Iterative Learning Fault Diagnosis of Fractional-Order Nonlinear Multi-Agent Systems with Initial State Learning and Switching Topology

Abstract

This paper proposes an iterative learning framework for a class of fractional-order nonlinear multi-agent systems operating under directed iteration-varying switching topologies. To suppress trial-to-trial fluctuations in initial states, a P-type initial condition learning mechanism is integrated into the update law, enabling each agent to actively compensate for its own startup offset in each iteration. The study designs a distributed iterative learning protocol using only local neighbor information, and this protocol can simultaneously achieve fault estimation and diagnosis. By constructing a fractional-order system model and adopting the contraction-mapping analysis method, sufficient conditions are derived in this paper, which guarantee that both the fault error and initial condition error converge asymptotically to zero as the number of iterations approaches infinity. The proposed scheme, based on iterative learning fault estimation, can effectively handle unknown nonlinearities without relying on an accurate system model. Numerical simulation results further verify the effectiveness of the designed fault observer in achieving fault estimation.

1. Introduction

Multi-agent systems (MASs) are playing an increasingly crucial role in cutting-edge fields such as modern control theory and artificial intelligence [1,2]. Traditionally, integer-order multi-agent systems have long dominated research and applications [3]. They are modeled based on integer-order calculus, and through agent collaboration, have achieved remarkable results in scenarios like robotic collaboration and intelligent transportation [4,5]. However, with the deepening research on complex systems, the limitations of integer-order systems have become prominent: integer-order models cannot accurately characterize the memory characteristics and specific state transition features of practical systems, and they are also insufficient in utilizing historical iteration information and addressing non-ideal repetitive tasks [6]. Against this backdrop, fractional calculus has opened up a new path for the modeling and analysis of MASs [7]. Fractional-order multi-agent systems (FOMASs) integrated with this theory can better solve complex system problems, gradually making fractional-order dynamics a research focus in relevant fields [8,9]. Although FOMASs can accurately characterize the memory properties and state changes of complex systems, they still face two key challenges when repeatedly performing periodic tasks. One is the accumulation of initial state deviations, and the other is the difficulty in achieving fast convergence of inter-iteration errors [10]. However, iterative learning control (ILC) excels at leveraging historical iteration information to optimize control strategies and can precisely address such error correction issues in repeated tasks [11]. Thus, it has become a critical technical approach for FOMASs to achieve high-precision cooperative control.

ILC was first proposed by Arimoto et al. in 1984 [12]. Since then, it has promoted innovations in system theories and methods within the field of finite-dimensional systems, and a comprehensive convergence analysis system has now been established [13]. Specifically, ILC has demonstrated significant advantages in handling initial state disturbances. To tackle the challenge of random initial state deviations in discrete-time systems, a robust ILC framework integrated with an adaptive compensation mechanism was proposed in [14], and this framework significantly enhanced the algorithm’s robustness against initial disturbances. Meanwhile, for linear motors with random initial positions, a segmented ILC strategy was developed in [15], which effectively solved the problem of accurate trajectory tracking during their continuous motion. ILC is a precise tracking control strategy specifically designed for systems that repeatedly perform the same tasks within fixed time intervals [16]. Its core advantage lies in the ability to leverage control input and tracking error information from historical iteration processes to continuously optimize tracking performance, without relying on an accurate system model throughout the entire process. In the frontier research on distributed control of MAS, the integration of ILC and distributed cooperation has further driven technological breakthroughs. One of the achievements is the construction of a distributed ILC framework, which achieves high-precision tracking of generalized MASs by accurately characterizing the system coupling characteristics under both fixed topologies and iteration-varying topologies [17]. Moreover, to overcome the reliance of traditional ILC on system models, scholars have proposed robust ILC methods, offering new directions for the research on linear system control [18]. However, in practical engineering scenarios, unknown faults frequently arise due to complex factors such as inherent system uncertainties, external environmental disturbances, and output constraints [19]. To eliminate the impact of these faults, the design of state observers and fault estimators is necessary; the consensus problem in such networks is termed fault diagnosis, whose objective is to ensure faults converge to zero within a finite time to realize system error convergence [20].

Breakthroughs have been achieved in multiple aspects of fault diagnosis and fault-tolerant control research. For photovoltaic modules, an efficient fault diagnosis method was constructed by integrating multi-angle feature expansion and visual neural networks [10]. The technical pathway for fuel cell fault diagnosis was innovated using ILC [21]. For time-varying faults of underwater thrusters, an adaptive iterative learning observer scheme was proposed to solve the coupling problem of fault isolation and dynamic compensation in nonlinear systems [22]. Notably, most existing studies focus on control problems in the continuous-time domain [23,24,25], yet practical engineering scenarios demand higher precision for control within finite time intervals [26]. In response, the iterative convergence rate of nonlinear discrete systems has been significantly improved through the ILC framework [27]. Meanwhile, researchers have also developed an efficient strategy optimization scheme [26], providing a new method for the control of nonlinear processes. To address the key issue of initial state errors, its control accuracy has been improved by an order of magnitude with the support of an adaptive error quantization mechanism [28]. From the perspective of a broader range of system types, the application of ILC in the field of fault diagnosis continues to expand. The application of ILC in the field of fault diagnosis continues to expand, and current research on FONMASs is also advancing in depth.

For FONMASs with stochastic repetitive dynamics involving Brownian motion, an ILC-based fault diagnosis theory has been established. By designing a fault estimation observer, deriving convergence conditions, and ensuring the stability of stochastic inputs in the closed-loop system, high-precision control of the system is ultimately achieved [29]. To address the time-varying fault control problem of autonomous underwater vehicle thrusters, a study proposes an adaptive iterative learning observer method [25]. With this method, direct reconstruction of the thrusters’ time-varying fault efficiency factors and estimation of the total disturbance term are made possible. It not only significantly improves tracking accuracy but also greatly reduces tracking errors. For distributed parameter systems described by second-order hyperbolic partial differential equations, fault-tolerant control was achieved through the employment of an adaptive iterative learning strategy [30]. Coordinated suppression of multiplicative and additive faults was realized via a compensation framework, and successful application of the strategy was achieved in active noise reduction scenarios. The application scenarios of ILC are further expanding. Beyond fault diagnosis and fault-tolerant control, its value in new research directions is emerging: a notable study applied ILC to the formation control of nonholonomic mobile robots [31], which not only solved the challenging problem of dynamic tracking for time-varying trajectories but also enriched the theoretical systems of ILC and multi-agent cooperative control via cross-domain technology transfer. While remarkable progress has been made in existing studies regarding the optimization of control strategies and the enhancement of MAS fault-tolerant performance, obvious deficiencies still exist in current research on FONMASs.

As noted in relevant works [25,31,32], the inherent memory effect of fractional-order models and its profound impact on fault diagnosis are often overlooked by traditional methods [33,34]. Moreover, when tasks are repeatedly executed within a finite time interval, the history-dependent property of fractional-order operators leads to the accumulation of initial state deviations. This accumulation makes it difficult for traditional iterative learning control to maintain stable fault estimation accuracy. To address this deficiency, an integrated iterative learning framework that combines initial state learning and fault diagnosis is proposed in this paper. The main contributions are as follows:

1.

