In recent decades, multi-agent systems (MASs) have been the subject of extensive research, resulting in substantial achievements on the field. MASs are utilized in many practical domains, including traffic management [1] and power systems [2,3]. These applications rely significantly on the consensus of MASs, which involves the states of agents converging to a desired state by the neighboring information [4]. Extensive studies are conducted on the consensus for MASs with integer-order differential models. However, integer-order models are inadequate for accurately representing some non-classical phenomena in various physical systems.

As the advancement of fractional calculus theory, fractional-order systems (FOSs) and fractional-order MASs (FOMASs) have emerged as a significant direction. Many achievements have been made in integer-order systems [5,6,7], but the utilization of fractional derivatives allows for a more comprehensive understanding of the characteristics of materials and systems exhibiting power-law, nonlocal, or long-term memory. FOSs offer enhanced capabilities for modelling and analysing complex systems, e.g., electrical systems [8,9], economic systems [10], motion models [11], and biological models [12,13]. The stability of control systems is fundamental problem, certainly including those involving FOSs. It is not possible to derive the stability criteria of FOSs directly from those of integer-order systems. Fortunately, in [14], authors establish the LMI-based stability criteria and design a method for robust feedback state stabilization control for commensurate FOSs with $\alpha \in (0,1)$ and $\alpha \in [1,2)$. In [15], a necessary and sufficient condition in unified LMIs formulation is provided to ensure the stability of FOSs within $\alpha \in (0,2)$. The aforementioned study makes a contribution to the consensus problem of FOMASs. In [16], the consensus of FOMASs with time delay is studied. Using the Lyapunov method, some consensus criteria are proposed to guarantee the consensus. Focusing on singular systems, the authors in [17] investigate the observer-based consensus problem of singular FOMASs. Then, the paper provides the corresponding control protocol and the calculation method of gain matrices. In [18], the distributed fixed-time consensus for FOMASs with a dynamic virtual leader under external disturbances is investigated, and a sliding-mode control protocol is designed. Meanwhile, the singular perturbation FOMASs are modeled and studied in [19], which provides a sufficient condition for consensus. Nevertheless, the consensus problem of FOMASs in these results is all based on state feedback. In most cases, only the measurement output of the agents is available, which indicates that the above method has certain limitations.

In comparison to the state feedback, the control via output feedback is a challenging problem, largely due to the affect of measurement matrices. Furthermore, the consensus of FOMASs via output feedback is a highly intricate issue. The stability criteria of FOSs with $\alpha \in \left(0,1\right)$ comprise two matrix variables, whereas in $\alpha \in \left[1,2\right)$, the matrix variable is related to the order number. In [20,21,22,23], static output feedback control is employed for FOSs, and a matrix exchange condition is utilized to integrate matrix variable with gain matrices. This approach utilizes singular value decomposition (SVD) of the measurement matrices. Nevertheless, this approach imposes constraints on the form of matrix variables and tends to be conservative. Authors in [24] also adopt some strong assumptions to get a feasible solution, which brings conservatism. Similarly, in the output feedback consensus of FOMASs, the limitation often exist. In [20], sufficient conditions are provided for the leader-following consensus of singular FOMASs with $\alpha \in \left(0,2\right)$. The authors in [25] also do the analogous study, and the SVD method is always indispensable. To complicate matters, disturbances persist in actual systems, and the output information transmitted between the agents contains the disturbances.

For FOMASs, the $H_{\infty }$ control method provides substantial benefits in addressing system uncertainty, robustness optimization, and other pertinent issues. The paper [26] derives the bounded real lemma for FOSs and establishes the foundation for $H_{\infty }$ control. In [27], a proposal is made to extend the application of $H_{\infty }$ control method from integer-order systems to FOSs. Robust fault-tolerant $H_{\infty }$ control for FOSs with actuator faults and uncertainties is addressed in [28] through the design of an output feedback controller. For singular FOSs, a state feedback control strategy is presented that guarantees the prescribed $H_{\infty }$ performance in [29]. These works form the foundation o the $H_{\infty }$ consensus of FOMASs. In [30], the admissible consensus of fuzzy singular FOMASs is considered, a sufficient condition for the system achieving admissible consensus while satisfying $H_{\infty }$ performance. The paper [31] investigates the $H_{\infty }$ consensus problem for discrete-time FOMASs. However, the relatively research is little. The output feedback $H_{\infty }$ consensus remains a challenging field. Meanwhile, the control input saturation is a common feature of practical engineering systems, due to physical limitations [32,33,34]. This renders the output feedback consensus for FOMASs a complex process.

The discussion provides the impetus for the ILMI algorithms towards the leader-following $H_{\infty }$ consensus of FOMASs with input saturation via output feedback. The contributions are as following:

(i) Based on the real bound lemma of FOSs, sufficient conditions for output feedback $H_{\infty }$ consensus of FOMASs in $\alpha \in \left(0,1\right)$ and $\left[1,2\right)$ are provided. The proposed method adopts a holistic analytical perspective to the entire system, which differs from the decomposition of the entire MASs.

