# Global Consensus for Heterogeneous Saturated Multi-Agent Systems via Sampled-Data Control

Juan Qian, Xiaoling Wang, Daniele Astolfi, Housheng Su, Guo-Ping Jiang

► To cite this version:

Juan Qian, Xiaoling Wang, Daniele Astolfi, Housheng Su, Guo-Ping Jiang. Global Consensus for Heterogeneous Saturated Multi-Agent Systems via Sampled-Data Control. 2024 IEEE 63rd Conference on Decision and Control (CDC), IEEE, Dec 2024, Milan, Italy. pp.4680-4685, 10.1109/CDC56724.2024.10886353. hal-04989471

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# Global Consensus for Heterogeneous Saturated Multi-Agent Systems via Sampled-Data Control

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

**Abstract**— This paper addresses the global consensus problem for multi-input multi-output saturated systems within a sampled-data framework, aiming to advance global consensus, manage heterogeneous actuator saturation across components, and preserve distributed characteristics by using sampled-data feedback. We propose a distributed control algorithm that incorporates a redesigned saturation function, represented as decentralized dynamic saturation levels. These levels for each agent's dimensions are autonomously updated through an adaptive strategy, which mitigates heterogeneous saturation by carefully selecting constant and time-varying saturation parameters in it. Lyapunov analysis proves that global consensus can be achieved under the proposed control law, provided the sampling periods of all agents remain below a calculated threshold. An example is given to demonstrate the effectiveness of this approach.

## I. INTRODUCTION

The research center of multi-agent systems (MASs) is distributed coordination control, with consensus being a fundamental challenge. As technology rapidly advances, consensus theory has been applied to various fields, such as robotic systems [1], unmanned aerial systems [2], wireless sensor networks [3], and other practical domains. However, despite its wide applicability, actuator saturation in real-world engineering systems remains a significant challenge, hindering the direct implementation of many consensus algorithms in engineering projects [4].

To date, considerable progress has been made in addressing the impact of saturation on agent dynamics, with studies

primarily focusing on first-order or second-order systems [5] to achieve global consensus, and higher-order systems [6]–[8] aimed at semi-global consensus. It is noteworthy that asymptotically null controllability with bounded controls (ANCBC) is a crucial precondition for solving global coordination control in saturated linear MASs [9]. Additionally, references [10], [11] highlight the necessity of nonlinear controllers for achieving global results in general ANCBC systems. For further insights into higher-order systems, readers may refer to our recent publication [12].

Existing research on saturated MASs, as previously discussed, predominantly addresses actuator saturation with homogeneous levels. However, in practical applications, heterogeneous saturation actuators are common. Heterogeneity in multi-input multi-output (MIMO) systems can manifest in four main aspects: asymmetric saturation, where the upper and lower bounds are unequal; non-uniform saturation levels among different agents; non-uniform saturation levels among different components; and non-uniform saturation levels among both agents and components. While there have been efforts to explore consensus in MASs with asymmetric input saturation, such as the algorithms proposed in [13] and [14], there is limited work addressing the coordination control of MASs with non-uniform saturation levels for distinct agents or components.

Moreover, sampled-data control offers easier implementation and greater energy efficiency in real networked environments. The principle involves the digital signal processor generating control signals for each agent, which are then converted into continuous signals using a zero-order hold [15] before being transmitted to the actuators for further action. This approach has driven research into distributed control protocols that use sampled-data to achieve consensus. Currently, the application of sampled-data control is relatively mature for both first-order and second-order systems. For general linear MASs, [16], [17] investigated the sampled-data leaderless mean square consensus problem. Furthermore, [18] proposed a sampled-data consensus control law for directed networks, providing explicit upper bounds on sampling periods for all agents based on a technical lemma. Building on this, [19] developed a hybrid systems approach, which results in larger upper bounds on sampling periods.

Despite the advances in sampled-data control, there remains a scarcity of literature addressing its application in heterogeneous saturated MASs, with [20] being a notable exception. However, it is important to recognize the limitations of this paper, specifically, the sampled-data distributed control protocol proposed in this reference is constrained to

<footnote>*This work was supported by the following grants: National Natural Science Foundation of China under Grants 62473207, 62425602 and 62373197; Natural Science Fund for Excellent Young Scholars of Jiangsu Province under Grant No. BK20220104; Fundamental Research Funds for the Central Universities under Grant No. 2024SMECP03; Foundation of Jiangsu Scientific Research and Innovation Program under Grant No. 4600KYCX221012; French ANR via the grant ALLIGATOR ANR-22-CE48-0009-01.</footnote>

<footnote n="1">¹Juan Qian, Xiaoling Wang and Guo-Ping Jiang are with College of Automation and the College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China; Jiangsu Engineering Center for IOT Intelligent Robots (IOTRobot), Nanjing 210023, China 2020050318@njupt.edu.cn, xiaolingwang@njupt.edu.cn, jianggp@njupt.edu.cn</footnote>

<footnote n="2">²Juan Qian and Daniele Astolfi are with Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43 boulevard du 11 novembre 1918, F-69100, Villeurbanne, France juan.qian@univ-lyon1.fr, daniele.astolfi@univ-lyon1.fr</footnote>

<footnote n="3">³Xiaoling Wang is also with Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China xiaolingwang@njupt.edu.cn</footnote>

<footnote n="4">⁴Housheng Su is with 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, China houshengsu@gmail.com</footnote>
single-order and single-dimensional dynamic models, which limits its ability to capture heterogeneity across different dimensions. Consequently, the study of MASs under heterogeneous input saturation constraints remains an open area for further research.

