# Research on the Attitude Control Strategy of TBM Digging with V-Shaped Propulsion System
## 3 TBM Digging Attitude Control Strategy  
### Nonlinear Controller Construction  
In analyzing the V-TBM propulsion system model, the system is identified as inherently complex, characterized by load disturbances and nonlinearities. Given the limitations of traditional linear controllers, which struggle to meet the required control accuracy and stability, this study develops a nonlinear PID controller based on a nonlinear tracking differentiator. The proposed controller accounts for system nonlinearity, minimizes control deviations by precisely tracking system state changes, enhances sensitivity to cumulative deviations, and exhibits robust anti-interference capabilities to ensure stable control under load disturbances.  
The mathematical model of the tracking differentiator is [14]:  
\[\begin{cases}\dot{x}_{1}=x_{2}\\ x_{2}=v,|v|\leq r\end{cases}. \tag{7}\]  
By incorporating a novel feedback control algorithm model, the differential equation governing the nonlinear tracking differentiator is derived as:  
\[\begin{cases}\dot{w}_{1}=w_{2}\\ \dot{w}_{2}=-rsign\Big{(}w_{1}+\frac{w_{2}|w_{2}|}{2r}-u\Big{)}\end{cases}, \tag{8}\]  
where  
\[\left[\begin{array}{c}fst(w_{1},w_{2})=-r\cdot sat(m(z),\delta)\\ sat(m(z),\delta)=\begin{cases}sign(m(z))&m(z)|>\delta\\ \frac{m(z)}{\delta}&|m(z)|\leq\delta\\ m(z)=\left\{\begin{array}{c}w_{2}(t)-sign(z)\frac{r}{2}\left(T-\sqrt{\frac{8|z |}{r}+T^{2}}\right)|z|>\delta\\ w_{2}(t)+\frac{r}{7}&|z|\leq\delta\end{array}\right.\end{cases}, \tag{10}\]  
where \(r\) represents the velocity factor, \(fst(w_{1},w_{2})\) denotes the optimal feedback control system, and \(T\) corresponds to the filtering factor. Consequently, the output of the discrete system is:  
\[\begin{cases}w_{1}(k+1)=w_{1}(k)+hw_{2}(k)\\ w_{2}(k+1)=w_{2}(k)+(-rh)sat(w_{1}(t),w_{2}(t))\end{cases}, \tag{11}\]  
where \(w_{1}\) represents the tracking signal of \(x_{1}\), and \(w_{2}\) corresponds to the differential signal of \(w_{1}\). The suppression of errors in nonlinear systems can be achieved by considering \(w_{2}\) as an approximate differential signal of \(x_{1}\).  
Combined with Figure 4, the system deviation signal is obtained as:  
\[\begin{cases}e_{1}=x_{1}(t)-y_{1}\\ e_{2}=x_{2}(t)-y_{2}.\\ e_{0}=\int_{0}^{t}e_{1}dt\end{cases} \tag{12}\]  
The nonlinear PID controller is built in conjunction with the system deviation signal:  
\[u(t)=K_{\text{p}}fal(e_{1},\alpha,\beta)+K_{\text{l}}fal(e_{0},\alpha,\beta) +K_{\text{D}}fal(e_{2},\alpha,\beta), \tag{13}\]  
where \(K_{\text{P}},K_{\text{l}},K_{\text{D}}\) represent the proportional, integral, and differential coefficients in the nonlinear controller, respectively, while \(fal(e,\alpha,\beta)\) denotes the nonlinear function of these gains. By comparing the output signal of the nonlinear tracking differentiator, the control deviation signal of the system is derived and introduced into the nonlinear function, yielding the following nonlinear function:  
\[fal(e,\alpha,\beta)=\begin{cases}\frac{e}{\beta^{1-\alpha}}&|e|\leq\beta\\ |e|^{4}sign(e)&|e|\textgreater{}\beta\end{cases}. \tag{14}\]  
Figure 4: System diagram of excavation attitude control strategy.