【Paper】2020_Anomaly Detection and Identification for Multiagent Systems Subjected to Physical Faults
Y. Li, H. Fang and J. Chen, “Anomaly Detection and Identification for Multiagent Systems Subjected to Physical Faults and Cyberattacks,” in IEEE Transactions on Industrial Electronics, vol. 67, no. 11, pp. 9724-9733, Nov. 2020, doi: 10.1109/TIE.2019.2952802.
| 符号 | 说明 | |
|---|---|---|
| xi(k)x_i(k)xi(k) | 状态 | |
| ui(k)u_i(k)ui(k) | 控制输入 | |
| yi(k)y_i(k)yi(k) | 测量输出向量 | |
| did_idi | 未知外界干扰信号 | |
| fif_ifi | 故障 | |
| yij(k)y_i^j(k)yij(k) | iii 从 jjj 这里获得的输出信息 | |
| bij(k)b_i^j(k)bij(k) | iii 从 jjj 这里获得的攻击信息 | |
| x^i(k)\hat{x}_i(k)x^i(k) | 状态的估计值 | |
| y^i(k)\hat{y}_i(k)y^i(k) | 输出的估计值 | |
| rij(k)r_i^j(k)rij(k) | iii 从 jjj 这里获得的残差信息 |
文章目录
- 1. Introduction
- 2. Preliminaries and Problem Formulation
- 2.1 System Description
- 2.2 Anomaly Detector Dynamics
- 2.3 Problem Formulation
- 3. Secure Scheme Design for MASs
- 3.1 Independent Anomaly Detection
1. Introduction
2. Preliminaries and Problem Formulation
2.1 System Description
xi(k+1)=Axi(k)+Bui(k)+Bddi(k)+Bffi(k)yi(k)=Cxi(k)+Dddi(k)+Dffi(k)(1)\begin{aligned} x_i(k+1) &= A x_i(k) + B u_i(k) + &B_d d_i(k) + B_f f_i(k) \\ y_i(k) &= C x_i(k) + &D_d d_i(k) + D_f f_i(k) \end{aligned}\tag{1}xi(k+1)yi(k)=Axi(k)+Bui(k)+=Cxi(k)+Bddi(k)+Bffi(k)Dddi(k)+Dffi(k)(1)
袭击模型表示为
yij(k)=yi(k)+bij(k)(3)y_i^j(k) = y_i(k) + b_i^j(k) \tag{3}yij(k)=yi(k)+bij(k)(3)
2.2 Anomaly Detector Dynamics
异常检测器：
x^ij(k+1)=Ax^ij(k)+Bui(k)+L(yij(k)−y^ij(k))y^ij(k)=Cx^ij(k)rij(k)=V(yij(k)−y^ij(k))(5)\begin{aligned} \hat{x}_i^j(k+1) &=A \hat{x}_i^j(k) + Bu_i(k) + L(y_i^j(k) - \hat{y}_i^j(k)) \\ \hat{y}^j_i(k) &=C \hat{x}^j_i(k) \\ r^j_i(k) &=V (y_i^j(k) - \hat{y}_i^j(k)) \end{aligned}\tag{5}x^ij(k+1)y^ij(k)rij(k)=Ax^ij(k)+Bui(k)+L(yij(k)−y^ij(k))=Cx^ij(k)=V(yij(k)−y^ij(k))(5)
残差信号 rijr_i^jrij 的 Z 变换为：
rij(z)=V[Gb(z)bij(z)+Gf(z)fi(z)+Gddi(z)]Gb(z)=I−C(zI−A+LC)−1LGf(z)=Df+C(zI−A+LC)−1(Bf−LDf)Gd(z)=Dd+C(zI−A+LC)−1(Bd−LDd)(7)\begin{aligned} r^j_i(z) &= V [G_b(z) b_{i}^{j}(z) + G_f(z) f_i(z) + G_d d_i(z)] \\ G_b(z) &= I-C(zI - A + LC)^{-1} L \\ G_f(z) &= D_f+C(zI - A + LC)^{-1} (B_f - L D_f) \\ G_d(z) &= D_d+C(zI - A + LC)^{-1} (B_d - L D_d) \\ \end{aligned}\tag{7}rij(z)Gb(z)Gf(z)Gd(z)=V[Gb(z)bij(z)+Gf(z)fi(z)+Gddi(z)]=I−C(zI−A+LC)−1L=Df+C(zI−A+LC)−1(Bf−LDf)=Dd+C(zI−A+LC)−1(Bd−LDd)(7)
