1. 系统描述

设我们有如下线性系统
$$\{\begin{array}{l}\begin{array}{rl}\dot{x}& =Ax+Bu+Ef\\ y& =Cx\end{array}\end{array}\text{\hspace{1em}(1)}$$其中$f$为外界扰动；$x\in {\mathbb{R}}^n,u\in {\mathbb{R}}^m,f\in {\mathbb{R}}^r,y\in {\mathbb{R}}^p$；$A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m},E\in {\mathbb{R}}^{n\times r},C\in {\mathbb{R}}^{p\times n}$。
作为扰动，$f$一般认为是有界的，即
$$\Vert \dot{f}\Vert \le \theta$$
容错控制的目的：设计一个算法，使得系统能够较为接近地估计扰动$f$，并在最终的输出中得到抵消了扰动作用后的输出值。

2. 估计器设计

设计一个中间变量/辅助变量
$$\xi =f+Ky,K\in {\mathbb{R}}^{r\times p}\text{\hspace{1em}(2)}$$则有$f=\xi -Ky$。对上式求导
$$\begin{array}{rl}\dot{\xi }& =\dot{f}+K\dot{y}\\ & =\dot{f}+KC\dot{x}\\ & =\dot{f}+KC\left(Ax+Bu+Ef\right)\\ & =\dot{f}+KC\left[Ax+Bu+E\left(\xi -Ky\right)\right]\\ & =\dot{f}+KC\left(Ax+Bu+E\xi -EKCx\right)\\ & =\dot{f}+KC\left[\left(A-EKC\right)x+Bu+E\xi \right]\end{array}\text{\hspace{1em}(3)}$$
设计如下估计器
$$\{\begin{array}{l}\begin{array}{rl}\dot{\hat{x}}& =A\hat{x}+Bu+E\hat{f}+L\left(y-\hat{y}\right)\\ \dot{\hat{\xi }}& =KC\left[\left(A-EKC\right)\hat{x}+Bu+E\hat{\xi }\right]\\ \hat{y}& =C\hat{x}\\ \hat{f}& =\hat{\xi }-K\hat{y}\end{array}\end{array}\text{\hspace{1em}(4)}$$设计的上述估计器能够正确估计扰动$f$，并使系统达到期望状态。证明如下。