Motivated by the aforementioned discussions, this paper addresses the challenge of imposing the heterogeneous input saturation constraints and aims at achieving global consensus by a sampled-data control framework, where the constraints involve symmetric saturation but non-uniform input saturation across different dimensions of each agent. Compared with [20], we treat general linear MASs, which contain the single-integrator MASs and the double-integrator MASs as special cases. Moreover, rather than the uniform symmetric input saturations in [7], [8], [21], we consider a practical case where the continuous-signal-driven system is controlled by sampled-data. In order to solve the global consensus problem of MASs with such kind of heterogeneous actuator saturation, a novel distributed sampled-data control protocol, which is subject to a re-designed saturation function with dynamic saturation level is constructed for each agent. This dynamic saturation level, inspired from the stubborn redesign approach proposed in [22], and unique to this manuscript, is proposed to be decentralized, with each component autonomously updating itself by following a first-order differential equation. In addition, our findings have a new feature of firstly giving an explicit upper bound for the sampling periods of all heterogeneous saturated agents in a MIMO system that guarantees the stability of the closed-loop system, which lead to difficulties of the consensus analysis schemes.

## II. MATHEMATICAL PRELIMINARIES

The notations are denoted 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. $\mathbb{N}$ denotes the set of 0 and all positive integers. $\| \cdot \|$ is the 2-norm of a vector or a matrix. Besides, $\otimes$ stands for the Kronecker product, the superscript “T” represents matrix transposition. We denote with $\mathbf{e}_k$ the unit column vector with compatible dimension, where the $k$-th entry is 1 and all the others are 0. Finally, $P > 0$ means that $P$ is a symmetric positive definite matrix.

**Graph theory:** We introduce the undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{A})$ to visually and concretely depict the topological relationships among $N$ agents within the networked system. In $\mathcal{G}$, $\mathcal{V}$ denotes the node set comprising all $N$ agents, while $\mathcal{E}$ represents the edge set, symbolizing the connections between these agents. Importantly, the adjacency matrix $\mathcal{A}$ vividly describes the flow of information between any two nodes through binary values, 0 and 1. If there exists a direct edge between the $i$-th and $j$-th agents, it is denoted by 1, i.e., $a_{ij} = 1$, where $i, j = 1, \dots, N$. The Laplacian matrix $L \in \mathbb{R}^{N \times N}$ associated with $\mathcal{A}$ is defined as $L = (l_{ij})$, where $i, j = 1, \dots, N$. Specifically, $l_{ij} = -a_{ij}$ for $i \neq j$, and $l_{ij} = \sum_{k=1, k \neq i}^{N} a_{ik}$ for $i = j$. Since $L$ is symmetric, there exists an orthogonal matrix $U \in \mathbb{R}^{N \times N}$ such that $L = U^T \Lambda U$, where $\Lambda = \text{diag}\{\lambda_1(L), \dots, \lambda_N(L)\} = \text{diag}\{\lambda_1, \dots, \lambda_N\}$.

**Lemma 1.** [23] For Laplacian matrix $L$ associated with an un-directed connected graph $\mathcal{G}$, zero is a simple eigenvalue of $L$ if and only if the graph is connected. Then one has $\lambda_2 \ge 2 [1 - \cos(\frac{\pi}{N})]$, $\lambda_N \le N$, where the strict equality holds if $\mathcal{G}$ is a chain or a complete graph, respectively.

**Lemma 2.** [24] For any given positive definite matrix $H \in \mathbb{R}^{n \times n}$, and vector function $W : [a, b] \to \mathbb{R}^n$, the following relation holds:

$$ \left( \int_a^b W(s) ds \right)^T G \left( \int_a^b W(s) ds \right) \le (b-a) \int_a^b W(s)^T G W(s) ds. \quad (1) $$

## III. MODEL DESCRIPTION & PROBLEM FORMULATION

Consider a group of $N$ identical agents with general linear dynamics being described as

$$ \dot{x}_i = Ax_i + B\mathbf{sat}_{\Delta_i}(u_i), \quad i = 1, \dots, N, \quad (2) $$

where $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$ are constant matrices with compatible dimensions, $x_i \in \mathbb{R}^n$ is the state, $u_i \in \mathbb{R}^m$ is the control input which will be designed later, and

$$ \mathbf{sat}_{\Delta_i}(u_i(t)) = [\mathbf{sat}_{\Delta_{i1}}(u_{i1}(t)), \dots, \mathbf{sat}_{\Delta_{im}}(u_{im}(t))]^T \in \mathbb{R}^m $$

is the saturation function of the physical actuator of the systems, which is described as follows:

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

where $k = 1, \dots, m$, $\Delta_{ik}$ is the heterogeneous saturation level of each agent in each dimension.

In this network (2), we suppose that the control actions are not continuous but satisfy a sampled-data input saturation reconstruction algorithm $u_i(t) = f(x(t_p))$, $t_p \le t < t_{p+1}$ with appropriate sampling period $T$ satisfying $\underline{T} \le T \le \bar{T}$, where $\underline{T} \le T$ are two positive real numbers [18] such that system (2) achieves global consensus in the sense of $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \forall i, j = 1, \dots, N$. Moreover, for achieving global (in terms of domain of attraction of the initial conditions) consensus, the MAS must satisfy the following condition.

**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.

## IV. SAMPLED-DATA CONTROL PROTOCOLS FOR HETEROGENEOUS SATURATED MAS

The main objective is to design nonlinear protocols within the sampled-data framework.

