## 4 Improved Control Method
### Controller Analysis
: _\(\mathbf{J}\) is a moment of inertia matrix composed of constants with physical meaning. In general, the moment of inertia parameters are positive real numbers, and they are different from each other. Thus, \(\mathbf{J}\) is a real matrix. In addition, it is also a positive definite matrix: \(\mathbf{J}\in\mathbb{R}^{3\times 3}\). Therefore, the matrix \(\mathbf{M}\) is a positive definite real matrix with three different eigenvalues, which is the implicit premise of the **Theorem 2**._  
The compensation loop is used for the compound disturbance compensation, and the system is closer to its normal model. The feedback control stability is given below. **Theorem 3**. : _The dynamics model is expressed as follows,_  
\[\mathbf{J}\dot{\Theta}^{*}(t)+\mathbf{B}^{*}\dot{\Theta}^{*}(t)+\mathbf{K}^{* }\Theta^{*}(t)=\boldsymbol{e}_{d}(t) \tag{65}\]  
_where the matrix \(\Theta^{*}(t)\) is the expansion matrix of \(\Theta(t)\), and \(\Theta^{*}(t)=[\theta_{b}(t)\ \theta_{e}(t)\ \theta_{l}(t)\ \theta_{b}^{*}(t)]^{T}\). The matrices are as follows,_  
\[\mathbf{B}^{*}= \begin{bmatrix}B_{b}&K_{p}K_{a}&0&K_{p}K_{a}\\ NB_{m}&B_{m}+NK_{p}K_{a}&B_{l}&NK_{p}K_{a}\\ -B_{s}&-B_{s}&B_{l}+B_{s}&0\end{bmatrix}  \tag{66}\]  
\[\mathbf{K}^{*}= \begin{bmatrix}0&K_{i}K_{a}&0&K_{i}K_{a}\\ 0&NK_{i}K_{a}&0&NK_{i}K_{a}\\ -K_{s}&-K_{s}&K_{s}&0\end{bmatrix}  \tag{67}\]  
_Along with the controller (28), when the parameters \(K_{p}\) and \(K_{i}\) are positive, the modified feedback controller is globally asymptotically stable._  
Proof of Theorem 3.: Because \(\epsilon_{d}\) is a small constant, the condition for the controller stability is that the matrix solution of the differential equations is negative. Therefore, the controller stability conditions are as follows,  
\[\mathbf{J}^{-1}\mathbf{B}^{*}>0,\quad\mathbf{J}^{-1}\mathbf{K}^{*}>0 \tag{68}\]  
To simplify the calculation, it is assumed that the system damping parameters have the same value. It is relatively easy to calculate the results of Equation (68), as well as to determine the conditions for which parameters \(K_{p}\) and \(K_{i}\) are positive, and the stability of the control system is achieved. In the real system, if the compensation loop can compensate the disturbance to ensure that **Assumption 1** is satisfied, the modified PI control can guarantee system stability only when \(K_{p}\) and \(K_{i}\) have positive values. \(K_{p}\) and \(K_{i}\) cannot be constantly increased because of the system energy limitations and the model high-frequency part. On the other hand, the composite anti-disturbance controller performance is verified from the point of view of solving the dynamics model. By substituting the composite controller (27) into the dynamics model (13), the model with the composite controller is as follows,  
\[\mathbf{J}\ddot{\Theta}^{*}(t)+\mathbf{B}^{*}\dot{\Theta}^{*}(t)+\mathbf{K}^{ *}\dot{\Theta}^{*}(t)=\mathbf{p}(t)+\mathbf{e}_{d}(t)+\mathbf{R}^{*}\mathbf{r }_{c}^{T}(t) \tag{69}\]  
where the expression for \(\mathbf{R}^{*}\) is presented below. \[\mathbf{R}^{*}= \begin{bmatrix}0&0&0\\ K_{i}K_{a}&K_{p}K_{a}+B_{m}K_{a}&J_{m}K_{a}\\ NK_{i}K_{a}&NK_{p}K_{a}+NB_{m}K_{a}&NJ_{m}K_{a}\end{bmatrix}  \tag{70}\]  
Based on the new dynamics model that uses the composite controller, under the same conditions in Section 2, the numerical analysis results are presented in Figure 4.  
Compared with Figure 3, the vibration disappears, the tracking performance is better, and there is almost no phase lag for the low-frequency command. The composite controller can suppress the vibrations and achieve a high performance.