【Paper】2015_异构无人机群鲁棒一致性协议设计_孙长银
2015_异构无人机群鲁棒一致性协议设计_孙长银
- 4 分布式鲁棒一致性协议设计
- 6 Simulations
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- Condition 1
- Condition 2
- Condition 3
- Condition 4
先把无人机的通信拓扑图放出来，这样在接下来的公式分析时能够便于理解。
4 分布式鲁棒一致性协议设计
Laplacian Matrix 是
L = [ 0 0 0 0 0 0 1 − 1 0 0 0 0 1 − 1 0 − 1 0 0 0 0 0 0 0 − 1 1 ] L= \left[\begin{matrix} 0 && 0 & 0 & 0 & 0 \\ \\ 0 && 1 & -1 & 0 & 0 \\ 0 && 0 & 1 & -1 & 0 \\ -1 && 0 & 0 & 0 & 0 \\ 0 && 0 & 0 & -1 & 1 \\ \end{matrix}\right] L=⎣⎢⎢⎢⎢⎢⎢⎡000−10010000−110000−10−100001⎦⎥⎥⎥⎥⎥⎥⎤
s 1 = [ s 1 , 1 T s 2 , 1 T ⋮ s M , 1 T ] s_1 = \left[\begin{matrix} s^T_{1,1} \\ s^T_{2,1} \\ \vdots\\ s^T_{M,1} \end{matrix}\right] s1=⎣⎢⎢⎢⎡s1,1Ts2,1T⋮sM,1T⎦⎥⎥⎥⎤
y 1 = [ y 1 T s 2 T ⋮ s M T ] y_1 = \left[\begin{matrix} y^T_{1} \\ s^T_{2} \\ \vdots\\ s^T_{M} \end{matrix}\right] y1=⎣⎢⎢⎢⎡y1Ts2T⋮sMT⎦⎥⎥⎥⎤
第1步：
s ˙ i , 1 = ∑ j = 1 M a i j ( y ˙ i − y ˙ j ) + b i ( y ˙ i − r ˙ ) = ∑ j = 1 M a i j ( g i ( Θ i ) x i , 2 − g j ( Θ j ) x j , 2 ) + b i ( g i ( Θ i ) x i 2 − r ˙ ) = − ∑ j = 1 M a i j g j ( Θ j ) x j , 2 − b i r ˙ + ( ∑ j = 1 M a i , j + b i ) g i ( Θ i ) x i , 2 = \begin{aligned} \dot{s}_{i,1} &= \sum_{j=1}^{M} a_{ij} (\dot{y}_i - \dot{y}_j) + b_i (\dot{y}_i - \dot{r}) \\ &= \sum_{j=1}^{M} a_{ij} (~g_i(\Theta_i)x_{i,2} - g_j(\Theta_j)x_{j,2}~) + b_i (~g_i(\Theta_i)x_{i_2} - \dot{r}~) \\ &= -\sum_{j=1}^{M} a_{ij} g_j (\Theta_j) x_{j,2} - b_i \dot{r} + (\sum_{j=1}^{M} a_{i,j} + b_i) g_i(\Theta_i)x_{i,2}\\ &= \end{aligned} s˙i,1=j=1∑Maij(y˙i−y˙j)+bi(y˙i−r˙)=j=1∑Maij( gi(Θi)xi,2−gj(Θj)xj,2 )+bi( gi(Θi)xi2−r˙ )=−j=1∑Maijgj(Θj)xj,2−bir˙+(j=1∑Mai,j+bi)gi(Θi)xi,2=
6 Simulations
M M M 个跟随者的动态模型为
x ˙ i , 1 = g i ( Θ i ) x i , 2 x ˙ i , 2 = u i + Φ i ( x i , 1 , x i , 2 , κ i ) y i = x i , 1 \begin{aligned} &{\dot{x}_{i,1}} = g_i(\varTheta_i) x_{i,2} \\ &\dot{x}_{i,2} = u_i + \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \\ &y_i = x_{i,1} \end{aligned} x˙i,1=gi(Θi)xi,2x˙i,2=ui+Φi(xi,1,xi,2,κi)yi=xi,1
