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SCIENCE CHINA Information Sciences, Volume 61, Issue 11: 112205(2018) https://doi.org/10.1007/s11432-017-9309-3

Distributed sensor fault diagnosis for a formation system with unknown constant time delays

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  • ReceivedJun 26, 2017
  • AcceptedNov 30, 2017
  • PublishedOct 10, 2018

Abstract

In this paper, a distributed velocity sensor fault diagnosis scheme is presented for a formation of a second-order multi-agent system with unknown constant communication time delays. An existing distributed proportion-derivation (DPD) formation control law is adopted and a delay-independent condition is proposed to guarantee the asymptotical formation stability of the formation system based on the Nyquist stability criterion. Then a distributed fault diagnosis scheme is developed. In each agent, a distributed fault detection residual generator (DFDRG) and a bank of distributed fault isolation residual generators (DFIRGs) are designed based on the closed-loop model of the whole system. Each DFIRG is built up on the basis of a reduced-order unknown input observer (UIO) which is robust to the fault of one neighboring agent. According to the robust relationship between DFIRGs and faults, distributed fault isolation can be achieved. Conditions are presented to guarantee that each agent is able to diagnose faults of itself and its neighbors despite the disturbance of time delays. Finally, outdoor experimental results illustrate the effectiveness of the proposed schemes.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61210012, 61490701, 61522309, 61473163), Tsinghua University Initiative Scientific Research Program, and Research Fund for the Taishan Scholar Project of Shandong Province of China.


Supplement

Appendix

Proof of Lemma lemctrl

proof The dynamic of the tracking error of the $i$th agent can be described as \begin{equation*}\begin{split} \ddot{e}_i(t) =-k_1e_i(t)-k_2\sum_{j\in \mathcal{N}_i}a_{ij}[e_i(t-\tau)-e_j(t-\tau)] -k_3\dot{e}_i(t)-k_4\sum_{j\in \mathcal{N}_i}a_{ij}[\dot{e}_i(t-\tau)-\dot{e}_j(t-\tau)], \\ \end{split}\end{equation*} where $i=1,2,\ldots,N$. It is also assumed that $r(t-\tau)=0$ and $\dot{r}(t-\tau)=0$ when $t\leq~\tau$.

Let $\boldsymbol{e}(t)=[e_1(t),\ldots,e_N(t),\dot{e}_1(t),\ldots,\dot{e}_N(t)]^{\rm~T}$. It follows that $\dot{\boldsymbol{e}}(t)=\boldsymbol{A}_1\boldsymbol{e}(t)+\boldsymbol{A}_2\boldsymbol{e}(t-\tau)$ holds, where \begin{equation*}\begin{split} \boldsymbol{A}_1=\begin{bmatrix} \boldsymbol{0} & \boldsymbol{I}_N \\ -k_1\boldsymbol{I}_N & -k_3\boldsymbol{I}_N \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k_1 & -k_3 \end{bmatrix}\otimes \boldsymbol{I}_N, \boldsymbol{A}_2=\begin{bmatrix} \boldsymbol{0} & \boldsymbol{0} \\ -k_2\boldsymbol{L}_g & -k_4\boldsymbol{L}_g \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ -k_2 & -k_4 \end{bmatrix} \otimes \boldsymbol{L}_g. \end{split}\end{equation*}

