# Pitch Channel Control of a REMUS AUV with Input Saturation and Coupling Disturbances
## 2 Model
### Pitch Channel Model for REMUS AUV via Taylor Formula Simplification  
The nonlinear model of REMUS AUV is examined to evaluate the performance of the proposed control strategy. The actuator of REMUS includes one propeller and four rudders, which are physically restricted to the ranges of 90 degrees. According to models introduced by [PERSON] [30], the governing equations of pitch channel can be extracted using Equation (5) [4].  
\[\begin{array}{l}z=-\mathrm{sin}\theta u+\cos\theta\sin\varphi v+\cos\theta \cos\varphi\omega\\ I_{y}\dot{q}+(I_{x}-I_{z})rp+m[z_{G}(\dot{u}-vr+\omega q)-x_{G}(\dot{\omega}- uq+vp)]=-(z_{G}W-z_{B}B)\sin\theta\\ \qquad-(x_{G}W-x_{B}B)\cos\theta\cos\varphi+M_{\omega|\omega|}\omega|\omega|+ M_{q|q}|q|+M_{\dot{\omega}}\dot{\omega}+M_{q}q+M_{uq}uq\\ \qquad+M_{vp}vp+M_{rp}rp+M_{uu}u\omega+M_{uu}u^{2}\delta_{e}\\ \theta=\cos\varphi q-\sin\varphi r\end{array} \tag{5}\]  
where \(\delta_{e}\) is elevator fins input.  
To obtain the equation for control, Equation (5) should be simplified around operating points \(\theta_{0}\), \(q_{0}\), \(u_{0}\), \(z_{0}\), using Taylor series, and the linear model is achieved by neglecting high order terms [4; 30; 31; 32]. The simplification model is as follows:  
\[\begin{array}{l}z=-u_{0}\cos\theta_{0}\theta\\ I_{y}\dot{q}+mx_{G}u_{0}q=-(z_{G}W-z_{B}B)\cos\theta_{0}\theta\\ \qquad+(x_{G}W-x_{B}B)\sin\theta_{0}\theta+2M_{|q|q}|q_{0}|q\\ \qquad+M_{\dot{q}}\dot{q}+M_{uq}u_{0}q+M_{uu}u_{0}^{2}\delta_{e}\\ \dot{\theta}=q\end{array} \tag{6}\]  
The simplified equation should then be transformed to the state space model with uncertainties added:  
\[\begin{array}{l}\left[\begin{array}{c}\dot{\theta}\\ \dot{q}\end{array}\right]=A_{m}\left[\begin{array}{c}\theta\\ q\end{array}\right]+B_{m}\delta_{e}\\ \qquad+\frac{1}{I_{y}-M_{\dot{q}}}\left[\begin{array}{ccc}0&0&0\\ \gamma&\lambda&\zeta\end{array}\right]\ \left[\begin{array}{c}u\\ q\\ \theta\end{array}\right]\end{array} \tag{7}\]  
where \(\gamma\), \(\lambda\), and \(\zeta\) are coefficients of uncertainties associated with \(u\), \(q\), and \(\theta\) in nonlinear terms, respectively. In addition, the roll state \(\phi\) and \(p\) are treated as disturbances in Equation (7). \(A_{m}\) and \(B_{m}\) in Equation (7) are defined below:  
\[A_{m}=\left[\begin{array}{cc}0&1\\ \frac{-(z_{G}W-z_{B}B)\cos\theta_{0}}{I_{y}-M_{\dot{q}}}&\frac{-mx_{G}u_{0}+2M _{|q|}|q_{0}|+M_{\mathrm{uq}}u_{0}}{I_{y}-M_{\dot{q}}}\end{array}\right]\]  
\[B_{m}=\left[\begin{array}{c}0\\ \frac{M_{uu}u_{0}^{2}}{I_{y}-M_{\dot{q}}}\end{array}\right]\]  
Parameters in Equation (7) are time varying, and unknown due to different operating conditions. Linearization approximations and nonlinearities of vehicle actuators also bring additional uncertainties and coupling disturbances. \(L_{1}\) Adaptive control technique is an effective means for dealing with underwater vehicle control with general uncertainties. However, the transient dynamics caused by actuator nonlinearities cannot be handled by \(L_{1}\) adaptive control. A compensator, therefore, should be introduced and integrated with adaptive control such as L1 AC. In Section 4, hypothetical linear plant as well as the nonlinear six DOF model of REMUS with uncertainties and nonlinearities of actuator are investigated in the simulation using the proposed approach.