### A. Design of $u_i(t)$

First, we reorder the sampling time instants of all agents in an ascending sequence as $0 = t_0 < t_1 < \dots < t_p < t_{p+1}$. Clearly, this time sequence has the property of $t_{p+1} - t_p = T$. Then for convenience, we denote $x(t_p) = [x_1(t_p)^T, \dots, x_N(t_p)^T]^T$ and define the distributed sampled-data state coupling term of the $i$-th agent as

$$ \xi_i(t_p) = -B^T P \sum_{j=1}^{N} l_{ij} x_j(t_p) = -(\mathbf{e}_i^T L \otimes B^T P) x(t_p), $$
which indicates intuitively that

$$ \xi_{ik}(t_p) = - \left( \mathbf{e}_i^T L \otimes \mathbf{e}_k^T B^T P \right) x(t_p), \quad k = 1, \dots, m, \quad (4) $$

with $P$ obtained according to the algebraic Riccati equation (where $0 < \mu < \frac{5}{4}$) in the form as follows

$$ A^T P + P A - 2\mu \left[ 1 - \cos\left(\frac{\pi}{N}\right) \right] P B B^T P + I_n = 0. \quad (5) $$

Next we design the dynamic input saturation reconstruction algorithm $u_i(t)$ based on the dynamic saturation level $\sqrt{\omega_{ik}(t)}$ with $\omega_i(t) = [\omega_{i1}(t), \dots, \omega_{im}(t)]^T \in \mathbb{R}^m$ as

$$ u_i(t) = \sigma \sqrt{\omega_i(t)} (\xi_i(t)), t \in [t_p, t_{p+1}), \quad (6) $$

$$ \dot{\omega}_{ik}(t) = -\theta_{ik}\omega_{ik}(t) + r_{ik}(t)\xi_{ik}(t_p)^2, t \in [t_p, t_{p+1}). \quad (7) $$

In the whole algorithm, $\theta_{ik} > 0$ is a constant saturation parameter, while $r_{ik}(t) > 0$ is a time-varying saturation parameter to be defined. Moreover,

$$ \sigma \sqrt{\omega_i}(\xi_i) = [\sigma \sqrt{\omega_{i1}}(\xi_{i1}), \dots, \sigma \sqrt{\omega_{ik}}(\xi_{ik})]^T \in \mathbb{R}^m, $$

specifically $\sigma \sqrt{\omega_{ik}}(\xi_{ik}) = \max\{-\sqrt{\omega_{ik}}, \min\{\sqrt{\omega_{ik}}, \xi_{ik}\}\}$, where the time argument is omitted for compact notation. Besides, $\omega_{ik}(0) \ge 0$ is constrained to evolve in the forward invariant set $\mathbb{R}$.

### B. A technical lemma for dynamic saturation level $\sqrt{\omega_{ik}(t)}$

In this subsection, we remark that for any function $r_{ik}$ and sequence of constant signals $\xi_{ik}(t_p)$ which are continuous on any interval of time $[t_p, t_{p+1}]$, the corresponding solution $\omega_{ik}$ is $C^0$ for all $t \ge 0$ and $C^1$ almost everywhere (i.e. it is differentiable on each interval $[t_p, t_{p+1}]$).

**Lemma 3.** Let the time-varying parameter $r_{ik} > 0$ regulate itself to guarantee $r_{ik}\xi_{ik}(t_p)^2 \le \theta_{ik}\Delta_{ik}^2$. Then for any $0 \le \omega_{ik}(0) \le \Delta_{ik}^2$, there holds

$$ 0 \le \omega_{ik}(t) \le \Delta_{ik}^2, \quad \forall t \ge 0. $$

*Proof.* By integrating equation (7) on $t \in [t_p, t_{p+1})$, one obtains

$$ \omega_{ik}(t) \le \Delta_{ik}^2 + e^{-\theta_{ik}(t-t_p)}(\omega_{ik}(t_p) - \Delta_{ik}^2). \quad (8) $$

Thus for $t \in [0, t_1)$,

$$ \omega_{ik}(t_1) = \lim_{t \to t_1^{-}} \omega_{ik}(t) \le \Delta_{ik}^2 + e^{-\theta_{ik}(t_1-t_0)}(\omega_{ik}(0) - \Delta_{ik}^2). $$

Since $\omega_{ik}(0) \le \Delta_{ik}^2$, we obtain $\omega_{ik}(t_1) \le \Delta_{ik}^2$. Then, for $t \in [t_1, t_2)$,

$$ \omega_{ik}(t_2) = \lim_{t \to t_2^{-}} \omega_{ik}(t) \le \Delta_{ik}^2 + e^{-\theta_{ik}(t_2-t_1)}(\omega_{ik}(t_1) - \Delta_{ik}^2). $$

Since $\omega_{ik}(t_1) \le \Delta_{ik}^2$, we also have $\omega_{ik}(t_2) \le \Delta_{ik}^2$. Hence, for any $t \in [t_p, t_{p+1})$,

$$ \omega_{ik}(t_{p+1}) = \lim_{t \to t_{p+1}^{-}} \omega_{ik}(t) \le \Delta_{ik}^2 + e^{-\theta_{ik}(t_{p+1}-t_p)}(\omega_{ik}(t_p) - \Delta_{ik}^2). $$

Since $\omega_{ik}(t_p) \le \Delta_{ik}^2$, we get $\omega_{ik}(t_{p+1}) \le \Delta_{ik}^2$. Therefore, $\omega_{ik}(t) \le \Delta_{ik}^2$ as $\omega_{ik}(0) \le \Delta_{ik}^2$, concluding the proof. $\square$

