4 分布式鲁棒一致性协议设计

Laplacian Matrix 是
$\left[$

\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}

$\begin{matrix} 0 && 0 & 0 & 0 & 0 \\ \\ 0 && 1 & -1 & 0 & 0 \\ 0 && 0 & 1 & -1 & 0 \\ -1 && 0 & 0 & 0 & 0 \\ 0 && 0 & 0 & -1 & 1 \\ \end{matrix}$ \right]

L = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 000 - 1 0 01000 0 - 1 100 00 - 1 0 - 1 00001 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤

$s_1 = \left[$

\begin{matrix} s_{1, 1}^{T} \\ s_{2, 1}^{T} \\ ⋮ \\ s_{M, 1}^{T} \end{matrix}

$\begin{matrix} s^T_{1,1} \\ s^T_{2,1} \\ \vdots\\ s^T_{M,1} \end{matrix}$ \right]

s_{1} = ⎣ ⎢ ⎢ ⎢ ⎡ s_{1, 1}^{T} s_{2, 1}^{T} ⋮ s_{M, 1}^{T} ⎦ ⎥ ⎥ ⎥ ⎤

$y_1 = \left[$

\begin{matrix} y_{1}^{T} \\ s_{2}^{T} \\ ⋮ \\ s_{M}^{T} \end{matrix}

$\begin{matrix} y^T_{1} \\ s^T_{2} \\ \vdots\\ s^T_{M} \end{matrix}$ \right]

y_{1} = ⎣ ⎢ ⎢ ⎢ ⎡ y_{1}^{T} s_{2}^{T} ⋮ s_{M}^{T} ⎦ ⎥ ⎥ ⎥ ⎤

第1步：

\begin{aligned} {\dot{s}}_{i, 1} & = \sum_{j = 1}^{M} a_{i j} ({\dot{y}}_{i} - {\dot{y}}_{j}) + b_{i} ({\dot{y}}_{i} - \dot{r}) \\ = \sum_{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}} - \dot{r}) \\ = - \sum_{j = 1}^{M} a_{i j} g_{j} (Θ_{j}) x_{j, 2} - b_{i} \dot{r} + (\sum_{j = 1}^{M} a_{i, j} + b_{i}) g_{i} (Θ_{i}) x_{i, 2} \\ = \end{aligned}

$\begin{aligned} \dot{s}_{i,1} &= \sum_{j=1}^{M} a_{ij} (\dot{y}_i - \dot{y}_j) + b_i (\dot{y}_i - \dot{r}) \\ &= \sum_{j=1}^{M} a_{ij} (~g_i(\Theta_i)x_{i,2} - g_j(\Theta_j)x_{j,2}~) + b_i (~g_i(\Theta_i)x_{i_2} - \dot{r}~) \\ &= -\sum_{j=1}^{M} a_{ij} g_j (\Theta_j) x_{j,2} - b_i \dot{r} + (\sum_{j=1}^{M} a_{i,j} + b_i) g_i(\Theta_i)x_{i,2}\\ &= \end{aligned}$

\overset{s}{˙}_{i, 1} = j = 1 \sum M a_{i j} (\overset{y}{˙}_{i} - \overset{y}{˙}_{j}) + b_{i} (\overset{y}{˙}_{i} - \overset{r}{˙}) = j = 1 \sum 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}} - \overset{r}{˙}) = - j = 1 \sum M a_{i j} g_{j} (Θ_{j}) x_{j, 2} - b_{i} \overset{r}{˙} + (j = 1 \sum M a_{i, j} + b_{i}) g_{i} (Θ_{i}) x_{i, 2} =

6 Simulations

$M$ 个跟随者的动态模型为

\begin{aligned} {\dot{x}}_{i, 1} = g_{i} (Θ_{i}) x_{i, 2} \\ {\dot{x}}_{i, 2} = u_{i} + Φ_{i} (x_{i, 1}, x_{i, 2}, κ_{i}) \\ y_{i} = x_{i, 1} \end{aligned}

$\begin{aligned} &{\dot{x}_{i,1}} = g_i(\varTheta_i) x_{i,2} \\ &\dot{x}_{i,2} = u_i + \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \\ &y_i = x_{i,1} \end{aligned}$