下面给出推导过程。
先将状态值与观测值做差值可得：
xi(k+1)−x^ij(k+1)=Axi(k)+Bui(k)+Bddi(k)+Bffi(k)−Ax^ij(k)−Bui(k)−L(yij(k)−y^ij(k))=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−Lyij(k)+Ly^ij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−Lyi(k)+Ly^ij(k)−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(yi(k)−y^ij(k))−Lbij(k)\begin{aligned} x_i(k+1) - \hat{x}_i^j(k+1) &= A x_i(k) + B u_i(k) + B_d d_i(k) + B_f f_i(k) \\ &-\red{A \hat{x}_i^j(k)} - Bu_i(k) - \blue{L(y_i^j(k) - \hat{y}_i^j(k))} \\ &= A (x_i(k)-\red{\hat{x}_i^j(k)}) + B_d d_i(k) + B_f f_i(k) - \blue{Ly_i^j(k) + L\hat{y}_i^j(k)} \\ &= A (x_i(k)-\hat{x}_i^j(k)) + B_d d_i(k) + B_f f_i(k) - \blue{Ly_i(k) + L\hat{y}_i^j(k) - L b_i^j(k)} \\ &= A (x_i(k)-\hat{x}_i^j(k)) + B_d d_i(k) + B_f f_i(k) - L(\green{y_i(k) -\hat{y}_i^j(k)}) - L b_i^j(k) \\ \end{aligned}xi(k+1)−x^ij(k+1)=Axi(k)+Bui(k)+Bddi(k)+Bffi(k)−Ax^ij(k)−Bui(k)−L(yij(k)−y^ij(k))=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−Lyij(k)+Ly^ij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−Lyi(k)+Ly^ij(k)−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(yi(k)−y^ij(k))−Lbij(k)
把其中输出值和输出值的观测器之间的差值 yi(k)−y^ij(k)y_i(k) -\hat{y}_i^j(k)yi(k)−y^ij(k) 单独拿出来。
yi(k)−y^ij(k)=Cxi(k)+Dddi(k)+Dffi(k)−Cx^ij(k)=C(xi(k)−x^ij(k))+Dddi(k)+Dffi(k)\begin{aligned} \green{y_i(k) -\hat{y}_i^j(k)} &= C x_i(k) + D_d d_i(k) + D_f f_i(k) - C \hat{x}^j_i(k) \\ &= C (x_i(k) - \hat{x}^j_i(k)) + D_d d_i(k) + D_f f_i(k) \\ \end{aligned}yi(k)−y^ij(k)=Cxi(k)+Dddi(k)+Dffi(k)−Cx^ij(k)=C(xi(k)−x^ij(k))+Dddi(k)+Dffi(k)
那么
xi(k+1)−x^ij(k+1)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(yi(k)−y^ij(k))−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(C(xi(k)−x^ij(k))+Dddi(k)+Dffi(k))−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−LC(xi(k)−x^ij(k))−LDddi(k)−LDffi(k))−Lbij(k)=(A−LC)(xi(k)−x^ij(k))+(Bd−LDd)di(k)+(Bf−LDf)fi(k)−Lbij(k)\begin{aligned} x_i(k+1) - \hat{x}_i^j(k+1) &= A (x_i(k)-\hat{x}_i^j(k)) + B_d d_i(k) + B_f f_i(k) - L(\green{y_i(k) -\hat{y}_i^j(k)}) - L b_i^j(k) \\ &= A (x_i(k)-\hat{x}_i^j(k)) + B_d d_i(k) + B_f f_i(k) - L(\green{C (x_i(k) - \hat{x}^j_i(k)) + D_d d_i(k) + D_f f_i(k)}) - L b_i^j(k) \\ &= A (x_i(k)-\hat{x}_i^j(k)) + B_d d_i(k) + B_f f_i(k) - LC (x_i(k) - \hat{x}^j_i(k)) - LD_d