Mean square consensus of heterogeneous multi-agent systems in the presence of actuator faults

Abstract

This paper presents the mean square consensus of multi-agent systems in the presence of actuator faults and stochastic disturbances. The dynamics of agents are considered to be heterogeneous. In addition, the information exchange between agents is modeled by a directed graph. The actuator faults are assumed as bias fault and loss of effectiveness fault. Using the adaptive control method and neural networks, a distributed controller is designed for the consensus problem of heterogeneous multi-agent systems with actuator faults and stochastic disturbances. Also, by applying Lyapunov theory, it is proved that the distributed controller guarantees the mean square consensus of heterogeneous multi-agent systems in the presence of bias faults, loss of effectiveness faults, and stochastic disturbances. Eventually, two numerical simulations are provided to show the ability of the distributed controller in the consensus problem.

Keywords

Mean square consensus multi-agent systems heterogeneous dynamics actuator faults stochastic disturbances

Introduction

The consensus means that a group of agents reach the same states or reach an agreement (Ren et al., 2016). The consensus problem has wide applications such as estimation problem, optimization problem, and control of unmanned air vehicles (Li et al., 2019; Liu et al., 2020; Najm et al., 2020). It should be noted that designing a proper distributed algorithm is a critical problem for the consensus (Zou et al., 2018). In the distributed consensus algorithm, all the agents can only access to the local information which is received from the neighbors. The consensus problem of multi-agent system is divided into leader-following (Maadani and Butcher, 2021) and leader-less (Shang et al., 2021). The aim of the leader-following consensus problem is that all the follower agents follow the leader states (Zou et al., 2018). However, the leader-less consensus problem aims to steer the agents to reach an unprescribed value. This unprescribed value is based on the initial state values of the agents. It should be mentioned that the leader-less consensus needs more energy than the leader-following consensus (Fan and Wu, 2018).

A network of agents with the same dynamics is called homogeneous multi-agent system. However, a heterogeneous multi-agent system consists of a group of agents which have different dynamics (Wang et al., 2020). The consensus problem of homogeneous multi-agent system is much simpler than the consensus problem for heterogeneous multi-agent system. However, it is more reasonable to study the consensus problem of heterogeneous multi-agent system because it is a more general issue to consider that the dynamics of agents are different. According to the above-mentioned discerptions, the consensus problem of heterogonous multi-agent system is studied in this paper.

In real systems, there exist stochastic factors such as stochastic communication noise, stochastic time delay, random missing measurements, and stochastic disturbances (Ren et al., 2016). These stochastic factors have adverse effects on the performance of the multi-agent system. Instability of the system is one of the adverse effects caused by stochastic factors (Tong et al., 2011). Hence, stochastic systems have been widely studied in the last few years. Disturbances may occur when random changes happen in the environment of the system. These random changes include failures, changes in the connections of subsystems, and the component repairs (Wang et al., 2009). According to the above-mentioned explanations, disturbances are inescapable in real applications. Therefore, stochastic disturbances are one of the most important stochastic factors in real systems. It should be mentioned that most existing works about the consensus of multi-agent system have not considered stochastic disturbances. In Ren et al. (2016), the consensus problem of stochastic multi-agent system with heterogeneous dynamics has been studied. In the mentioned reference, the undirected graph has been used for modeling the communication network. An event-triggered controller has been proposed for the mean square consensus of nonlinear stochastic multi-agent system with time delay and external disturbances (Sun et al., 2021). It should be noted that a simple nonlinear model has been considered in the mentioned reference, in which the model is the sum of a linear model with nonlinearities. Therefore, the nonlinear model studied in this paper is much more comprehensive than the model considered in the mentioned reference. In Luo et al. (2021), the mean square consensus has been studied for heterogeneous multi-agent systems in the presence of communication delays and system noises. It should be emphasized that the multi-agent system with linear dynamics has been considered in the mentioned reference.

Occurring fault is unavoidable in real applications such as multi-agent systems. Actuator faults, communication faults, and sensor faults are types of faults which are considered to occur in the multi-agent system. Actuator faults are the most important type of fault because they have detrimental effects on the system performance and may even cause instability (Isermann, 2005). Hence, fault-tolerant control (FTC) has been extensively studied for the consensus issue of multi-agent system because it can maintain the reliability against actuator faults. In Zou et al. (2020), the consensus issue has been investigated for nonlinear multi-agent system with actuator faults. In the mentioned reference, the agent dynamics are assumed to be switched and heterogeneous. It should be noted that stochastic disturbances have not been considered in the mentioned reference. Two sliding mode controllers have been designed for the mean square consensus of homogeneous multi-agent system with actuator faults (Ren et al., 2021). In the mentioned reference, loss of effectiveness and bias faults are considered as actuator faults. The proper choosing of the controller parameter to avoid chattering and achieve consensus with high accuracy is the disadvantage of the controller designed in Ren et al. (2021). It should be noted that the mean square consensus has not been investigated for heterogeneous multi-agent system with actuator faults and stochastic disturbances. Therefore, this is the first time that a controller is proposed for the mean square consensus of heterogonous multi-agent system in the presence of actuator faults which are assumed as loss of effectiveness faults and bias faults.