4. 稳定性证明（重难点！！）

对状态$x$与中间变量$\xi$求其真实值与估计值之间的误差
$$\begin{gathered} e_x=x-hat,e_{\xi }=\xi -\hat{\xi }, \\ {e}_f=f-\hat{f}=\xi -Ky-\hat{\xi }+K\hat{y}=e_{\xi }-KC{e}_x \end{gathered}$$并求其各自导数（参考式(3)(4)）：
$$\begin{array}{rl}{\dot{e}}_x& =\dot{x}-\dot{\hat{x}}\\ & =Ax+Bu+Ef-\left[A\hat{x}+Bu+E\hat{f}+L\left(y-\hat{y}\right)\right]\\ & =A{e}_x+E{e}_f-LC{e}_x\\ & =\left(A-LC\right)e_x+E{e}_f\\ & =\left(A-LC\right)e_x+E\left(e_{\xi }-KC{e}_x\right)\\ & =\left(A-LC-EKC\right)e_x+E{e}_{\xi }\end{array}\text{\hspace{1em}(5)}$$$$\begin{array}{rl}{\dot{e}}_{\xi }& =\dot{\xi }-\dot{\hat{\xi }}\\ & =\dot{f}+KC\left[\left(A-EKC\right)x+Bu+E\xi \right]-KC\left[\left(A-EKC\right)\hat{x}+Bu+E\hat{\xi }\right]\\ & =\dot{f}+KC\left(A-EKC\right)e_x+KCE{e}_{\xi }\end{array}\text{\hspace{1em}(6)}$$设正定对称矩阵$P\in {\mathbb{R}}^{n\times n}$，标量$\delta$，由于$P$为正定对称的，故$P^T=P$。设计李雅普诺夫函数如下：
$$V=e_x^{T}P{e}_x^T+\delta e_{\xi }^T{e}_{\xi }\text{\hspace{1em}(7)}$$对其求导（考虑式(5)(6)）
$$\begin{array}{rl}\dot{V}& ={\dot{e}}_x^{T}P{e}_x+e_x^{T}P{\dot{e}}_x+\delta {\dot{e}}_{\xi }^T{e}_{\xi }+\delta e_{\xi }^T{\dot{e}}_{\xi }\\ & ={\left[\left(A-LC-EKC\right)e_x+E{e}_{\xi }\right]}^{T}P{e}_x+e_x^{T}P\left[\left(A-LC-EKC\right)e_x+E{e}_{\xi }\right]\\ & +{\left[\dot{f}+KC\left(A-EKC\right)e_x+KCE{e}_{\xi }\right]}^T{e}_{\xi }\\ & +\delta e_{\xi }^T\left[\dot{f}+KC\left(A-EKC\right)e_x+KCE{e}_{\xi }\right]\\ & =e_x^T{\left(A-LC-EKC\right)}^{T}P{e}_x+e_{\xi }^T{E}^{T}P{e}_x+e_x^{T}P\left(A-LC-EKC\right)e_x\\ & +e_x^{T}PE{e}_{\xi }+\delta {\dot{f}}^T{e}_{\xi }+\delta e_x^T{\left(A-EKC\right)}^T{C}^T{K}^T{e}_{\xi }+\delta e_{\xi }^T{E}^T{C}^T{K}^T{e}_{\xi }\\ & +\delta e_{\xi }^T\dot{f}+\delta e_{\xi }^{T}KC\left(A-EKC\right)e_x+\delta e_{\xi }^{T}KCE{e}_{\xi }\\ & =e_x^T\left[P\left(A-LC-EKC\right)+{\left(A-LC-EKC\right)}^{T}P\right]e_x\\ & +e_x^T\left[PE+\delta {\left(A-EKC\right)}^T{C}^T{K}^T\right]e_{\xi }+e_{\xi }^T\left[E^{T}P+\delta KC\left(A-EKC\right)\right]e_x\\ & +e_{\xi }^T\left(\delta E^T{C}^T{K}^T+\delta KCE\right)e_{\xi }+2\delta e_{\xi }^T\dot{f}\end{array}\text{\hspace{1em}(8)}$$对于(8)式中最后一项$2\delta e_{\xi }^T\dot{f}$，$2\delta e_{\xi }^T\dot{f}\in {\mathbb{R}}^{1\times r\cdot r\times 1}={\mathbb{R}}^{1\times 1}$为标量，根据绝对不等式，并考虑到$f$的有界性：