d_i(k) - LD_f f_i(k)) - L b_i^j(k) \\ &= (A-LC) (x_i(k)-\hat{x}_i^j(k)) + (B_d-LD_d) d_i(k) + (B_f-LD_f) f_i(k) - L b_i^j(k) \\ \end{aligned}xi(k+1)−x^ij(k+1)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(yi(k)−y^ij(k))−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−L(C(xi(k)−x^ij(k))+Dddi(k)+Dffi(k))−Lbij(k)=A(xi(k)−x^ij(k))+Bddi(k)+Bffi(k)−LC(xi(k)−x^ij(k))−LDddi(k)−LDffi(k))−Lbij(k)=(A−LC)(xi(k)−x^ij(k))+(Bd−LDd)di(k)+(Bf−LDf)fi(k)−Lbij(k)
做 Z 变换
xi(k+1)−x^ij(k+1)=(A−LC)(xi(k)−x^ij(k))+(Bd−LDd)di(k)+(Bf−LDf)fi(k)−Lbij(k)zxi(z)−zx^ij(z)=(A−LC)(xi(z)−x^ij(z))+(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)z(xi(z)−x^ij(z))=(A−LC)(xi(z)−x^ij(z))+(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)(zI−A+LC)(xi(z)−x^ij(z))=(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)(xi(z)−x^ij(z))=(zI−A+LC)−1(Bd−LDd)di(z)+(zI−A+LC)−1(Bf−LDf)fi(z)−(zI−A+LC)−1Lbij(z)C(xi(z)−x^ij(z))=C(zI−A+LC)−1(Bd−LDd)di(z)+C(zI−A+LC)−1(Bf−LDf)fi(z)−C(zI−A+LC)−1Lbij(z)\begin{aligned} x_i(k+1) - \hat{x}_i^j(k+1) &= (A-LC) (x_i(k)-\hat{x}_i^j(k)) + (B_d-LD_d) d_i(k) + (B_f-LD_f) f_i(k) - L b_i^j(k) \\ z x_i(z) - z \hat{x}_i^j(z) &= (A-LC) (x_i(z)-\hat{x}_i^j(z)) + (B_d-LD_d) d_i(z) + (B_f-LD_f) f_i(z) - L b_i^j(z) \\ z (x_i(z) - \hat{x}_i^j(z)) &= (A-LC) (x_i(z)-\hat{x}_i^j(z)) + (B_d-LD_d) d_i(z) + (B_f-LD_f) f_i(z) - L b_i^j(z) \\ (zI-A+LC) (x_i(z) - \hat{x}_i^j(z)) &= (B_d-LD_d) d_i(z) + (B_f-LD_f) f_i(z) - L b_i^j(z) \\ (x_i(z) - \hat{x}_i^j(z)) &= (zI-A+LC)^{-1}(B_d-LD_d) d_i(z) + (zI-A+LC)^{-1}(B_f-LD_f) f_i(z) - (zI-A+LC)^{-1}L b_i^j(z) \\ C(x_i(z) - \hat{x}_i^j(z)) &= C(zI-A+LC)^{-1}(B_d-LD_d) d_i(z) + C(zI-A+LC)^{-1}(B_f-LD_f) f_i(z) - C(zI-A+LC)^{-1}L b_i^j(z) \\ \end{aligned}xi(k+1)−x^ij(k+1)zxi(z)−zx^ij(z)z(xi(z)−x^ij(z))(zI−A+LC)(xi(z)−x^ij(z))(xi(z)−x^ij(z))C(xi(z)−x^ij(z))=(A−LC)(xi(k)−x^ij(k))+(Bd−LDd)di(k)+(Bf−LDf)fi(k)−Lbij(k)=(A−LC)(xi(z)−x^ij(z))+(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)=(A−LC)(xi(z)−x^ij(z))+(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)=(Bd−LDd)di(z)+(Bf−LDf)fi(z)−Lbij(z)=(zI−A+LC)−1(Bd−LDd)di(z)+(zI−A+LC)−1(Bf−LDf)fi(z)−(zI−A+LC)−1Lbij(z)=C(zI−A+LC)−1(Bd−LDd)di(z)+C(zI−A+LC)−1(Bf−LDf)fi(z)−C(zI−A+LC)−1Lbij(z)