Motivated by the above discussions, the consensus problem of the heterogeneous multi-agent system in the presence of actuator faults and stochastic disturbances is investigated in this study. A directed graph is considered for describing the communication between agents. In this study, actuator faults are time-varying and unknown. Furthermore, actuator faults are assumed as bias faults and loss of effectiveness faults. A distributed controller based on the adaptive control method and radial basis function neural networks (RBFNNs) is proposed for the mean square consensus of heterogeneous multi-agent systems. By applying the proper Lyapunov functions and Young’s inequality, the distributed control method ensures the mean square consensus of heterogeneous multi-agent systems with unknown and time-varying actuator faults and stochastic disturbances.

The remainder of this study is organized as follows. In section “Problem formulation and preliminaries,” graph theory, neural network description, stochastic system model, and necessary lemmas and assumptions are presented, in which all these concepts will be used in the controller design. In section “Controller design,” a distributed controller is designed for the mean square consensus of heterogeneous multi-agent system with actuator faults and stochastic disturbances. In section “Stability analysis,” the stability of the distributed controller is analyzed using Lyapunov theory. For demonstrating the ability of the distributed control method, the simulation results of two examples are given in section “Simulation results.” The last section of this study contains the conclusions.

Notations: In this paper, $R$ , $| | . | |$ , and $E [.]$ denote the real number set, the Euclidean norm, and the mathematical expectation, respectively.

Problem formulation and preliminaries

In this section, first, the concepts related to the communication network of the multi-agent system are stated. Then, the mathematical equations of RBFNNs are described in detail. Finally, the stochastic system model for agents and actuator faults is described.

Graph theory

The information exchange between agents is modeled by the graph. Therefore, a directed graph is considered for the communication topology of multi-agent system. Let $G = (V, E, A)$ be the directed graph which contains node set $V = [V_{1}, V_{2}, \dots, V_{N}]$ , edge set $E \subset V \times V$ , and adjacency matrix $A = {a_{i, j}}_{i, j = 1, . ., N} \in R^{N \times N}$ . If the $i th$ follower cannot receive data from the $j th$ follower, then $a_{i, j} = 0$ ; otherwise, $a_{i, j} = 1$ . It is assumed that $a_{i, i} = 0, \forall i = 1, \dots, N$ .

Laplacian matrix associated with graph $G$ is determined as

L = {l_{i, j}}_{i, j = 1, . ., N} \in R^{N \times N}, l_{i, j} = {\begin{matrix} - a_{i, j}, if i \neq j \\ \sum_{j = 1}^{N} a_{i, j}, if i = j \end{matrix}

(1)

According to equation (1), it is obvious that $L 1_{N} = 0, 1_{N} = [1, 1, \dots, 1]^{T} \in R^{N}$ .

Denote $B = diag {b_{1}, \dots, b_{N}}$ as the leader adjacency matrix. If the follower $i$ has access to the leader’s data, then $b_{i} > 0$ ; otherwise, $b_{i} = 0$ .

Definition 1 (Qin et al., 2016): In the directed graph, if there exists a root node which has a directed path to all other nodes, then the graph contains a directed spanning tree.

Definition 2 (Ji et al., 2017): In the directed graph, if there is a directed path from any node to any other node, then the graph is strongly connected.

The essential condition for consensus is satisfied when the graph associated with followers and the leader consists of a directed spanning tree. Therefore, the following assumption is provided for this essential condition.

Assumption 1. Let $\bar{G}$ be the communication graph between the leader and the followers. The graph $\bar{G}$ contains at least one directed spanning tree.

Lemma 1 (Zhang and Lewis, 2012): If the digraph $\bar{G}$ has a directed spanning tree, then matrix $L + B$ is invertible.