Therefore, for any selection of $r_{ik}$ satisfying the previous lemma, system (2) for $i = 1, \dots, N$ is equivalent to

$$ \begin{aligned} \dot{x}_i(t) &= Ax_i(t) + B \mathbf{sat}_{\Delta_i} (\sigma \sqrt{\omega_i(t)}(\xi_i(t_p))) \\ &= Ax_i(t) + B \sigma \sqrt{\omega_i(t)}(\xi_i(t_p)), \quad t \in [t_p, t_{p+1}). \end{aligned} \quad (9) $$

In particular, this can be achieved if one selects the time-varying parameters $r_{ik}$ as follows:

$$ r_{ik}(t) = \max \left\{ \psi_{ik}(t) \in \left(0, \frac{\theta_{ik}}{2}\right] : \psi_{ik}(t) \xi_{ik}(t_p)^2 \le \theta_{ik} \Delta_{ik}^2 \right\}. \quad (10) $$

Note that with such a choice, $r_{ik}(t)$ is guaranteed to be $r_{ik}(t_p)$ on any interval of time $[t_p, t_{p+1})$.

### C. Theoretical analysis

This section shows that the global consensus of continuous-signal-driven system with heterogeneous saturation can be achieved by sampled-data based control law.

**Theorem 1.** Consider the network of $N$ agents (2), and the dynamic input saturation reconstruction protocol $u_i(t)$ in (6) with distributed sampled-data $\xi_{ik}(t_p)$ in (4) as well as the dynamic saturation level $\omega_{ik}(t)$ in (7) with $r_{ik}(t)$ selected as in (10). Suppose that Assumption 1 holds and that the sampling period $T$ satisfies

$$ \left\{ \begin{array}{l} \underline{T} \le T \le \bar{T} \text{ with } 0 < \underline{T} \le \bar{T} \le \sqrt{\frac{5-4\mu}{32\|A\|^2}}, \text{ if } \|A\| \ne 0, \\ T > 0, \text{ if } \|A\| = 0. \end{array} \right. \quad (11) $$

Then, for any $\theta_{ik} > 2 + 8 (\|A\| + N \|BB^T P\|)$, the global consensus problem of the MAS in (2) is solved, namely $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \forall i, j = 1, \dots, N$.

*Proof.* For $i = 1, \dots, N$, let $\tilde{x}_i(t) = x_i(t) - x_i(t_p)$, $t \in [t_p, t_{p+1})$, and $\zeta = [\zeta_1^T, \dots, \zeta_N^T]^T$. Then, there is $\zeta_i(t) \in \mathbb{R}^m$ satisfying

$$ \begin{aligned} \zeta_i(t) &= -B^T P \sum_{j=1, j \ne i}^{N} a_{ij}(\tilde{x}_i(t) - \tilde{x}_j(t)) \\ &= \xi_i(t) - \xi_i(t_p), \quad t \in [t_p, t_{p+1}), \end{aligned} \quad (12) $$

and for $k = 1, \dots, m$, $\zeta_{ik}(t)$ is an element on the $k$-th dimension. Furthermore, $x(t), t \in [t_p, t_{p+1})$ meets the following compact form

$$ \dot{x}(t) = (I_N \otimes A)x(t) + (I_N \otimes B)\sigma \sqrt{\omega(t)}(\xi(t_p)), \quad (13) $$

$$ \text{where } \xi(t_p) = -(L \otimes B^T P)x(t_p). $$

Consider now the Lyapunov function on $t \in [t_p, t_{p+1})$ for the $x(t)$ and $\omega_{ik}(t)$ dynamics as follows

$$ V = \epsilon_1 x^T(t) (L \otimes P) x(t) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \begin{aligned} & \frac{1}{2} \omega_{ik}(t) \\ & + \max \left\{ \frac{2r_{ik}(t)}{\theta_{ik}} (\xi_{ik}(t) - \zeta_{ik}(t))^2 - \omega_{ik}(t), 0 \right\} \\ & + 4\epsilon_1 \bar{T} \int_{-\bar{T}}^{0} \int_{t+r}^{t} \dot{x}^T(s)(L^2 \otimes PBB^T P)\dot{x}(s)dsdr, \end{aligned} \right) \quad (14) $$

where $\epsilon_1 > 0$ is a sufficient small constant. It can be verified that $V$ is positive definite. Then, we split $N$ agent elements into two sets, that is, let $\mathbb{I} \subset \{1, \dots, N\} \times \{1, \dots, m\}$ be the subset of indexes $\langle i, k \rangle$ for which $\omega_{ik}(t) \ge \frac{2r_{ik}(t)}{\theta_{ik}} \xi_{ik}(t_p)^2$ and $\mathbb{J} = (\{1, \dots, N\} \times \{1, \dots, m\}) \setminus \mathbb{I}$ be the set of indexes
$\langle i, k \rangle$ for which $\omega_{ik}(t) < \frac{2r_{ik}(t)}{\theta_{ik}} \xi_{ik}(t_p)^2$. Continuously, there holds

$$
\begin{align}
V ={}& \underbrace{\epsilon_1 x^T(t) (L \otimes P) x(t)}_{V_1} + \underbrace{\sum_{\langle i,k \rangle \in \mathbb{I}} \frac{1}{2} \omega_{ik}(t) + \sum_{\langle i,k \rangle \in \mathbb{J}} (-\frac{1}{2}) \omega_{ik}(t)}_{V_2} \nonumber \\
& + \underbrace{\sum_{\langle i,k \rangle \in \mathbb{J}} \frac{2r_{ik}(t)}{\theta_{ik}} \xi_{ik}^2(t_p)}_{V_3} \nonumber \\
& + \underbrace{4\epsilon_1 \bar{T} \int_{-\bar{T}}^{0} \int_{t+r}^{t} \dot{x}^T(s) (L^2 \otimes PBB^T P) \dot{x}(s) dsdr}_{V_4}. \tag{15}
\end{align}
$$

