| 符号 | 说明 | |
|---|---|---|
| x i ( k ) x_i(k) xi?(k) | 状态 | |
| u i ( k ) u_i(k) ui?(k) | 控制输入 | |
| y i ( k ) y_i(k) yi?(k) | 测量输出向量 | |
| d i d_i di? | 未知外界干扰信号 | |
| f i f_i fi? | 故障 | |
| y i j ( k ) y_i^j(k) yij?(k) | i i i 从 j j j 这里获得的输出信息 | |
| b i j ( k ) b_i^j(k) bij?(k) | i i i 从 j j j 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 | |
| r i j ( k ) r_i^j(k) rij?(k) | i i i 从 j j j 这里获得的残差信息 |
文章目录
1. Introduction
2. Preliminaries and Problem Formulation
2.1 System Description
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 ) (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)+?Bd?di?(k)+Bf?fi?(k)Dd?di?(k)+Df?fi?(k)?(1)
袭击模型表示为
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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：
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\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 r i j r_i^j rij? 的 Z 变换为：
r i j ( 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 ( z I ? A + L C ) ? 1 L G f ( z ) = D f + C ( z I ? A + L C ) ? 1 ( B f ? L D f ) G d ( z ) = D d + C ( z I ? A + L C ) ? 1 ( B d ? L D d ) (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)+Gd?di?(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：
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\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)+Bd?di?(k)+Bf?fi?(k)?Ax^ij?(k)?Bui?(k)?L(yij?(k)?y^?ij?(k))=A(xi?(k)?x^ij?(k))+Bd?di?(k)+Bf?fi?(k)?Lyij?(k)+Ly^?ij?(k)=A(xi?(k)?x^ij?(k))+Bd?di?(k)+Bf?fi?(k)?Lyi?(k)+Ly^?ij?(k)?Lbij?(k)=A(xi?(k)?x^ij?(k))+Bd?di?(k)+Bf?fi?(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
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y_i(k) -\hat{y}_i^j(k)
yi?(k)?y^?ij?(k) 单独拿出来。
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\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)+Dd?di?(k)+Df?fi?(k)?Cx^ij?(k)=C(xi?(k)?x^ij?(k))+Dd?di?(k)+Df?fi?(k)?
那么
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\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))+Bd?di?(k)+Bf?fi?(k)?L(yi?(k)?y^?ij?(k))?Lbij?(k)=A(xi?(k)?x^ij?(k))+Bd?di?(k)+Bf?fi?(k)?L(C(xi?(k)?x^ij?(k))+Dd?di?(k)+Df?fi?(k))?Lbij?(k)=A(xi?(k)?x^ij?(k))+Bd?di?(k)+Bf?fi?(k)?LC(xi?(k)?x^ij?(k))?LDd?di?(k)?LDf?fi?(k))?Lbij?(k)=(A?LC)(xi?(k)?x^ij?(k))+(Bd??LDd?)di?(k)+(Bf??LDf?)fi?(k)?Lbij?(k)?
做 Z 变换
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\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)?
r i j ( z ) = V ( y i j ( z ) ? y ^ i j ( z ) ) = V ( y i ( z ) + b i j ( z ) ? y ^ i j ( z ) ) = V ( C x i ( z ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ? C x ^ i j ( z ) ) = V [ C x i ( z ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ? C x ^ i j ( z ) ] = V [ C ( x i ( z ) ? x ^ i j ( z ) ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ] = V [ C ( z I ? A + L C ) ? 1 ( B d ? L D d ) d i ( z ) + C ( z I ? A + L C ) ? 1 ( B f ? L D f ) f i ( z ) ? C ( z I ? A + L C ) ? 1 L b i j ( z ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ] = V [ C ( z I ? A + L C ) ? 1 ( B d ? L D d ) d i ( z ) + D d d i ( z ) ????? + C ( z I ? A + L C ) ? 1 ( B f ? L D f ) f i ( z ) + D f f i ( z ) ????? ? C ( z I ? A + L C ) ? 1 L b i j ( z ) + b i j ( 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)+Dd?di?(z)+Df?fi?(z)+bij?(z)?Cx^ij?(z))=V[Cxi?(z)+Dd?di?(z)+Df?fi?(z)+bij?(z)?Cx^ij?(z)]=V[C(xi?(z)?x^ij?(z))+Dd?di?(z)+Df?fi?(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)+Dd?di?(z)+Df?fi?(z)+bij?(z)]=V[C(zI?A+LC)?1(Bd??LDd?)di?(z)+Dd?di?(z)?????+C(zI?A+LC)?1(Bf??LDf?)fi?(z)+Df?fi?(z)??????C(zI?A+LC)?1Lbij?(z)+bij?(z)]?