Neural network

Neural networks are extensively applied to approximate unknown functions. RBFNN is the most commonly used neural network. Using RBFNN, the unknown function $F (Z) : R^{n} \to R$ can be determined as the following equation

F (Z) = θ^{*^{T}} φ (Z) + ε (Z), \forall Z \in Ω_{Z} \in R^{n}

(2)

where $θ^{*} = [θ_{1}, θ_{2}, \dots, θ_{l}]^{T} \in R^{l}$ is the ideal constant weight vector, $φ (Z) = [φ_{1} (Z), φ_{2} (Z), \dots, φ_{l} (z)]^{T} \in R^{l}$ is the Gaussian function vector, $l > 1$ is the number of neurons, and $ε (Z)$ is the approximation error. The ideal constant weight vector and the Gaussian function are defined as follows

θ^{*} = \arg min_{θ \in R^{l}} sup_{Z \in Ω_{Z}} {| θ^{T} φ (Z) - F (Z) |}

(3)

φ_{m} (Z) = \exp (- \frac{{(z - χ_{m})}^{T} (z - χ_{m})}{ξ_{m}^{2}}), m = 1, 2, \dots, l

(4)

where $χ_{m} = [χ_{m, 1}, χ_{m, 2}, \dots, χ_{m, n}]^{T}$ and $ξ_{m}$ are the center and width of the Gaussian function.

Assumption 2 (Hashemi and Shahgholian, 2018): For the approximation error and the ideal constant weight vector over compact set $Ω_{Z} \in R^{n}$ , the following inequalities can be written

| ε (Z) | \leq ε^{*}, ‖ θ^{*^{T}} ‖ \leq Ψ^{*}, Z \in Ω_{Z} \in R^{n}, ε^{*}, Ψ^{*} \in R^{+}

(5)

Remark 1: It should be noted that the uses of neural networks in control are different from their uses in other applications, such as digital signal processing (Lewis et al., 1998). In control applications, the neural network must provide that a dynamical plant is stable while simultaneously learning (Lewis et al., 1998). Adaptive control methods are required to solve this problem. Therefore, adaptive laws will be used for online training of the neural network.

Stochastic system model

In this paper, it is assumed that stochastic multi-agent system has $N$ followers and one leader. The leader dynamics of stochastic multi-agent system is modeled as follows

{\overset{\cdot}{x}}_{0} = f_{0} (x_{0})

(6)

The dynamics of the follower $i$ is given by

d x_{i} = (f_{i} (x_{i}) + u_{i}) dt + g_{i} (x_{i}) d ω

(7)

where $x_{i} \in R^{n}$ and $u_{i} \in R^{n}$ are state vector and actuator output of the follower $i$ , respectively. Unknown smooth nonlinear function $f_{i} (x_{i}) : R^{n} \to R^{n}$ represents the dynamics of follower $i$ . $g_{i} (x_{i}) : R^{n} \to R^{n \times r}$ and $ω$ are unknown nonlinear function and standard Wiener process, respectively.

Remark 2. Without loss of generality, it is assumed that the agents have first-order dynamics. Stability analysis of higher-order dynamics is similar to first-order dynamics, but only Kronecker product and Young’s inequality for vector mode should be used. The controller presented in this paper can be generalized to higher-order agents, in which the simulation results of second example show the performance of the proposed controller in this case.

Assumption 3. Unknown constant $G$ exists such that $| g_{i} (x_{i}) | \leq G, i = 1, \dots, N$ .

Remark 3. Stochastic dynamic model can be used for describing the motion of mechanical systems in the environments with stochastic vibration (Ren et al., 2016). Therefore, investigating on the multi-agent system with stochastic dynamics is of particular importance.

The actuator output with loss of effectiveness fault and bias fault is modeled by

u_{i} = ρ_{i} (t) v_{i} + r_{i} (t)

(8)

where $v_{i}$ , $ρ_{i} (t)$ , and $r_{i} (t)$ are the control input, the severity of loss of effectiveness fault, and the severity of bias fault, respectively. It should be noted that the two types of actuator faults are considered to be unknown and time-varying.

Assumption 4: For the severity of loss of effectiveness fault and severity of bias fault, the following inequalities hold

0 < ρ_{i} (t) \leq 1, | r_{i} (t) | \leq \bar{r} < \infty

(9)

where $\bar{r}$ is an unknown constant.

Two definitions and one lemma are given in the following which will be used in the controller design section and the stability analysis section.

Definition 3 (Tong et al., 2011): If for any $ϵ > 0$ and any initial condition $x (t_{0}) = [x_{1} (t_{0}), \dots, x_{N} (t_{0})]^{T}$ , there is a finite time $T (x (t_{0}), ϵ)$ such that $E {{| x_{i} (t) - x_{0} (t) |}^{2}} < ϵ, \forall t > T (x (t_{0}), ϵ)$ , then the mean square consensus of stochastic multi-agent system equations (6) and (7) is achieved.

