \begin{bmatrix} 2.8962 & -1.2674 & 1.3769 & -0.5067 \\ -1.2674 & 4.8470 & -0.6157 & 1.7267 \\ 1.3769 & -0.6157 & 2.3049 & 0.2894 \\ -0.5067 & 1.7267 & 0.2894 & 2.7325 \end{bmatrix}.$$

The inherent heterogeneous asymmetric saturation levels are chosen as $\Delta_1^+ = [1, 1.5]^T$, $\Delta_2^+ = [2, 2.5]^T$, $\Delta_3^+ = [3, 3.5]^T$, $\Delta_4^+ = [4, 4.5]^T$, $\Delta_5^+ = [5, 5.5]^T$, $\Delta_6^+ = [6, 6.5]^T$, $\Delta_1^- = [4, 4.5]^T$, $\Delta_2^- = [5, 5.5]^T$, $\Delta_3^- = [6, 6.5]^T$, $\Delta_4^- = [7, 7.5]^T$, $\Delta_5^- = [8, 8.5]^T$, $\Delta_6^- = [9, 9.5]^T$.

Moreover, following Theorem 2, we have $\underline{T} = 0.001$, $\overline{T} = 0.1093$ and we choose $T_p = 0.1$. The initial states for all agents are randomly chosen from the interval $[-20, 20] \times [-20, 20] \times [-20, 20] \times [-20, 20]$, and the initial values for $\underline{\omega}_{ik}$ and $\overline{\omega}_{ik}$ are all chosen to be 0. Besides, we choose $\underline{\theta}_{ik} = 36$ and $\overline{\theta}_{ik} = 36$ for $i = 1, \dots, 6, k = 1, 2$.

As shown in Figure 1, under the steering of $u_i(t)$ in (23), the state of $\zeta_i(t) = -B^T Q \sum_{j \in \mathcal{N}_i} a_{ij}(x_j(t) - x_j(t))$ can converge to 0 which indicates the consensus of the multi-agent system. Figures 2 and 3 depict their trajectories for $\underline{\bar{r}}_{ik}$ and $\overline{\underline{r}}_{ik}$ defined in (28). As it is shown in Figures 2 and 3, $\underline{\bar{r}}_{ik}$ and $\overline{\underline{r}}_{ik}$ are time-varying at the beginning and then turn to $\frac{1}{2}$ as time goes by. This is because that $\zeta_{ik}^2(t_p)$ would be very large at the beginning due to the global selection of the initial state, which leads to $\frac{\Delta_{ik}^2}{\underline{\varepsilon}_{ik}^2(t_p)}$, but as time goes to infinity, $\zeta_{ik}^2(t_p)$ approaches to 0 so that $\underline{\bar{r}}_{ik}(t) = \frac{1}{2}$ and $\overline{\underline{r}}_{ik}(t) = \frac{1}{2}$.

Figure 4 displays the trajectories of $u_i(t)$ in (23), where $-\Delta_{ik}^- \le u_{ik}(t) \le \Delta_{ik}^+$ can be ensured for $i = 1, \dots, 6$ and $k = 1, 2$. For the dynamic saturation level associated with the re-designed saturation function, the trajectories of $\sqrt{\underline{\omega}_{ik}}(t)$ and $-\sqrt{\overline{\omega}_{ik}}(t)$ are given in Figure 5. As shown in this figure, $\sqrt{\underline{\omega}_{ik}}(t) \le \Delta_{ik}^+$ and $\sqrt{\overline{\omega}_{ik}}(t) \le \Delta_{ik}^-$ can be guaranteed for $i = 1, \dots, 6$ and $k = 1, 2$, which verifies Lemma 3. In accordance with (24) and the global consensus verified by Figure 1, as time goes to infinity, $\sqrt{\underline{\omega}_{ik}}(t)$ and $\sqrt{\overline{\omega}_{ik}}(t)$ approach to 0. Overall, the five figures validate the theoretical findings on global consensus achieved via the proposed sampled-data based dynamic saturation reconstruction approach. In particular, the simulations confirm that consensus can be reached in systems subject to asymmetric and heterogeneous input saturation constraints, while ensuring that all control inputs remain within their prescribed bounds and that the sampling period approaches its admissible upper limit. These observations demonstrate the effectiveness and practical applicability of the proposed control strategy for handling saturation-constrained consensus problems within a sampled-data framework.
<figure><img src="image_1.png" /><figcaption>**Figure 1** Convergence of $\zeta_i(t)$ in which $x_i(t)$ is steered by (1) with $u_i(t)$ in (23).</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>**Figure 2** Trajectories (12 in total) of $\frac{\bar{r}_{ik}}{\bar{\theta}_{ik}}$ defined in (28).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 3** Trajectories (12 in total) of $\frac{r_{ik}}{\theta_{ik}}$ defined in (28).</figcaption></figure>