Because $\boldsymbol{L}_g$ is an Hermitian matrix, there exists an orthogonal matrix $\boldsymbol{U}$ such that $\boldsymbol{U}^{\rm~T}\boldsymbol{L}_g\boldsymbol{U}={\rm~diag}\{\lambda_1,\ldots,\lambda_N\}$. Let $\boldsymbol{T}_1=\boldsymbol{I}_2\otimes~\boldsymbol{U}^{\rm~T}$ and $\boldsymbol{T}_2=[\boldsymbol{i}_1,\boldsymbol{i}_{N+1},\boldsymbol{i}_2,\boldsymbol{i}_{N+2},\ldots,\boldsymbol{i}_N,\boldsymbol{i}_{2N}]^{\rm~T}$, where $\boldsymbol{i}_k$ is the $k$th column of the matrix $\boldsymbol{I}_{2N}$. It follows that \begin{equation*}\begin{split} &\tilde{\boldsymbol{A}}_1=\boldsymbol{T}_2\boldsymbol{T}_1\boldsymbol{A}_1\boldsymbol{T}_1^{-1}\boldsymbol{T}_2^{-1}=\boldsymbol{I}_N\otimes\begin{bmatrix} 0 & 1 \\ -k_1 & -k_3 \end{bmatrix}, \tilde{\boldsymbol{A}}_2=\boldsymbol{T}_2\boldsymbol{T}_1\boldsymbol{A}_2\boldsymbol{T}_1^{-1}\boldsymbol{T}_2^{-1}={\rm diag}\{\lambda_1,\ldots,\lambda_N\}\otimes\begin{bmatrix} 0 & 0 \\ -k_2 & -k_4 \end{bmatrix}. \end{split}\end{equation*}

Let $\tilde{\boldsymbol{e}}=\boldsymbol{T}_2\boldsymbol{T}_1\boldsymbol{e}$. It can be derived that $\dot{\tilde{\boldsymbol{e}}}(t)=\tilde{\boldsymbol{A}}_1\tilde{\boldsymbol{e}}(t)+\tilde{\boldsymbol{A}}_2\tilde{\boldsymbol{e}}(t-\tau) $. Furthermore, it follows that \begin{equation} \begin{split} \dot{\boldsymbol{\eta}}_i(t)&= \begin{bmatrix} 0 & 1 \\ -k_1 & -k_3 \end{bmatrix}\boldsymbol{\eta}_i(t)+\lambda_i \begin{bmatrix} 0 & 0 \\ -k_2 & -k_4 \end{bmatrix}\boldsymbol{\eta}_i(t-\tau), \end{split} \tag{23}\end{equation} where $i=1,2,\ldots,N$. $\boldsymbol{\eta}_i\in~\mathbb{R}^2$ consists of the $(2i-1)$th and $(2i)$th elements of $\tilde{\boldsymbol{e}}$.

The characteristic equation of the system 23 is $s^2+k_3s+k_1+(k_4s+k_2)\lambda_i{\rm~e}^{-\tau~s}=0$. If all roots of the above equation lie in the left half complex plane, the system 23 is stable. Define a function $G_i(s)$ as follows. \begin{equation}G_i(s)=\frac{(k_4s+k_2)\lambda_i{\rm e}^{-\tau s}}{s^2+k_3s+k_1}, i=1,2,\ldots,N. \tag{24}\end{equation}

Let $s={\rm~j}w$, where ${\rm~j}^2=-1$ and $w\in~\mathbb{R}$. It follows that \begin{equation} G_i({\rm j}w)=\frac{(k_2+{\rm j}wk_4)\lambda_i{\rm e}^{-{\rm j}w\tau}}{k_1-w^2+{\rm j}k_3w}, i=1,2,\ldots,N. \tag{25}\end{equation}

Since $k_1>0$ and $k_2>0$, it follows that all roots of the equation $s^2+k_3s+k_1=0$ lie in the left half complex plane. According to the Nyquist stability criterion and the literature 2) 3), the number of the roots of $s^2+k_3s+k_1+(k_4s+k_2)\lambda_i{\rm~e}^{-\tau~s}=0$ with positive real part is equal to the number of the times for which the Nyquist curve of $G_i({\rm~j}w)$ encloses the point $(-1,0)$ as the $w$ increases from $0$ to $\infty$. Therefore, the condition on the asymptotical formation stability can be gained by analyzing the characteristic of the Nyquist curve of $G_i({\rm~j}w)$.

