Synchronous fault-tolerant consensus control and disturbance rejection for leader-following multi-agent systems with fault of polynomial form

Abstract

This article studies the design problem of fault-tolerant and disturbance rejection control for leader-following multi-agent systems with polynomial form fault. First, a novel observer is designed to estimate polynomial form faults and external disturbances in the tracker. Second, a fault-tolerant control protocol is designed for each follower based on the obtained estimates; the designed distributed controller can effectively achieve both the goal of fault-tolerant control and the effect of disturbance rejection. Finally, a simulation result of a practical example is presented to verify the feasibility of the proposed scheme.

Keywords

Multi-agent systems fault-tolerant control observer disturbance rejection consensus

Introduction

In the past couple of decades, the consensus control problem of multi-agent systems (MASs) has received extensive attention.^1–9 The consensus control method of MASs has been widely used in biology, physics, and engineering and other related disciplines, such as the attitude alignment of satellites, the collaborative control of unmanned aircraft, and the analysis of animal aggregation behavior.^10–17 However, the problems of actuator fault and external disturbance cannot be avoided during the operation of the system. Once the above problems occur, the entire mission may fail. Therefore, in this article, the problems of the disturbance rejection performance and fault-tolerant control (FTC) for MASs have become research hotspots.

FTC problems can be roughly divided into two categories according to different design ideas: passive FTC^18,19 and active FTC.^20,21 In passive FTC, the controller has fixed parameters and structure, the controller is designed for a specific type of fault, and it does not need to detect the fault information online. In active FTC, there are components that can detect and identify faults automatically, and actively react to failures by reconfiguring control actions. Wang et al.²² studied the problem of fully distributed inclusion control for MASs with dual integrator dynamics-directed topology. Gong et al.²³ investigated the problem of fault-tolerant consistency control for non-homogeneous nonlinear fractional multi-intelligent body systems with general-directed topology. Yadegar and Meskin²⁴ proposed an adaptive FTC scheme based on a virtual actuator framework, which takes into account the time-varying additive actuator failures in nonlinear heterogeneous multi-agent systems (HMASs).

On the other hand, in the actual project operation, the external disturbances have different degrees of influence on the performance of the whole system, which will further increase the management complexity of the whole system and also affect the stability of the whole system, and even lead to the failure of the whole project operation. The method to deal with such problems is to construct a disturbance estimation observer to estimate the disturbance, and then design the corresponding controller based on some estimated values obtained from the observer, so as to produce suppression of external disturbances. Han et al.’s²⁵ control method based on disturbance observer estimates the disturbance of external system and deals with the control problem of MASs with external disturbance. Pei²⁶ studied the consensus tracking problem for HMASs with interference and directed graphs. Wang et al.²⁷ developed an interference observer-based method to estimate the interference generated by external systems.

Guided by the research described above, in this article, we consider design problem of fault-tolerant and disturbance rejection control for leader-following MASs with fault in polynomial form. The main contributions of this article are generalized as follows: (1) a neoteric observer is designed to get estimations of faults and disturbances in followers. To be more general, faults which are in followers can be represented as a polynomial function of time and assume that the unknown perturbation is a harmonic perturbation. (2) Each follower is designed with an FTC protocol according to the obtained estimates. The designed distributed controller can not only effectively accomplish the goal of FTC, but also effectively suppress the influence of disturbance.

The rest of this article is arranged as follows: In section “Preliminaries and problem formulation,” some necessary basic knowledge is stated and the research question is elaborated. In section “Main results,” the main design approach and results are presented. In section “Simulation results,” a simulation result of a practical example is presented to verify the feasibility of the proposed scheme. Finally, the conclusions are given in section “Conclusion.”

Notations

$R^{n \times m}$ stands for the $n \times m$ matrix. $I$ denotes a unit matrix with the appropriate number of dimensions, $0$ denotes the zero matrix. $A^{T}$ denotes the transposed matrix of $A$ . $A \otimes B$ is defined as the Kronecker product of matrices $A, B$ . The matrix $diag {\dots}$ denotes as a diagonal block matrix.

