## 4 Improved Control Method
### Controller Analysis  
First, the feedforward controller does not affect the stability of the control system. By substituting the feedforward controller \(u_{f}(t)\) into the dynamics model (13), only the matrix \(\mathbf{R}^{\prime}\) is changed, which has nothing to do with system stability. Second, the compensate loop and outer loop stability are proven separately [14; 24; 40]. The model after the disturbance compensation of the inner loop is very close to the system nominal model. Thus, in the practical application process, the controller is designed from the inner loop to the outer loop. The velocity loop controller parameters, i.e., \(K_{p}\) and \(K_{i}\), are tuned by considering only the nominal dynamics model because the disturbances are well attenuated by the compound anti-disturbance controller in the compensate loop. The friction model (42) is in compliance with the actual system friction, and the residual disturbance \(\mathbf{d}(t)\) and its deviation \(\dot{\mathbf{d}}(t)\) are bounded:  
\[\|\mathbf{d}(t)\|\leq\delta_{d}\text{ and }\|\dot{\mathbf{d}}(t)\|\leq\epsilon_{d} \tag{49}\]  
where \(\delta_{d}\) and \(\epsilon_{d}\) are small constants, \(\delta_{d},\epsilon_{d}\in\mathbb{R}^{+}\), \(\|\cdot\|\) is the 2-norm function. As the NDOB discussed above is the minimum order observer, we make a practical assumption and give the convergence theorem of this NDOB. For the dynamics model (45) with the NDOB (46) and (47), when the gain \(L\) of the NDOB is strictly positive and Assumption 1 is satisfied, the disturbance estimation error \(\mathbf{e}_{d}\) of residual NDOB is convergent, and there is a positive \(K\in\mathbb{R}\), which makes it converge to a small positive constant. \[\lim_{t\to\infty}\|\mathbf{e}_{d}(t)\|=\frac{\epsilon_{d}}{K} \tag{50}\]  
The disturbance estimation error has the following expression. \[\mathbf{e}_{d}(t)=\hat{\mathbf{d}}(t)-\mathbf{d}(t) \tag{51}\]  
The time derivative of (51) is derived as follows,  
\[\dot{\mathbf{e}}_{d}(t)=\dot{\hat{\mathbf{d}}}(t)-\dot{\mathbf{d}}(t) \tag{52}\]  
Substituting (46) and (47) into (52) leads to the following equation. \[ \begin{split}\dot{\mathbf{e}}_{d}(t)&=L\mathbf{J}^{-1}( \mathbf{p}(t)-\mathbf{B}\hat{\Theta}(t)-\mathbf{K}\Theta(t)-\hat{\mathbf{d}}(t))-L\hat {\Theta}(t)-\dot{\mathbf{d}}(t)\\ &=L\mathbf{J}^{-1}(\mathbf{p}(t)-\mathbf{B}\hat{\Theta}(t)-\mathbf{K} \Theta(t)-\hat{\mathbf{d}}(t)-\mathbf{J}\hat{\Theta}(t))-\dot{\mathbf{d}}(t)\\ &=-L\mathbf{J}^{-1}\mathbf{e}_{d}(t)-\dot{\mathbf{d}}(t)\end{split}  \tag{53}\]  
Make \(\mathbf{M}=L\mathbf{J}^{-1}\), the corresponding first-order differential Equation (53) is as follows,  
\[\dot{\mathbf{e}}_{d}(t)+\mathbf{M}\mathbf{e}_{d}(t)+\dot{\mathbf{d}}(t)=0 \tag{54}\]  
When \(\dot{\mathbf{d}}(t)=\mathbf{0}\), by choosing \(\frac{1}{2}\mathbf{e}_{d}(t)^{T}\mathbf{e}_{d}(t)>0\) as the Lyapunov function \(V(t)\), the time derivative of \(V(t)\) is obtained:  
\[ \begin{split}\dot{V}(t)&=\frac{1}{2}\dot{\mathbf{e}}_{d} (t)^{T}\mathbf{e}_{d}(t)+\frac{1}{2}\mathbf{e}_{d}(t)^{T}\dot{\mathbf{e}}_{d}(t)\\ &=\mathbf{e}_{d}(t)^{T}\dot{\mathbf{e}}_{d}(t)\\ &=\mathbf{e}_{d}(t)^{T}(-\mathbf{M})\mathbf{e}_{d}(t)\end{split}  \tag{55}\]According to the Equation (15), the eigenvalues of \(\mathbf{J}\) can be obtained by \(|\;\lambda\mathbf{I}-\mathbf{J}\;|=0\), where \(\mathbf{I}\) is an unit diagonal matrix. The three eigenvalues of \(\mathbf{J}\) are \(J_{b},J_{m}\), and \(J_{l}\). In the model of SISP, the three parameters are positive definite real numbers with physical meaning. Thus, the square matrix \(\mathbf{J}\) is real and positive definite. As \(\mathbf{J}\) is positive, if \(L\) is positive, \(\hat{V}(t)<0\) is obtained. It shows that asymptotic stability is achieved if \(L\) is strictly positive. When \(\dot{\mathbf{d}}(t)\
eq\mathbf{0}\) and **Assumption 1** are satisfied, the general solution of linear time-invariant differential Equation (54) is proposed.