Definition 4 (Tong et al., 2011): Consider the following stochastic equation

dx = f (x) dt + g (x) d ω

(10)

where $x \in R^{n}$ , $f (x) : R^{n} \to R^{n}$ , $g (x) : R^{n} \to R^{n \times r}$ , and $ω$ is r-D standard Wiener process. For function $V (x) : R^{n} \to R^{+}$ , an infinitesimal generator is defined as follows

L V (x) = \frac{\partial V (x)}{\partial x} f (x) + \frac{1}{2} trace (g^{T} (x) \frac{\partial^{2} V (x)}{\partial x^{2}} g (x))

(11)

Lemma 2 (Tong et al., 2011): If there are two positive constants $C_{1}$ and $C_{2}$ such that

α_{1} (‖ x ‖) \leq V (x) \leq α_{2} (‖ x ‖)

(12)

L V (x) \leq - C_{1} V (x) + C_{2}

(13)

where $α_{1} (\cdot)$ and $α_{2} (\cdot)$ are two class $k_{\infty}$ functions, then the following inequality holds for $\forall x_{0} \in R^{n}$ and $\forall t \geq t_{0}$

E [V (x)] \leq V (x_{0}) e^{- C_{1} t} + \frac{C_{2}}{C_{1}}

(14)

Controller design

In this section, a distributed controller based on the adaptive control method is proposed for the stochastic multi-agent system with actuator faults and stochastic disturbances to guarantee the mean square consensus.

First, the tracking error of follower $i$ is determined as

e_{i} = \sum_{j = 1}^{N} a_{ij} (x_{i} - x_{j}) + b_{i} (x_{i} - x_{0})

(15)

According to equations (6)–(8), the derivative of the tracking error becomes

\begin{array}{l} d e_{i} = \sum_{j = 1}^{N} a_{i j} (d x_{i} - d x_{j}) + b_{i} (d x_{i} - d x_{0}) \\ = [h_{i} (f_{i} (x_{i}) + ρ_{i} v_{i} + r_{i}) - c_{i} (f_{j} (x_{j}) + ρ_{j} v_{j} + r_{j}) - b_{i} f_{0} (x_{0})] d t \\ + [h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})] d ω \end{array}

(16)

where $h_{i} = c_{i} + b_{i}$ and $c_{i} = \sum_{j = 1}^{N} a_{ij}$ .

Also, the derivative of the tracking error can be rewritten as follows

d e_{i} = [h_{i} ρ_{i} v_{i} + F_{i} (Z_{i})] dt + [h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})] d ω

(17)

where $Z_{i} = [x_{i}, x_{j,} x_{0}]^{T}$ and function $F_{i} (Z_{i})$ is given in the following

F_{i} (Z_{i}) = h_{i} (f_{i} (x_{i}) + r_{i}) - c_{i} (f_{j} (x_{j}) + ρ_{j} v_{j} + r_{j}) - b_{i} f_{0} (x_{0})

(18)

Using RBFNN, unknown function $F_{i} (Z_{i})$ is estimated as follows

F_{i} (Z_{i}) = θ_{i}^{* T} φ_{i} (Z_{i}) + ε_{i} (Z_{i})

(19)

According to equations (17) and (19), the derivative of the tracking error can be rewritten as

d e_{i} = [h_{i} ρ_{i} v_{i} + θ_{i}^{* T} φ_{i} + ε_{i}] dt + [h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})] d ω

(20)

Therefore, the control input is defined as follows

v_{i} = - \frac{{\hat{K}}_{i}}{h_{i}} e_{i} v_{0 i}

(21)

where ${\hat{K}}_{i}$ is the approximation of $K_{i}^{*}$ defined as $K_{i}^{*} ρ_{i} = 1$ . Parameter $v_{0 i}$ is determined by

v_{0 i} = (μ_{i} + \frac{1}{2 τ_{i}} {\hat{W}}_{i})

(22)

in which $μ_{i}$ and $τ_{i}$ are positive constants. ${\hat{W}}_{i}$ is the approximation of $W_{i}^{*}$ defined as $W_{i}^{*} = {(θ_{i}^{* T} φ_{i})}^{2}$ .

The derivative of ${\hat{K}}_{i}$ and ${\hat{W}}_{i}$ is described as

{\overset{\cdot}{\hat{K}}}_{i} = γ'_{i} e_{i}^{2} v_{0 i} - γ'_{i} σ'_{i} {\hat{K}}_{i}

(23)

\dot{{\hat{W}}_{i}} = \frac{1}{2 τ_{i}} γ_{i} e_{i}^{2} - γ_{i} σ_{i} {\hat{W}}_{i}

(24)

where $γ'_{i}$ , $σ'_{i}$ , $γ_{i}$ , and $σ_{i}$ are positive constants.