According to 25, the amplitude of $G_i({\rm~j}w)$ is \begin{equation} |G_i({\rm j}w)|=\frac{\lambda_i\sqrt{k_2^2+(wk_4)^2}}{\sqrt{(k_1-w^2)^2+(wk_3)^2}}, i=1,2,\ldots,N. \tag{26}\end{equation}

When $i=1$, it follows that $\lambda_i=0$ and $|G_i({\rm~j}w)|=0$. It is obvious that the Nyquist curve of $G_1({\rm~j}w)$ dose not enclose the point $(-1,0)$ and the system 23 is stable when $i=1$. When $i\in~\{2,3,\ldots,N\}$, since $k_1>k_2\lambda_N$, it follows that $k_1>k_2\lambda_i$ holds. Since $k_3^2-2k_1-k_4^2\lambda_N^2>0$, it is obvious that $k_3^2-2k_1-k_4^2\lambda_i^2>0$ holds, where $i=2,3,\ldots,N$. Then, the following equation can be obtained. \begin{equation} w^4+(k_3^2-2k_1-k_4^2\lambda_i^2)w^2+k_1^2-k_2^2\lambda_i^2>0, i=2,3,\ldots,N. \tag{27}\end{equation}

According to 26 and 27, it can be obtained that $|G_i({\rm~j}w)|<1$ holds, where $i=2,3,\ldots,N$. Hence, the Nyquist curve of $G_i({\rm~j}w)$ dose not enclose the point $(-1,0)$, $i=1,2,\ldots,N$. The system 23 is stable when $i=2,3,\ldots,N$. Overall, the formation system is stable and the asymptotical formation stability is achieved. This ends the proof.

Ansoff H, Krumhansl J. A general stability criterion for linear oscillating systems with constant time lag. Quart Appl Math, 1948, 6: 337–341.

Ansoff H. Stability of linear oscillating systems with constant time lag. J Appl Mech-Trans ASME, 1949, 16: 158–164.

