1. Introduction
Understanding of the marine environment is becoming increasingly important across several domains, including oceanic energy [
1], ecosystems [
2], raw materials [
3], and understanding the ocean’s influence on climate [
4], weather patterns, etc. [
5]. Consequently, monitoring different ocean parameters holds significant importance, prompting endeavors in academia and industry to develop devices that are cost-effective and robust and require minimal human intervention [
6]. In the design of ocean monitoring devices, achieving energetic autonomy is a key focus, as it not only dictates mission types and durations but also influences the range of sensors and measurements that can be employed, with enhanced energy availability enabling broader capabilities [
5,
7].
In this context, underwater gliders [
8] emerge as notable and successful solutions, capable of completing missions spanning months without maintenance or support. The basic mechanism behind the low energetic consumption of gliders lies in its Variable Buoyancy Module (VBM), which allows travelling at the expense of low energy buoyancy changes. The utilization of variable flotation engines is also central to well-established solutions like the Argo floats [
9]. In this growing energy demand scenario, the use of VBM as auxiliary engines for thruster-powered devices becomes increasingly appealing, leading to the so-called hybrid AUVs [
10]. In this approach, the VBM may be used to provide neutral buoyancy for vertical displacements while the propellers handle motion in other directions. One relevant scenario is the operation in estuaries, where the changes in water density cause large variations in the vehicle buoyancy. This process is the reverse of the one in underwater gliders, where hybrid gliders now include a thruster to act whenever high currents or tight maneuvers are required [
11]. Beyond energy constraints, the use of VBM for depth control holds great importance for specific missions that require a minimal acoustic signature, such as military missions [
6] or marine observation to avoid disturbing studied species [
12]. However, despite this significance, few studies in the literature have focused on determining the circumstances favoring VBM use over propellers. The authors have previously contributed to this area by developing static models, allowing the calculation of the power consumed by a VBM, both in electric and hydraulic solutions [
13], as well as dynamic models based on experimental data for electrical actuated VBMs [
14].
Nevertheless, the comparative energy consumption of propeller and VBM solutions under closed-loop control remains largely unexplored in the literature. Specifically, the development of control laws targeting low energy consumption and the trade-offs involved in balancing energy consumption and depth control performance require further investigation. Some studies can be found on depth control, like, for instance, refs. [
15,
16]. In [
15], three depth controllers are developed and tested for a profiling float specifically designed for monitoring thermoclines. The float uses a linear electric actuator to achieve buoyancy change. The proposed segment PD control method is based on switching a velocity PD controller to a depth PD controller when a given error band is reached. Simulation and experimental results show that a maximum depth error of around 0.3 m is achieved, for depth steps up to 60 meters. However, no insights are given regarding the absolute or relative energy consumption of each control method. In [
16], a combinational controller including a PID controller, linear quadratic regulator (LQR), and sliding mode controller (SMC) is designed for the depth control of a bellows-like electrically driven VBM. It is shown, through simulations and experiments for several depth steps, that the proposed combinational controller outperforms all individual ones regarding control performance measured by rise time, overshoot, and settling time. However, once again, there is no remark regarding the energy spent by each controller or by the proposed combinational one. The only study the authors could find presenting a quantitative comparison between energy consumption of two VBM controllers was [
17]. In this study, a linear quadratic regulator (LQR) is compared against a two-stage cascaded proportional-derivative controller (2S-PD) using depth and vertical velocity in the feedback loops. It is shown that the LQR with a wide deadband in both depth and vertical velocity can significantly reduce the energy consumption in comparison to the 2S-PD controller. However, no deadband was implemented in the 2S-PD controller, so a full comprehension of the benefits of the LQR controller is not possible. Also, results do not consider any buoyancy disturbance, so the robustness of the controllers is not clear.
As far as the authors could ascertain, energy consumption figures are primarily available in studies combining propellers and VBMs, within hybrid control strategies. For instance, study [
11] introduced a Slocum hybrid glider equipped with folding propeller blades to reduce drag during buoyancy-driven flight. The study highlights several advantages of this hybrid approach, including enhanced efficiency in horizontal flight due to neutral buoyancy at depth, improved performance in overcoming strong currents and surfacing in low-density waters, and the ability to optimize the efficiency of the pump-driven VBM by employing the propeller when necessary. However, the study lacks specific energy gain figures and detailed insights into control strategy benefits. In another study [
7], a vehicle incorporating both propellers and a hydraulic VBM was explored, testing four control strategies: propeller-only, VBM-only, and two hybrid approaches. The hybrid strategies involved (i) sequential control, where the propellers initially stabilize the depth followed by transferring control to the VBM, and (ii) simultaneous control using both actuation systems with different controllers. While the study indicates some advantages of hybrid strategies, uncertainties remain regarding the generalizability of results to different depth steps and the absence of experiments including VBM deactivation or buoyancy disturbances. Bi et al. [
18] proposed a hybrid depth control strategy integrating an on–off hydraulic VBM for swift depth adjustments and fin control for energy conservation during cruising with propellers. The strategy enables efficient diving and surfacing without propeller usage and effective hovering control with low power consumption. However, the study lacks energy consumption estimates or measurements, studies on robustness to external disturbances, and a detailed description of the propeller control strategy.