In contrast to the iterative learning-based fault diagnosis studies for integer-order multi-agent systems [35,36], the system considered in this paper is a FONMAS, and a model-free iterative learning fault diagnosis framework is proposed herein.

2.

To synchronize with iterative learning, a novel state observer integrating real-time accurate estimation and fault diagnosis is designed. The observer-based protocol uses local information, adapts to topologies, and boosts fault diagnosis for the FONMASs.

3.

Under the fault diagnosis framework of iterative learning, the contraction mapping method achieves fault diagnosis by constructing an fault observer, with rigorous convergence conditions and estimation consistency established under both fixed and iteration-varying topologies.

The remainder of this paper is organized as follows: Section 2 elaborates on the system description, the fault estimator, and the relevant theories required for the theoretical analysis. Section 3 presents the convergence proof for the system under fixed topology. Section 4 extends the analysis to fault diagnosis for systems with switching topology. Section 5 verifies the effectiveness of the method through numerical simulations. Section 6 summarizes the full paper and prospects future research directions. Theoretical analysis shows that the proposed method can ensure the asymptotic convergence of initial state errors and fault estimation errors. Simulation results confirm that this method exhibits superior performance in improving diagnostic accuracy.

Notations. ${\mathbb{R}}^n$ represents the Euclidean space with n dimensions. ${\mathbb{R}}^{n\times n}$ denotes the $n\times n$-dimensional matrix set. $I_n$ represents the $n\times n$-dimension identity matrix. ${(*)}^T$ indicates the transpose of a vector or matrix. $\mathrm{diag}(*)$ denotes a diagonal matrix. The symbols ‘⊗’ denotes the Kronecker product operators.

2. Preliminaries and Problem Statement

This section provides essential background knowledge to offer stronger support for the subsequent analysis of fault diagnosis presented in this paper.

2.1. Graph Theory

A multi-agent system is composed of N agents. The graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ is a communication topology graph, where $\mathcal{A}$ represents the adjacency matrix, $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ denotes the directed edge set, and $\mathcal{V}=\{1,2,\cdots ,N\}$ is a set of points that make up the agent system. $(v_j,v_i)\in \mathcal{E}$ indicates that agent i can receive the information of agent j. $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ denotes the adjacency matrix of graph $\mathcal{G}$, if $(v_j,v_i)\in \mathcal{E}$, $a_{ij}\ne 0$, otherwise $a_{ij}=0$. Due to the consideration of open-loop graphs, that is, $a_{ii}=0$. $\mathcal{D}=\mathrm{diag}\{d_1,d_2,\cdots ,d_N\}$ is the degree matrix and $d_i={\displaystyle \sum _{j=1}^N}\left|a_{ij}\right|$. The Laplacian matrix is defined as $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$ and $\mathcal{L}=\mathcal{D}-\mathcal{A}$. For the i-th follower, to account for access to reference fault information, define $b_i>0$ if agent i has direct access to such information; otherwise, set $b_i=0$. On this basis, define the leader–follower communication weight matrix as $\mathcal{B}=\mathrm{diag}\{b_1,b_2,\dots ,b_N\}$, where $\mathrm{diag}\{·\}$ denotes the diagonal matrix construction operation, meaning $\mathcal{B}$ is a diagonal matrix with $b_1$, $b_2$, …, $b_N$ as its diagonal elements in sequence.

Definition 1

([37]). For $t>0$ and $\alpha >0$, the Riemann–Liouville fractional integral of order α of a function h is defined by

$${\mathrm{I}}^{\alpha }h\left(t\right)\triangleq {\displaystyle \frac{1}{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}h\left(\tau \right)\mathrm{d}\tau ,$$

where $\mathsf{\Lambda }(·)$ denotes the Gamma function.

Definition 2

([38]). Let $m\in \mathbb{N}$ satisfy $m-1<\alpha <m$ and $t>t_0$. The Caputo fractional derivative of order α of a function h is defined by

$$\begin{array}{cc}\hfill \stackrel{C}{t_0}{\mathcal{D}}_t^{\alpha }h\left(t\right)& \triangleq {\mathrm{I}}^{m-\alpha }{\mathcal{D}}^{m}h\left(t\right)\hfill \\ & ={\displaystyle \frac{1}{\mathsf{\Lambda }(m-\alpha )}}{\int }_{t_0}^t{\displaystyle \frac{h^{\left(m\right)}\left(\tau \right)}{{(t-\tau )}^{\alpha -m+1}}}\mathrm{d}\tau \hfill \end{array}$$

Throughout this paper, the Caputo definition for fractional differentiation is adopted. For notational brevity, the following notation is used:

$${\mathcal{D}}_t^{\alpha }f\left(t\right)\equiv \stackrel{C}{t_0}{\mathcal{D}}_t^{\alpha }f\left(t\right).$$

Definition 3.

Let $z\left(t\right):[0,T]\to {\mathbb{R}}^n$ be a vector-valued function defined on the finite time interval $[0,T]$, where T denotes the duration of a single iterative task. For a given positive weighting coefficient $\lambda >0$, the λ-norm (exponentially weighted supremum norm) of $z\left(t\right)$ is defined as [39]:

$${\parallel z\parallel }_{\lambda }=\{\underset{t\in [0,T]}{sup}∥z\left(t\right)∥e^{-\lambda t}\}$$

where ${sup}_{t\in [0,T]}$ denotes the supremum (least upper bound) of the function over the interval $[0,T]$; $∥z\left(t\right)∥$ represents the Euclidean norm of the vector $z\left(t\right)$ at time t, i.e., $∥z\left(t\right)∥=\sqrt{z_1{\left(t\right)}^2+z_2{\left(t\right)}^2+\cdots +z_n{\left(t\right)}^2}$; and $e^{-\lambda t}$ is the exponential decay factor, which assigns smaller weights to the function values in the later stage of the time interval, enhancing the characterization of early convergence in iterative learning processes.

2.2. System Description and Fault Observation

Considering a fractional-order nonlinear multi-agent system composed of N agents. The dynamics of the i-th agent can be formulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }{x}_i\left(t\right)& =A{x}_i\left(t\right)+B{u}_i\left(t\right)+g(x_i\left(t\right),t)+D{f}_i\left(t\right),\hfill \\ \hfill y_i\left(t\right)& =C{x}_i\left(t\right),i\in \mathcal{V},\hfill \end{array}\right.\end{array}$$

(1)

where $\alpha$ denotes the order of the fractional order, $\alpha \in (0,1)$. $t\in [0,T]$ represents time variable. $g_i(·)$ is nonlinear function and $f_i(·)$ is an unknown nonlinear fault. A, B, C, and D are real matrix with appropriate dimensions. $y_i\left(t\right)\in {\mathbb{R}}^n$, $x_i\left(t\right)\in {\mathbb{R}}^n$ and $u_i\left(t\right)\in {\mathbb{R}}^m$ represent the output, the state and control input, respectively.

Thus, a novel iterative learning-based fault observer is designed for the FONMASs to achieve accurate fault diagnosis and ensure system stability.