Remark 4. The ideal constant weight vector is unknown and should be approximated. In addition, the parameters of Gaussian functions should be selected properly. It should be noted that the choosing of Gaussian function parameters is a time-consuming and difficult process. In common methods, the Gaussian functions are determined at first and then adaptive law is defined to estimate the ideal constant vector $θ^{*}$ . In Shahvali et al. (2020) and Hashemi and Shahgholian (2018), contrary to common methods, an adaptive law has been designed to estimate the parameter $π_{i}^{*} = Λ_{i} | | θ_{i}^{*} | |^{2}$ where $Λ_{i} \geq φ_{i}^{T} (Z) φ_{i} (Z)$ . Therefore, the mentioned problem for selecting the parameters of the Gaussian functions is solved by choosing a novel adaptive parameter. In this paper, using Lyapunov theory and Young’s inequality, the adaptive law (equation (24)) is designed to estimate $W_{i}^{*}$ which is defined as $W_{i}^{*} = (θ_{i}^{* T} φ_{i})^{2}$ . It should be emphasized that the conservative condition $Λ_{i} \geq φ_{i}^{T} (Z) φ_{i} (Z)$ is not used in this paper.

Remark 5. The bias fault is approximated by neural networks and the adaptive control method is used to compensate the loss of effectiveness fault. Thus, $K_{i}^{*}$ is defined as $K_{i}^{*} ρ_{i} = 1$ where $0 < ρ_{i} (t) \leq 1$ . It is obvious that $K_{i}^{*} ρ_{i} v_{i} = v_{i}$ . Therefore, the loss of effectiveness fault is compensated by applying $K_{i}^{*}$ . It should be noted that $K_{i}^{*}$ is unknown because the two types of actuator faults are considered to be unknown, or in other words, $ρ_{i}$ is unknown. Hence, the approximation of $K_{i}^{*}$ is determined as ${\hat{K}}_{i}$ and the adaptive law (equation (23)) is designed to approximate $K_{i}^{*}$ .

The Lyapunov function is selected as follows

V_{i} = \frac{1}{2} e_{i}^{2} + \frac{1}{2 γ_{i}} {\tilde{W}}_{i}^{2} + \frac{1}{2 {γ'}_{i}} ρ_{i} {\tilde{K}}_{i}^{2}

(25)

where ${\tilde{W}}_{i} = {\hat{W}}_{i} - W_{i}^{*}$ and ${\tilde{K}}_{i} = {\hat{K}}_{i} - K_{i}^{*}$ .

Using Definition 4, one has

\begin{matrix} L V_{i} = e_{i} [h_{i} ρ_{i} v_{i} + θ_{i}^{* T} φ_{i} + ε_{i}] + \frac{1}{2} {[h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})]}^{2} \\ + \frac{1}{γ_{i}} {\tilde{W}}_{i} \dot{{\hat{W}}_{i}} + \frac{1}{{γ'}_{i}} ρ_{i} {\tilde{K}}_{i} {\overset{\cdot}{\hat{K}}}_{i} \end{matrix}

(26)

By substituting equations (23) and (24) into equation (26), one can obtain

\begin{matrix} L V_{i} = e_{i} [h_{i} ρ_{i} v_{i} + θ_{i}^{* T} φ_{i} + ε_{i}] + \frac{1}{2} {[h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})]}^{2} \\ + \frac{1}{γ_{i}} {\tilde{W}}_{i} [\frac{1}{2 τ_{i}} γ_{i} e_{i}^{2} - γ_{i} σ_{i} {\hat{W}}_{i}] \\ + \frac{1}{{γ'}_{i}} ρ_{i} {\tilde{K}}_{i} [{γ'}_{i} e_{i} v_{0 i} - {γ'}_{i} {σ'}_{i} {\hat{K}}_{i}] \end{matrix}

(27)

The following inequalities can be written using Young’s inequality

e_{i} θ_{i}^{* T} φ_{i} \leq \frac{1}{2 τ_{i}} e_{i}^{2} W_{i}^{*} + \frac{1}{2} τ_{i}

(28)

e_{i} ε_{i} \leq \frac{1}{2} e_{i}^{2} + \frac{1}{2} ε_{i}^{2}

(29)

In addition, the following inequality can be written using Assumption 3

\frac{1}{2} {[h_{i} g_{i} (x_{i}) - c_{i} g_{j} (x_{j})]}^{2} \leq β_{i} G^{2}

(30)

where $β_{i}$ is defined as

\begin{matrix} β_{i} = [\sum_{l = 1}^{N} a_{i, l} \sum_{j = 1}^{N} a_{i, j} + b_{i} \sum_{j = 1}^{N} a_{i, j}] \\ - [\sum_{l = 1}^{N} a_{i, l} \sum_{j = 1}^{N} a_{i, j} + b_{i} \sum_{j = 1}^{N} a_{i, j}] + \frac{1}{2} b_{i}^{2} \end{matrix}

(31)

