## 1 Introduction
\begin{array}{c}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&-\gamma\mathbf{I}\end{array} \right]\left[ \begin{array}{c}\mathbf{ \varepsilon}\\ \mathbf{\phi}\end{array} \right]+\dot{V}(t)\right)dt\] \[=\int_{0}^{\tau}\left(\left[ \begin{array}{c}\mathbf{\varepsilon} \\ \mathbf{\phi}\end{array} \right]^{\mathbf{T}}\left[ \begin{array}{c}\mathbf{A}_{e}^{ \mathbf{T}}\mathbf{P}_{1}+\mathbf{P}_{1}\mathbf{A}_{e}+\mathbf{I}&\mathbf{P}_{1} \mathbf{B}_{e}\\ \mathbf{B}_{e}^{\mathbf{T}}\mathbf{P}_{1}&-\gamma\mathbf{I}\end{array} \right] \left[ \begin{array}{c}\mathbf{\varepsilon}\\ \mathbf{\phi}\end{array} \right]\right)dt\]  
Using condition Equation (7), it is obvious that  
\[\int_{0}^{\tau}\left(\left[ \begin{array}{c}\mathbf{\varepsilon}\\ \mathbf{\phi}\end{array} \right]^{\mathbf{T}}\left[ \begin{array}{c}\mathbf{A}_{e}^{ \mathbf{T}}\mathbf{P}_{1}+\mathbf{P}_{1}\mathbf{A}_{e}+\mathbf{I}&\mathbf{P}_{1} \mathbf{B}_{e}\\ \mathbf{B}_{e}^{\mathbf{T}}\mathbf{P}_{1}&-\gamma\mathbf{I}\end{array} \right] \left[ \begin{array}{c}\mathbf{\varepsilon}\\ \mathbf{\phi}\end{array} \right]\right)dt<0,\]  
so that  
\[\int_{0}^{\tau}\Big{(}\mathbf{\varepsilon}^{\mathbf{T}}\mathbf{\varepsilon}-\gamma\bm {\phi}^{\mathbf{T}}\mathbf{\phi}\Big{)}dt=\left\|\mathbf{\varepsilon}\right\|^{2}- \gamma\left\|\mathbf{\phi}\right\|^{2}<V(0).\]  
Thus, the Theorem 1 is proven. When there are the system noise and the observation noise \(\mathbf{w}(t),\mathbf{v}(t)\), the state observer (4) is considered, then the error system is easily obtained as follows  
\[\left[ \begin{array}{c}\mathbf{\dot{e}}_{x}\\ \mathbf{\dot{e}}_{f}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{A}-\mathbf{K}_{p} \mathbf{C}&\mathbf{F}_{x}-\mathbf{K}_{p}\mathbf{F}_{s}\\ -\mathbf{K}_{i}\mathbf{C}&\mathbf{G}-\mathbf{K}_{i}\mathbf{F}_{s}\end{array} \right]\left[ \begin{array} []{c}\mathbf{e}_{x}\\ \mathbf{e}_{f}\end{array} \right] \tag{9}\]  
In order to organize it into the matrix form, define  
\[\mathbf{\varepsilon}=\left[ \begin{array}{c}\mathbf{e}_{x}\\ \mathbf{e}_{f}\end{array} \right],\mathbf{\phi}=\left[ \begin{array}{c}\mathbf{f}\\ \mathbf{f}\end{array} \right],\mathbf{D}=\left[ \begin{array}{c}\mathbf{K}_{p}\mathbf{v}(t)- \mathbf{w}(t)\\ \mathbf{K}_{i}\mathbf{v}(t)\end{array} \right],\]  
Equation (9) can be rewritten as follows  
\[\dot{\mathbf{\varepsilon}}=\mathbf{A}_{e}\mathbf{\varepsilon}+\mathbf{B}_{e}\mathbf{\phi}+\mathbf{D}, \tag{10}\]  
where  
\[\mathbf{A}_{e}=\left[ \begin{array}{cc}\mathbf{A}-\mathbf{K}_{p}\mathbf{C}&\mathbf{F}_{x}-\mathbf{K} _{p}\mathbf{F}_{s}\\ -\mathbf{K}_{i}\mathbf{C}&\mathbf{G}-\mathbf{K}_{i}\mathbf{F}_{s}\end{array} \right],\mathbf{B}_{e}= \left[ \begin{array}{cc}\mathbf{0}&\mathbf{0}\\ \mathbf{G}&-\mathbf{I}\end{array} \right].\]  
According to the comparison theorem, when the condition in Equation (7) is true, from Equation (8), we can obtain the bound that satisfies  
\[\left\|\mathbf{\varepsilon}\right\|^{2}\leq\gamma\left\|\mathbf{f}\right\|^{2}+\gamma \left\|\hat{\mathbf{f}}\right\|^{2}+V(0)+\left\|p \begin{bmatrix}\mathbf{K}_{p}\\ \mathbf{K}_{l}\end{bmatrix} (\hat{\mathbf{g}}(t)-\mathbf{y}(t))\right\|^{2}.\]  
**Remark:** Equation (15) only holds when \(\left\|(\hat{\mathbf{g}}(t)-\mathbf{y}(t))\right\|\) is large. When \(\left\|(\hat{\mathbf{g}}(t)-\mathbf{y}(t))\right\|\) is small, choosing \(p<1\) may lead to the amplification of the estimation error, and \(p>1\) should be considered. At this point, it is proved that if the original PI observer in Equation (12) is stable, the nonlinear observer in Equation (13) can improve the observation effect by selecting appropriate parameter. Furthermore, it is a universal method, and a more general nonlinear observer can be considered  
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