Preliminaries and problem formulation

Graph theory

Consider a network consisting of $n$ agents, the network topology is represented by the graph $G = (V, E)$ , where $V = {v_{1}, v_{2}, \dots, v_{n}}$ is the node set, $E \subseteq V \times V$ is the edge set, denote the neighbor node set of agent $i$ as $N_{i} = {v_{i} \in V$ : $(v_{i}, v_{j}) \in E}$ . $A = [a_{ij}] \in R^{n \times n}$ is the associated adjacency matrix of $G$ . If $(v_{i}, v_{j}) \in E$ , then $a_{ij} = 1$ , otherwise $a_{ij} = 0$ . For undirected graph $G$ , $a_{ij} = a_{ji}$ . Denote the degree matrix of a node as $D = diag {d_{1}, d_{2}, \dots, d_{n}}$ , where $d_{i} = \sum_{j = 1, j \neq i}^{n} a_{ij}$ . Then the Laplacian matrix $L$ of the graph $G$ can be defined as $L = D - A$ . Define $B = diag {b_{1}, \dots, b_{n}}$ as the leader adjacency matrix related to $G$ , if agent $i$ can receive the leader’s information, then $b_{i} > 0$ , otherwise $b_{i} = 0$ .

System description

Consider the following MASs. The dynamic behavior of the leader is as follows

{\begin{matrix} {\overset{\cdot}{x}}_{0} (t) = A_{0} x_{0} (t) \\ y_{0} (t) = C_{0} x_{0} (t) \end{matrix}

(1)

and the dynamic behavior of the $i th$ agent is represented as follows

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = A_{0} x_{i} (t) + B_{u} u_{i} (t) + B_{d} d_{i} (t) + E_{f} f_{i} (t) \\ y_{i} (t) = C_{0} x_{i} (t), i = 1, 2 \dots, n \end{matrix}

(2)

where $A_{0} \in R^{n \times n}, B_{u} \in R^{n \times m}, B_{d} \in R^{n \times m}, E_{f} \in R^{n \times m}, C_{0} \in R^{p \times n},$ while $x (t) \in R^{n}$ is the state, the control input and output of each agent are denoted by $u (t) \in R^{m}$ and $y (t) \in R^{p}$ , respectively, $d (t) \in R^{q}$ is the external disturbance, $f_{i} (t) \in R^{q}$ represents the fault of the $i th$ follower.

In this article, the fault $f_{i} (t)$ is represented by a polynomial function of time as follows

f_{i} (t) = k_{0} + k_{1} t + k_{2} t^{2} + \dots + k_{m - 1} t^{m - 1}

(3)

where $k_{i}$ are unknown constants, and $m$ is known.

$f^{(i)}$ is defined as the $i th$ derivative of $f$ . Equation (3) can be reformulated as follows

\underset{[\begin{matrix} \overset{\cdot}{z} \\ \overset{\cdot}{f} \end{matrix}]}{\underset{︸}{[\begin{matrix} f^{(m)} \\ f^{(m - 1)} \\ f^{(m - 2)} \\ ⋮ \\ f^{(2)} \\ f^{(1)} \end{matrix}]}} = \underset{[\begin{matrix} {\bar{A}}_{1} & 0 \\ {\bar{A}}_{2} & 0 \end{matrix}]}{\underset{︸}{[\begin{matrix} 0 & 0 & \dots & 0 & 0 & 0 \\ I & 0 & \dots & 0 & 0 & 0 \\ 0 & I & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & I & 0 & 0 \\ 0 & 0 & \dots & 0 & I & 0 \end{matrix}]}} \underset{[\begin{matrix} z \\ f \end{matrix}]}{\underset{︸}{[\begin{matrix} f^{(m - 1)} \\ f^{(m - 2)} \\ f^{(m - 3)} \\ ⋮ \\ f^{(1)} \\ f \end{matrix}]}}

(4)

From equation (4), equation (2) can be reformulated as follows

{\begin{matrix} \begin{matrix} {\overset{\cdot}{x}}_{i} (t) = A_{0} x_{i} (t) + B_{u} u_{i} (t) + B_{d} d_{i} (t) + E_{f} f_{i} (t) \\ y_{i} (t) = C_{0} x_{i} (t) \\ {\overset{\cdot}{z}}_{i} (t) = {\bar{A}}_{1} {\overset{\cdot}{z}}_{i} (t) \\ {\overset{\cdot}{f}}_{i} (t) = {\bar{A}}_{2} {\overset{\cdot}{z}}_{i} (t) \end{matrix} \end{matrix}

(5)

The unknown external interference $d (t)$ is produced by a system described by

\begin{matrix} {\overset{\cdot}{ω}}_{i} (t) = \tilde{W} ω_{i} (t) \\ d_{i} (t) = \tilde{V} ω_{i} (t) \end{matrix}

(6)

where $\tilde{W} \in R^{p \times p}, \tilde{V} \in R^{m \times p}$ , while $ω_{i} (t)$ represents the state variable.