转换成矩阵写法：
[ x ˙ 1 , 1 x ˙ 2 , 1 x ˙ 3 , 1 x ˙ 4 , 1 ] = [ x 1 , 2 x 2 , 2 x 3 , 2 x 4 , 2 ] [ x ˙ 1 , 2 x ˙ 2 , 2 x ˙ 3 , 2 x ˙ 4 , 2 ] = [ u 1 u 2 u 3 u 4 ] + [ Φ 1 ( x 1 , 1 , x 1 , 2 , κ 1 ) Φ 2 ( x 2 , 1 , x 2 , 2 , κ 2 ) Φ 3 ( x 3 , 1 , x 3 , 2 , κ 3 ) Φ 4 ( x 4 , 1 , x 4 , 2 , κ 4 ) ] \begin{aligned}& \left[\begin{matrix} \dot{x}_{1,1} \\ \dot{x}_{2,1} \\ \dot{x}_{3,1} \\ \dot{x}_{4,1} \\ \end{matrix}\right]= \left[\begin{matrix} {x}_{1,2} \\ {x}_{2,2} \\ {x}_{3,2} \\ {x}_{4,2} \\ \end{matrix}\right] \\& \left[\begin{matrix} \dot{x}_{1,2} \\ \dot{x}_{2,2} \\ \dot{x}_{3,2} \\ \dot{x}_{4,2} \\ \end{matrix}\right]= \left[\begin{matrix} {u}_{1} \\ {u}_{2} \\ {u}_{3} \\ {u}_{4} \\ \end{matrix}\right] + \left[\begin{matrix} {\varPhi}_{1}(x_{1,1}, x_{1,2}, \kappa_1) \\ {\varPhi}_{2}(x_{2,1}, x_{2,2}, \kappa_2) \\ {\varPhi}_{3}(x_{3,1}, x_{3,2}, \kappa_3) \\ {\varPhi}_{4}(x_{4,1}, x_{4,2}, \kappa_4) \\ \end{matrix}\right] \end{aligned} ⎣⎢⎢⎡x˙1,1x˙2,1x˙3,1x˙4,1⎦⎥⎥⎤=⎣⎢⎢⎡x1,2x2,2x3,2x4,2⎦⎥⎥⎤⎣⎢⎢⎡x˙1,2x˙2,2x˙3,2x˙4,2⎦⎥⎥⎤=⎣⎢⎢⎡u1u2u3u4⎦⎥⎥⎤+⎣⎢⎢⎡Φ1(x1,1,x1,2,κ1)Φ2(x2,1,x2,2,κ2)Φ3(x3,1,x3,2,κ3)Φ4(x4,1,x4,2,κ4)⎦⎥⎥⎤
整个一致性协议为：
u i = − ρ s i , 2 + f w i w i = − ( 1 + ρ s ) s i , 2 s i , 1 = ∑ j = 1 M a i j ( y i − y j ) + b i ( y i − r ) s i , 2 = x i , 2 − 1 ( d i + b i ) g i − 1 ( Θ i ) ( − ρ s i , 1 + ∑ j = 1 M a i j g j ( Θ j ) x j , 2 + b i r ˙ ) \begin{aligned} & u_i = -\rho \red{s_{i,2}} + f \blue{w_i} \\ & \blue{w_i} = - (1+\frac{\rho}{s}) \red{s_{i,2}} \\ & \green{s_{i,1}} = \sum_{j=1}^{M} a_{ij} (y_i - y_j) + b_i(y_i - r) \\ & \red{s_{i,2}} = x_{i,2} - \frac{1}{(d_i + b_i)} g_i^{-1}(\varTheta_i) (-\rho \green{s_{i,1}} + \sum_{j=1}^{M} a_{ij} g_j (\varTheta_j) x_{j,2} + b_i \dot{r}) \end{aligned} ui=−ρsi,2+fwiwi=−(1+sρ)si,2si,1=j=1∑Maij(yi−yj)+bi(yi−r)si,2=xi,2−(di+bi)1gi−1(Θi)(−ρsi,1+j=1∑Maijgj(Θj)xj,2+bir˙)