rij(z)=V(yij(z)−y^ij(z))=V(yi(z)+bij(z)−y^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V[Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z)]=V[C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z)]=V[C(zI−A+LC)−1(Bd−LDd)di(z)+C(zI−A+LC)−1(Bf−LDf)fi(z)−C(zI−A+LC)−1Lbij(z)+Dddi(z)+Dffi(z)+bij(z)]=V[C(zI−A+LC)−1(Bd−LDd)di(z)+Dddi(z)+C(zI−A+LC)−1(Bf−LDf)fi(z)+Dffi(z)−C(zI−A+LC)−1Lbij(z)+bij(z)]\begin{aligned} r^j_i(z) &= V (\red{y_i^j(z)} - \hat{y}_i^j(z)) \\ &= V (\red{y_i(z) + b_i^j(z)} - \blue{\hat{y}_i^j(z)}) \\ &= V (\red{C x_i(z) + D_d d_i(z) + D_f f_i(z) + b_i^j(z)} - \blue{C \hat{x}^j_i(z)}) \\ &= V [C x_i(z) + D_d d_i(z) + D_f f_i(z) + b_i^j(z) - C \hat{x}^j_i(z)] \\ &= V [\red{C (x_i(z) - \hat{x}^j_i(z))} + D_d d_i(z) + D_f f_i(z) + b_i^j(z)] \\ &= V [\red{C(zI-A+LC)^{-1}(B_d-LD_d) d_i(z) + C(zI-A+LC)^{-1}(B_f-LD_f) f_i(z) - C(zI-A+LC)^{-1}L b_i^j(z)} + D_d d_i(z) + D_f f_i(z) + b_i^j(z)] \\ &= V[ C(zI-A+LC)^{-1}(B_d-LD_d) d_i(z) + D_d d_i(z) \\ &~~~~~+ C(zI-A+LC)^{-1}(B_f-LD_f) f_i(z) + D_f f_i(z) \\ &~~~~~- C(zI-A+LC)^{-1}L b_i^j(z) + b_i^j(z) ] \\ \end{aligned}rij(z)=V(yij(z)−y^ij(z))=V(yi(z)+bij(z)−y^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V[Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z)]=V[C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z)]=V[C(zI−A+LC)−1(Bd−LDd)di(z)+C(zI−A+LC)−1(Bf−LDf)fi(z)−C(zI−A+LC)−1Lbij(z)+Dddi(z)+Dffi(z)+bij(z)]=V[C(zI−A+LC)−1(Bd−LDd)di(z)+Dddi(z) +C(zI−A+LC)−1(Bf−LDf)fi(z)+Dffi(z) −C(zI−A+LC)−1Lbij(z)+bij(z)]
再令
Gb(z)=C(zI−A+LC)−1(Bd−LDd)+DdG_b(z) = C(zI-A+LC)^{-1}(B_d-LD_d) + D_dGb(z)=C(zI−A+LC)−1(Bd−LDd)+Dd
Gf(z)=C(zI−A+LC)−1(Bf−LDf)+DfG_f(z) = C(zI-A+LC)^{-1}(B_f-LD_f) + D_fGf(z)=C(zI−A+LC)−1(Bf−LDf)+Df
Gd(z)=I−C(zI−A+LC)−1LG_d(z) = I - C(zI-A+LC)^{-1}LGd(z)=I−C(zI−A+LC)−1L
即为论文中的公式（7）。
下面是一些推导过程，暂时保留，最后再将未用到的删除。
zxi(z)=Axi(z)+Bui(z)+Bddi(z)+Bffi(z)zxi(z)−Axi(z)=Bui(z)+Bddi(z)+Bffi(z)(zI−A)xi(z)=Bui(z)+Bddi(z)+Bffi(z)xi(z)=(zI−A)−1(Bui(z)+Bddi(z)+Bffi(z))yi(z)=Cxi(z)+Dddi(z)+Dffi(z)yi(z)=C(zI−A)−1(Bui(z)+Bddi(z)+Bffi(z))+Dddi(z)+Dffi(z)yij(z)=yi(z)+bij(z)\begin{aligned} z x_i(z) &= A x_i(z) + B u_i(z) + B_d d_i(z) + B_f f_i(z) \\ z x_i(z) - A x_i(z) &= B u_i(z) + B_d d_i(z) + B_f f_i(z) \\ (zI-A) x_i(z) &= B u_i(z) + B_d d_i(z) + B_f f_i(z) \\ x_i(z) &= (zI-A)^{-1} (B u_i(z) + B_d d_i(z) + B_f f_i(z) ) \\ \\ y_i(z) &= C x_i(z) + D_d d_i(z) + D_f