$$\begin{array}{rl}2\delta e_{\xi }^T\dot{f}& =2\sqrt{{\delta }^2\cdot \Vert e_{\xi }^T\dot{f}{\Vert }^2}\\ & =2\sqrt{{\delta }^2\cdot e_{\xi }^T{e}_{\xi }\cdot {\dot{f}}^T\dot{f}}\\ & \le \frac{1}{\epsilon }{e}_{\xi }^T{e}_{\xi }+\epsilon {\delta }^2{\dot{f}}^T\dot{f}\\ & =\frac{1}{\epsilon }{e}_{\xi }^T{e}_{\xi }+\epsilon {\delta }^2\Vert \dot{f}{\Vert }^2\\ & \le \frac{1}{\epsilon }{e}_{\xi }^T{e}_{\xi }+\epsilon {\delta }^2{\theta }^2\end{array}$$代入式(8)有
$$\begin{array}{rl}\dot{V}& \le e_x^T\left[P\left(A-LC-EKC\right)+{\left(A-LC-EKC\right)}^{T}P\right]e_x\\ & +e_x^T\left[PE+\delta {\left(A-EKC\right)}^T{C}^T{K}^T\right]e_{\xi }+e_{\xi }^T\left[E^{T}P+\delta KC\left(A-EKC\right)\right]e_x\\ & +e_{\xi }^T\left(\delta E^T{C}^T{K}^T+\delta KCE\right)e_{\xi }+\left(\frac{1}{\epsilon }{e}_{\xi }^T{e}_{\xi }+\epsilon {\delta }^2{\theta }^2\right)\\ & =e_x^T{G}_{11}{e}_x+e_x^T{G}_{12}{e}_{\xi }+e_{\xi }^T{G}_{21}{e}_x+e_{\xi }^T{G}_{22}{e}_{\xi }+\epsilon {\delta }^2{\theta }^2\end{array}\text{\hspace{1em}(9)}$$其中
$$\begin{array}{rl}{G}_{11}& =P\left(A-LC-EKC\right)+{\left(A-LC-EKC\right)}^{T}P\\ G_{12}& =PE+\delta {\left(A-EKC\right)}^T{C}^T{K}^T\\ G_{21}& =E^{T}P+\delta KC\left(A-EKC\right)\\ G_{22}& =\delta E^T{C}^T{K}^T+\delta KCE+\frac{1}{\epsilon }{I}_{r\times r}\end{array}\text{\hspace{1em}(10)}$$注意到$G_{12}^T=G_{21}$。设拓展状态量$e_t=\left[\begin{array}{c}{e}_x\\ e_{\xi }\end{array}\right]$，则式(9)可进一步简化为
$$\dot{V}\le e_t^T{G}_1{e}_t+\alpha \text{\hspace{1em}(11)}$$其中
$$G_1=\left[\begin{array}{cc}{G}_{11}& G_{12}\\ G_{21}& G_{22}\end{array}\right],\alpha =\epsilon {\delta }^2{\theta }^2\text{\hspace{1em}(12)}$$可见，为满足李雅普诺夫稳定性，矩阵$G_1$负定即可，即$G_1\le 0$。$G_{11}\in {\mathbb{R}}^{n\times n},G_{12}\in {\mathbb{R}}^{n\times r},G_{21}\in {\mathbb{R}}^{r\times n},G_{22}\in {\mathbb{R}}^{r\times r}$；$G_1\in {\mathbb{R}}^{\left(n+r\right)\times \left(n+r\right)}$。