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }{\widehat{x}}_{k,i}\left(t\right)& =A{\widehat{x}}_{k,i}\left(t\right)+B{u}_i\left(t\right)+g({\widehat{x}}_{k,i}\left(t\right),t)+D{\widehat{f}}_{k,i}\left(t\right)+L(y_i\left(t\right)-{\widehat{y}}_{k,i}\left(t\right)),\hfill \\ \hfill {\widehat{y}}_{k,i}\left(t\right)& =C{\widehat{x}}_{k,i}\left(t\right),\hfill \end{array}\right.$$

(2)

where k denoting the iteration index, ${\widehat{x}}_{k,i}\left(t\right)$, ${\widehat{y}}_{k,i}\left(t\right)$, $g\left({\widehat{x}}_{k,i}\left(t\right),t\right)$ and ${\widehat{f}}_{k,i}\left(t\right)$ represent the iterative estimates of $x_i\left(t\right)$, $y_i\left(t\right)$, $g\left(x_i\left(t\right),t\right)$ and $f_i\left(t\right)$, respectively. The observer gain matrix L is selected such that all eigenvalues of $A-LC$ lie strictly in the left half-plane of the complex plane.

By means of the proposed fault observer in this study, as the iteration index $k\to \infty$, the multi-agents achieve fault diagnosis within the finite time interval $t\in [0,T]$, that is,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}∥{\tilde{f}}_{k,i}\left(t\right)∥=\left|\right|f_i\left(t\right)-{\widehat{f}}_{k,i}\left(t\right)\left|\right|=0,i\in \mathcal{V}.\end{array}$$

Remark 1.

The state matrix $A\in {\mathbb{R}}^{n\times n}$ has full rank n with $rank\left(A\right)=n$ to ensure the non-singularity of the state transition dynamics, while the output matrix $C\in {\mathbb{R}}^{p\times n}$ has full column rank p with $rank\left(C\right)=p$ to guarantee that the output contains sufficient information for state reconstruction.

2.3. Iterative Learning for Fault Diagnosis of FONMAS

To simplify the expression, this section defines some variables

$$\begin{array}{cc}& {\tilde{x}}_{k,i}\left(t\right)=x_i\left(t\right)-{\widehat{x}}_{k,i}\left(t\right),{\tilde{y}}_{k,i}\left(t\right)=y_i\left(t\right)-{\widehat{y}}_{k,i}\left(t\right),\hfill \\ & {\tilde{g}}_{k,i}\left(t\right)=g(x_i\left(t\right),t)-g({\widehat{x}}_{k,i}\left(t\right),t),{\tilde{f}}_{k,i}\left(t\right)=f_i\left(t\right)-{\widehat{f}}_{k,i}\left(t\right).\hfill \end{array}$$

To achieve fault diagnosis through iterative learning in FONMASs, we define the system’s estimation error.

$$\begin{array}{cc}\hfill e_{k,i}\left(t\right)& =\sum _{j=1}^N{a}_{ij}\left[\left(y_j\left(t\right)-{\widehat{y}}_{k,j}\left(t\right)\right)-\left(y_i\left(t\right)-{\widehat{y}}_{k,i}\left(t\right)\right)\right]-b_i\left(y_i\left(t\right)-{\widehat{y}}_{k,i}\left(t\right)\right)\hfill \end{array}$$

2.4. Design of Fault Estimator Based on Iterative Learning

To drive the fault estimation error to converge, this paper introduces the following fault observer design:

$${\widehat{f}}_{k+1,i}\left(t\right)={\widehat{f}}_{k,i}\left(t\right)+\tilde{\mathsf{\Gamma }}{\mathcal{D}}_t^{\alpha }{e}_{k,i}\left(t\right),$$

where $\tilde{\mathsf{\Gamma }}$ represents learning gain. For a compact representation of the N-agent vectors, we adopt the following notations:

$$\begin{array}{cc}\hfill x\left(t\right)& ={\left[x_1^T\left(t\right),x_2^T\left(t\right),\dots ,x_N^T\left(t\right)\right]}^T,\hfill \\ \hfill u\left(t\right)& ={\left[u_1^T\left(t\right),u_2^T\left(t\right),\dots ,u_N^T\left(t\right)\right]}^T,\hfill \\ \hfill y\left(t\right)& ={\left[y_1^T\left(t\right),y_2^T\left(t\right),\dots ,y_N^T\left(t\right)\right]}^T,\hfill \\ \hfill f\left(t\right)& ={\left[f_1^T\left(t\right),f_2^T\left(t\right),\dots ,f_N^T\left(t\right)\right]}^T.\hfill \end{array}$$

Similarly, ${\tilde{x}}_k\left(t\right)$, ${\widehat{x}}_k\left(t\right)$, ${\widehat{y}}_k\left(t\right)$, ${\tilde{y}}_k\left(t\right)$, ${\widehat{f}}_k\left(t\right)$, ${\tilde{f}}_k\left(t\right)$, $g({\widehat{x}}_k\left(t\right),t)$, $g\left(x\right(t),t)$ and $g({\tilde{x}}_k\left(t\right),t)$ undergo vector lifting in the same way. Hence, the compact form of $e_{k,i}\left(t\right)$ can be written as:

$$e_k\left(t\right)=[-(\mathcal{L}+\mathcal{B})\otimes I_n]{\tilde{y}}_k\left(t\right),$$

(3)

among them, $I_n$ denotes the n-th order identity matrix and $e_k\left(t\right)=\left[e_{k,1}^T\left(t\right),e_{k,2}^T\left(t\right),\dots ,\right.$${\left.e_{k,N}^T\left(t\right)\right]}^T$.

Therefore, the fault observer can also be expressed in the following compact form:

$${\widehat{f}}_{k+1}\left(t\right)={\widehat{f}}_k\left(t\right)-[(\mathcal{L}+\mathcal{B})\otimes \tilde{\mathsf{\Gamma }}C]{\mathcal{D}}_t^{\alpha }{\tilde{x}}_k\left(t\right),$$

(4)

with $\tilde{\mathsf{\Gamma }}=\mathrm{diag}\{{\tilde{\mathsf{\Gamma }}}_1,{\tilde{\mathsf{\Gamma }}}_2,\cdots ,{\tilde{\mathsf{\Gamma }}}_N\}$.

2.5. Design of Initial Value Learning

To address the fault diagnosis problem in FONMASs, this paper make the following assumptions:

Assumption 1.

$g_i(·)$ represents the nonlinear function which satisfies the Lipschitz condition, i.e.,

$$\begin{array}{c}\hfill \forall x_1,x_2\in {\mathbb{R}}^n,\exists l_f>0: \parallel g_i\left(x_1\right)-g_i\left(x_2\right)\parallel ⩽l_f\parallel x_1-x_2\parallel .\end{array}$$

Assumption 2

([40]). The FONMASs defined by the parameter matrices (A, C) is observable, and both matrix A and matrix C are of full rank.

Assumption 3.

At the k-th iteration, each agent is initialized with a distinct initial state,

$${\widehat{x}}_{k,i}\left(0\right)\ne x_i\left(0\right),i\in \mathcal{V}.$$

To correct for initial-state discrepancies, a P-type learning update law for the initial condition is introduced as follows:

$$\begin{array}{c}\hfill {\widehat{x}}_{k+1,i}\left(0\right)={\widehat{x}}_{k,i}\left(0\right)+\mathsf{\Gamma }{e}_{k,i}\left(0\right),\end{array}$$

where Γ is a learning gain matrix for initial value learning.