Assumption 1

The undirected topology graph $G$ is connected, and the information sent by the leader can be received by at least one follower.

Assumption 2

$(A_{0}, B_{u})$ is controllable, $(\tilde{W}, B_{d} \tilde{V})$ is observable.

Assumption 3

Rank $(B_{u}, E_{f})$ = rank $(B_{u})$ .

Remark 1

The faults considered in this article are time-varying polynomial faults. In practical systems, external disturbances are unavoidable, so we also introduce disturbances in the system modeling process. Usually, the system may have both faults and external disturbances, so in the next part, we will design observers to observe faults and disturbances simultaneously.

Main results

In this part, the problem of fault-tolerant consensus for system (2) with disturbance and polynomial fault will be researched. First, the design method of the observer will be given, and then based on each estimate obtained, a distributed fault-tolerant controller will be designed to achieve the consistency of the leader-follower.

Estimation observer design

The leader is defined to be fault-free, so the design of the observer is only for each follower. Consider the following fault estimation observer for system (2)

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i} (t) = A_{0} {\hat{x}}_{i} (t) + B_{u} u_{i} (t) + B_{d} {\hat{d}}_{i} (t) + E_{f} {\hat{f}}_{i} (t) - L_{1} ζ_{i} (t) \\ {\overset{\cdot}{\hat{z}}}_{i} (t) = {\bar{A}}_{1} {\hat{z}}_{i} (t) - L_{2} ζ_{i} (t) \\ {\overset{\cdot}{\hat{ω}}}_{i} (t) = \tilde{W} {\hat{ω}}_{i} (t) - L_{3} ζ_{i} (t) \\ {\overset{\cdot}{\hat{f}}}_{i} (t) = {\bar{A}}_{2} {\hat{z}}_{i} (t) - L_{4} ζ_{i} (t) \end{matrix}

(7)

where $L_{i}$ is the unknown observer gain matrix, the relative output error can be written as follows

\begin{matrix} ζ_{i} = \sum_{j \in N_{i}} a_{ij} (({\hat{y}}_{i} (t) - y_{i} (t)) - ({\hat{y}}_{j} (t) - y_{j} (t))) \\ + b_{i} (({\hat{y}}_{i} (t) - y_{i} (t)) - ({\hat{y}}_{0} (t) - y_{0} (t))) \end{matrix}

(8)

where $a_{ij}, b_{i}$ are elements of the adjacency matrix $A$ and $B$ , ${\hat{x}}_{i} (t)$ is defined as the estimation of $x_{i} (t)$ , ${\hat{z}}_{i} (t)$ is defined as the estimation of $z_{i} (t)$ , ${\hat{ω}}_{i} (t)$ is defined as the estimation of $ω_{i} (t)$ , ${\hat{f}}_{i} (t)$ is defined as the estimation of $f_{i} (t)$ , ${\hat{y}}_{i} (t)$ is defined as the estimation of $y_{i} (t)$ . For simplicity, the state of the leader is known by default, which is ${\hat{x}}_{0} (t) = x_{0} (t)$ , thus obtain ${\hat{y}}_{0} (t) = y_{0} (t)$ .

According to the estimate of the $i th$ follower, let $e_{xi} = {\hat{x}}_{i} (t) - x_{i} (t)$ , $e_{zi} = {\hat{z}}_{i} (t) - z_{i} (t)$ , $e_{ω i} = {\hat{ω}}_{i} (t) - ω_{i} (t)$ , and $e_{fi} = {\hat{f}}_{i} (t) - f_{i} (t)$ represent the estimation error.