\overset{x}{˙}_{i, 1} = g_{i} (Θ_{i}) x_{i, 2} \overset{x}{˙}_{i, 2} = u_{i} + Φ_{i} (x_{i, 1}, x_{i, 2}, κ_{i}) y_{i} = x_{i, 1}

转换成矩阵写法：

\begin{aligned} [\begin{matrix} {\dot{x}}_{1, 1} \\ {\dot{x}}_{2, 1} \\ {\dot{x}}_{3, 1} \\ {\dot{x}}_{4, 1} \end{matrix}] = [\begin{matrix} x_{1, 2} \\ x_{2, 2} \\ x_{3, 2} \\ x_{4, 2} \end{matrix}] \\ [\begin{matrix} {\dot{x}}_{1, 2} \\ {\dot{x}}_{2, 2} \\ {\dot{x}}_{3, 2} \\ {\dot{x}}_{4, 2} \end{matrix}] = [\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}] + [\begin{matrix} 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}) \end{matrix}] \end{aligned}

$\begin{aligned}& \left[\begin{matrix} \dot{x}_{1,1} \\ \dot{x}_{2,1} \\ \dot{x}_{3,1} \\ \dot{x}_{4,1} \\ \end{matrix}\right]= \left[\begin{matrix} {x}_{1,2} \\ {x}_{2,2} \\ {x}_{3,2} \\ {x}_{4,2} \\ \end{matrix}\right] \\& \left[\begin{matrix} \dot{x}_{1,2} \\ \dot{x}_{2,2} \\ \dot{x}_{3,2} \\ \dot{x}_{4,2} \\ \end{matrix}\right]= \left[\begin{matrix} {u}_{1} \\ {u}_{2} \\ {u}_{3} \\ {u}_{4} \\ \end{matrix}\right] + \left[\begin{matrix} {\varPhi}_{1}(x_{1,1}, x_{1,2}, \kappa_1) \\ {\varPhi}_{2}(x_{2,1}, x_{2,2}, \kappa_2) \\ {\varPhi}_{3}(x_{3,1}, x_{3,2}, \kappa_3) \\ {\varPhi}_{4}(x_{4,1}, x_{4,2}, \kappa_4) \\ \end{matrix}\right] \end{aligned}$

⎣ ⎢ ⎢ ⎡ \overset{x}{˙}_{1, 1} \overset{x}{˙}_{2, 1} \overset{x}{˙}_{3, 1} \overset{x}{˙}_{4, 1} ⎦ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ x_{1, 2} x_{2, 2} x_{3, 2} x_{4, 2} ⎦ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎡ \overset{x}{˙}_{1, 2} \overset{x}{˙}_{2, 2} \overset{x}{˙}_{3, 2} \overset{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}) ⎦ ⎥ ⎥ ⎤

整个一致性协议为：

\begin{aligned} u_{i} = - ρ \red s_{i, 2} + f \blue w_{i} \\ \blue w_{i} = - (1 + \frac{ρ}{s}) \red s_{i, 2} \\ \green s_{i, 1} = \sum_{j = 1}^{M} a_{i j} (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} (Θ_{i}) (- ρ \green s_{i, 1} + \sum_{j = 1}^{M} a_{i j} g_{j} (Θ_{j}) x_{j, 2} + b_{i} \dot{r}) \end{aligned}

$\begin{aligned} & u_i = -\rho \red{s_{i,2}} + f \blue{w_i} \\ & \blue{w_i} = - (1+\frac{\rho}{s}) \red{s_{i,2}} \\ & \green{s_{i,1}} = \sum_{j=1}^{M} a_{ij} (y_i - y_j) + b_i(y_i - r) \\ & \red{s_{i,2}} = x_{i,2} - \frac{1}{(d_i + b_i)} g_i^{-1}(\varTheta_i) (-\rho \green{s_{i,1}} + \sum_{j=1}^{M} a_{ij} g_j (\varTheta_j) x_{j,2} + b_i \dot{r}) \end{aligned}$

u_{i} = - ρ s_{i, 2} + f w_{i} w_{i} = - (1 + \frac{ρ}{s}) s_{i, 2} s_{i, 1} = j = 1 \sum M a_{i j} (y_{i} - y_{j}) + b_{i} (y_{i} - r) s_{i, 2} = x_{i, 2} - \frac{1}{( d _{i} + b _{i} )} g_{i}^{- 1} (Θ_{i}) (- ρ s_{i, 1} + j = 1 \sum M a_{i j} g_{j} (Θ_{j}) x_{j, 2} + b_{i} \overset{r}{˙})