(ii) For solving the QMIs, the ILMI algorithms are provided, which propose a calculation method for initial values. Based on the stability region of FOSs, the iterative condition are designed to guarantee the consensus condition of FOMASs. This paper delves deeper into the issue of the input saturation, which is reframed as an LMI-based optimisation problem. The ILMI algorithms circumvent the necessity for matrix exchange conditions from the SVD method. No strong assumptions are required for feasible solutions, and it reduces the conservatism.

Notations: Given a matrix A, sym$\left(A\right)$ denotes $A^T+A$, where $A^T$ is the transpose of A, and her$\left(A\right)=A^{*}+A$, where $A^{*}$ is conjugate transpose of matrix A. $A>0$ and $A<0$ represent that A is positive definite and negative definite, respectively. Then $A⩾0$ denotes that A is negative semi-definite and symbol ★ stands for the symmetric part. The Kronecker product is represented by ⊗. arg(·) denotes the argument of complex numbers, and spec(A) indicates the spectrum of matrix A. ${\mathbb{R}}^{n\times m}$ stands for sets of $n\times m$ real matrices. diag(·) represents a diagonal matrix. With $\alpha \in (0,2)$, denote $a=sin\left(\alpha {\displaystyle \frac{\pi }{2}}\right)$, $b=cos\left(\alpha {\displaystyle \frac{\pi }{2}}\right)$.

An MAS consists of $\mathcal{N}$ followers represented by the undirected graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E},\mathcal{A}\right)$, where $\mathcal{V}=\left\{1,2,\dots ,\mathcal{N}\right\}$ presents a set of $\mathcal{N}$ followers, and $\mathcal{E}=\left\{\left(i,j\right):i,j\in \mathcal{V}\right\}$ is the edge set. The adjacency matrix is $\mathcal{A}={\left[a_{ij}\right]}_{\mathcal{N}\times \mathcal{N}}$, and if $\left(i,j\right)\in \mathcal{E}$, $a_{ij}>0$, otherwise $a_{ij}=0$ .

Furthermore, denote the communication graph between the leader and followers as $\tilde{\mathcal{G}}$. Define $\mathrm{diag}\left(h_1,h_2,\cdots ,h_{\mathcal{N}}\right)$ for representing the communication. If follower i receives the information from the leader, then $h_i>0$, otherwise $h_i=0$. The Laplacian matrix is defined as $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{\mathcal{N}\times \mathcal{N}}$, where

Consider the FOMAS under actuator saturation, and the $\mathcal{N}$ followers are described by where $x_i\left(t\right)\in {\mathbb{R}}^n$, $u_i\left(t\right)\in {\mathbb{R}}^{n_u}$, $y_i\left(t\right)\in {\mathbb{R}}^m$, $z_i\left(t\right)\in {\mathbb{R}}^{n_z}$, and $w_i\left(t\right)\in {\mathbb{R}}^{n_w}$ denote the state, control input, measured output, controlled output, and disturbances, respectively; A, B, $D_1$, $D_2$, $D_3$, C, and $C_y$ are constant real matrices; $D^{\alpha }$ represents the Caputo fractional derivative of $f\left(t\right)$ as and $\mathsf{\Gamma }\left(·\right)$ is the Euler Gamma function; the saturation function sat$(·)$ for $u\left(t\right)$ is denoted as The leader is described by where $x_0\left(t\right)\in {\mathbb{R}}^n$ is the state, and $y_0\left(t\right)\in {\mathbb{R}}^n$ is the output.

For achieving consensus of the FOMAS in (1) and (2), a distributed consensus protocol is carried out by

The initial values are considered as $x_i\left(0\right)\in {\mathbb{R}}^n$. Denote the state trajectory as $T(x_i\left(0\right),t)$, and the domain of attraction is For $F\in {\mathbb{R}}^{n_u\times m}$, let where $f_s$ denotes the s-th row of the matrix F; $\mathcal{L}\left(F\right)$ denotes the region where $F{C}_{y}x\left(t\right)$ does not saturate.

With the consensus protocol, the FOMAS in (1) and (2) is written as where

$x\left(t\right)={\left[\begin{array}{ccc}{x}_1^T\left(t\right)& \cdots & x_{\mathcal{N}}^T\left(t\right)\end{array}\right]}^T,$$z\left(t\right)={\left[\begin{array}{ccc}{z}_1^T\left(t\right)& \cdots & z_{\mathcal{N}}^T\left(t\right)\end{array}\right]}^T,$$w\left(t\right)={\left[\begin{array}{ccc}{w}_1^T\left(t\right)& \cdots & w_{\mathcal{N}}^T\left(t\right)\end{array}\right]}^T,$$\tilde{K}=I_{\mathcal{N}}\otimes {\displaystyle \sum _{l=1}^{2^{n_u}}}{h}_l({\Gamma }_{l}K+{\Gamma }_l^{-}F),\mathcal{H}=\mathcal{L}+\mathrm{diag}(h_1,h_2,\cdots ,h_{\mathcal{N}})$.

This section introduces the iterative algorithms for solving $H_{\infty }$ consensus of FOMASs via output feedback.