Thus, the following inequality can be written by substituting equations (28)–(30) into equation (27)

\begin{matrix} L V_{i} \leq e_{i} h_{i} ρ_{i} v_{i} + \frac{1}{2 τ_{i}} e_{i}^{2} W_{i}^{*} + \frac{1}{2} τ_{i} + \frac{1}{2} e_{i}^{2} + \frac{1}{2} ε_{i}^{2} + β_{i} G^{2} \\ + \frac{1}{2 τ_{i}} e_{i}^{2} {\tilde{W}}_{i} - σ_{i} {\tilde{W}}_{i} {\hat{W}}_{i} + ρ_{i} {\tilde{K}}_{i} e_{i} v_{0 i} \\ - ρ_{i} {σ'}_{i} {\tilde{K}}_{i} {\hat{K}}_{i} \end{matrix}

(32)

By substituting equations (21) and (22) into equation (32), one has

L V_{i} \leq - (μ_{i} - \frac{1}{2}) e_{i}^{2} + \frac{1}{2} ε_{i}^{2} + \frac{1}{2} τ_{i} + β_{i} G^{2} - σ_{i} {\tilde{W}}_{i} {\hat{W}}_{i} - ρ_{i} σ'_{i} {\tilde{K}}_{i} {\hat{K}}_{i}

(33)

The following inequalities can be written using Young’s inequality

{\tilde{W}}_{i} {\hat{W}}_{i} \leq \frac{3}{2} {\tilde{W}}_{i}^{2} + \frac{1}{2} W_{i}^{* 2}

(34)

{\tilde{K}}_{i} {\hat{K}}_{i} \leq \frac{3}{2} {\tilde{K}}_{i}^{2} + \frac{1}{2} K_{i}^{* 2}

(35)

Finally, by substituting equations (34) and (35) into equation (33), one can obtain

\begin{matrix} L V_{i} \leq - (μ_{i} - \frac{1}{2}) e_{i}^{2} - \frac{3}{2} σ_{i} {\tilde{W}}_{i}^{2} - \frac{3}{2} ρ_{i} σ'_{i} {\tilde{K}}_{i}^{2} + \frac{1}{2} ε_{i}^{* 2} + \frac{1}{2} τ_{i} \\ + β_{i} G^{2} - \frac{1}{2} σ_{i} W_{i}^{* 2} - \frac{1}{2} ρ_{i} σ'_{i} K_{i}^{* 2} \end{matrix}

(36)

where $| ε_{i} | \leq ε_{i}^{*}$ .

Remark 6: The methods of FTC are classified into two types which are active and passive FTC. In active FTC, the structure and parameters of controller are adjusted by the online information coming from fault diagnosis units (Benosman and Lum, 2009). Designing a proper controller to compensate the actuator faults is the main purpose of passive FTC (Benosman and Lum, 2009). Passive FTC does not need to change the controller structure and fault diagnosis units. It should be noted that passive FTC is the best method when actuator faults are not discoverable (Ren et al., 2021). Hence, the proposed controller is designed based on passive FTC. Therefore, one of the advantages of the proposed controller is that it can compensate actuator faults without the fault detection and isolation unit.

Remark 7: For compensating actuator faults, each follower only requires local information. Thus, the distributed FTC is used in the design of the offered controller. It should be mentioned that the distributed FTC has many advantages over the centralized FTC. Communication saving, scalability, and robustness are the advantages of distributed FTC (Qin et al., 2016).

Stability analysis

In this section, the stability of the proposed controller is investigated. Hence, Theorem 1 is presented for the consensus problem of the heterogeneous multi-agent system in the presence of actuator faults and stochastic disturbances.

Theorem 1: Under Assumptions 1–4, the multi-agent system defined in equations (6) and (7) with the control input (equation (21)) reaches the mean square consensus if the inequality $μ_{i} \geq \frac{1}{2}$ holds for $i = 1, \dots, N$ .

Proof: The Lyapunov function is considered as follows

V = \sum_{i = 1}^{N} V_{i}

(37)

By substituting equation (36) into equation (37), one has

\begin{matrix} L V \leq \sum_{i = 1}^{N} - (μ_{i} - \frac{1}{2}) e_{i}^{2} - \frac{3}{2} σ_{i} {\tilde{W}}_{i}^{2} - \frac{3}{2} ρ_{i} {σ'}_{i} {\tilde{K}}_{i}^{2} + \frac{1}{2} ε_{i}^{* 2} + \frac{1}{2} τ_{i} \\ + β_{i} G^{2} - \frac{1}{2} σ_{i} W_{i}^{* 2} - \frac{1}{2} ρ_{i} {σ'}_{i} K_{i}^{* 2} \end{matrix}

(38)

Parameters $C_{1}$ and $C_{2}$ are selected as follows

C_{1} = min_{i = 1, \dots, N} {2 (μ_{i} - \frac{1}{2}), 3 σ_{i} γ_{i}, 3 {σ'}_{i} {γ'}_{i}}