f_i(z) \\ y_i(z) &= C (zI-A)^{-1} (B u_i(z) + B_d d_i(z) + B_f f_i(z) ) + D_d d_i(z) + D_f f_i(z) \\ \\ y_i^j(z) &= y_i(z) + b_i^j(z) \\ \end{aligned}zxi(z)zxi(z)−Axi(z)(zI−A)xi(z)xi(z)yi(z)yi(z)yij(z)=Axi(z)+Bui(z)+Bddi(z)+Bffi(z)=Bui(z)+Bddi(z)+Bffi(z)=Bui(z)+Bddi(z)+Bffi(z)=(zI−A)−1(Bui(z)+Bddi(z)+Bffi(z))=Cxi(z)+Dddi(z)+Dffi(z)=C(zI−A)−1(Bui(z)+Bddi(z)+Bffi(z))+Dddi(z)+Dffi(z)=yi(z)+bij(z)
zx^ij(z)=Ax^ij(z)+Bui(z)+L(yij(z)−y^ij(z))(zI−A)x^ij(z)=Bui(z)+L(yij(z)−y^ij(z))x^ij(z)=(zI−A)−1(Bui(z)+Lyij(z)−Ly^ij(z))x^ij(z)=(zI−A)−1(Bui(z)+Lyij(z))−(zI−A)−1Ly^ij(z)(zI−A)−1LCx^ij(z)+x^ij(z)=(zI−A)−1(Bui(z)+Lyij(z))((zI−A)−1LC+I)x^ij(z)=(zI−A)−1(Bui(z)+Lyij(z))((zI−A)−1LC+I)x^ij(z)=(zI−A)−1(Bui(z)+Lyi(z)+Lbij(z))((zI−A)−1LC+I)x^ij(z)=(zI−A)−1(Bui(z)+LCxi(z)+Lbij(z))y^ij(z)=Cx^ij(z)=C(zI−A)−1(Bui(z)+L(yij(z)−y^ij(z)))=C(zI−A)−1(Bui(z)+Lyij(z)−Ly^ij(z))=C(zI−A)−1(Bui(z)+Lyij(z))−C(zI−A)−1Ly^ij(z)y^ij(z)+C(zI−A)−1Ly^ij(z)=C(zI−A)−1(Bui(z)+Lyij(z))(C(zI−A)−1L+I)y^ij(z)=C(zI−A)−1(Bui(z)+Lyij(z))(C(zI−A)−1L+I)y^ij(z)=C(zI−A)−1Bui(z)+C(zI−A)−1Lyij(z)(C(zI−A)−1L+I)y^ij(z)=C(zI−A)−1Bui(z)+C(zI−A)−1Lyij(z)rij(z)=V(yij(z)−y^ij(z))\begin{aligned} z\hat{x}_i^j(z) &=A \hat{x}_i^j(z) + Bu_i(z) + L(y_i^j(z) - \hat{y}_i^j(z)) \\ (zI-A) \hat{x}_i^j(z) &= Bu_i(z) + L(y_i^j(z) - \hat{y}_i^j(z)) \\ \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z) - L \hat{y}_i^j(z)) \\ \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) - (zI-A)^{-1} L \hat{y}_i^j(z) \\ (zI-A)^{-1} L C \hat{x}_i^j(z) + \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) \\ ((zI-A)^{-1} L C + I) \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) \\ ((zI-A)^{-1} L C + I) \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + Ly_i(z) + L b_i^j(z)) \\ ((zI-A)^{-1} L C + I) \hat{x}_i^j(z) &= (zI-A)^{-1} (Bu_i(z) + L C x_i(z) + L b_i^j(z)) \\ \\ \hat{y}^j_i(z) &=C \hat{x}^j_i(z) \\ &= C (zI-A)^{-1} (Bu_i(z) + L(y_i^j(z) - \hat{y}_i^j(z))) \\ &= C (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z) - L\hat{y}_i^j(z)) \\ &= C (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) - C (zI-A)^{-1} L\hat{y}_i^j(z) \\ \hat{y}^j_i(z) + C (zI-A)^{-1} L\hat{y}_i^j(z) &= C (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) \\ (C (zI-A)^{-1} L + I )\hat{y}_i^j(z) &= C (zI-A)^{-1} (Bu_i(z) + Ly_i^j(z)) \\ (C (zI-A)^{-1} L + I )\hat{y}_i^j(z) &= C (zI-A)^{-1} Bu_i(z) + C (zI-A)^{-1} Ly_i^j(z) \\ (C (zI-A)^{-1} L + I )\hat{y}_i^j(z) &= C (zI-A)^{-1} Bu_i(z) + C (zI-A)^{-1} Ly_i^j(z) \\ \\ r^j_i(z) &=V (y_i^j(z) - \hat{y}_i^j(z)) \end{aligned}zx^ij(z)(zI−A)x^ij(z)x^ij(z)x^ij(z)(zI−A)−1LCx^ij(z)+x^ij(z)((zI−A)−1LC+I)x^ij(z)((zI−A)−1LC+I)x^ij(z)((zI−A)−1LC+I)x^ij(z)y^ij(z)y^ij(z)+C(zI−A)−1Ly^ij(z)(C(zI−A)−1L+I)y^ij(z)(C(zI−A)−1L+I)y^ij(z)(C(zI−A)−1L+I)y^ij(z)rij(z)=Ax^ij(z)+Bui(z)+L(yij(z)−y^ij(z))=Bui(z)+L(yij(z)−y^ij(z))=(zI−A)−1(Bui(z)+Lyij(z)−Ly^ij(z))=(zI−A)−1(Bui(z)+Lyij(z))−(zI−A)−1Ly^ij(z)=(zI−A)−1(Bui(z)+Lyij(z))=(zI−A)−1(Bui(z)+Lyij(z))=(zI−A)−1(Bui(z)+Lyi(z)+Lbij(z))=(zI−A)−1(Bui(z)+LCxi(z)+Lbij(z))=Cx^ij(z)=C(zI−A)−1(Bui(z)+L(yij(z)−y^ij(z)))=C(zI−A)−1(Bui(z)+Lyij(z)−Ly^ij(z))=C(zI−A)−1(Bui(z)+Lyij(z))−C(zI−A)−1Ly^ij(z)=C(zI−A)−1(Bui(z)+Lyij(z))=C(zI−A)−1(Bui(z)+Lyij(z))=C(zI−A)−1Bui(z)+C(zI−A)−1Lyij(z)=C(zI−A)−1Bui(z)+C(zI−A)−1Lyij(z)=V(yij(z)−y^ij(z))
rij(z)=V(yij(z)−y^ij(z))=V(yi(z)+bij(z)−y^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V(C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z))=V(C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z))\begin{aligned} r^j_i(z) &= V (\red{y_i^j(z)} - \hat{y}_i^j(z)) \\ &= V (\red{y_i(z) + b_i^j(z)} - \blue{\hat{y}_i^j(z)}) \\ &= V (\red{C x_i(z) + D_d d_i(z) + D_f f_i(z) + b_i^j(z)} - \blue{C \hat{x}^j_i(z)}) \\ &= V (\red{C x_i(z) + D_d d_i(z) + D_f f_i(z) + b_i^j(z)} - \blue{C \hat{x}^j_i(z)}) \\ &= V (\red{C (x_i(z) - \blue{\hat{x}^j_i(z)}) + D_d d_i(z) + D_f f_i(z) + b_i^j(z)}) \\ &= V (\red{C (x_i(z) - \blue{\hat{x}^j_i(z)}) + D_d d_i(z) + D_f f_i(z) + b_i^j(z)}) \\ \end{aligned}rij(z)=V(yij(z)−y^ij(z))=V(yi(z)+bij(z)−y^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V(Cxi(z)+Dddi(z)+Dffi(z)+bij(z)−Cx^ij(z))=V(C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z))=V(C(xi(z)−x^ij(z))+Dddi(z)+Dffi(z)+bij(z))
2.3 Problem Formulation
3. Secure Scheme Design for MASs
3.1 Independent Anomaly Detection
下述仿真和分析对应于程序 Main_2020_AnomalyDetector.m
首先看下不发生异常时，系统（如论文中公式（1））正常运行，运行效果如下图左半部分所示。
同时构建的观测器都能成功观测到系统的状态值，观测效果如下图右半部分所示。
接下来加入攻击信号，b12b_1^2b12 意味着信息从智能体 2 传递给 1 时出现了攻击。
至此，完成了对论文中 Algorithm 1 的实现。
【Paper】2020_Anomaly Detection and Identification for Multiagent Systems Subjected to Physical Faults相关推荐