Parameters in the experiment

begintiny \begin{equation*}\begin{split} & \boldsymbol{G}_1^0=\begin{bmatrix} 3 & 0 & 0 & 1 & 0 & 0 \\ 6 & 6 & 0 & 1 & 1 & 0 \\ 9 & 0 & 9 & 1 & 0 & 1 \\ -3 & 0.17 & 0.17 & 9 & 0.37 & 0.37 \\ -3 & -3.34 & 0.17 & 12 & 11.26 & 0.37 \\ -3 & 0.17 & -3.34 & 15 & 0.37 & 14.26 \\ \end{bmatrix}, \boldsymbol{G}_2^0=\begin{bmatrix} 2 & 2 & 0 & 1 & 1 & 00 \\ 4 & 0 & 0 & 1 & 0 & 0 \\ 6 & 0 & 6 & 1 & 0 & 1 \\ -3 & -3.34 & 0.17 & 5 & 4.26 & 0.37 \\ -3 & 0.17 & 0.17 & 7 & 0.37 & 0.37 \\ -3 & 0.17 & -3.34 & 9 & 0.37 & 8.26 \\ \end{bmatrix}, \boldsymbol{G}_3^0=\begin{bmatrix} 2 & 2 & 0 & 1 & 1 & 0 \\ 4 & 0 & 4 & 1 & 0 & 1 \\ 6 & 0 & 0 & 1 & 0 & 0 \\ -3 & -3.34 & 0.17 & 5 & 4.26 & 0.37 \\ -3 & 0.17 & -3.34 & 7 & 0.37 & 6.26 \\ -3 & 0.17 & 0.17 & 9 & 0.37 & 0.37 \\ \end{bmatrix}, \\ & \boldsymbol{H}_1^2=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}, \boldsymbol{F}_1^2=\begin{bmatrix} -10.0047 & -0.0000 & 4.4964 & 0.5801 & 5.6948 \\ 0.0000 & -10.0000 & 0 & -0.0000 & 0.0000 \\ -1.6615 & -0.0000 & -2.3022 & 0.3298 & 2.2370 \\ -0.3855 & -0.0000 & 0.4673 & -5.9239 & 0.7048 \\ -1.4658 & -0.0000 & 1.6772 & 0.4651 & -1.7692 \\ \end{bmatrix}, \boldsymbol{M}_1^2=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.9904 & 0.0970 & -0.0096 \\ 0.0970 & 0.0192 & 0.0970 \\ -0.0096 & 0.0970 & 0.9904 \\ \end{bmatrix}, \\ &\boldsymbol{S}_1^2=\begin{bmatrix} 10.4287 & 10.0047 & 0.0000 & -10.1081 & 0 & -5.6533 \\ 10.0000 & -0.0000 & 10.0000 & 1.0000 & 0 & 1.0000 \\ -1.9049 & 1.5042 & 0.2169 & -3.3315 & 0 & -1.8314 \\ 5.7273 & 0.3544 & -0.3043 & 0.3506 & 0 & -0.4386 \\ -2.2373 & 1.3084 & -3.2931 & -3.3315 & 0 & -1.9486 \\ \end{bmatrix}, \boldsymbol{J}_1^2=\begin{bmatrix} 0 & -1.0000 & -0.0980 & 0.9904 & -0.0980 \\ 1.0000 & 0 & 0.0980 & -0.9904 & 0.0980 \\ 0 & 0 & -0.0980 & 0.9904 & -0.0980 \\ 0 & 0 & 1.0000 & -0.0000 & -1.0000 \\ \end{bmatrix}, \\ &\boldsymbol{F}_1^3=\begin{bmatrix} -16.0002 & -0.0260 & 0.0000 & -0.0000 & 0.0000 \\ -0.3347 & -31.9998 & 5.0541 & 5.0541 & 1.0000 \\ -0.5072 & -23.2880 & -4.0763 & 3.9144 & 0.8184 \\ -0.5072 & -23.2880 & 3.9144 & -3.9096 & -0.0242 \\ -0.1004 & -4.6078 & 0.8184 & -0.0242 & -4.0141 \\ \end{bmatrix}, \boldsymbol{M}_1^3=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.9904 & -0.0096 & 0.0970 \\ -0.0096 & 0.9904 & 0.0970 \\ 0.0970 & 0.0970 & 0.0192 \\ \end{bmatrix}, \\ &\boldsymbol{H}_1^3=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}, \boldsymbol{S}_1^3=\begin{bmatrix} 16.0262 & 16.0002 & 0.0260 & 1.0000 & 1.0000 & 0 \\ 32.3345 & 0.3347 & 31.9998 & -10.1081 & -5.0541 & 0 \\ 19.7353 & 0.7241 & 23.1306 & -3.3315 & -3.5571 & 0 \\ 20.5861 & -2.7859 & 23.1306 & -3.3315 & 0.2402 & 0 \\ 8.1215 & -0.2040 & 4.5767 & 0.3506 & 0.1014 & 0 \\ \end{bmatrix}, \\ &\boldsymbol{J}_1^3=\begin{bmatrix} 0 & -1.0000 & -0.0980 & -0.0980 & 0.9904 \\ 1.0000 & 0 & 0.0980 & 0.0980 & -0.9904 \\ 0 & 0 & 1.0000 & -1.0000 & -0.0000 \\ 0 & 0 & -0.0980 & -0.0980 & 0.9904 \\ \end{bmatrix}, \\ & \boldsymbol{F}_2^1=\begin{bmatrix} -3.0000 & 0.0000 & 0.0000 & -0.0000 & -0.0000 \\ -0.0000 & -15.0000 & -0.0000 & 0.0000 & 0.0000 \\ 0.0000 & -0.0000 & -21.6599 & 7.2461 & 7.2461 \\ 0.0000 & -0.0000 & -8.4386 & -2.6700 & 3.3300 \\ 0.0000 & -0.0000 & -8.4386 & 3.3300 & -2.6700 \\ \end{bmatrix}, \boldsymbol{S}_2^1=\begin{bmatrix} 3.0000 & -0.0000 & -0.0000 & 1.0000 & 0 & -0.0000 \\ 15.0000 & 0.0000 & 15.0000 & 1.0000 & 0 & 1.0000 \\ 22.2311 & 22.8398 & -0.3043 & -10.6504 & 0 & -5.3252 \\ 5.1884 & 8.2645 & 0.2169 & -2.2431 & 0 & -2.0666 \\ 5.1884 & 8.2645 & -3.2931 & -2.2431 & 0 & -0.1766 \\ \end{bmatrix}, \boldsymbol{H}_2^1=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}, \\ &\boldsymbol{M}_2^1=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.0192 & 0.0970 & 0.0970 \\ 0.0970 & 0.9904 & -0.0096 \\ 0.0970 & -0.0096 & 0.9904 \\ \end{bmatrix}, \boldsymbol{J}_2^1=\begin{bmatrix} 1.0000 & -1.0000 & 0 & 0 & 0 \\ -1.0000 & 0 & 0.9904 & -0.0980 & -0.0980 \\ 1.0000 & 0 & 0 & 0 & 0 \\ 0 & 0 & -0.0000 & 1.0000 & -1.0000 \\ \end{bmatrix}, \boldsymbol{F}_2^3=\begin{bmatrix} -12.0000 & 0.0110 & -0.0000 & -0.0000 & 0.0000 \\ 0.1622 & -24.0000 & 5.0541 & 5.0541 & 1.0000 \\ 0.1868 & -13.0993 & -3.0576 & 2.9419 & 0.5849 \\ 0.1868 & -13.0993 & 2.9419 & -2.9435 & 0.0085 \\ 0.0370 & -2.5918 & 0.5849 & 0.0085 & -2.9989 \\ \end{bmatrix}, \\ &\boldsymbol{M}_2^3=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.9904 & -0.0096 & 0.0970 \\ -0.0096 & 0.9904 & 0.0970 \\ 0.0970 & 0.0970 & 0.0192 \\ \end{bmatrix}, \boldsymbol{H}_2^3=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}, \boldsymbol{S}_2^3=\begin{bmatrix} 11.9890 & -0.0000 & -0.0110 & 1.0000 & 0.0000 & 0 \\ 23.8378 & 0.0000 & 24.0000 & -10.1081 & -5.0541 & 0 \\ 9.0884 & -3.8837 & 12.9420 & -3.3315 & -0.7700 & 0 \\ 9.6705 & 0.2084 & 12.9420 & -3.3315 & -2.6024 & 0 \\ 4.9432 & 2.7238 & 2.5607 & 0.3506 & 0.3822 & 0 \\ \end{bmatrix}, \end{split}\end{equation*} \begin{equation*}\begin{split} &\boldsymbol{J}_2^3=\begin{bmatrix} 1.0000 & -1.0000 & 0 & 0 & 0 \\ -1.0000 & 0 & -0.0980 & -0.0980 & 0.9904 \\ 0 & 0 & -1.0000 & 1.0000 & 0.0000 \\ 1.0000 & 0 & 0 & 0 & 0 \\ \end{bmatrix}, \boldsymbol{F}_3^1=\begin{bmatrix} -3.0044 & -0.0000 & -0.0000 & -0.0808 & 0.0808 \\ -0.0000 & -15.0000 & -0.0000 & 0.0000 & 0.0000 \\ -0.0000 & -0.0000 & -21.6599 & 7.2461 & 7.2461 \\ -0.0808 & -0.0000 & -8.4386 & -2.6679 & 3.3278 \\ 0.0808 & -0.0000 & -8.4386 & 3.3278 & -2.6679 \\ \end{bmatrix}, \boldsymbol{H}_3^1=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}, \\ &\boldsymbol{M}_3^1=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.0192 & 0.0970 & 0.0970 \\ 0.0970 & 0.9904 & -0.0096 \\ 0.0970 & -0.0096 & 0.9904 \\ \end{bmatrix}, \boldsymbol{S}_3^1=\begin{bmatrix} 3.0044 & 0.0000 & 3.0044 & 1.0000 & 0 1.0808 \\ 15.0000 & 0.0000 & 0.0000 & 1.0000 & 0 -0.0000 \\ 22.2311 & 22.8398 & -0.3043 & -10.6504 & 0 -5.3252 \\ 5.2692 & 8.2645 & -3.2122 & -2.2431 & 0 -0.1788 \\ 5.1075 & 8.2645 & 0.1361 & -2.2431 & 0 -2.0644 \\ \end{bmatrix}, \boldsymbol{J}_3^1=\begin{bmatrix} -1.0000 & 1.0000 & 0 & 0 & 0 \\ 0 & -1.0000 & 0.9904 & -0.0980 & -0.0980 \\ 0 & 1.0000 & 0 & 0 & 0 \\ 0 & 0 & 0.0000 & -1.0000 & 1.0000 \\ \end{bmatrix}, \\ &\boldsymbol{F}_3^2=\begin{bmatrix} -21.0000 & -0.0000 & 5.0541 & 1.0000 & 5.0541 \\ -0.0000 & -15.0000 & 0.0000 & -0.0000 & -0.0000 \\ -10.4794 & -0.0000 & -2.8595 & -0.1612 & 2.8913 \\ -2.0735 & -0.0000 & -0.1612 & -3.0961 & 0.7738 \\ -10.4794 & -0.0000 & 2.8913 & 0.7738 & -3.0444 \\ \end{bmatrix}, \boldsymbol{M}_3^2=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0.9904 & 0.0970 & -0.0096 \\ 0.0970 & 0.0192 & 0.0970 \\ -0.0096 & 0.0970 & 0.9904 \\ \end{bmatrix}, \boldsymbol{H}_3^2=\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}, \\ &\boldsymbol{S}_3^2=\begin{bmatrix} 21.0000 & -0.0000& 21.0000& -10.1081& -5.0541& 0 \\ 15.0000 & 0.0000& 0.0000& 1.0000& -0.00007& 0 \\ 7.4086 & -3.1304& 10.3220& -3.3315& -0.8943& 0 \\ 4.5599 & 2.8219& 2.0423& 0.3506& 1.1379& 0 \\ 6.4646 & -0.5644& 10.3220& -3.3315& -2.6276& 0 \\ \end{bmatrix}, \boldsymbol{J}_3^2=\begin{bmatrix} -1.0000 & 1.0000 & 0 & 0 & 0 \\ 0 & -1.0000 & -0.0980 & 0.9904 & -0.0980 \\ 0 & 0 & -1.0000 & 0.0000 & 1.0000 \\ 0 & 1.0000 & 0 & 0 & 0 \\ \end{bmatrix}. \end{split}\end{equation*} endtiny