After lifting the vector, one has

$$\begin{array}{c}\hfill {\widehat{x}}_{k+1}\left(0\right)={\widehat{x}}_k\left(0\right)+(I_N\otimes \mathsf{\Gamma })e_k\left(0\right),\end{array}$$

(5)

with $\mathsf{\Gamma }=\mathit{diag}\{{\mathsf{\Gamma }}_1,{\mathsf{\Gamma }}_2,\dots ,{\mathsf{\Gamma }}_N\}$.

Lemma 1

([41]). Suppose $\left\{a_k\right\}$, $\left\{b_k\right\}$ are two nonnegative real sequences satisfying

$$a_{k+1}\le \rho a_k+b_k,0\le \rho <1,$$

If $\underset{k\to \infty }{lim}{b}_k=0$, then $\underset{k\to \infty }{lim}{a}_k=0$.

Lemma 2

([42]). Let $t\in [0,T]$ and $\lambda >0$. Then, the following inequality holds:

$${\int }_0^t{\displaystyle \frac{e^{\lambda \tau }}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau <{\displaystyle \frac{e^{\lambda t}}{{\lambda }^{\alpha }}}\mathsf{\Lambda }\left(\alpha \right).$$

3. Main Results on Fault Diagnosis

This section presents the core research findings derived from theoretical analysis.

Fault Diagnosis of FONMASs with Fixed Topology

Lemma 3.

By choosing the constant gain matrix Γ so that the $\left|\right|I_N-(I_N\otimes \mathsf{\Gamma }C)\left|\right|<1$, it follows that

$${\displaystyle \underset{k\to \infty }{lim}\parallel {\tilde{x}}_k\left(0\right)\parallel =0.}$$

Proof. 

It follows from Equation (5) that

$$\begin{array}{cc}\hfill {\tilde{x}}_{k+1}\left(0\right)& =x\left(0\right)-{\widehat{x}}_{k+1}\left(0\right),\hfill \\ \hfill & =x\left(0\right)-{\widehat{x}}_k\left(0\right)-(I_N\otimes \mathsf{\Gamma })e_k\left(0\right),\hfill \end{array}$$

Further simplification can yield

$$\begin{array}{cc}\hfill {\tilde{x}}_{k+1}\left(0\right)& ={\tilde{x}}_k\left(0\right)-(I_N\otimes \mathsf{\Gamma })(I_N\otimes C){\tilde{x}}_k\left(0\right),\hfill \\ \hfill & =\left(I_{Nn}-(I_N\otimes \mathsf{\Gamma }C)\right){\tilde{x}}_k\left(0\right),\hfill \end{array}$$

(6)

Taking the norm of Equation (6), one has

$$\begin{array}{c}\hfill \left|\right|{\tilde{x}}_{k+1}\left(0\right)\left|\right|\le \left|\right|I_{Nn}-(I_N\otimes \mathsf{\Gamma }C)\left|\right|\left|\right|{\tilde{x}}_k\left(0\right)\left|\right|,\end{array}$$

(7)

Therefore, if $\left|\right|I_{Nn}-(I_N\otimes \mathsf{\Gamma }C)\left|\right|<1$, then

$$\underset{k\to \infty }{lim}\parallel {\tilde{x}}_k\left(0\right)\parallel =0.$$

Hence, the proof of Lemma 3 is complete. □

Remark 2.

The proposed P-type rule breaks this constraint via an active compensation mechanism: it utilizes the previous iteration’s initial error ${\tilde{x}}_{k-1}\left(0\right)$ and learning gain matrix Γ to dynamically update the current initial state ${\widehat{x}}_{k+1}\left(0\right)={\widehat{x}}_k\left(0\right)+(I_N\otimes \mathsf{\Gamma })e_k\left(0\right)$, enabling asymptotic convergence of $\left|\right|{\tilde{x}}_k\left(0\right)\left|\right|\to 0$ without relying on natural repetition of initial conditions, which aligns with the core idea of ILC that optimizes through iterative error accumulation.

Theorem 1.

For a class of FONMASs, if there exists a learning gain matrix $\tilde{\mathsf{\Gamma }}$ such that the following two constraints hold:

$$\begin{array}{cc}\hfill {\rho }_1& =\left|\right|I_N+\left[(\mathcal{L}+\mathcal{B})\otimes \tilde{\mathsf{\Gamma }}C\right](I_N\otimes D)\left|\right|=\left|\right|I_N+\mathcal{K}(I_N\otimes D)\left|\right|<1,\hfill \end{array}$$

Among them, $\mathcal{K}$ denotes a matrix given by $\mathcal{K}=\left[(\mathcal{L}+\mathcal{B})\otimes \tilde{\mathsf{\Gamma }}C\right]$.

With the fault observer and initial-condition learning mechanism properly designed and implemented, the fault estimation error of the FONMASs will converge to zero, i.e.,

$$\underset{k\to \infty }{lim}∥{\tilde{f}}_k\left(t\right)∥=0.$$

Proof. 

First, Equation (1) is reformulated in a vectorized manner.

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }x\left(t\right)& =(I_N\otimes A)x\left(t\right)+(I_N\otimes B)u\left(t\right)+g(x\left(t\right),t)+(I_N\otimes D)f\left(t\right),\hfill \\ \hfill y\left(t\right)& =(I_N\otimes C)x\left(t\right),\hfill \end{array}\right.$$

(8)

Next, Equation (2) is similarly reformulated in a vectorized manner.

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }{\widehat{x}}_k\left(t\right)& =(I_N\otimes A){\widehat{x}}_k\left(t\right)+(I_N\otimes B)u\left(t\right)+g_k({\widehat{x}}_k\left(t\right),t)+(I_N\otimes D){\widehat{f}}_k\left(t\right)\hfill \\ & +(I_N\otimes L){\tilde{y}}_k\left(t\right),\hfill \\ \hfill {\widehat{y}}_k\left(t\right)& =(I_N\otimes C){\widehat{x}}_k\left(t\right),\hfill \end{array}\right.$$

(9)

It follows directly from Equation (4) that:

$${\tilde{f}}_{k+1}\left(t\right)={\tilde{f}}_k\left(t\right)+[(\mathcal{L}+\mathcal{B})\otimes \tilde{\mathsf{\Gamma }}C]{\mathcal{D}}_t^{\alpha }{\tilde{x}}_k\left(t\right).$$

(10)

The subtraction of Equation (10) from Equation (9) enables the following result to be obtained:

$$\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }{\tilde{x}}_k\left(t\right)& =(I_N\otimes A){\tilde{x}}_k\left(t\right)+g({\tilde{x}}_k\left(t\right),t)+(I_N\otimes D){\tilde{f}}_k\left(t\right)\hfill \\ & -(I_N\otimes L){\tilde{y}}_k\left(t\right).\hfill \end{array}$$

(11)

