### 文章目录

 Symbols Description x i x_i xi u i F u_i^F uiF 执行器出现故障之后的控制器 P i P_i Pi 执行器失效系数矩阵

## 第二节 问题描述

x ˙ 0 = A x 0 (3.2.1) \dot{x}_0 = A x_0 \tag{3.2.1} x˙0​=Ax0​(3.2.1)

x ˙ i = A x i + B u i F (3.2.2) \dot{x}_i = A x_i + B u_i^F \tag{3.2.2} x˙i​=Axi​+BuiF​(3.2.2)

u i F u_i^F uiF​ 的具体表达形式如下：

u i F = ( I − P i ) u i (3.2.3) u_i^F = (I - P_i) u_i \tag{3.2.3} uiF​=(I−Pi​)ui​(3.2.3)

## 第三节 容错一致性控制器设计与分析

J i = ∫ 0 C [ ( u i F ( t ) ) T H u i F ( t ) + ∑ j   i n N i ( x i ( t ) − x j ( t ) ) T Q ∑ j   i n N i ( x i ( t ) − x j ( t ) ) ] d t (3.3.1) J_i = \int_0^C [ (u_i^F(t))^\text{T} H u_i^F(t) + \sum_{j\ in N_i}(x_i(t)-x_j(t))^\text{T} Q \sum_{j\ in N_i}(x_i(t)-x_j(t)) ] dt \tag{3.3.1} Ji​=∫0C​[(uiF​(t))THuiF​(t)+j inNi​∑​(xi​(t)−xj​(t))TQj inNi​∑​(xi​(t)−xj​(t))]dt(3.3.1)

u i ∗ ( t ) = − ∣ N i ∣ 2 ( I − P i ) − 1 H − 1 B T λ i ∗ ( t ) , i = 1 , 2 , ⋯   , N (3.3.6) u_i^*(t) = - \frac{|N_i|}{2} (I - P_i)^{-1} H^{-1} B^\text{T} \lambda_i^*(t), \quad i=1,2,\cdots,N \tag{3.3.6} ui∗(t)=−2∣Ni∣(I−Pi)−1H−1BTλi∗(t),i=1,2,⋯,N(3.3.6)