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  • Figure 1

    (Color online) The framework of the distributed fault diagnosis scheme for the formation system with time delays.

  • Figure 4

    (Color online) The control framework of a quadrotor along $x$ axis.

  • Figure 5

    (Color online) The formation results of all quadrotors. (a) The trajectories of three quadrotors; (b) the tracking errors of three quadrotors.

  • Figure 6

    (Color online) The fault diagnosis results of all quadrotors. (a1) The distributed fault detection results in quadrotor 1; (a2) the self-fault detection results in quadrotor 1; (a3) the distributed fault isolation results in quadrotor 1; (b1) the distributed fault detection results in quadrotor 2; (b2) the self-fault detection results in quadrotor 2; (b3) the distributed fault isolation results in quadrotor 2; (c1) the distributed fault detection results in quadrotor 3; (c2) the self-fault detection results in quadrotor 3; (c3) the distributed fault isolation results in quadrotor 3.

  •   

    Algorithm 1 Fault detection logic of the $i$th agent

    if $J_i^0(t)\geq~J_{i,{\rm~th}}^0$ then

    There is a fault in the system.

    else

    There is no fault in the system.

    end if

  •   

    Algorithm 2 Fault isolation logic of the $i$th agent

    if $J_i^i(t)\geq~J_{i,{\rm~th}}^i$ then

    The $i$th agent is faulty.ELSIF$\exists~k\in~\mathcal{N}_i$, $\forall~p\in~\mathcal{N}_i\backslash\{k\}$, $[J_i^k(t)<~J_{i,{\rm~th}}^k]~~\wedge~~[J_i^p(t)\geq~J_{i,{\rm~th}}^p]$

    The $k$th agent is faulty.ELSIF$\forall~k\in~\mathcal{N}_i$, $J_i^k(t)\geq~J_{i,{\rm~th}}^k$

    The faulty agent belongs to $\mathcal{V}\backslash~\bar{\mathcal{N}}_i$.

    end if

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