# State Observer Based on an Accelerometer for an Elastic Joint with Nonlinear Friction
## 3 Stability and Performance Analysis  
In order to prove Theorem 1, we obtain that the nonlinear function \(\Omega(x)\) of (6) has the following property. Lemma 1: \(\Omega(x)\) of (6) is globally Lipschitz. In other words, there exists a positive number \(\gamma\) (the so-called Lipschitz constant) that satisfies the following Lipschitz condition:  
\[\|\Omega(x)-\Omega(\hat{x})\|\leq\gamma\|x-\hat{x}\|,\quad^{\forall}x,\hat{x }\in\mathbb{R}^{4}.\]  
In addition, the Lipschitz constant is expressed as:  
\[\gamma=\int_{m}^{-1}(6k_{2}\theta_{B}^{2}+\alpha f_{c}). \tag{10}\]  
Proof: The Jacobian matrix of \(\Omega\) at \(x\) is given by:  
\[\frac{\partial\Omega}{\partial x}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ -\int_{m}^{-1}\frac{\partial\hat{p}(x_{1},x_{2})}{\partial x_{1}}&0&-\int_{m} ^{-1}\frac{\partial\hat{p}(x_{1},x_{2})}{\partial x_{3}}&-\int_{m}^{-1}\frac{ \partial\hat{p}(x_{4})}{\partial x_{4}}\end{bmatrix} \]  
where  
\[ \begin{split}\frac{\phi(x_{1},x_{3})}{\partial x_{1}}&= \left\{\begin{array}{ll}\frac{\phi_{1}(x_{1},x_{3})}{\alpha x_{1}}&=-3k_{2} \theta_{B}^{2},&\text{if }x_{3}-x_{1}<-\theta_{B},\\ \frac{\phi_{2}(x_{1},x_{3})}{\alpha x_{1}}&=-3k_{2}(x_{3}-x_{1})^{2},&\text{if } \|x_{3}-x_{1}\|\leq\theta_{B},\\ \frac{\phi_{3}(x_{1},x_{3})}{\alpha x_{1}}&=-3k_{2}\theta_{B}^{2},&\text{if }x_{3}-x_{1}> \theta_{B},\end{array}\right.\\ \frac{\phi(x_{1},x_{3})}{\alpha x_{3}}&=\left\{\begin{array}{ll}\frac{\phi_{ 1}(x_{1},x_{3})}{\alpha x_{1}}&=3k_{2}\theta_{B}^{2},&\text{if }x_{3}-x_{1}<- \theta_{B},\\ \frac{\phi_{2}(x_{1},x_{3})}{\alpha x_{3}}&=3k_{2}(x_{3}-x_{1})^{2},&\text{if } \|x_{3}-x_{1}\|\leq\theta_{B},\\ \frac{\phi_{3}(x_{1},x_{3})}{\alpha x_{3}}&=3k_{2}\theta_{B}^{2},&\text{if }x_{3}-x_{1}> \theta_{B},\end{array}\right.\\ \frac{\partial\psi(x_{4})}{\partial x_{4}}&=\alpha f_{c}\mu_{ \text{sech}}\mathbb{E}^{2}(\alpha x_{4})\\ &\quad+f_{c}(1-u_{k})\Big{(}-\beta\text{sech}(\beta x_{4})\tanh(\beta x_{4}) \tanh(ax_{4})+\alpha\text{sech}(\beta x_{4})\text{sech}^{2}(\alpha x_{4}) \Big{)}\\ &\leq\alpha f_{c}\mu_{k}+f_{c}(1-\mu_{k})\alpha=\alpha f_{c}.\end{split} \]Thus, the function \(\Omega(\cdot)\) is continuously differentiable on \(\mathbb{R}^{4}\). In addition, the \(\frac{\partial\Omega}{\partial x}\) is uniformly bounded as follows:  
\[\left\|\frac{\partial\Omega}{\partial x}\right\|_{\infty}=J_{m}^{-1}\bigg{(} \bigg{|}\frac{\phi(x_{1},x_{3})}{\partial x_{1}}\bigg{|}+\bigg{|}\frac{\phi(x_ {1},x_{3})}{\partial x_{3}}\bigg{|}+\bigg{|}\frac{\partial\psi(x_{4})}{ \partial x_{4}}\bigg{|}\bigg{)}\leq J_{m}^{-1}\Big{(}\delta k_{2}\theta_{B}^{2} +\alpha f_{c}\Big{)}.\]  
Therefore, the proof is complete from ([17] Lemma 3.3). Now the dynamics for the estimation error \(e(t)\) can be expressed as follows:  
\[\dot{e} =\dot{x}-\dot{x}\] \[=(A-LC)e+(\Omega(x)-\Omega(\dot{x})). \tag{11}\]  
Consider a quadratic Lyapunov function candidate  
\[V(t)=e^{\top}(t)P_{1}e(t).\]  
Then, it follows from Lemma 1 that the derivative of \(V(t)\) is given by:  
\[\dot{V} =e^{\top}((A-LC)^{\top}P_{1}+P_{1}(A-LC))e+2e^{T}P_{1}(\Omega(x)- \Omega(\hat{x}))\] \[\leq e^{\top}((A-LC)^{\top}P_{1}+P_{1}(A-LC))e+2\|P_{1}e\|\|( \Omega(x)-\Omega(\hat{x}))\|\] \[\leq e^{\top}((A-LC)^{\top}P_{1}+P_{1}(A-LC))e+2\gamma\|P_{1}e\| \|e\|\] \[\leq e^{\top}((A-LC)^{\top}P_{1}+P_{1}(A-LC))e+\gamma^{2}e^{T}P_{1 }P_{1}e+e^{T}e\] \[=e^{\top}((A-LC)^{\top}P_{1}+P_{1}(A-LC)+\gamma^{2}P_{1}P_{1}+I)e. \tag{12}\]  
Next, let \(P_{1}=\gamma^{-1}P\); it then follows from the Schur complement that the LMI (9) can be written as:  
\[A^{\top}P+PA+\gamma^{2}I-\frac{1}{e}C^{\top}C+PP<0\] \[\dot{e} =\dot{x}-\dot{x}\]0  
where \(L=\frac{p^{-1}C^{\top}}{2e}\). From Equations (12) and (13), \(\dot{V}<0\), and the error dynamics (11) are asymptotically stable by ([17] Theorem 4.1), i.e., \(\lim_{t\rightarrow\infty}e(t)=0\). In actual system, the acceleration \(y_{2}\) of the vector \(Y\) contains measurement noise. On the other hand, the encoder noise for motion position \(y_{1}\) is small enough to be ignored.