Firstly, for the derivative of $V_1$, define $\tilde{x}(t) = (U \otimes I_n)x(t)$ and $\mathbf{dz}_{\sqrt{\omega(t)}}(\xi(t_p)) = \xi(t_p) - \sigma_{\sqrt{\omega(t)}}(\xi(t_p))$, then there hold $\sigma_{\sqrt{\omega(t)}}(\xi(t_p)) = -(L \otimes B^T P)x(t_p) - \mathbf{dz}_{\sqrt{\omega(t)}}(\xi(t_p))$ and $(L \otimes B^T P)x(t) = (U^T \Lambda \otimes B^T P)\dot{x}(t) \in \mathbb{R}^{mN}$. Combining Young inequality [25], and recalling the definition of $\mathbb{I}$ and $\mathbb{J}$ given at the beginning of the proof, one has

$$
\begin{equation}
\begin{aligned}
\dot{V}_1 \le \epsilon_1 \left\{ \sum_{i=2}^{N} \lambda_i \tilde{x}_i(t)^T \left( PA + A^T P - \frac{5}{4} \lambda_i PBB^T P \right) \tilde{x}_i(t) \right. \\
\left. \qquad + \sum_{\langle i,k \rangle \in \mathbb{I}} \left( \frac{2\theta_{ik}}{r_{ik}(t)} + 4 \right) \omega_{ik}(t) \right. \\
\left. \qquad + \sum_{\langle i,k \rangle \in \mathbb{J}} \left( 4 + \frac{8r_{ik}(t)}{\theta_{ik}} \right) \xi_{ik}^2(t_p) + 4\zeta^T(t)\zeta(t) \right\}.
\end{aligned}
\tag{16}
\end{equation}
$$

Continuously, For the derivative of $V_2$, substitute (7), one obtains

$$
\begin{equation}
\begin{aligned}
\dot{V}_2 \le & -\frac{1}{2} \sum_{\langle i,k \rangle \in \mathbb{I}} \theta_{ik} \omega_{ik}(t) + \sum_{\langle i,k \rangle \in \mathbb{I}} \frac{\theta_{ik}}{4r_{ik}(t)} r_{ik}(t) \omega_{ik}(t) \\
& - \sum_{\langle i,k \rangle \in \mathbb{J}} \left(-\frac{1}{2}\right) \theta_{ik} \omega_{ik}(t) - \sum_{\langle i,k \rangle \in \mathbb{J}} \frac{1}{2} r_{ik}(t) \xi_{ik}^2(t_p).
\end{aligned}
\tag{17}
\end{equation}
$$

Thirdly, in view of the definition of $\frac{r_{ik}(t)}{\theta_{ik}}$ in (10), which is a constant in $t \in [t_p, t_{p+1})$, and according to the equation $\|\sigma_{\sqrt{\omega(t)}}(\xi(t_p))\|^2 \le \|\xi(t_p)\|^2$, one obtains

$$
\dot{V}_3 \le 4 (\|A\| + N \|BB^T P\|) \sum_{\langle i,k \rangle \in \mathbb{J}} \frac{r_{ik}(t)}{\theta_{ik}} \xi_{ik}^2(t_p). \quad (18)
$$

Finally, for the derivative of $V_4$, there is

$$
\begin{align}
& \frac{d}{dt} \left( \bar{T} \int_{-\bar{T}}^{0} \int_{t+r}^{t} \dot{x}^T(s) (L^2 \otimes PBB^T P) \dot{x}(s) dsdr \right) \nonumber \\
& = \bar{T}^2 \dot{x}^T(t) (L^2 \otimes PBB^T P) \dot{x}(t) \tag{19} \\
& \quad - \bar{T} \int_{t-\bar{T}}^{t} \dot{x}^T(s) (L^2 \otimes PBB^T P) \dot{x}(s) ds. \nonumber
\end{align}
$$

Then adopting (13), Young inequality in [25] and $t - t_p = T \le \bar{T}$ as well as Lemma 2, we can get

$$
\begin{align}
\dot{V}_4 \le {}& 4\epsilon_1 (2\bar{T}^2 \|A\|^2 x^T(t) (L^2 \otimes PBB^T P)x(t) \nonumber \\
& + 2\bar{T}^2 N^2 \|B^T PB\|^2 \sum_{\langle i,k \rangle \in \mathbb{I}} \omega_{ik}(t) + 2\bar{T}^2 N^2 \|B^T PB\|^2 \nonumber \\
& \qquad \sum_{\langle i,k \rangle \in \mathbb{J}} \frac{2r_{ik}(t)}{\theta_{ik}} \xi_{ik}^2(t_p) - \zeta^T(t)\zeta(t)). \tag{20}
\end{align}
$$