(39)

C_{2} = \sum_{i = 1}^{N} {\frac{1}{2} ε_{i}^{* 2} + \frac{1}{2} τ_{i} + β_{i} G^{2} - \frac{1}{2} σ_{i} W_{i}^{* 2} - \frac{1}{2} ρ_{i} {σ'}_{i} K_{i}^{* 2}}

(40)

Therefore, equation (38) can be rewritten as

L V (x) \leq - C_{1} V (x) + C_{2}

(41)

According to Lemma 2, the following inequality can be written as

E [V (x)] \leq V (0) e^{- C_{1} t} + \frac{C_{2}}{C_{1}}, \forall t > 0

(42)

The tracking error vector is determined as $e = [e_{1}, e_{2}, \dots, e_{N}]^{T}$ . Therefore, equation (15) can be rewritten as follows

e = (L + B) (x - 1_{N} x_{0})

(43)

where $x = [x_{1}, x_{2}, \dots, x_{N}]^{T}$ and $1_{N} = [1, 1, \dots, 1]^{T} \in R^{N}$ .

By considering Lemma 1 and equation (43), one can obtain

E {{‖ x - 1_{N} x_{0} ‖}^{2}} \leq ‖ {(L + B)}^{- 1} ‖^{2} E {{‖ e ‖}^{2}}

(44)

Besides, the following inequality can be written using norm properties and equation (25)

E {{‖ e ‖}^{2}} \leq 2 E {V}

(45)

By combining equations (42), (44), and (45), the following inequality can be written as

E {{‖ x - 1_{N} x_{0} ‖}^{2}} \leq 2 V (0) e^{- C_{1} t} + \frac{2 C_{2}}{C_{1}}

(46)

According to equation (40), it can be concluded that if parameters are correctly selected then $C_{2}$ will be a small positive constant. As a result, for any $ϵ' > 0$ , there exists a finite time $T (x (0), ϵ')$ such that the following inequality holds

E {{‖ x - 1_{N} x_{0} ‖}^{2}} < ϵ', \forall t > T (x (0), ϵ')

(47)

Therefore, according to Definition 3, the multi-agent system (equations (6) and (7)) reaches the mean square consensus in the presence of two types of actuator faults and stochastic disturbances. As a result, the proof of Theorem 1 is completed.▪

At the end of this section, to better understand the design procedure of the proposed controller, its structure for the multi-agent system in the presence of actuator faults and stochastic disturbances is shown in Figure 1.

Figure 1.

The distributed control structure for the multi-agent system.

Simulation results

In this section, two simulation examples are given to prove the effectiveness of the offered controller. For both examples, five followers and one leader are considered. The communication topology of multi-agent system is depicted in Figure 2. In Figure 2, L and F represent the leader and follower, respectively. As depicted in Figure 2, the graph contains a directed spanning tree and only Follower 1 has the access to the leader’s information. For both simulation examples, the simulation time and the simulation step are 60 and 0.001 seconds, respectively.

Figure 2.

Communication topology.

Example 1: In this example, the leader dynamics is modeled as follows (Ren et al., 2016)

{\overset{\cdot}{x}}_{0} = \frac{\sqrt{2}}{2} \sin (t)

The dynamics of the followers are also considered as follows

d x_{1} = (x_{1} + x_{1}^{2} + u_{1}) dt + (0.8 e^{- t} \cos (x_{1})) d ω

d x_{2} = (x_{2}^{1 / 3} + u_{2}) dt + (0.7 e^{- 0.8 t} \sin (x_{2} / 3)) d ω

d x_{3} = (x_{3} e^{- 0.5 x_{3}} + u_{3}) dt + (0.1 co s^{2} (x_{3})) d ω

d x_{4} = (x_{4}^{4} + u_{4}) dt + (0.1 \sin (x_{4})) d ω

d x_{5} = (| x_{5} | + u_{5}) dt + (0.2 e^{- 0.5 t} + 0.1 \sin (x_{5} / 2)) d ω

In this example, the actuator faults are assumed as follows

u_{1} (t) = 0.2 v_{1} (t) + 1, t \geq 0

u_{2} (t) = {\begin{matrix} 0.2 v_{2} (t), t < 40 \\ 0.1 v_{2} (t) + 2, t \geq 40 \end{matrix}

Initial conditions of the leader and the followers are selected as $x_{0} (0) = 0.5$ and $x (0) = [0.5, 0.9, 0.1, 0.6, 0.1]$ . To illustrate the influence of stochastic disturbances on the system, the open-loop trajectory of multi-agent system is shown in Figure 3. As shown in Figure 3, the agents have not reached the consensus. Thus, the proposed controller is applied to multi-agent system, and the simulation results of this example are shown in Figures 4 –6 which represent the controller performance. The states of the leader and the followers are shown in Figure 4.

