## 4 Improved Control Method
### Controller Analysis
\[\mathbf{e}_{d}(t)=\exp(-\mathbf{M}t)\mathbf{e}_{d}(0)+\int_{0}^{t}\exp(- \mathbf{M}\cdot(t-\tau))(-\dot{\mathbf{d}}(t))d\tau \tag{56}\]  
where \(\mathbf{e}_{d}(0)\) is the initial error vector at a zero initial time. \[\|\mathbf{e}_{d}(t)\| \leq\|\exp(-\mathbf{M}t)\mathbf{e}_{d}(0)\|+\|\int_{0}^{t}\exp(- \mathbf{M}\cdot(t-\tau))(-\dot{\mathbf{d}}(t))d\tau\| \tag{57}\] \[\leq\|\exp(-\mathbf{M}t)\|\cdot\|\mathbf{e}_{d}(0)\|+\int_{0}^{t} \|\exp(-\mathbf{M}\cdot(t-\tau))(-\dot{\mathbf{d}}(t))\|d\tau\] \[\leq\|\exp(-\mathbf{M}t)\|\cdot\|\mathbf{e}_{d}(0)\|+\int_{0}^{t} \|\exp(-\mathbf{M}\cdot(t-\tau))\|\cdot\|\dot{\mathbf{d}}(t)\|d\tau\]  
where \(\exp(-\mathbf{M}t)\) is the exponential function of matrix \(\mathbf{M}\). Because \(\mathbf{J}\) is a real positive definite square matrix, \(\mathbf{M}=L\mathbf{J}^{-1}\) is also a real positive definite square matrix. \[\mathbf{M}= \begin{bmatrix}\frac{L}{I\beta}&0&0\\ -\frac{NL}{I\beta}&\frac{L}{Jm}&-\frac{L}{Jm}\\ 0&0&\frac{L}{\beta l}\end{bmatrix}  \tag{58}\]  
We can easily get the three positive eigenvalues of \(\mathbf{M},\lambda_{1}=\frac{L}{I\beta},\lambda_{2}=\frac{L}{Jm},\lambda_{3}= \frac{L}{I\beta},\lambda_{1},\lambda_{2},\lambda_{3}\in\mathbb{R}^{+}\). Subsequently, the results of the matrix exponential of \(\mathbf{M}\) has three cases, according to the eigenvalues of \(\mathbf{M}\). However, in the actual system, the three inertia parameters \(J_{b},J_{m},J_{l}\) usually do not appear to be equal, so this paper mainly considers the eigenvalues which are different from each other. \[\exp(-\mathbf{M}t)=\mathbf{P}_{\lambda}\mathbf{M}\mathbf{P}_{\lambda}^{-1}= \mathbf{P}_{\lambda} \begin{bmatrix}\exp(-\lambda_{1}t)&0&0\\ 0&\exp(-\lambda_{2}t)&0\\ 0&0&\exp(-\lambda_{3}t)\end{bmatrix} \mathbf{P}_{\lambda}^{-1},\;\;\;\lambda_{ 1}\
eq\lambda_{2},\lambda_{2}\
eq\lambda_{3},\lambda_{1}\
eq\lambda_{3} \tag{59}\]  
where \(\mathbf{P}_{\lambda}\) is a nonsingular transformation matrix from \(\bar{\mathbf{M}}\) to \(\exp(-\mathbf{M}t)\). \(\bar{\mathbf{M}}\) is a diagonal form. Using the norm inequality theorem, we can obtain the following inequality. \[\|\exp(-\mathbf{M}t)\|\leq\|\mathbf{P}_{\lambda}\|\cdot\|\bar{\mathbf{M}}\| \cdot\|\mathbf{P}_{\lambda}^{-1}\| \tag{60}\]  
As \(\|\bar{\mathbf{M}}\|\) is a real diagonal matrix, its 2-norm is as follows,  
\[\|\bar{\mathbf{M}}\| =\sqrt{f_{max}(\bar{\mathbf{M}}^{H}*\bar{\mathbf{M}})} \tag{61}\] \[=\sqrt{f_{max}(\bar{\mathbf{M}}*\bar{\mathbf{M}})}\] \[=\sqrt{\exp(-2\lambda_{min}t)}\] \[=\exp(-\lambda_{min}t)\]where \(\bar{\mathbf{M}}^{H}\) is a conjugate transposition matrix of \(\bar{\mathbf{M}}\), \(f_{max}(\cdot)\) presents the largest eigenvalue of the matrix \(\bar{\mathbf{M}}^{H}*\bar{\mathbf{M}}\), \(\lambda_{min}\) is the minimum eigenvalue of matrix \(\mathbf{M}\). Therefore, the inequality (60) is reduced as follows,  
\[\|\exp(-\mathbf{M}t)\|\leq\|\mathbf{P}_{\lambda}\|\exp(-\lambda_{min}t)\| \mathbf{P}_{\lambda}^{-1}\|=K_{\lambda}\exp(-\lambda_{min}t) \tag{62}\]  
where \(K_{\lambda}=\|\mathbf{P}_{\lambda}\|\cdot\|\mathbf{P}_{\lambda}^{-1}\|\) is a positive constant, related to the eigenvectors of matrix \(\mathbf{M}\). Similarly, the norm of \(\exp(-\mathbf{M}\cdot(t-\tau))\) is obtained. \[\|\exp(-\mathbf{M}\cdot(t-\tau))\|\leq K_{\lambda}\exp(-\lambda_{min}\cdot(t- \tau)) \tag{63}\]  
Thus, the inequality (57) can be written as follows,  
\[\|\mathbf{e}_{d}(t)\| \leq K_{\lambda}\exp(-\lambda_{min}t)\|\mathbf{e}_{d}(0)\|+ \epsilon_{d}\int_{0}^{t}K_{\lambda}\exp(-\lambda_{min}\cdot(t-\tau))d\tau \tag{64}\] \[=K_{\lambda}\exp(-\lambda_{min}t)\|\mathbf{e}_{d}(0)\|+\frac{K_{ \lambda}\epsilon_{d}}{\lambda_{min}}(1-\exp(-\lambda_{min}\cdot t))\]  
If the positive gain \(L\) is larger, the minimum eigenvalue \(\lambda_{min}\) of \(\mathbf{M}\) is larger. According to (64), the convergence rate and the accuracy of disturbance estimation can be simply improved by the NDOB gain \(L\). As \(t\to 0\), the disturbance observer error converges to \(\frac{K_{\lambda}\epsilon_{d}}{\lambda_{min}}\). **Remark 1**.