转换成矩阵写法：

\begin{aligned} [\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}] & = - ρ [\begin{matrix} s_{1, 2} \\ s_{2, 2} \\ s_{3, 2} \\ s_{4, 2} \end{matrix}] + f [\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \\ w_{4} \end{matrix}] \\ [\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \\ w_{4} \end{matrix}] & = - (1 + \frac{ρ}{s}) [\begin{matrix} s_{1, 2} \\ s_{2, 2} \\ s_{3, 2} \\ s_{4, 2} \end{matrix}] \\ [\begin{matrix} s_{1, 1} \\ s_{2, 1} \\ s_{3, 1} \\ s_{4, 1} \end{matrix}] & = L \cdot [\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{matrix}] + [\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}] \\ = L \cdot [\begin{matrix} x_{1, 1} \\ x_{2, 1} \\ x_{3, 1} \\ x_{4, 1} \end{matrix}] + [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}] [\begin{matrix} x_{1, 1} - r \\ x_{2, 1} - r \\ x_{3, 1} - r \\ x_{4, 1} - r \end{matrix}] \\ [\begin{matrix} s_{1, 2} \\ s_{2, 2} \\ s_{3, 2} \\ s_{4, 2} \end{matrix}] & = [\begin{matrix} x_{1, 2} \\ x_{2, 2} \\ x_{3, 2} \\ x_{4, 2} \end{matrix}] - [\begin{matrix} \frac{1}{d_{1} + b_{1}} (- ρ s_{1, 1} + x_{j, 2} + b_{1} \dot{r}) \\ \frac{1}{d_{2} + b_{2}} (- ρ s_{2, 1} + x_{j, 2} + b_{2} \dot{r}) \\ \frac{1}{d_{3} + b_{3}} \\ \frac{1}{d_{4} + b_{4}} \end{matrix}] \end{aligned}

$\begin{aligned} \left[\begin{matrix} {u}_{1} \\ {u}_{2} \\ {u}_{3} \\ {u}_{4} \\ \end{matrix}\right]&=-\rho \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right]+f \left[\begin{matrix} {w}_{1} \\ {w}_{2} \\ {w}_{3} \\ {w}_{4} \\ \end{matrix}\right] \\ \left[\begin{matrix} {w}_{1} \\ {w}_{2} \\ {w}_{3} \\ {w}_{4} \\ \end{matrix}\right]&=-(1+\frac{\rho}{s}) \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right] \\ \left[\begin{matrix} {s}_{1,1} \\ {s}_{2,1} \\ {s}_{3,1} \\ {s}_{4,1} \\ \end{matrix}\right]&= L \cdot \left[\begin{matrix} {y}_{1} \\ {y}_{2} \\ {y}_{3} \\ {y}_{4} \\ \end{matrix}\right]+ \left[\begin{matrix} b_1 \cdot ({y}_{1}-r) \\ b_2 \cdot ({y}_{2}-r) \\ b_3 \cdot ({y}_{3}-r) \\ b_4 \cdot ({y}_{4}-r) \\ \end{matrix}\right] \\&= L \cdot \left[\begin{matrix} {x}_{1,1} \\ {x}_{2,1} \\ {x}_{3,1} \\ {x}_{4,1} \\ \end{matrix}\right]+ \left[\begin{matrix} {b}_{1} \\ &{b}_{2} \\ &&{b}_{3} \\ &&&{b}_{4} \\ \end{matrix}\right] \left[\begin{matrix} {x}_{1,1}-r \\ {x}_{2,1}-r \\ {x}_{3,1}-r \\ {x}_{4,1}-r \\ \end{matrix}\right] \\ \left[\begin{matrix} {s}_{1,2} \\ {s}_{2,2} \\ {s}_{3,2} \\ {s}_{4,2} \\ \end{matrix}\right]&= \left[\begin{matrix} {x}_{1,2} \\ {x}_{2,2} \\ {x}_{3,2} \\ {x}_{4,2} \\ \end{matrix}\right]- \left[\begin{matrix} \frac{1}{d_1+b_1}(-\rho s_{1,1} + x_{j,2} + b_1 \dot{r}) \\ \frac{1}{d_2+b_2}(-\rho s_{2,1} + x_{j,2} + b_2 \dot{r}) \\ \frac{1}{d_3+b_3} \\ \frac{1}{d_4+b_4} \\ \end{matrix}\right] \end{aligned}$