Subsequently, substituting Equation (11) into Equation (10), it has

$$\begin{array}{cc}\hfill {\tilde{f}}_{k+1}\left(t\right)& ={\tilde{f}}_k\left(t\right)+[(\mathcal{L}+\mathcal{B})\otimes \tilde{\mathsf{\Gamma }}C][(I_N\otimes A){\tilde{x}}_k\left(t\right)\hfill \\ \hfill & +g({\tilde{x}}_k\left(t\right),t)+(I_N\otimes D){\tilde{f}}_k\left(t\right)-(I_N\otimes L){\tilde{y}}_k\left(t\right)]\hfill \\ \hfill & =\mathcal{K}\left[\left(I_N\otimes (A-LC)\right){\tilde{x}}_k\left(t\right)+g({\tilde{x}}_k\left(t\right),t)\right]\hfill \\ \hfill & +\left(I_N+\mathcal{K}(I_N\otimes D)\right){\tilde{f}}_k\left(t\right)\hfill \end{array}$$

(12)

After taking the norm of Equation (12) and then the $\lambda$-norm with the aid of Assumption 1, the following result can be obtained:

$$\begin{array}{cc}\hfill \left|\right|{\tilde{f}}_{k+1}{\left|\right|}_{\lambda }& \le \left[\left|\right|\mathcal{K}\left(I_N\otimes (A-LC)\right)\left|\right|+l_f\right]\left|\right|{\tilde{x}}_k{\left|\right|}_{\lambda }+\left|\right|I_N+\mathcal{K}(I_N\otimes D)\left|\right|\left|\right|{\tilde{f}}_k{\left|\right|}_{\lambda }.\hfill \end{array}$$

(13)

According to Definition 1, a fractional-order integral is performed on Equation (11), it has

$$\begin{array}{cc}\hfill {\tilde{x}}_k\left(t\right)& ={\tilde{x}}_k\left(0\right)+{\displaystyle \frac{1}{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\widehat{A}{\tilde{x}}_k\left(\tau \right)}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau +{\displaystyle \frac{1}{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{g({\tilde{x}}_k\left(\tau \right),\tau )}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\widehat{D}{\tilde{f}}_k\left(\tau \right)}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau -{\displaystyle \frac{1}{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\widehat{L}{\tilde{y}}_k\left(\tau \right)}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau ,\hfill \end{array}$$

(14)

with $\widehat{A}=I_N\otimes A$, $\widehat{D}=I_N\otimes D$, $\widehat{L}=I_N\otimes L$, $\widehat{C}=I_N\otimes C.$

By taking the norm on both sides of Equation (14), applying Lemma 2, and further combining Assumption 1—under which the nonlinear term $g({\tilde{x}}_k\left(t\right),t)$ satisfies the global Lipschitz condition, we first derive

$$\begin{array}{cc}\hfill \parallel {\tilde{x}}_k\left(t\right)\parallel & \le {\displaystyle \frac{\parallel \widehat{A}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{x}}_k\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau +{\displaystyle \frac{\parallel \widehat{F}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel g({\tilde{x}}_k\left(\tau \right),\tau )\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \\ \hfill & +{\displaystyle \frac{\parallel \widehat{D}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{f}}_{k-1}\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau +{\displaystyle \frac{\parallel \widehat{L}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{y}}_k\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau ,\hfill \\ \hfill & \le {\displaystyle \frac{\parallel \widehat{A}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{x}}_k\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau +{\displaystyle \frac{l_f\parallel \widehat{F}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{x}}_k\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \\ \hfill & +{\displaystyle \frac{\parallel \widehat{D}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{f}}_{k-1}\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau +{\displaystyle \frac{\parallel \widehat{L}\widehat{C}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{\parallel {\tilde{x}}_k\left(\tau \right)\parallel }{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \end{array}$$

By introducing the exponential weighting and supremum norm, the result can be further rewritten as

$$\begin{array}{cc}\hfill \parallel {\tilde{x}}_k\left(t\right)\parallel & \le {\displaystyle \frac{\parallel \widehat{A}\parallel +l_f+\parallel \widehat{L}\widehat{C}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{e^{\lambda \tau }}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \\ \hfill & \times \underset{t\in [0,T]}{sup}\left\{e^{-\lambda t}\parallel {\tilde{x}}_k\left(t\right)\parallel \right\}+{\displaystyle \frac{\parallel \widehat{D}\parallel }{\mathsf{\Lambda }\left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{e^{\lambda \tau }}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \hfill \\ \hfill & \times \underset{t\in [0,T]}{sup}\left\{e^{-\lambda t}\parallel {\tilde{f}}_{k-1}\left(t\right)\parallel \right\},\hfill \\ \hfill & ={\int }_0^t{\displaystyle \frac{e^{\lambda \tau }}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \times \left[{\displaystyle \frac{b_{\widehat{A}}+l_f+b_{\widehat{LC}}}{\mathsf{\Lambda }\left(\alpha \right)}}{\parallel {\tilde{x}}_k\parallel }_{\lambda }\right.+\left.{\displaystyle \frac{b_{\widehat{D}}}{\mathsf{\Lambda }\left(\alpha \right)}}{\parallel {\tilde{f}}_k\parallel }_{\lambda }\right]\hfill \end{array}$$

(15)

with $b_{\widehat{A}}=\parallel \widehat{A}\parallel$, $b_{\widehat{LC}}=\parallel \widehat{L}\widehat{C}\parallel$.

For the integral term in Equation (15), it follows from Lemma 2 that

$$\begin{array}{cc}\hfill \parallel {\tilde{x}}_k\left(t\right)\parallel & \le {\int }_0^t{\displaystyle \frac{e^{\lambda \tau }}{{(t-\tau )}^{1-\alpha }}}\mathrm{d}\tau \times \left[{\displaystyle \frac{b_{\widehat{A}}+l_f+b_{\widehat{LC}}}{\mathsf{\Lambda }\left(\alpha \right)}}\parallel {\tilde{x}}_k{\parallel }_{\lambda }+{\displaystyle \frac{b_{\widehat{D}}}{\mathsf{\Lambda }\left(\alpha \right)}}{\parallel {\tilde{f}}_k\parallel }_{\lambda }\right]\hfill \\ \hfill & <{\displaystyle \frac{e^{\lambda t}}{{\lambda }^{\alpha }}}\left[(b_{\widehat{A}}+l_f+b_{\widehat{LC}})\parallel {\tilde{x}}_k{\parallel }_{\lambda }+b_{\widehat{D}}{\parallel {\tilde{f}}_k\parallel }_{\lambda }\right]\hfill \end{array}$$

(16)

Multiplying both sides of Equation (16) by $e^{-\lambda t}$ and taking the supremum over time t, it follows that

$$\begin{array}{cc}\hfill {∥{\tilde{x}}_k∥}_{\lambda }& <{\displaystyle \frac{1}{{\lambda }^{\alpha }}}\left[(b_{\widehat{A}}+l_f+b_{\widehat{LC}})\left|\right|{\tilde{x}}_k{\left|\right|}_{\lambda }+b_{\widehat{D}}\left|\right|{\tilde{f}}_k{\left|\right|}_{\lambda }\right]\hfill \end{array}$$

(17)