Let augmented estimation error ${\bar{e}}_{i} = {[\begin{matrix} e_{xi}^{T} & e_{zi}^{T} & e_{ω i}^{T} & e_{fi}^{T} \end{matrix}]}^{T}$ , the error system for follower $i$ can be obtained by equations (5)–(7)

{\begin{matrix} {\overset{\cdot}{e}}_{xi} = {\overset{\cdot}{\hat{x}}}_{i} (t) - {\overset{\cdot}{x}}_{i} (t) = A_{0} e_{xi} + B_{u} \tilde{V} e_{ω i} + E_{f} e_{fi} - L_{1} ζ_{i} (t) \\ {\overset{\cdot}{e}}_{zi} = {\overset{\cdot}{\hat{z}}}_{i} (t) - {\overset{\cdot}{z}}_{i} (t) = {\bar{A}}_{1} e_{zi} - L_{2} ζ_{i} (t) \\ {\overset{\cdot}{e}}_{ω i} = {\overset{\cdot}{\hat{ω}}}_{i} (t) - {\overset{\cdot}{ω}}_{i} (t) = \tilde{W} e_{ω i} - L_{3} ζ_{i} (t) \\ {\overset{\cdot}{e}}_{fi} = {\overset{\cdot}{\hat{f}}}_{i} (t) - {\overset{\cdot}{f}}_{i} (t) = {\bar{A}}_{2} e_{zi} - L_{4} ζ_{i} (t) \end{matrix}

(9)

Further summarized and sorted out

{\overset{\cdot}{\bar{e}}}_{i} = \underset{\bar{A}}{\underset{︸}{[\begin{matrix} A_{0} & 0 & B_{d} \cdot \tilde{V} & E_{f} \\ 0 & {\bar{A}}_{1} & 0 & 0 \\ 0 & 0 & \tilde{W} & 0 \\ 0 & {\bar{A}}_{2} & 0 & 0 \end{matrix}]}} \bar{e} + \underset{L}{\underset{︸}{[\begin{matrix} L_{1} \\ L_{2} \\ L_{3} \\ L_{4} \end{matrix}]}} ζ_{i}

(10)

Therefore, the augmented error system is as follows

{\overset{\cdot}{\bar{e}}}_{i} = \bar{A} {\bar{e}}_{i} - L ζ_{i}

(11)

Define $\bar{e} = [{\bar{e}}_{1}^{T}, \dots, {\bar{e}}_{n}^{T}]^{T}$ . Then the global error system can be expressed as follows

\overset{\cdot}{\bar{e}} = (I_{n} \otimes \bar{A} - (L + B) \otimes (L \bar{C})) \bar{e}

(12)

where $\bar{C} = [\begin{matrix} C_{0} & 0 & 0 & 0 \end{matrix}]$ .

As of now, the global estimation error system can be obtained. The sufficient condition to ensure the convergence of system (12) will be given in the following sections.

Theorem 1

Consider MASs (1) and (2) under the conditions of Assumption 1, if a positive definite matrix $P_{1}$ and matrix $R$ exists, the following inequalities are satisfied

\begin{matrix} I_{n} \otimes (P_{1} \bar{A} + {\bar{A}}^{T} P_{1}) - (L + B) \otimes \\ R \bar{C} - (L + B)^{T} \otimes ({\bar{C}}^{T} R^{T}) < 0 \end{matrix}

(13)

where $R = P_{1} L$ .

Then system (12) with fault estimation observer is asymptotically stable.

Proof

Consider the Lyapunov function as follows

\bar{V} = {\bar{e}}^{T} (I_{n} \otimes P_{1}) \bar{e}

(14)

The derivative of the above function $\bar{V}$ along equation (12) with respect to time is calculated as follows

\begin{matrix} \overset{\cdot}{\bar{V}} = 2 {\bar{e}}^{T} (I_{n} \otimes P_{1}) (I_{n} \otimes \bar{A} - (L + B) \otimes L \bar{C}) \bar{e} \\ {\bar{e}}^{T} (I_{n} \otimes (P_{1} \bar{A} + {\bar{A}}^{T} P_{1}) - (L + B) \otimes P_{1} L \bar{C} \\ - {(L + B)}^{T} \otimes ({\bar{C}}^{T} L^{T} P_{1})) \bar{e} \end{matrix}

(15)

According to equation (13), one can obtain $\overset{\cdot}{\bar{V}} < 0$ . Consequently, the global estimation error system (12) is asymptotically stable. This proof has been completed.

Remark 2

In this section, an observer is designed, which is used to estimate faults and interference simultaneously. At the next stage, a control protocol will be designed in order that can guarantee the stability of the system when faults and disturbances exist simultaneously.

