# Saturation Based Nonlinear FOPD Motion Control Algorithm Design for Autonomous Underwater Vehicle
## 5 Simulations  
The stability of AUV motion control system with the proposed controller is analyzed in the last section. Sine we prefer the pitch and roll angles of AUV to be close to zero in many applications, only yaw and depth motions are considered in this section [32].  
Firstly, the yaw motion control performance of AUV is presented; the main features of the presented AUV can be found in [34]. To take the control performance and effect as well as energy efficiency into consideration, the controller parameters of the proposed FOPD and traditional PD controller are tuned by the following objective function \(F\)[35].  
\[F = w_{1}\times ITAE+w_{2}\times ISCO, \tag{16}\]  
where \(ITAE=\int_{0}^{\infty}t\left|e(t)\right|dt\) stands for the integrated time absolute error criterion, \(ISCO=\int_{0}^{\infty}u^{2}(t)dt\) stands for the control effort respectively, and \(w_{1},\ w_{2}\) are their weights which can help adjust the control performance. The set-point control performance comparison of the proposed nonlinear FOPD controller and a traditional nonlinear PD controller with \(\delta=5\) is shown in Figure 2. The parameters (\(k_{p},\ k_{d}\)) of traditional PD controller are (0.9464, 4.832), (\(k_{p},\ k_{d},\ \mu\)) of FOPD controller are (1.125, 5.453, 0.617), and \(w_{1},\ w_{2}\) in Equation (16) are set as 1.0. These parameters are achieved using the Nelder-Mead simplex optimization method with the objective function \(F\)[36]. It can be seen that the set-point control performance of the proposed FOPD controller is much better than traditional PD controller with smaller overshoot and rising time, which can ensure the rapidity and smoothness of the AUV yaw control performance.  
Then, to show the flexibility of the proposed controller, the weight pairs (\(w_{1},\ w_{2}\)) are tuned to satisfy different control requirements. Because that dynamic control performance and robustness are usually more important than relatively low control signal in most control requirements, \(w_{1}\) and \(w_{2}\) are set to 1 and 0:0.1:1, respectively. Figure 3 shows the set-point performance comparison of different weight pairs. The corresponding evaluation indicator trends of set-point control performance, namely overshoot, settling time and rising time, are illustrated in Figure 4. Clearly, the control performance is not very satisfactory when \(w_{2}=0\), which may be caused by the large control signal without restriction. This indicates that the ISCO criterion is necessary in the design of the proposed nonlinear PD controller. With the increase of \(w_{2}\) from 0.1, almost all the evaluation indicators increase sharply at first and then  
Figure 2: Set-point control performance comparison.  
slightly afterwards. This example shows that the set-point regulation performance of AUV yaw system can be tuned to satisfy different control requirements.  
Finally, the trajectory tracking performance of AUV yaw and depth system are shown in Figures 5 and 6 with different weight pairs, respectively. The control signal of the yaw system when \(w_{2}=0\) oscillates severely, which accords with the results in set-point control performance. Except from the control signal oscillation, all the trajectory tracking performance of AUV yaw system are quite accurate, as shown in Figure 5. However, the overshoots and settling times of the AUV depth tracking performance are relatively large when \(w_{2}\) is quite small and decrease with the increase of \(w_{2}\), which also accords with the results in yaw control system. Moreover, both the control signal and tracking error also decrease along with the increase of \(w_{2}\).  
Figure 4: Control performance indicators trends.  
Figure 3: Set-point control performance.