Figure 3.

The open-loop trajectory of multi-agent system.

Figure 4.

States of the leader and the followers.

Figure 5.

Outputs of faulty actuators.

Figure 6.

Adaptive parameters.

In Figure 5, the outputs of faulty actuators are shown. Adaptive parameters are shown in Figure 6. From Figures 4 –6, it can be seen that the mean square consensus of heterogeneous multi-agent system is accomplished in the presence of actuator faults and stochastic disturbances.

Example 2: The leader dynamics are defined as follows (Ren et al., 2016)

m_{0} {\overset{\cdot\cdot}{q}}_{0} + d_{0} {\overset{\cdot}{q}}_{0} + k_{0} q_{0} = u_{0}

where ${\overset{\cdot}{q}}_{0}$ and $q_{0}$ are the angular speed and the angular, respectively. The control input for the leader is given by

u_{0} = (k_{0} - 4 m_{0}) \sin (2 t) + 2 d_{0} \cos (2 t)

The follower dynamics are considered as

d q_{1, i} = (q_{2, i} + u_{1 i}) dt + (0.1 \cos (q_{1, i}) + 0.1 \sin (q_{1, i})) d ω

\begin{matrix} d q_{2, i} = J_{i}^{- 1} (B_{i}^{r} q_{2, i} - M_{i} g l_{i} \sin (q_{1, i}) + u_{2 i}) dt \\ + (l_{i} co s^{2} (q_{2, i} / 4) - l_{i} si n^{2} (q_{1, i} / 4)) d ω \end{matrix}

where $q_{i} = [q_{1, i}, q_{2, i}]^{T}$ is the state vector. Parameters $J_{i}$ , $B_{i}^{r}$ , $M_{i}$ , $l_{i}$ , and $g$ are the total inertia of the motor and the link, the total mass, the damping coefficient, the distance from the link center of mass to the join axis, and the gravitational acceleration, respectively.

The values of the system parameters are determined in the following equation

m_{0} = M_{i} = 1, d_{0} = 1, k_{0} = 0.5, B_{i}^{r} = 1.5, g = 0.98

{[l_{1}, l_{2}, l_{3}, l_{4}, l_{5}]}^{T} = {[0.1 0.1 0.1 0.8 0.9]}^{T}

{[J_{1}, J_{2}, J_{3}, J_{4}, J_{5}]}^{T} = [0.7 0.5 0.1 0.3 0.25]^{T}

For this example, actuator faults are considered as follows

u_{2, 2} (t) = 0.1 v_{2, 2} (t) + 2, t \geq 0

u_{2, 3} (t) = {\begin{matrix} v_{2, 3} (t), t < 10 \\ 0.4 v_{2, 3} (t) + 1, t \geq 10 \end{matrix}

In the simulation, initial conditions are selected as

\begin{matrix} q_{0} (0) = 1.5, {\overset{\cdot}{q}}_{0} (0) = 1, q_{1} (0) = [1.2, 0.15, 1.2, 0.9, 0.7], \\ q_{2} (0) = [1.5, 0.1, 0.9, 0.8, 1.2] \end{matrix}

In Figures 7 –9, the simulation results are displayed. Figures 7 and 8 show the states of the leader and the followers. From Figures 7 and 8, it can be concluded that all the followers accurately track the leader of multi-agent system.

Figure 7.

Trajectories of $q_{0}$ and $q_{1, i}$

Figure 8.

Trajectories of ${\overset{\cdot}{q}}_{0}$ and $q_{2, i}$

Figure 9.

Outputs of faulty actuators.

The outputs of the two faulty actuators are shown in Figure 9. According to Figures 7 –9, the heterogeneous multi-agent system with two types of actuator faults and stochastic disturbances has reached the mean square consensus. As a result, the efficiency of the theoretical analysis is proved using simulation results.

Conclusion

The consensus problem of heterogeneous multi-agent systems in the presence of unknown and time-varying actuator faults and stochastic disturbances has been studied in this paper. A distributed controller has been designed based on the adaptive control method and RBFNNs. RBFNNs have been used to approximate the unknown nonlinear uncertainties. These uncertainties include unknown nonlinear functions and actuator faults. The distributed controller has compensated two types of actuator faults without the fault detection and isolation unit. By applying the proper Lyapunov functions and Young’s inequality, it has been proved that all the agents with heterogeneous dynamics reach the mean square consensus in the presence of actuator faults and stochastic disturbances. Finally, the simulation results demonstrate the ability of the proposed control method in the consensus problem of heterogeneous multi-agent systems. Future works will be focused on the distributed control of time-delay systems with actuator faults and stochastic disturbances.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Mohsen Shafieirad

Seyed Ali Zahiripour

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