- 【Paper】2020_Event-Triggered Time-Varying Formation Control for Discrete-Time Multi-Agent Systems wit
Z. Yan, L. Han, X. Li, X. Dong, Q. Li and Z. Ren, "Event-Triggered Time-Varying Formation Contr ...
- 【Paper】2021_Optimal Distributed Leader-following Consensus of Linear Multi-agent Systems: A Dynamic
Ren Y, Wang Q, Duan Z. Optimal Distributed Leader-following Consensus of Linear Multi-agent Systems: ...
- 【Paper】2022_Fixed-Time Cooperative Tracking for Delayed Disturbed Multi-Agent Systems Under Dynamic
2022_Fixed-Time Cooperative Tracking for Delayed Disturbed Multi-Agent Systems Under Dynamic Event-T ...
- 【Paper】2019_Distributed bipartite leader-following consensus of linear multi-agent systems with inpu
2019_Distributed bipartite leader-following consensus of linear multi-agent systems with input time ...
- 【Paper】2021_Distributed Consensus Tracking of Networked Agent Systems Under Denial-of-Service Attack
Y. Wan, G. Wen, X. Yu and T. Huang, "Distributed Consensus Tracking of Networked Agent Systems ...
- 【Paper】2017_水下潜航器编队海洋勘测的协调控制方法研究
友情链接:[paper]2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and T ...
- 【Paper】2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and Time D
Zhang W, Zeng J, Yan Z, et al. Consensus control of multiple AUVs recovery system under switching to ...
- 【Paper】2009_Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective 精炼版
详细版请参考:[Paper]2009_Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective 文章目录 5. ...
- 【Paper】2015_El H_Decentralized Control Architecture for UAV-UGV Cooperation
Decentralized Control Architecture for UAV-UGV Cooperation 1 Introduction 2 Problem Statement and Ar ...
最新文章
- spark 获取广播变量_Spark流式程序中广播变量和累加器为何使用单例模式
- 自己写的python软件可以在哪发布-如何发布一个Python命令行工具
- 成本中心的费用计划/KP06
- linux boost 卸载,Ubuntu下boost库的编译安装步骤及卸载方法详解
- 北邮OJ 85. Three Points On A Line
- linux利用grep查看打印匹配的下几行或前后几行的命令
- IllegalArgumentException：argument type mismatch
- 蓝桥杯评分标准_蓝桥杯比赛要求
- Vue中CSS模块化最佳实践
- 微软Skype推出Slack整合预览版
- 典型传动机构的动力学模型
- ftfind 桌面搜索引擎的设计文档 (2)
- 百度文库免费下载含豆丁、道客巴巴等
- sipp basic call 脚本
- 红蓝眼睛（答案在文章中找）
- c语言中-next是什么意思,Java中关于nextInt()、next()和nextLine()的理解
- 【人工智能II】实验2 强化学习Q-Learning算法
- 如何加声调口诀_汉语拼音声调标注口诀 涨知识了
- java excel 模板 替换_java替换Excel字符
- 【小甲鱼编程题05】打印杨辉三角形