Overall, existing studies suggest potential gains in energy efficiency with appropriate depth control strategies, but a systematic approach is lacking. Further research is needed to quantify energy efficiency, address control robustness, and evaluate performance under various environmental conditions. This work contributes to this endeavor by providing tools for systematic comparison between propeller- and VBM-driven depth controllers, aiming to address key questions surrounding their usage and performance trade-offs: When using a VBM, should one use one single controller for depth control or two cascaded ones, one for depth and another for volume control? Should deadbands be used in those controllers? If so, for depth control, for volume control, or for both? How can (internal and external) buoyancy disturbances be counteracted? What is the trade-off between depth control performance and energy consumption for a given control law?
The contributions of this work include (i) the development of a linearized model, accounting for real-world nonlinearities, of depth motion for a propeller or VBM actuated sensor platform; (ii) the development and simulation testing of several PID-based controller structures for this platform, including the robustness to buoyancy disturbances; (iii) examples illustrating the trade-offs between depth control performance and energy consumption for the developed controllers.
This work is organized as follows:
Section 2 presents the prototype previously developed by the authors. After a short description of the prototype modules in
Section 2.1, the several partial models are presented in
Section 2.2,
Section 2.3 and
Section 2.4. The parameters of the presented models are determined in
Section 2.5.
Section 3 presents in detail the controllers proposed in this work and
Section 4 presents the simulation results when the controllers developed in
Section 3 are applied to the models developed in
Section 2 in different operational scenarios. Finally,
Section 5 draws the main conclusions obtained in this work.
3. Controllers
To control the depth of the simulated prototype, different control architectures were devised for the PM and VBM. The PM only requires a standard architecture, with a single depth controller (please check
Figure 12), because there is a static unequivocal relation between the current
and the force on the vertical motion model input
. This is not the case with the VBM, since there is a second-order, type 1 dynamic relation between the voltage applied to the VBM and the corresponding volume obtained. For this reason, an inner control loop was devised to control the VBM output (please check
Figure 13). Since one of the main goals of the present work is to study the energy efficiency of both the VBM and PM, deadbands were implemented in each controller.
Regarding both
Figure 12 and
Figure 13, notice that each feedback branch has a Zero-Order Hold (ZOH) to model the signal acquisition between Arduino loops and a Quantizer block, accounting for sensor acquisition resolution.
is the depth error between the reference depth
and the depth signal
calculated using the pressure read by the pressure sensor. In each control architecture, there is a depth controller
. The output of
for the PM model is the current
and the output of
for the VBM controller is the reference volume
. The input for
is
, the volume error between
and the volume signal read by the position sensor
. The output from
is the control action
for the VBM.
To counteract possible disturbances acting on the prototype, each controller requires an integral action. To avoid integrator windup, an anti-windup scheme was implemented as presented in
Figure 14. In this figure,
,
, and
are the proportional, derivative, and integral control actions, respectively;
is the maximum value that the control action
can take; and
is the integral of the error. Whenever the control value equals or exceeds
for two consecutive time instants (
), the value of
is limited to
and the value of
is calculated so that
is also
. Integrator windup is thus prevented.
was chosen to be a PI controller. The block diagram for
is presented in
Figure 15.
In
Figure 15,
and
are the
integral and proportional gains, respectively, and
is the sampling time. To save energy, a volume deadband
is included, such that when the volume error
is smaller than
, the control signal
rises to 1 and the VBM is switched off. When this happens, the integrator input is set to zero to freeze the integrator. The PI controller action
is saturated to avoid exceeding the linear actuator input voltage range. Also, the ZOH block was added to simulate the holding of the control action
calculated value between Arduino sampling instants.
Regarding
, PID and I-PD controllers were tested. The block diagrams for the VBM
controllers are presented in
Figure 16 and
Figure 17.
