# Design and Depth Control of a Buoyancy-Driven Profiling Float
## 3 Depth Control System
### Control Strategy  
When designing the controller, the following four aspects should be considered. First, the cross-sectional area of the float is very small, and the characteristics of seawater make the float system have low damping and large inertia in seawater. Second, due to the limited putter speed, the system has an inevitable time delay. Third, in order to quickly deploy floats, the controller should be capable of adapting to different depths without changing the parameters. Finally, the sampling frequency of most ocean observation sensors is relatively low, and seawater parameters vary dramatically in the thermocline region. Therefore, to ensure the quality of the data collected by floats, it is generally required that the diving or floating speed of floats should not be greater than 0.2 m/s. The traditional PID control system is the most widely used control method in the industry. The differential part of a traditional PID control can solve the system characteristics of time delay. In contrast, the integral part will increase the time delay and overshoot in the position control process. Therefore, the comparison control algorithm and the control algorithm proposed in this paper are based on PD control. For a system with large inertia and time delay like a float, the differential term overcomes this difficulty to a certain extent and has good robustness. However, it may be necessary to reset the parameters when the target depth changes, which will be an enormous waste of time. In addition, when tracking the target depth, the speed of the float cannot be effectively controlled. Therefore, when the targeted depth cannot be reached, and the floating speed will be too fast, which will affect the quality of data acquisition. In essence, the direct output result by the controller is expressed as changing the static buoyancy of the float, that is, the acceleration of the float. Therefore, inertia will be greater when the depth error is used as the PD controller input than when the velocity error is used as the PD controller input. In order to reduce the inertia, the velocity error can be considered as the input of the PD controller. The actual velocity can be obtained by the difference approximation of the depth sensor, and the target velocity can be obtained by designing a curve of the target velocity changing with the depth error through the function relationship. This curve should be an odd function curve that takes the error between the target depth and the actual depth as the x-axis and the target velocity as the y-axis and passes through the origin. When the depth error is greater than zero, the target speed must be greater than zero. There are various functions that satisfy this condition. In this study, Equation (10) was adopted. \[v_{d}=\frac{e_{d}}{a(b+|e_{d}|)} \tag{10}\]  
where \(e_{d}\) is the error between the target depth and the current depth, and \(a\) and \(b\) are two constant parameters. The value of \(a\) can be determined by the desired speed when the depth is far from the target. The value of \(b\) can be determined by the expected velocity at a given position. The values of the parameters \(a\) and \(b\) in the subsequent simulation and experiment were 5 and 5.67, respectively. Theoretically, depth control can be achieved by the above method. However, as the float speed determination depends on the depth sensor and is limited by the program's \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Value** & **Units** \\ \hline \(M\) & Float mass & 6.29 & kg \\ \(C_{r}\) & Coefficient of drag & 0.66 & - \\ \(A\) & Cross sectional area & 0.00785 & m\({}^{3}\) \\ \(L\) & Electron bin length & 51.5 & cm \\ \(R_{o}\) & Outer diameter of electronic bin & 0.05 & m \\ \(R_{i}\) & Internal diameter of the electron bin & 0.04 & m \\ \(V_{0}\) & Initial float volume & 0.00839 & m\({}^{3}\) \\ \(\mu\) & Poisson ratio & 0.32 & - \\ \(E\) & Elasticity modulus & 3.16 & GPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Float dynamic parameters.