Fault-tolerant controller design

The control protocol will be introduced in this section. The relative output information among the neighbors of the $i th$ follower can be written as follows

V_{i} = \sum_{j \in N_{i}} a_{ij} (y_{i} (t) - y_{j} (t)) + b_{i} (y_{i} (t) - y_{0} (t))

(16)

Assumption 3 shows that $span (E_{f}) \subseteq span (B_{u})$ , that is, there exists a nonzero transformation matrix $E_{f}$ $\in R^{n \times m}$ such that $E_{f} = B_{u} E_{f}$ . And since $B_{u}$ is a full-rank matrix, the generalized inverse of $B_{u}$ exists and is unique, that is, there is a matrix $B^{*} \in R^{m \times n}$ ²⁸ such that

(I_{n} - B_{u} B^{*}) E_{f} = 0 .

(17)

Since the fault observer of the design can estimate the disturbance and fault at the same time, the FTC law based on the fault observer is as follows

u_{i} (t) = K_{v} V_{i} - B^{*} E_{f} {\hat{f}}_{i} (t) - B^{*} B_{d} {\hat{d}}_{i} (t)

(18)

where ${\hat{f}}_{i} (t)$ is the fault estimation, ${\hat{d}}_{i} (t)$ is the disturbances estimation, $K_{v}$ is the feedback gain matrix.

Substituting equation (18) into equation (2) yields

\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = A_{0} x_{i} (t) + B_{u} K_{v} V_{i} - B_{u} B^{*} E_{f} {\hat{f}}_{i} (t) - B_{u} B^{*} B_{d} {\hat{d}}_{i} (t) \\ + E_{f} f_{i} (t) + B_{d} d_{i} (t) \\ = A_{0} x_{i} (t) + B_{u} K_{v} V_{i} - E_{f} e_{fi} - B_{d} e_{di} \end{matrix}

(19)

Define $e_{i} = x_{i} (t) - x_{0} (t)$ . From equations (1) and (19), the closed-loop dynamics system is as follows

\begin{matrix} {\overset{\cdot}{e}}_{i} = A_{0} e_{i} + B_{u} K_{v} V_{i} - E_{f} e_{fi} - B_{d} e_{di} \end{matrix}

(20)

Let $e = [e_{1}^{T}, \dots, e_{n}^{T}]$ . Then the global tracking error system can be written as follows

\begin{matrix} \overset{\cdot}{e} = (I_{n} \otimes A_{0} + (L + B) \otimes B_{u} K_{v} C_{0}) e \\ - I_{n} \otimes E_{f} e_{f} - I_{n} \otimes B_{d} e_{d} \end{matrix}

(21)

Combining equations (12) and (21)

[\begin{matrix} \overset{\cdot}{\bar{e}} \\ \overset{\cdot}{e} \end{matrix}] = [\begin{matrix} I_{n} \otimes \bar{A} - (L + B) \otimes (L \bar{C}) & 0 \\ I_{n} \otimes \bar{B} & I_{n} \otimes A_{0} + (L + B) \otimes B_{u} K_{v} C_{0} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}]

(22)

where $\bar{B} = [\begin{matrix} 0 & 0 & - B_{d} \cdot \tilde{V} & - E_{f} \end{matrix}]$ .

Next, sufficient conditions for the leader equation (1) and the follower equation (2) in a MASs to reach consensus will be given.

Theorem 2

Considering MASs (1) and (2) under the condition of FTC protocol equation (18), if there are positive definite matrices $P_{2}$ and $K_{v}$ , the following inequalities are satisfied

[\begin{matrix} Φ & {(L + B)}^{T} \otimes {\bar{B}}^{T} P_{2} \\ (L + B) \otimes P_{2} \bar{B} & Ψ \end{matrix}] < 0

(23)

where $Ψ = (L + B) \otimes (P_{2} A_{0} + A_{0}^{T} P_{2}) + (L + B)^{2} \otimes$ $(P_{2} B_{u} K_{v} C_{0} + (P_{2} B_{u} K_{v} C_{0})^{T})$ , $Φ = I_{n} \otimes (P_{1} \bar{A} + {\bar{A}}^{T} P_{1}) -$ $(L + B) \otimes (R \bar{C} + {\bar{C}}^{T} R^{T})$ , and $R = P_{1} L$ .

Then the closed-loop system (22) is stable.