⎣ ⎢ ⎢ ⎡ u_{1} u_{2} u_{3} u_{4} ⎦ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎡ w_{1} w_{2} w_{3} w_{4} ⎦ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎡ s_{1, 1} s_{2, 1} s_{3, 1} s_{4, 1} ⎦ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎡ s_{1, 2} s_{2, 2} s_{3, 2} s_{4, 2} ⎦ ⎥ ⎥ ⎤ = - ρ ⎣ ⎢ ⎢ ⎡ s_{1, 2} s_{2, 2} s_{3, 2} s_{4, 2} ⎦ ⎥ ⎥ ⎤ + f ⎣ ⎢ ⎢ ⎡ w_{1} w_{2} w_{3} w_{4} ⎦ ⎥ ⎥ ⎤ = - (1 + \frac{ρ}{s}) ⎣ ⎢ ⎢ ⎡ s_{1, 2} s_{2, 2} s_{3, 2} s_{4, 2} ⎦ ⎥ ⎥ ⎤ = L \cdot ⎣ ⎢ ⎢ ⎡ y_{1} y_{2} y_{3} y_{4} ⎦ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎡ b_{1} \cdot (y_{1} - r) b_{2} \cdot (y_{2} - r) b_{3} \cdot (y_{3} - r) b_{4} \cdot (y_{4} - r) ⎦ ⎥ ⎥ ⎤ = L \cdot ⎣ ⎢ ⎢ ⎡ 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 ⎦ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ x_{1, 2} x_{2, 2} x_{3, 2} x_{4, 2} ⎦ ⎥ ⎥ ⎤ - ⎣ ⎢ ⎢ ⎡ \frac{1}{d _{1} + b _{1}} (- ρ s_{1, 1} + x_{j, 2} + b_{1} \overset{r}{˙}) \frac{1}{d _{2} + b _{2}} (- ρ s_{2, 1} + x_{j, 2} + b_{2} \overset{r}{˙}) \frac{1}{d _{3} + b _{3}} \frac{1}{d _{4} + b _{4}} ⎦ ⎥ ⎥ ⎤

仿真部分也给出了干扰的表述方式：

\begin{aligned} Φ_{i} (x_{i, 1}, x_{i, 2}, κ_{i}) \leq | x_{i, 1} | + | x_{i, 2} | + rand (5) \end{aligned}

$\begin{aligned} \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \le |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \end{aligned}$

Φ_{i} (x_{i, 1}, x_{i, 2}, κ_{i}) \leq ∣ x_{i, 1} ∣ + ∣ x_{i, 2} ∣ + rand (5)

把两个公式合并一下（忽略转换矩阵的作用）：

\begin{aligned} {\dot{x}}_{i, 1} & = x_{i, 2} \\ {\dot{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 + \frac{ρ}{s}) s_{i, 2}] + | x_{i, 1} | + | x_{i, 2} | + rand (5) \end{aligned}

$\begin{aligned} \dot{x}_{i,1} &= x_{i,2} \\ \dot{x}_{i,2} &= u_i + \varPhi_i(x_{i,1}, x_{i,2}, \kappa_i) \\ &= u_i + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ &= -\rho s_{i,2} + f w_i + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ &= -\rho s_{i,2} + f [-(1+\frac{\rho}{s})s_{i,2}] + |x_{i,1}| + |x_{i,2}| + \text{rand}(5) \\ \end{aligned}$