Combined with (16), (17), (18) and (20), it obtains that

$$
\begin{equation}
\begin{aligned}
\dot{V} \le \epsilon_1 \left\{ \sum_{i=2}^{N} \lambda_i \tilde{x}_i(t)^T \left( PA + A^T P - \left( \frac{5}{4} - 8\bar{T}^2 \|A\|^2 \right) \right. \right. \\
\left. \left. \lambda_i PBB^T P \right) \tilde{x}_i(t) \right\} - \sum_{\langle i,k \rangle \in \mathbb{I}} \varphi_{ik}(t) \omega_{ik}(t) && (21) \\
- \sum_{\langle i,k \rangle \in \mathbb{I}} (-\frac{1}{2}) \theta_{ik} \omega_{ik}(t) - \sum_{\langle i,k \rangle \in \mathbb{J}} \rho_{ik}(t) \xi_{ik}^2(t_p),
\end{aligned}
\end{equation}
$$

where

$$
\begin{align*}
\varphi_{ik}(t) &= \frac{1}{2}\theta_{ik} - \frac{\theta_{ik}}{4r_{ik}(t)}r_{ik}(t) - \left(\frac{2\theta_{ik}}{r_{ik}(t)} + 4\right) \epsilon_1 \\
&\quad - 8\epsilon_1 \bar{T}^2 N^2 \|B^T PB\|^2, \\
\rho_{ik}(t) &= \frac{1}{2}r_{ik}(t) - 4 \left( \|A\| + N \|BB^T P\| \right) \frac{r_{ik}(t)}{\theta_{ik}} \\
&\quad - \left( 4 + \frac{8r_{ik}(t)}{\theta_{ik}} \right) \epsilon_1 - 16\epsilon_1 \bar{T}^2 N^2 \|B^T PB\|^2 \frac{r_{ik}(t)}{\theta_{ik}}.
\end{align*}
$$

Substitute (5), $\bar{T}$ in (11) and $\theta_{ik} > 2 + 8 (\|A\| + N \|BB^T P\|)$ into the above equation, there exists $\epsilon_1 \le \frac{r_{ik}(t)}{4\theta_{ik} + 8r_{ik}(t) + 16\bar{T}^2 N^2 \|B^T PB\|^2 r_{ik}(t)}$ satisfying

$$
\begin{equation}
\begin{aligned}
\dot{V} \le {}& -2\epsilon_1 \left(1 - \cos\left(\frac{\pi}{N}\right)\right) \|x\|^2 - \sum_{\langle i,k \rangle \in \mathbb{I}} \varphi_{ik}(t) \omega_{ik}(t) \\
& - \sum_{\langle i,k \rangle \in \mathbb{J}} \left(-\frac{1}{2}\right) \theta_{ik} \omega_{ik}(t) - \sum_{\langle i,k \rangle \in \mathbb{J}} \rho_{ik}(t) \xi_{ik}^2(t_p),
\end{aligned}
\tag{22}
\end{equation}
$$

with $\varphi_{ik}(t) > 0$, $\rho_{ik}(t) > 0$.

In view of $V$ in (15) and $\dot{V}$ in (22), one can get

$$
\dot{V} \le -\eta(V_1 + V_2 + V_3), \quad (23)
$$

which indicates that $V$ is asymptotically stable with

$$
\eta = \min \left\{ \frac{2 \left(1 - \cos\left(\frac{\pi}{N}\right)\right)}{N \|P\|}, 2\varphi_{ik}(t), \theta_{ik}(t), \frac{\rho_{ik}(t)}{2 \frac{r_{ik}(t)}{\theta_{ik}}} \right\}
$$

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

From the construction of $V(t)$, one has

$$
\begin{align}
V(t_{p+1}^-) &= V_1(t_{p+1}) + V_4(t_{p+1}) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2} \omega_{ik}(t_{p+1}) \right. \nonumber \\
&\qquad \left. + \max \left\{ \frac{2r_{ik}(t_p)}{\theta_{ik}} \xi_{ik}^2(t_p) - \omega_{ik}(t_{p+1}), 0 \right\} \right), \nonumber \\
V(t_{p+1}^+) &= V_1(t_{p+1}) + V_4(t_{p+1}) + \sum_{i=1}^{N} \sum_{k=1}^{m} \left( \frac{1}{2} \omega_{ik}(t_{p+1}) \right. \nonumber \\
&\qquad \left. + \max \left\{ \frac{2r_{ik}(t_p)}{\theta_{ik}} \xi_{ik}^2(t_p) - \omega_{ik}(t_{p+1}) \right. \right. \nonumber \\
&\qquad \left. \left. + \left( \frac{2r_{ik}(t_{p+1})}{\theta_{ik}} \xi_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}} \xi_{ik}^2(t_p) \right), 0 \right\} \right). \tag{24}
\end{align}
$$

By the method of contradiction, suppose that $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}} \xi_{ik}^2(t_{p+2}) > \frac{2r_{ik}(t_{p+1})}{\theta_{ik}} \xi_{ik}^2(t_{p+1}) > \frac{2r_{ik}(t_p)}{\theta_{ik}} \xi_{ik}^2(t_p) > 0$, there must be an interval $[t_p, t_{p+1}]$ that makes $\begin{cases} \frac{2r_{ik}(t_p)}{\theta_{ik}} \xi_{ik}^2(t_p) < \Delta_{ik}^2, & \text{where } \frac{r_{ik}(t_p)}{\theta_{ik}} = \frac{1}{2}, \frac{r_{ik}(t_{p+1})}{\theta_{ik}} \le \frac{1}{2}, \\ \frac{2r_{ik}(t_{p+1})}{\theta_{ik}} \xi_{ik}^2(t_{p+1}) = \Delta_{ik}^2, & \text{where } \frac{r_{ik}(t_p)}{\theta_{ik}} = \frac{1}{2}, \frac{r_{ik}(t_{p+1})}{\theta_{ik}} \ge \frac{1}{2}, \end{cases}$
and $\xi_{ik}^2(t_p) < \xi_{ik}^2(t_{p+1}) < \xi_{ik}^2(t_{p+2})$. According to the relationship of parameters in (10), one has $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}}\xi_{ik}^2(t_{p+2}) = \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\xi_{ik}^2(t_{p+1}) = \Delta_{ik}^2$. That's the opposite of $\frac{2r_{ik}(t_{p+2})}{\theta_{ik}}\xi_{ik}^2(t_{p+2}) > \frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\xi_{ik}^2(t_{p+1})$. Thus, we have $\frac{2r_{ik}(t_{p+1})}{\theta_{ik}}\xi_{ik}^2(t_{p+1}) - \frac{2r_{ik}(t_p)}{\theta_{ik}}\xi_{ik}^2(t_p) \le 0$, leading to $V(t_{p+1}^+) \le V(t_{p+1})$.