Proof

Let

\tilde{V} = {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} I_{n} \otimes P_{1} & 0 \\ 0 & (L + B) \otimes P_{2} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}]

(24)

The derivative of the above function $\tilde{V}$ along the closed-loop system (22) with respect to time is calculated as follows

\begin{matrix} \overset{\cdot}{\tilde{V}} = {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} I_{n} \otimes P_{1} & 0 \\ 0 & (L + B) \otimes P_{2} \end{matrix}] [\begin{matrix} \overset{\cdot}{\bar{e}} \\ \overset{\cdot}{e} \end{matrix}] \\ + {[\begin{matrix} \overset{\cdot}{\bar{e}} \\ \overset{\cdot}{e} \end{matrix}]}^{T} [\begin{matrix} I_{n} \otimes P_{1} & 0 \\ 0 & (L + B) \otimes P_{2} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}] \\ = {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} I_{n} \otimes P_{1} & 0 \\ 0 & (L + B) \otimes P_{2} \end{matrix}] \\ [\begin{matrix} Φ_{1} & 0 \\ I_{n} \otimes \bar{B} & Ψ_{1} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}] \\ + {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} Φ_{1}^{T} & I_{n} \otimes {\bar{B}}^{T} \\ 0 & Ψ_{1}^{T} \end{matrix}] [\begin{matrix} I_{n} \otimes P_{1} & 0 \\ 0 & (L + B) \otimes P_{2} \end{matrix}] \\ [\begin{matrix} \bar{e} \\ e \end{matrix}] \\ = {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} Φ_{2} & 0 \\ (L + B) \otimes P_{2} \tilde{B} & Ψ_{2} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}] \\ + {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} Φ^{T} & (L + B) \otimes {\tilde{B}}^{T} P_{2} \\ 0 & Ψ_{2}^{T} \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}] \\ = {[\begin{matrix} \bar{e} \\ e \end{matrix}]}^{T} [\begin{matrix} Φ & {(L + B)}^{T} \otimes {\bar{B}}^{T} P_{2} \\ (L + B) \otimes P_{2} \bar{B} & Ψ \end{matrix}] [\begin{matrix} \bar{e} \\ e \end{matrix}] \end{matrix}

(25)

where ${\begin{array}{l} Φ_{1} = I_{n} \otimes \bar{A} - (ℒ + ℬ) \otimes L \bar{C}, \\ Ψ_{1} = I_{n} \otimes A_{0} + (ℒ + ℬ) \otimes B_{u} K_{v} C_{0}, \\ Φ_{2} = I_{n} \otimes P_{1} \bar{A} - (ℒ + ℬ) \otimes R \bar{C}, \\ Ψ_{2} = (ℒ + ℬ) \otimes P_{2} A_{0} + {(ℒ + ℬ)}^{2} \otimes P_{2} B_{u} K_{v} C_{0} \end{array}$ .

According to equation (24), one can obtain $\overset{\cdot}{\tilde{V}} < 0$ . Therefore, the consensus can be achieved between the leader and the followers. This proof has been completed.

Simulation results

In this section, an aircraft model in the work by Yang et al.²⁹ will be used to demonstrate the feasibility of the scheme described in this article; this aircraft model consists of one leader and four followers. The parameters are given as follows

A_{0} = [\begin{matrix} - 0.0366 & 0.0271 & 0.0188 & - 0.4555 \\ 0.0482 & - 1.0100 & 0.0024 & - 4.0208 \\ 0.1002 & 0.3681 & - 0.7070 & 1.4200 \\ 0 & 0 & 1.0000 & 0 \end{matrix}]

\begin{matrix} x_{1} (0) & = [\begin{matrix} - 0.1000 \\ 2.5000 \\ - 2.5000 \\ 0.5000 \end{matrix}], & x_{2} (0) & = [\begin{matrix} 0.8745 \\ - 0.4000 \\ 0.6071 \\ - 0.1090 \end{matrix}] \\ x_{3} (0) & = [\begin{matrix} 0.1557 \\ 0.8193 \\ 0.3083 \\ 0.3800 \end{matrix}], & x_{4} (0) & = [\begin{matrix} 2.9000 \\ - 2.5000 \\ - 2.5000 \\ - 1.4000 \end{matrix}] \\ B_{u} & = [\begin{matrix} 0.1761 \\ 7.5922 \\ 4.4900 \\ 0 \end{matrix}], C_{0} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ B_{d} & = {[\begin{matrix} 0.1 & 0.1 & 0.1 & 0.1 \end{matrix}]}^{T} \end{matrix}

where $x_{1} (0), x_{2} (0), x_{3} (0)$ , and $x_{4} (0)$ represent the initial state of each multi-agent, respectively.