There exists $\lambda >0$ and sufficiently large $\lambda$ such that

$${\lambda }^{\alpha }-b_{\widehat{A}}-l_f-b_{\widehat{LC}}>0,$$

and thus it follows that

$$\begin{array}{c}\hfill {∥{\tilde{x}}_k∥}_{\lambda }<{\displaystyle \frac{{\lambda }^{\alpha }{b}_{\widehat{D}}}{{\lambda }^{\alpha }-b_{\widehat{A}}-l_f-b_{\widehat{LC}}}}{∥{\tilde{f}}_k∥}_{\lambda }.\end{array}$$

(18)

Then, Equation (18) can be simplified as

$$\begin{array}{c}\hfill {∥{\tilde{x}}_k∥}_{\lambda }<b_{\widehat{D}}{o}_3\left({\lambda }^{-1}\right){∥{\tilde{f}}_k∥}_{\lambda },\end{array}$$

(19)

with $o_3\left({\lambda }^{-1}\right)={\displaystyle \frac{{\lambda }^{\alpha }}{{\lambda }^{\alpha }-b_{\widehat{A}}-l_f-b_{\widehat{LC}}}}.$

By substituting Equation (19) into Equation (13), one obtains

$$\begin{array}{cc}\hfill \parallel {\tilde{f}}_{k+1}{\parallel }_{\lambda }& \le \left[\parallel I_N+\mathcal{K}(I_N\otimes D)\parallel +\parallel \mathcal{K}(I_N\otimes (A-LC))\parallel +l_f\right]\times b_{\widehat{D}}{o}_3\left({\lambda }^{-1}\right){\parallel {\tilde{f}}_k\parallel }_{\lambda },\hfill \\ \hfill & \le {\rho }_2{\parallel {\tilde{f}}_k\parallel }_{\lambda }.\hfill \end{array}$$

(20)

with

$$\begin{array}{cc}\hfill {\rho }_2& =\parallel I_N+\mathcal{K}(I_N\otimes D)\parallel +\left[\parallel \mathcal{K}\left(I_N\otimes (A-LC)\right)\parallel +l_f\right]b_{\widehat{D}}{o}_3\left({\lambda }^{-1}\right).\hfill \end{array}$$

It can be seen from Theorem 1 that ${\rho }_1=\parallel I_N+\mathcal{K}(I_N\otimes D)\parallel <1$. Therefore, we can choose a sufficiently large $\lambda$ such that

$$\begin{array}{cc}\hfill {\rho }_2& =\left[\parallel I_N+\mathcal{K}(I_N\otimes D)\parallel +\parallel \mathcal{K}(I_N\otimes (A-LC))\parallel +l_f\right]\times b_{\widehat{D}}{o}_3\left({\lambda }^{-1}\right)<1.\hfill \end{array}$$

(21)

By applying Lemma 3, one obtains:

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{∥{\tilde{f}}_k∥}_{\lambda }=0.\end{array}$$

(22)

According to $\lambda$-norm, Equation (22) can be further rewritten as

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}∥{\tilde{f}}_k\left(t\right)∥=\underset{k\to \infty }{lim}{e}^{\lambda T}{∥{\tilde{f}}_k∥}_{\lambda }=0.\end{array}$$

Hence, the proof of Theorem 1 is complete. □

Remark 3.

For a class of FONMASs described by Equation (2), suppose that Assumptions 1–3 hold, where the nonlinear function $g_i(x_i\left(t\right),t)$ satisfies the Lipschitz condition, the pair $(A,C)$ is observable with A and C being of full rank, respectively, and each agent possesses a distinct initial state at the k-th iteration.

4. Fault Diagnosis of FONMASs with Switching Topology

In practice, it is challenging to maintain a fixed communication topology in real FONMASs. By contrast, switching topologies are far more common. Therefore, the above results are extended from a fixed topology to a switching one, in order to highlight the advantages of the fault observer when the network structure changes. The corresponding time-varying communication graph can be represented as:

$${\mathcal{L}}_k={\mathcal{D}}_k-{\mathcal{A}}_k,$$

where ${\mathcal{L}}_k$ is denoted as the Laplacian matrix at the k-th iteration, ${\mathcal{D}}_k$ as the degree matrix at the k-th iteration, and ${\mathcal{A}}_k$ as the adjacency matrix at the k-th iteration. Thus, under the switching topology, the observation error in Equation (3) and the fault observer in Equation (10) can be, respectively, written as

$$e_k\left(t\right)=[({\mathcal{L}}_k+{\mathcal{B}}_k)\otimes I_n]{\tilde{y}}_k\left(t\right).$$

(23)

$${\widehat{f}}_{k+1}\left(t\right)={\widehat{f}}_k\left(t\right)-[({\mathcal{L}}_k+{\mathcal{B}}_k)\otimes \tilde{\mathsf{\Gamma }}C]{\mathcal{D}}_t^{\alpha }{\widehat{x}}_k\left(t\right).$$

(24)

Under an iteration-varying topology, the following Corollary 1 is obtained.

Corollary 1.

Consider the continuous FONMASs (2) equipped with the fault observer (10). Assume that $\underset{k}{sup}\parallel {\mathcal{L}}_k\parallel <\infty$ and Assumptions 1–3 hold.

Defining ${\mathcal{K}}_k=({\mathcal{L}}_k+{\mathcal{B}}_k)\otimes \tilde{\mathsf{\Gamma }}C$ and the learning gain $\tilde{\mathsf{\Gamma }}$ satisfies

$$\begin{array}{c}\hfill {\rho }_3=\underset{k}{sup}\left|\right|I_N+{\mathcal{K}}_k(I_N\otimes D)\left|\right|<1,\end{array}$$

the fault estimation error converges to 0.

Proof. 

It follows from Equation (24) that

$${\tilde{f}}_{k+1}\left(t\right)={\tilde{f}}_k\left(t\right)+[({\mathcal{L}}_k+{\mathcal{B}}_k)\otimes \tilde{\mathsf{\Gamma }}C]{\mathcal{D}}_t^{\alpha }{\tilde{x}}_k\left(t\right).$$

(25)

Similar to the proof method of Equation (12), the following result can be obtained in this section

$$\begin{array}{cc}\hfill {\tilde{f}}_{k+1}\left(t\right)& =(I_N+{\mathcal{K}}_k(I_N\otimes D)){\tilde{f}}_k\left(t\right)\hfill \\ & +{\mathcal{K}}_k\left[\left(I_N\otimes (A-LC)\right)\tilde{x}\left(t\right)+g({\tilde{x}}_k\left(t\right),t)\right].\hfill \end{array}$$

(26)

By employing a similar reasoning as in the proof of fault diagnosis for FONMASs under fixed topologies, the following result can be derived.

$$\begin{array}{cc}\hfill \parallel {\tilde{f}}_{k+1}{\parallel }_{\lambda }& \le [\parallel I_N+{\mathcal{K}}_k(I_N\otimes D)\parallel +\parallel {\mathcal{K}}_k(I_N\otimes (A\hfill \\ \hfill & -LC\left)\right)\parallel b_{\widehat{D}}{o}_4\left({\lambda }^{-1}\right)+l_f{b}_{\widehat{D}}{o}_4\left({\lambda }^{-1}\right)]{\parallel {\tilde{f}}_k\parallel }_{\lambda },\hfill \\ \hfill & \le {\rho }_4{\parallel {\tilde{f}}_k\parallel }_{\lambda }.\hfill \end{array}$$