令$G_2=-G_1$，则$G_2$正定，那么
$$\dot{V}\le -e_t^T{G}_2{e}_t+\alpha \text{\hspace{1em}(13)}$$
接下来证明为什么只要$G_1$负定（$G_2$正定），系统就稳定。
对于李雅普诺夫函数$V$，可以探究其有界性：
V = e x T P e x + δ e ξ T e ξ ≤ λ max ⁡ { P } e x T e x + δ e ξ T e ξ = λ max ⁡ { P } ∥ e x ∥ 2 + δ ∥ e ξ ∥ 2 ≤ max ⁡ { λ max ⁡ { P } , δ } ( ∥ e x ∥ 2 + ∥ e ξ ∥ 2 ) = max ⁡ { λ max ⁡ { P } , δ } ∥ e t ∥ 2 (14) \begin{aligned} V &= e_x^T P e_x + \delta e_\xi^T e_\xi \\ &\leq \lambda_{\max} \left\{ P \right\} e_x^T e_x + \delta e_\xi^T e_\xi \\ &= \lambda_{\max} \left\{ P \right\} \big \Vert e_x \big \Vert ^2 + \delta \big \Vert e_\xi \big \Vert ^2 \\ &\leq \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} \left( \big \Vert e_x \big \Vert ^2 + \big \Vert e_\xi \big \Vert ^2 \right) \\ &= \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} \big \Vert e_t \big \Vert ^2 \end{aligned} \tag{14} V​=exT​Pex​+δeξT​eξ​≤λmax​{P}exT​ex​+δeξT​eξ​=λmax​{P} ​ex​ ​2+δ ​eξ​ ​2≤max{λmax​{P},δ}( ​ex​ ​2+ ​eξ​ ​2)=max{λmax​{P},δ} ​et​ ​2​(14)其中 λ max ⁡ { P } \lambda_{\max} \left\{ P \right\} λmax​{P}是矩阵$P$的最大特征值。
另一方面，对于$e_t^T{G}_2{e}_t$有
λ min ⁡ { G 2 } e t T e t ≤ e t T G 2 e t ≤ λ max ⁡ { G 2 } e t T e t \lambda_{\min} \left\{ G_2 \right\} e_t^T e_t \leq e_t^T G_2 e_t \leq \lambda_{\max} \left\{ G_2 \right\} e_t^T e_t λmin​{G2​}etT​et​≤etT​G2​et​≤λmax​{G2​}etT​et​代入到式(13)有
V ˙ ≤ − e t T G 2 e t + α ≤ − λ min ⁡ { G 2 } e t T e t + α = − λ min ⁡ { G 2 } ⋅ max ⁡ { λ max ⁡ { P } , δ } max ⁡ { λ max ⁡ { P } , δ } ∥ e t ∥ 2 + α ≤ − λ min ⁡ { G 2 } max ⁡ { λ max ⁡ { P } , δ } ⋅ V + α = − κ V + α (15) \begin{aligned} \dot V &\leq -e_t^T G_2 e_t + \alpha \\ &\leq -\lambda_{\min} \left\{ G_2 \right\} e_t^T e_t + \alpha \\ &= -\lambda_{\min} \left\{ G_2 \right\} \cdot \frac{ \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} }{ \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} } \big \Vert e_t \big \Vert ^2 + \alpha \\ &\leq -\frac{ \lambda_{\min} \left\{ G_2 \right\} }{ \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} } \cdot V + \alpha \\ &= - \kappa V + \alpha \end{aligned} \tag{15} V˙​≤−etT​G2​et​+α≤−λmin​{G2​}etT​et​+α=−λmin​{G2​}⋅max{λmax​{P},δ}max{λmax​{P},δ}​ ​et​ ​2+α≤−max{λmax​{P},δ}λmin​{G2​}​⋅V+α=−κV+α​(15)其中
κ = λ min ⁡ { G 2 } max ⁡ { λ max ⁡ { P } , δ } , α = ε δ 2 θ 2 (16) \kappa = \frac{ \lambda_{\min} \left\{ G_2 \right\} }{ \max \left\{ \lambda_{\max} \left\{ P \right\}, \delta \right\} }, \qquad \alpha = \varepsilon \delta^2 \theta^2 \tag{16} κ=max{λmax​{P},δ}λmin​{G2​}​,α=εδ2θ2(16)。
假设有集合
S = { e t ∣ min ⁡ { λ min ⁡ { P } , δ } ⋅ ∥ e t ∥ 2 ≥ α κ } \mathcal{S} = \left\{ e_t \bigg\vert \min \left\{ \lambda_{\min} \left\{ P \right\}, \delta \right\} \cdot \big \Vert e_t \big \Vert ^2 \geq \frac{\alpha}{\kappa} \right\} S={et​ ​min{λmin​{P},δ}⋅ ​et​ ​2≥κα​}那么当$e_t\in \mathcal{S}$时， V ≥ min ⁡ { λ min ⁡ { P } , δ } ⋅ ∥ e t ∥ 2 ≥ α κ V \geq \min \left\{ \lambda_{\min} \left\{ P \right\}, \delta \right\} \cdot \big \Vert e_t \big \Vert ^2 \geq \frac{\alpha}{\kappa} V≥min{λmin​{P},δ}⋅ ​et​ ​2≥κα​，即$-V\le -\frac{\alpha }{\kappa }$。则
$$\dot{V}\le -\kappa V+\alpha \le \kappa \cdot \left(-\frac{\alpha }{\kappa }\right)+\alpha =0$$即$e_t\in \mathcal{S}$时，$\dot{V}\le 0$，系统一致稳定。且李雅普诺夫函数在时域上的解为
$$V\left(t\right)=\frac{\alpha }{\kappa }-\frac{\alpha }{\kappa }{e}^{-\kappa t}\text{\hspace{1em}(17)}$$系统一致有界。
对于集合$\mathcal{S}$的补集
S ˉ = { e t ∣ min ⁡ { λ min ⁡ { P } , δ } ⋅ ∥ e t ∥ 2 < α κ } \bar { \mathcal{S} } = \left\{ e_t \bigg\vert \min \left\{ \lambda_{\min} \left\{ P \right\}, \delta \right\} \cdot \big \Vert e_t \big \Vert ^2 < \frac{\alpha}{\kappa} \right\} \\ Sˉ={et​ ​min{λmin​{P},δ}⋅ ​et​ ​2<κα​}李雅普诺夫函数在时域上的解（式(17)）依然成立，依然是一个衰减函数，其将逐渐以速度$e^{-\kappa t}$衰减到下限值并过渡进入集合$\mathcal{S}$。

5. 线性矩阵不等式（LMI）与舒尔补定理（Schur Complement）

根据式(12)，只要满足$G_1<0$，系统即为稳定的。下面引入舒尔补定理：
对于拥有式(12)形式的对称矩阵$G_1$，以下式子等价：

1. $$G_1=\left[\begin{array}{cc}{G}_{11}& G_{12}\\ G_{21}& G_{22}\end{array}\right]<0$$
2. $$G_{11}-G_{12}{G}_{22}^{-1}{G}_{21}<0$$
3. $$G_{22}-G_{21}{G}_{11}^{-1}{G}_{12}<0$$因此可以将式(12)化简成上述三式的其中一种求解即可。求解得出的$K,L,P,\epsilon$即可以正确估计系统中的扰动。