转换成矩阵写法：
[ u 1 u 2 u 3 u 4 ] = − ρ [ s 1 , 2 s 2 , 2 s 3 , 2 s 4 , 2 ] + f [ w 1 w 2 w 3 w 4 ] [ w 1 w 2 w 3 w 4 ] = − ( 1 + ρ s ) [ s 1 , 2 s 2 , 2 s 3 , 2 s 4 , 2 ] [ s 1 , 1 s 2 , 1 s 3 , 1 s 4 , 1 ] = L ⋅ [ y 1 y 2 y 3 y 4 ] + [ b 1 ⋅ ( y 1 − r ) b 2 ⋅ ( y 2 − r ) b 3 ⋅ ( y 3 − r ) b 4 ⋅ ( y 4 − r ) ] = L ⋅ [ x 1 , 1 x 2 , 1 x 3 , 1 x 4 , 1 ] + [ b 1 b 2 b 3 b 4 ] [ x 1 , 1 − r x 2 , 1 − r x 3 , 1 − r x 4 , 1 − r ] [ s 1 , 2 s 2 , 2 s 3 , 2 s 4 , 2 ] = [ x 1 , 2 x 2 , 2 x 3 , 2 x 4 , 2 ] − [ 1 d 1 + b 1 ( − ρ s 1 , 1 + x j , 2 + b 1 r ˙ ) 1 d 2 + b 2 ( − ρ s 2 , 1 + x j , 2 + b 2 r ˙ ) 1 d 3 + b 3 1 d 4 + b 4 ] \begin{aligned} \left[\begin{matrix} {u}_{1} \\ {u}_{2} \\ {u}_{3} \\ {u}_{4} \\ \end{matrix}\right]&=-\rho \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right]+f \left[\begin{matrix} {w}_{1} \\ {w}_{2} \\ {w}_{3} \\ {w}_{4} \\ \end{matrix}\right] \\ \left[\begin{matrix} {w}_{1} \\ {w}_{2} \\ {w}_{3} \\ {w}_{4} \\ \end{matrix}\right]&=-(1+\frac{\rho}{s}) \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right] \\ \left[\begin{matrix} {s}_{1,1} \\ {s}_{2,1} \\ {s}_{3,1} \\ {s}_{4,1} \\ \end{matrix}\right]&= L \cdot \left[\begin{matrix} {y}_{1} \\ {y}_{2} \\ {y}_{3} \\ {y}_{4} \\ \end{matrix}\right]+ \left[\begin{matrix} b_1 \cdot ({y}_{1}-r) \\ b_2 \cdot ({y}_{2}-r) \\ b_3 \cdot ({y}_{3}-r) \\ b_4 \cdot ({y}_{4}-r) \\ \end{matrix}\right] \\&= L \cdot \left[\begin{matrix} {x}_{1,1} \\ {x}_{2,1} \\ {x}_{3,1} \\ {x}_{4,1} \\ \end{matrix}\right]+ \left[\begin{matrix} {b}_{1} \\ &{b}_{2} \\ &&{b}_{3} \\ &&&{b}_{4} \\ \end{matrix}\right] \left[\begin{matrix} {x}_{1,1}-r \\ {x}_{2,1}-r \\ {x}_{3,1}-r \\ {x}_{4,1}-r \\ \end{matrix}\right] \\ \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right]&= \left[\begin{matrix} {x}_{1,2} \\ {x}_{2,2} \\ {x}_{3,2} \\ {x}_{4,2} \\ \end{matrix}\right]- \left[\begin{matrix} \frac{1}{d_1+b_1}(-\rho s_{1,1} + x_{j,2} + b_1 \dot{r}) \\ \frac{1}{d_2+b_2}(-\rho s_{2,1} + x_{j,2} + b_2 \dot{r}) \\ \frac{1}{d_3+b_3} \\ \frac{1}{d_4+b_4} \\ \end{matrix}\right] \end{aligned} ⎣⎢⎢⎡u1u2u3u4⎦⎥⎥⎤⎣⎢⎢⎡w1w2w3w4⎦⎥⎥⎤⎣⎢⎢⎡s1,1s2,1s3,1s4,1⎦⎥⎥⎤⎣⎢⎢⎡s1,2s2,2s3,2s4,2⎦⎥⎥⎤=−ρ⎣⎢⎢⎡s1,2s2,2s3,2s4,2⎦⎥⎥⎤+f⎣⎢⎢⎡w1w2w3w4⎦⎥⎥⎤=−(1+sρ)⎣⎢⎢⎡s1,2s2,2s3,2s4,2⎦⎥⎥⎤=L⋅⎣⎢⎢⎡y1y2y3y4⎦⎥⎥⎤+⎣⎢⎢⎡b1⋅(y1−r)b2⋅(y2−r)b3⋅(y3−r)b4⋅(y4−r)⎦⎥⎥⎤=L⋅⎣⎢⎢⎡x1,1x2,1x3,1x4,1⎦⎥⎥⎤+⎣⎢⎢⎡b1b2b3b4⎦⎥⎥⎤⎣⎢⎢⎡x1,1−rx2,1−rx3,1−rx4,1−r⎦⎥⎥⎤=⎣⎢⎢⎡x1,2x2,2x3,2x4,2⎦⎥⎥⎤−⎣⎢⎢⎡d1+b11(−ρs1,1+xj,2+b1r˙)d2+b21(−ρs2,1+xj,2+b2r˙)d3+b31d4+b41⎦⎥⎥⎤
仿真部分也给出了干扰的表述方式：
Φ i ( x i , 1 , x i , 2 , κ i ) ≤ ∣ x i , 1 ∣ + ∣ x i , 2 ∣ + rand ( 5 ) \begin{aligned} \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \le |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \end{aligned} Φi(xi,1,xi,2,κi)≤∣xi,1∣+∣xi,2∣+rand(5)
把两个公式合并一下（忽略转换矩阵的作用）：
x ˙ i , 1 = x i , 2 x ˙ i , 2 = u i + Φ i ( x i , 1 , x i , 2 , κ i ) = u i + ∣ x i , 1 ∣ + ∣ x i , 2 ∣ + rand ( 5 ) = − ρ s i , 2 + f w i + ∣ x i , 1 ∣ + ∣ x i , 2 ∣ + rand ( 5 ) = − ρ s i , 2 + f [ − ( 1 + ρ s ) s i , 2 ] + ∣ x i , 1 ∣ + ∣ x i , 2 ∣ + rand ( 5 ) \begin{aligned} \dot{x}_{i,1} &= x_{i,2} \\ \dot{x}_{i,2} &= u_i + \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \\ &= u_i + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ &= -\rho s_{i,2} + f w_i + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ &= -\rho s_{i,2} + f [-(1+\frac{\rho}{s})s_{i,2}] + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ \end{aligned} x˙i,1x˙i,2=xi,2=ui+Φi(xi,1,xi,2,κi)=ui+∣xi,1∣+∣xi,2∣+rand(5)=−ρsi,2+fwi+∣xi,1∣+∣xi,2∣+rand(5)=−ρsi,2+f[−(1+sρ)si,2]+∣xi,1∣+∣xi,2∣+rand(5)
Condition 1
Condition 2
Condition 3
Condition 4
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