## 6 | CONCLUSION

This paper designs distributed control algorithms in the sampled-data framework for realizing the consensus of the multi-agent systems, which are in the presence of asymmetric heterogeneous actuator saturation. The first method improves the classical low-gain approach to accommodate saturation asymmetry and heterogeneity, achieving semi-global consensus. However, it is sensitive to gain selection and imposes restrictions on saturation structure. To overcome these limitations, a dynamic saturation reconstruction method is further developed, which ensures global consensus with explicit sampling bounds, guaranteed input constraints, and reduced update frequency. This approach significantly enhances the scalability and practicality of distributed control under saturation. However, the results are derived under undirected communication graphs, where the symmetry of the Laplacian matrix ensures the feasibility of Lyapunov analysis and distributed parameter design. Due to the loss of this structure in directed topologies, the proposed methods cannot be directly extended to such cases. Thus, developing effective distributed control laws to achieve consensus in directed multi-agent systems affected by asymmetric heterogeneous actuator saturation remains an open problem.
<figure><img src="image_4.png" /><figcaption>**Figure 4** Trajectories of $u_i(t)$ in (23).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 5** Trajectories of $\sqrt{\omega_{ik}(t)}$ with $\bar{\omega}_{ik}(t)$ and $\underline{\omega}_{ik}(t)$ defined in (24).</figcaption></figure>

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## AUTHOR BIOGRAPHY

<figure><img src="image_6.png" /></figure>

**Juan Qian** received the B.E. degree in automation from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2020. She is currently pursuing the PhD degree in information and control with the College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, China. Since October 2023, she has been starting a joint-PhD program at the LAGEEP Laboratory of University Claude Bernard Lyon 1 (Lyon, France). Her research interests include coordinated control of saturated multi-agent systems and adaptive control of nonlinear systems.

<figure><img src="image_7.png" /></figure>

**Xiaoling Wang** is an Associate Professor at Nanjing University of Posts and Telecommunications. She received her Ph.D. degree in control science and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2017. In 2015–2017, she was a Research Assistant with the City University of Hong Kong and the University of Hong Kong. Her research interests include coordinated control of uncertain multi-agent systems, distributed observer design and distributed interval state estimation.

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**Daniele Astolfi** received his BSc (2009) and MSc (2012) degrees in Automation Engineering with honors from the University of Bologna, Italy, and obtained a joint Ph.D. degree in Control Theory from the University of Bologna (Italy) and from Mines ParisTech (France), in 2016. Afterwards he spent two years at CRAN, Nancy (France) as a postdoc researcher. Since 2018, he is a CNRS Researcher at LAGEPP, Lyon, France. He obtained the “Habilitation à Diriger des Recherches” (HDR) in 2023 from Universitéé Claude Bernard Lyon 1. D. Astolfi was the recipient of the 2016 Best Italian Ph.D. Thesis Award from SIDRA and was nominated for the best student paper award at ECC16 and the best paper award at NOLCOS2019.

He has served as an Associate Editor for Automatica since 2018 and for European Journal of Control since 2023. His research interests lie in the areas of nonlinear control systems, observers, output regulation and multi-agent systems.

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**Housheng Su** received his B.S. degree in automatic control and his M.S. degree in control theory and control engineering from Wuhan University of Technology, Wuhan, China, in 2002 and 2005, respectively, and his Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2008. From December 2008 to January 2010, he was a Postdoctoral researcher with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. Since November 2014, he has been a full professor with the School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China. He is an Associate Editor of *IEEE Transactions on Systems, Man, and Cybernetics: Systems* and *IET Control Theory and Applications*. His research interests lie in the areas of multi-agent coordination control theory and its applications to autonomous robotics and mobile sensor networks.
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**Guo-Ping Jiang** received the B.E. degree in electrical engineering from Hohai University, Nanjing, China, in 1988 and the Ph.D. degree in control theory and engineering from Southeast University, Nanjing, in 1997.

From 1988 to 1992, he worked as an Assistant Lecturer in the Department of Electrical Engineering at Hohai University. From 1997 to 2005, he worked in the Department of Electronic Engineering, Nanjing University of Posts and Telecommunications. In 2001, 2002 and 2005, he worked as a Research Assistant and Research Fellow at the Department of Electronic Engineering, City University of Hong Kong for several times. From August to December 2003, he was a Visiting Fellow at the School of Quantitative Methods and Mathematical Sciences (QMMS), University of Western Sydney. He is currently a Professor and Vice President of Nanjing University of Posts and Telecommunications. Prof. Jiang was awarded the New Century Excellent Talents by the Ministry of Education in 2006 and the "333 Project" High Level Talents by Jiangsu Province in 2011.

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