The interconnection topology of the aircraft is displayed in Figure 1; from this figure, the Laplacian matrix $L$ and the leader adjacency matrix $B$ can be obtained as follows

L = [\begin{matrix} 2 & 0 & - 1 & - 1 \\ 0 & 1 & - 1 & 0 \\ - 1 & - 1 & 2 & 0 \\ - 1 & 0 & 0 & 1 \end{matrix}], B = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

To verify the stability of the system under external disturbances, $d_{i} (t)$ is the unknown external interference described by equation (6), and the parameters are as follows

\tilde{W} = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}], \tilde{V} = [\begin{matrix} 2 & 0 \end{matrix}]

We assume that the fault distribution matrix satisfies $E_{f} = B_{u}$ .³⁰ Furthermore, the polynomial fault of time is considered as follows

f (t) = {\begin{matrix} 0, & 0 s \leq t < 15 s \\ 0.1 (t - {3)}^{2} * ε (t - 3), & 15 s \leq t < 20 s \\ 0, & t \geq 20 s \end{matrix}

(26)

where $ε (t)$ is the unit step function.

Figure 1.

Interconnection topology.

The gain matrix of the observer can be obtained from Theorem 1 as follows

L = [\begin{matrix} 1.2534 & 0.6213 \\ 12.4448 & 26.1529 \\ 9.1201 & 27.2253 \\ 3.1202 & 9.7464 \\ 0.6234 & 1.9672 \\ 2.1137 & 5.8296 \\ 0.6424 & 1.5410 \\ - 0.2625 & 0.0297 \\ 2.7914 & 8.2103 \end{matrix}]

We can get $\hat{f}$ by the error estimation system and the obtained observer gain matrix. As shown in Figure 2, when t is 15–20 s, the fault was detected by the observer, where the solid line represents the fluctuation of the fault and the dashed line represents the estimation for it. From Figure 2 we can clearly see that with the progress of time, after about 10 s, the estimated fault value is almost completely consistent with the actual fault value, which proves that the observer designed in this article can effectively estimate the fault.

Figure 2.

Fault and its estimation.

Assuming that the $d (t)$ is the harmonic disturbance, represented by the solid line in Figure 3, and the estimate of disturbance obtained from the fault estimation observer designed in this article is indicated by dashed line. It can be seen from the figure that the estimated value of the disturbance and the actual value of the disturbance tend to be consistent in a short time, which indicates that the perturbation estimation observation scheme is feasible and the results are effective. According to Theorem 2, by solving equations (17) and (23), the matrices $K_{v}$ and $B^{*}$ can be obtained as follows

\begin{matrix} K_{v} = [\begin{matrix} 0.1773 & - 2.6267 \end{matrix}] \\ B^{*} = [\begin{matrix} 0.0023 & - 0.0975 & 0.0577 & 0 \end{matrix}] \end{matrix}

The tracking errors of followers 1, 2, 3 and 4 are shown in Figure 4. We can clearly see that the tracking error of each follower converges to zero in a short time, so it can be seen that followers 1–4 can track the leader and maintain consistency in a short time. Therefore, the fault-tolerant controller designed in this article has good performance for synchronous FTC and disturbance rejection in MASs with polynomial fault, thus also proves the effectiveness of the scheme.

Figure 3.

Disturbance and its estimation.

Figure 4.

Tracking error with control protocol in followers 1–4.

Conclusion

In this article, the design problem of fault-tolerant and disturbance rejection control is investigated for leader-following MASs with fault in polynomial form. First, a novel observer is designed to get estimations of faults and disturbances in followers. To be more general, faults which are in followers can be expressed as polynomial function with respect to time. Then, based on the estimations of disturbance and fault, the control protocol has been designed. FTC problem is solved by analyzing the stability of the closed-loop system. Finally, a numerical example of an aircraft is given to verify the effectiveness of the proposed method. In the future work, the fault-tolerant consensus problem of MASs with both fault and disturbance under reduced order observer will be considered.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Taishan Scholar Program of Shandong Province, the Natural Science Foundation Program of Shandong Province under grant no. ZR2019YQ28, the Development Plan of Youth Innovation Team of University in Shandong Province under grant no. 2019KJN007, and the National Natural Science Foundation of China under grant nos 61773193 and 62103177.

ORCID iD

Guochen Pang

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