Taking all the above proofs into account, we can conclude that $\lim_{t \to \infty} \|x_i(t) - x_j(t)\| = 0, \forall i, j = 1, \dots, N$. $\square$

**Remark 1.** According to literature [20], the imposition of more stringent constraints on the system or controller obviates the requirement for a specific choice of sampling period. The extremely special system characterized by the zero norm of system matrix A in Theorem 1 of this paper clearly demonstrates this.

## V. NUMERICAL SIMULATION

To explain the effectiveness of the sampled-data input saturation reconstruction algorithm, system (2) of a network with $N = 6$ agents is adopted for simulation with the matrices satisfying Assumption 1 given by

$$A = \begin{bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}. \quad (25)$$

The connected communication topology associated with $N = 6$ agents is described by the Laplacian matrix

$$L = \begin{bmatrix} 2 & -1 & 0 & 0 & 0 & -1 \\ -1 & 3 & -1 & 0 & 0 & -1 \\ 0 & -1 & 3 & -1 & -1 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & -1 & -1 & 3 & -1 \\ -1 & -1 & 0 & 0 & -1 & 3 \end{bmatrix}.$$

The inherent heterogeneous saturation levels are chosen as $\Delta_{11} = 1, \Delta_{12} = 1.5, \Delta_{21} = 2, \Delta_{22} = 2.5, \Delta_{31} = 3, \Delta_{32} = 3.5, \Delta_{41} = 4, \Delta_{42} = 4.5, \Delta_{51} = 5, \Delta_{52} = 5.5, \Delta_{61} = 6, \Delta_{62} = 6.5$. Solving algebraic Riccati equation (5) with $\mu = 1$, it yields

$$P = \begin{bmatrix} 2.8962 & -1.2674 & 1.3769 & -0.5067 \\ -1.2674 & 4.8470 & -0.6157 & 1.7267 \\ 1.3769 & -0.6157 & 2.3049 & 0.2894 \\ -0.5067 & 1.7267 & 0.2894 & 2.7325 \end{bmatrix}.$$

Moreover, following Theorem 1, we have $\underline{T} = 0.001, \overline{T} = 0.1093$ and we choose $T = 0.1$. 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 $\omega_{ik}$ are all chosen to be 1. By simple calculation, we choose $\theta_{ik} = 360$ for $i = 1, \dots, 6, k = 1, 2$.

As shown in Figure 1, under the steering of $u_i(t)$ in (6), the state of $\xi_i(t) = -B^T P \sum_{j=1, j \neq i}^N a_{ij}(x_i(t) - x_j(t))$ can converge to 0 which indicates the consensus of the MAS. Figure 2 depicts its trajectory for $\frac{r_{ik}(t)}{\theta_{ik}}$ defined in (10). As it is shown in Figure 2, $\frac{r_{ik}(t)}{\theta_{ik}}$ is a constant in $t \in [t_p, t_{p+1})$ and turns to $\frac{1}{2}$ as time goes by. This is because that $\xi_{ik}^2(t_p)$ would be very large at the beginning due to the global selection of the initial state, which leads to $\frac{\Delta_{ik}^2}{\xi_{ik}^2(t_p)}$, but as time goes to infinity, $\xi_{ik}(t_p)$ approaches to 0 so that $\frac{r_{ik}(t)}{\theta_{ik}} = \frac{1}{2}$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Convergence of $\xi_i(t)$ in which $x_i(t)$ is steered by (2) with $u_i(t)$ in (6).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Trajectories (12 in total) of $\frac{r_{ik}(t)}{\theta_{ik}}$ defined in (10).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Trajectories of $u_i(t)$ in (6).</figcaption></figure>

Figure 3 displays the trajectories of $u_i(t)$ in (6), where $|u_{ik}(t)| \le \Delta_{ik}$ 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 $\sqrt{\omega_{ik}(t)}$ are given in Figure 4. As shown in this figure, $\sqrt{\omega_{ik}(t)} \le \Delta_{ik}$ can be guaranteed for $i = 1,...,6$ and $k = 1,2$, which verifies Lemma 3. In accordance with (7) and the global consensus verified by Figure 1 as time goes to infinity, $\sqrt{\omega_{ik}(t)}$ approaches to 0. The four figures support all the sampled-data state feedback based results.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Trajectories of $\sqrt{\omega_{ik}(t)}$ with $\omega_{ik}(t)$ defined in (7).</figcaption></figure>

## VI. CONCLUSIONS

This paper addresses the global consensus of the MASs, whose input saturation constraint is in a symmetric saturation but non-uniform across different components form. The sampled-data control framework is designed for realizing the result. However, the obtained upper bound of the sampling periods is a little conservative due to all agents share a common value. Thus, how to develop approach to reduce the number of sampling is worth of further investigation.

## REFERENCES

[1] A. Amirkhani and A. H. Barshooi, "Consensus in multi-agent systems: a review," *Artificial Intelligence Review*, vol. 55, no. 5, pp. 3897-3935, 2022.

[2] P. Li, J. Lam, R. Lu, and H. Li, "Variable-parameter-dependent saturated robust control for vehicle lateral stability," *IEEE Transactions on Control Systems Technology*, vol. 30, no. 4, pp. 1711-1722, 2021.