(27)

with

$$\begin{array}{cc}\hfill {\rho }_4& =∥I_N+{\mathcal{K}}_k\left(I_N\otimes D\right)∥+∥{\mathcal{K}}_k\left(I_N\otimes (A-LC)\right)∥+b_{\widehat{D}}{o}_4\left({\lambda }^{-1}\right)+l_f{b}_{\widehat{D}}{o}_4\left({\lambda }^{-1}\right),\hfill \\ \hfill b_{\widehat{D}}& =\parallel \widehat{D}\parallel ,o_4\left({\lambda }^{-1}\right)={\displaystyle \frac{1}{{\lambda }^{\alpha }-b_{\widehat{A}}-l_f-b_{\widehat{LC}}}},\hfill \\ \hfill b_{\widehat{A}}& =\parallel \widehat{A}\parallel ,b_{\widehat{LC}}=\parallel \widehat{L}\widehat{C}\parallel .\hfill \end{array}$$

It can be readily obtained from Equation (27),

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{∥{\tilde{f}}_k∥}_{\lambda }=0.\end{array}$$

(28)

Finally, it can obtain that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}∥{\tilde{f}}_k\left(t\right)∥=\underset{k\to \infty }{lim}{e}^{\lambda T}{∥{\tilde{f}}_k∥}_{\lambda }=0.\end{array}$$

This completes the proof of Corollary 1. □

5. Numerical Simulation

To verify the effectiveness of the proposed fault diagnosis method, a simulation experiment is conducted in this section, and the core parameters are specified. The specific parameter configuration and construction logic are as follows: the duration of a single run is set to $5\mathrm{s}$ with a discrete time step $0.01\mathrm{s}$. The total number of iterations is 40, and the number of agents is 4.

The coefficient matrices for the accumulated terms and the prediction terms in the discrete fractional derivative are constructed via the Gamma function, ensuring an accurate characterization of the fractional dynamical behavior. The desired initial state vector of the FONMAS is set to $x_0={\left[\begin{array}{c}1.5,1.5,2,2,3,3,4,4\end{array}\right]}^T$. The fault observer is described as follows:

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }{\widehat{x}}_{k,i}\left(t\right)& =A{\widehat{x}}_{k,i}\left(t\right)+B{u}_i\left(t\right)+g({\widehat{x}}_{k,i}\left(t\right),t)\hfill \\ & +D{\widehat{f}}_{k,i}\left(t\right)+L(y_i\left(t\right)-{\widehat{y}}_{k,i}\left(t\right)),\hfill \\ \hfill {\widehat{y}}_{k,i}\left(t\right)& =C{\widehat{x}}_{k,i}\left(t\right).\hfill \end{array}\right.$$

where the fractional order $\alpha =0.85$, parameter matrix is

$$A=\left[\begin{array}{cc}-2& 3\\ 0& -1\end{array}\right],B=\left[\begin{array}{cc}1.5& 0\\ 0& 1\end{array}\right],C=\left[\begin{array}{cc}1.2& 0\\ 0& 0.9\end{array}\right],$$

$$D=\left[\begin{array}{cc}0.8& 0\\ 0& 0.9\end{array}\right],L=\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right].$$

The Laplacian matrix is

$$\mathcal{L}=\left[\begin{array}{cccc}2& -1& -1& 0\\ -1& 1& 0& 0\\ 0& 0& 1& -1\\ 0& -1& 0& 1\end{array}\right]$$

The directed communication graph corresponding to the Laplacian matrix is a visual representation of the information interaction relationships among agents in a multi-agent system. Figure 1 shows this directed communication graph. The nodes numbered 1–4 represent the four agents in the system. The arrows indicate the direction of information transmission. Different colors represent different intelligent agents.

The open-loop $D^{\alpha }$-type iterative learning structure is adopted, with the learning gain matrix $\widehat{\mathsf{\Gamma }}=\left[\begin{array}{cc}0.95& 0\\ 0& 1.1\end{array}\right]$. The global learning law for FONMASs is constructed using the Kronecker Product, which satisfies the convergence condition $\parallel I_8-I_4\otimes \widehat{\mathsf{\Gamma }}CD\parallel =0.109<1$. This ensures the continuous optimization of control inputs during the iteration process.

The fault estimator is

$${\widehat{f}}_{k+1,i}\left(t\right)={\widehat{f}}_{k,i}\left(t\right)+\widehat{\mathsf{\Gamma }}{\mathcal{D}}_t^{\alpha }{e}_{k,i}\left(t\right)$$

To reflect the diversity of faults in real-world engineering applications, time-varying fault profiles are assigned to the four agents.

$$\begin{array}{cc}\hfill \mathrm{MAS}1:& f_1\left(t\right)=\left\{\begin{array}{cc}{t}^{1.2},\hfill & t\in [1,3]\cup [3.5,4.5],\hfill \\ 0,\hfill & t\in [0,1)\cup (3,3.5)\cup (4.5,5].\hfill \end{array}\right.\hfill \\ \hfill \mathrm{MAS}2:& f_2\left(t\right)=\left\{\begin{array}{c}{e}^{{\displaystyle \frac{2}{t+0.5}}},t\in [0.5,2.5]\cup [3.5,4.5],\hfill \\ 0,t\in [0,0.5)\cup (2.5,3.5)\cup (4.5,5].\hfill \end{array}\right.\hfill \\ \hfill \mathrm{MAS}3:& f_3\left(t\right)=\left\{\begin{array}{c}2sin\left(2\pi t\right),t\in [0.5,1.5]\cup [2.5,3.5]\cup [4.5,5],\hfill \\ 1,t\in [0,0.5)\cup (1.5,2.5)\cup (3.5,4.5).\hfill \end{array}\right.\hfill \\ \hfill \mathrm{MAS}4:& f_4\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{2}{t+0.1}},t\in [0.2,1.2]\cup [1.5,2.5]\cup [3.5,4.5],\hfill \\ 0,t\in [0,0.2)\cup (1.2,1.5)\cup (2.5,3.5)\cup (4.5,5].\hfill \end{array}\right.\hfill \end{array}$$

To eliminate initial state deviations, a P-type initial learning strategy is designed with the learning gain $\mathsf{\Gamma }=\left[\begin{array}{cc}1.5& 0\\ 0& 1.5\end{array}\right]$. The initial state is updated via the formula $x_k\left(0\right)=x_{k-1}\left(0\right)+(I_4\otimes \mathsf{\Gamma }C){\tilde{x}}_{k-1}\left(0\right)$, ensuring that the initial error $\parallel {\tilde{x}}_k\left(0\right)\parallel$ approaches zero as the number of iterations increases.

The initial value error is shown in Figure 2. Under the strategy of initial value learning, as the number of iterations increases, the error at the initial moment gradually converges to 0.

Figure 3 demonstrates the fault estimation errors of FONMAS1–FONMAS4. Under the action of the fault estimator, the fault estimation errors converge to $7.90\times {10}^{-2}$ at the 30th iteration. The above results verify the effectiveness of the proposed iterative learning protocol.