再令
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G_b(z) = C(zI-A+LC)^{-1}(B_d-LD_d) + D_d
Gb?(z)=C(zI?A+LC)?1(Bd??LDd?)+Dd?
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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。
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\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)+Bd?di?(z)+Bf?fi?(z)=Bui?(z)+Bd?di?(z)+Bf?fi?(z)=Bui?(z)+Bd?di?(z)+Bf?fi?(z)=(zI?A)?1(Bui?(z)+Bd?di?(z)+Bf?fi?(z))=Cxi?(z)+Dd?di?(z)+Df?fi?(z)=C(zI?A)?1(Bui?(z)+Bd?di?(z)+Bf?fi?(z))+Dd?di?(z)+Df?fi?(z)=yi?(z)+bij?(z)?
z x ^ i j ( z ) = A x ^ i j ( z ) + B u i ( z ) + L ( y i j ( z ) ? y ^ i j ( z ) ) ( z I ? A ) x ^ i j ( z ) = B u i ( z ) + L ( y i j ( z ) ? y ^ i j ( z ) ) x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ? L y ^ i j ( z ) ) x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ? ( z I ? A ) ? 1 L y ^ i j ( z ) ( z I ? A ) ? 1 L C x ^ i j ( z ) + x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ( ( z I ? A ) ? 1 L C + I ) x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ( ( z I ? A ) ? 1 L C + I ) x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L y i ( z ) + L b i j ( z ) ) ( ( z I ? A ) ? 1 L C + I ) x ^ i j ( z ) = ( z I ? A ) ? 1 ( B u i ( z ) + L C x i ( z ) + L b i j ( z ) ) y ^ i j ( z ) = C x ^ i j ( z ) = C ( z I ? A ) ? 1 ( B u i ( z ) + L ( y i j ( z ) ? y ^ i j ( z ) ) ) = C ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ? L y ^ i j ( z ) ) = C ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ? C ( z I ? A ) ? 1 L y ^ i j ( z ) y ^ i j ( z ) + C ( z I ? A ) ? 1 L y ^ i j ( z ) = C ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ( C ( z I ? A ) ? 1 L + I ) y ^ i j ( z ) = C ( z I ? A ) ? 1 ( B u i ( z ) + L y i j ( z ) ) ( C ( z I ? A ) ? 1 L + I ) y ^ i j ( z ) = C ( z I ? A ) ? 1 B u i ( z ) + C ( z I ? A ) ? 1 L y i j ( z ) ( C ( z I ? A ) ? 1 L + I ) y ^ i j ( z ) = C ( z I ? A ) ? 1 B u i ( z ) + C ( z I ? A ) ? 1 L y i j ( z ) r i j ( z ) = V ( y i j ( z ) ? y ^ i j ( 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))?
r i j ( z ) = V ( y i j ( z ) ? y ^ i j ( z ) ) = V ( y i ( z ) + b i j ( z ) ? y ^ i j ( z ) ) = V ( C x i ( z ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ? C x ^ i j ( z ) ) = V ( C x i ( z ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ? C x ^ i j ( z ) ) = V ( C ( x i ( z ) ? x ^ i j ( z ) ) + D d d i ( z ) + D f f i ( z ) + b i j ( z ) ) = V ( C ( x i ( z ) ? x ^ i j ( z ) ) + D d d i ( z ) + D f f i ( z ) + b i j ( 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)+Dd?di?(z)+Df?fi?(z)+bij?(z)?Cx^ij?(z))=V(Cxi?(z)+Dd?di?(z)+Df?fi?(z)+bij?(z)?Cx^ij?(z))=V(C(xi?(z)?x^ij?(z))+Dd?di?(z)+Df?fi?(z)+bij?(z))=V(C(xi?(z)?x^ij?(z))+Dd?di?(z)+Df?fi?(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， b 1 2 b_1^2 b12? 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。