[3] C. Song, L. Liu, G. Feng, Y. Fan, and S. Xu, "Coverage control for heterogeneous mobile sensor networks with bounded position measurement errors," *Automatica*, vol. 120, p. 109118, 2020.

[4] H. Dong, Z. Wang, J. Lam, and H. Gao, "Distributed filtering in sensor networks with randomly occurring saturations and successive packet dropouts," *International Journal of Robust and Nonlinear Control*, vol. 24, no. 12, pp. 1743-1759, 2014.

[5] Y. Xie and Z. Lin, "Global optimal consensus for multi-agent systems with bounded controls," *Systems & Control Letters*, vol. 102, pp. 104-111, 2017.

[6] H. Su, M. Z. Chen, X. Wang, and J. Lam, "Semiglobal observer-based leader-following consensus with input saturation," *IEEE Transactions on Industrial Electronics*, vol. 61, no. 6, pp. 2842-2850, 2013.

[7] X. Wang, H. Su, X. Wang, and G. Chen, "Robust semiglobal swarm tracking of coupled harmonic oscillators with input saturation and external disturbance," *International Journal of Robust and Nonlinear Control*, vol. 28, no. 5, pp. 1566-1582, 2018.

[8] J. Qian, X. Wang, G.-P. Jiang, and H. Su, "Observer-based semi-global containment of saturated multi-agent systems with uncertainties," *Journal of the Franklin Institute*, vol. 358, no. 15, pp. 7740-7760, 2021.

[9] Q. Song, F. Liu, H. Su, and A. V. Vasilakos, "Semi-global and global containment control of multi-agent systems with second-order dynamics and input saturation," *International Journal of Robust and Nonlinear Control*, vol. 26, no. 16, pp. 3460-3480, 2016.

[10] H. J. Sussmann and Y. Yang, "On the stabilizability of multiple integrators by means of bounded feedback controls," in [*1991 Proceedings of the 30th IEEE Conference on Decision and Control*]. IEEE, 1991, pp. 70-72.

[11] A. T. Fuller, "In-the-large stability of relay and saturating control systems with linear controllers," *International Journal of Control*, vol. 10, no. 4, pp. 457-480, 1969.

[12] X. Wang, J. Qian, H. Su, X. Lu, and J. Lam, "Distributed global consensus of Itti mass with heterogeneous actuator saturation and communication noises," *SIAM Journal on Control and Optimization*, vol. 62, no. 3, pp. 1690-1716, 2024.

[13] X. He, Z. Wang, C. Gao, and D. Zhou, "Consensus control for multi-agent systems under asymmetric actuator saturations with applications to mobile train lifting jack systems," *IEEE Transactions on Industrial Informatics*, 2023.

[14] J. Qian, X. Wang, H. Su, and G.-P. Jiang, "Robust cooperative tracking of multiagent systems with asymmetric saturation actuator via output feedback," *International Journal of Robust and Nonlinear Control*, vol. 34, no. 1, pp. 506-529, 2024.

[15] X. Ge, Q.-L. Han, D. Ding, X.-M. Zhang, and B. Ning, "A survey on recent advances in distributed sampled-data cooperative control of multi-agent systems," *Neurocomputing*, vol. 275, pp. 1684-1701, 2018.

[16] W. Zhang, Y. Tang, T. Huang, and J. Kurths, "Sampled-data consensus of linear multi-agent systems with packet losses," *IEEE Transactions on Neural Networks and Learning Systems*, vol. 28, no. 11, pp. 2516-2527, 2017.

[17] Y. Zhang and Y.-P. Tian, "Allowable sampling period for consensus control of multiple general linear dynamical agents in random networks," *International Journal of Control*, vol. 83, no. 11, pp. 2368-2377, 2010.

[18] W. Liu and J. Huang, "Leader-following consensus for linear multiagent systems via asynchronous sampled-data control," *IEEE Transactions on Automatic Control*, vol. 65, no. 7, pp. 3215-3222, 2019.

[19] G. Zhao and C. Hua, "Leader-following consensus of multiagent systems via asynchronous sampled-data control: A hybrid system approach," *IEEE Transactions on Automatic Control*, vol. 67, no. 5, pp. 2568-2575, 2021.

[20] Y. Zou, Z. Zuo, K. Xia, and M. V. Basin, "Saturated sampled-data distributed control for interval consensus of multi-agent systems," *IEEE Transactions on Signal and Information Processing over Networks*, vol. 8, pp. 1024-1036, 2022.

[21] H. Su, M. Z. Chen, and G. Chen, "Robust semi-global coordinated tracking of linear multi-agent systems with input saturation," *International Journal of Robust and Nonlinear Control*, vol. 25, no. 14, pp. 2375-2390, 2015.

[22] G. Casadei, D. Astolfi, A. Alessandri, and L. Zaccarian, "Synchronization of interconnected linear systems via dynamic saturation redesign," *IFAC-PapersOnLine*, vol. 52, no. 16, pp. 622-627, 2019.

[23] X. Wang, H. Su, M. Z. Chen, and X. Wang, "Observer-based robust coordinated control of multiagent systems with input saturation," *IEEE Transactions on Neural Networks and Learning Systems*, vol. 29, no. 5, pp. 1933-1946, 2017.

[24] Y. Wu, H. Su, P. Shi, Z. Shu, and Z.-G. Wu, "Consensus of multiagent systems using aperiodic sampled-data control," *IEEE Transactions on Cybernetics*, vol. 46, no. 9, pp. 2132-2143, 2015.

[25] C. D. Meyer, *Matrix analysis and applied linear algebra*. SIAM, 2023.