Figure 4 and Figure 5 present 3D curve comparisons between the actual outputs $f_i$ of different agents in the FONMAS and the estimated outputs ${\widehat{f}}_{i,k}$ at iterations $k=2$, $k=5$, and $k=40$ with time varying. As the number of iterations increases from 2 to 40, the fitting degree between the estimated and actual curves gradually improves, reflecting continuous enhancement in estimation accuracy. This result confirms that the adopted estimation strategy exhibits favorable iterative convergence and can effectively approach the actual outputs of each agent.

Figure 6 and Figure 7 illustrate the output trajectories of the FONMASs under different numbers of iterations, with the integration of the fault estimator and the initial value learning strategy. It only takes 40 iterations for the system’s actual output to achieve accurate tracking over the interval $[0,5]$.

Next, a simulation verification of the switched topology is carried out. To characterize the cooperative behavior of FONMASs under dynamic topologies, this study designs four Laplacian matrices that switch cyclically with the number of iterations k, denoted as ${\mathcal{L}}_1$ to ${\mathcal{L}}_4$, respectively. Their specific definitions and topological characteristics are presented as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_1& =\left[\begin{array}{cccc}3& -2& -1& 0\\ -1& 2& -1& 0\\ 0& 0& 2& -2\\ 0& -1& 0& 1\end{array}\right],{\mathcal{L}}_2=\left[\begin{array}{cccc}2& -1& 0& -1\\ -1& 1& 0& 0\\ 0& -1& 1& 0\\ -1& 0& 0& 1\end{array}\right],\hfill \\ \hfill {\mathcal{L}}_3& =\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& -1& 0\\ 0& 0& 1& -1\\ -1& -1& 0& 2\end{array}\right],{\mathcal{L}}_4=\left[\begin{array}{cccc}4& -1& -1& -2\\ -1& 3& -1& -1\\ -1& -1& 3& -1\\ -2& -1& -1& 4\end{array}\right].\hfill \end{array}$$

All nodes are bidirectionally connected to each other. The diagonal elements have relatively large weights, while the off-diagonal elements are evenly distributed, ensuring direct information interaction channels between any pair of nodes.

In the above matrices, the diagonal elements represent the total weight of outgoing edges from the corresponding node. All matrices satisfy the condition of strong directed connectivity, enabling dynamic topology switching and cooperation of FONMASs during the iteration process.

In Figure 8, the four Laplacian matrices ${\mathcal{L}}_1$ to ${\mathcal{L}}_4$ represent different communication topologies, respectively. During the iterative learning process, a topology structure is switched in each iteration. This topology cyclic switching mechanism enables the FONMAS to adapt to the changing communication structure during the iteration process, which is crucial for fault diagnosis of the FONMAS.

Figure 9 shows the variation of fault estimation errors for FONMAS1–FONMAS4. As the number of iterations k increases, the fault estimation errors have converged to $8.63\times {10}^{-2}$ at the 30th iteration, and the errors of FONMAS1–FONMAS4 all exhibit a significant decreasing trend, eventually gradually tending to stabilize. This indicates that the fault estimation accuracy of the FONMASs continuously improves and reaches a good stable state, verifying the effectiveness of the proposed method in FONMAS fault estimation.

Figure 10 and Figure 11 present the actual output curves of $f_1$, $f_2$, $f_3$, and $f_4$ in a FONMAS, along with the estimated output curves at the 2nd, 5th, and 40th iterations. It can be observed that as the number of iterations increases from 2 to 40, the the consistency between the estimated and actual curves gradually improves, indicating that the accuracy of the estimation method continuously enhances with increasing iterations.

The design of the control law and the fault estimation update law is as follows:

$$\begin{array}{cc}\hfill u\left(t\right)& =B^{†}\left(-Ax\left(t\right)-g\left(x\left(t\right)\right)+k_{p}e\left(t\right)+D\widehat{f}\left(t\right)+\lambda r\left(t\right)\right),\hfill \\ \hfill \dot{\widehat{f}}\left(t\right)& =-\beta \widehat{f}\left(t\right)+\gamma \left(e\left(t\right)+\lambda r\left(t\right)\right)\hfill \end{array}$$

where $k_p=1.5$, $\gamma =80$, $\lambda =0.3$, and $\beta =0.05$ are the tunable adaptive gain parameters, $B^{†}$ denotes the Moore–Penrose pseudoinverse of the input matrix B, A denotes the system state matrix, $x\left(t\right)$ denotes the continuous-time system state vector, $g\left(x\right(t\left)\right)$ denotes the state-dependent nonlinear term, $e\left(t\right)$ denotes the tracking error, D denotes the fault coupling matrix, $\widehat{f}\left(t\right)$ denotes the estimated fault, $\widehat{f}\left(0\right)$ denotes the initial value of fault estimation, $r\left(t\right)$ represents the system residual, $\tau$ denotes the integral dummy variable, and $u\left(t\right)$ denotes the continuous-time control input.

Figure 12 and Figure 13 adopt an adaptive fault estimation algorithm based mainly on real-time state feedback and residual compensation. Simulation results demonstrate that the proposed ILC algorithm outperforms traditional adaptive control in complex time-varying fault diagnosis: ILC-based fault estimation shows strong congruence with the real fault dynamics, enabling accurate tracking even under conditions of fault mutation and high-frequency fluctuations. In contrast, the adaptive method exhibits persistent estimation errors, lag, and oscillation at fault mutation points. Notably, ILC shows clear iterative convergence, with the 20th iteration yielding significantly higher accuracy than the 10th, whereas adaptive control relies solely on current-time information and cannot improve with repeated operations. Thus, ILC offers higher accuracy, stronger robustness, and iterative domain convergence, making it a superior solution for fault diagnosis in fractional-order multi-agent systems under repetitive operation. The simulation results show that as the number of iterations increases, the initial state error gradually converges to 0. At the 5th iteration, the fault estimation curves of the four agents can already well approximate the actual faults. At the 40th iteration, the consistency between the estimated fault values and the actual fault values is significantly improved, and the fault estimation error converges effectively. Moreover, in the scenario of topology switching, the system still maintains excellent convergence performance, verifying the advantages of this fault diagnosis method in terms of accuracy and topological adaptability.

6. Conclusions

A novel iterative learning paradigm is proposed in this paper for FONMASs with directed switching topologies and initial state deviation. By embedding a P-type initial-state learning mechanism into the learning law, each agent can autonomously compensate for its own initial deviation in each iteration, thereby achieving rapid asymptotic convergence of both trajectory approximation errors and initial-state errors. The designed iterative learning protocol only relies on local neighborhood information and employs an observer to online estimate unknown nonlinearities, enabling simultaneous fault diagnosis and consensus approximation without requiring an accurate system model. With the aid of vector model lifting and tailored fractional-order Lyapunov-contraction mapping analysis, sufficient conditions are rigorously derived to guarantee that all errors converge to zero as the number of iterations tends to infinity. The numerical results show that with 30 iterations under the fixed topology, the fault estimation error drops to $7.90\times {10}^{-2}$. Under the switching topology, it stabilizes at $8.63\times {10}^{-2}$ after 30 iterations. However, this model is limited by its reliance on the global Lipschitz condition and does not consider the impact of external noise. Future research will consider the event-triggering mechanism, optimize the learning gain to improve convergence performance, and explore practical applications in complex engineering scenarios.