In
Figure 16 and
Figure 17,
,
, and
are the integral, proportional, and derivative gains, respectively. A depth deadband
was implemented in the VBM
: when the absolute value of the depth error
is smaller than
, the control signal
rises to 1 and the integrator is frozen. Unlike the
, in this situation, the controller output is not zero but rather
in the PID or
in the I-PD. This ensures that if there is a constant disturbance, the integral part that counteracts it is still affecting the
input. To limit the maximum value of the derivative action in
Figure 16, due to possible excessive values whenever, for example, there is a step reference input, its value is limited to
.
The PM
controllers are similar to the VBM depth controllers with some adjustments due to the differences in control architecture. The block diagrams for the PM
controllers are presented in
Figure 18 and
Figure 19.
As seen in
Figure 18 and
Figure 19, a
was also implemented in the PM depth controller, inside which the integrator is frozen. Since, in this case, the
output is the PM model input, when inside the deadband, the PM is switched off to save power. There is an anti-windup scheme to avoid saturation and integrator windup and the maximum value of the derivative action in
Figure 18 is limited to
. The calculated value is again held with a ZOH to simulate the discrete time Arduino behavior.
Table 4 lists the 12 different control structures tested.
4. Simulation Results
In this section, the models developed in
Section 2 are simulated with the controllers presented in
Section 3. Each control structure presented in
Table 4 was simulated with two sets of controller tunings: (a) to increase performance and (b) to reduce energy consumption. These sets of parameters were obtained in a trial-and-error procedure, after intensive simulation runs. For
, the same gain values were used for every structure,
= 1 × 10
5 V
m
−3 and
= 1 × 10
5 V
m
−3 s
−1. The VBM gains for
are presented in
Table 5 and for the PM in
Table 6. Regarding the deadbands, the values used were
= 0.1 m and
= 3.5 × 10
−5 m
3.
The simulation trials were conducted in Matlab Simulink with a reference signal comprising several equal amplitude steps. Three types of tests were conducted: (1) without disturbances; (2) with a constant buoyancy disturbance throughout the entire test; and (3) with disturbances at middle points of each reference step in addition to the constant disturbance used in type 2 tests. Disturbances in type 2 tests simulated a neutral buoyancy trimming error, while those in type 3 tests simulated loading or unloading weights, or sudden density changes, as encountered in estuaries. The reference signal and the disturbances are depicted in
Figure 20. Disturbances in type 3 tests are increasing steps of 35 cc up to 12,600 s and decreasing thereafter.
The energy consumption results for the prototype controlled by each structure and tuning are presented in
Table 7. Results indicate that the PM requires less energy than any VBM control strategy, irrespective of the control structure used, in tests without disturbances (type 1 tests). However, when disturbances are present, as in type 2 and 3 tests, the energy consumption of the PM solution increases considerably, no longer offering the lowest consumption solution. It is also noticeable that the control strategy used with the PM does not seem to significantly affect the overall energy consumption for each type of test. In contrast, in the VBM case, the energy consumption does not significantly increase with disturbances; in fact, in many cases, it decreases. Additionally, for both the PM and the VBM, whether the depth controller is a PID or an I-PD, the usage of a depth deadband (even number control structures) does not seem to reduce the energy consumption of the prototype. However, in the VBM case, using a volume deadband led to considerable energy savings as it contributes to reducing the number of control action switchings.
According to the results presented in
Table 7, three structures were selected with the lowest average energy consumption in the three types of tests. Among PID structures 1 to 4, tuning b of structure 3 was selected; among I-PD structures 5 to 8, tuning b of structure 7 was selected. As previously mentioned, no significant changes between the results of the different controller structures in PM were found, so among structures 9 to 12, only one controller (tuning b of structure 11) was selected.
Figure 21,
Figure 22 and
Figure 23 show the simulated prototype depth during the three tests with the selected control structures. No significant overshoot is noticed in any of the responses and the VBM is faster than the PM. A delay in the first step is observed in type 2 and 3 trials. This is caused by the fact that for the first step, the integral part of the controller takes time to reach the value that will counteract the constant disturbance. In the case of controller 11b, this is particularly noticeable due to the low integral gain tuning required for low energy consumption.
To assess the performance of the selected control structures, conventional linear behavior metrics could not be used as the deadbands induce a nonlinear behavior. For this reason, several average performance metrics obtained in each test were adopted in this work: average overshoot
, average maximum error magnitude
, average settling times
and
, and average disturbance error magnitude
.
Figure 24 presents a graphical representation of these performance metrics. The number of inflection points is counted after the system response crosses the target reference.
is defined as the maximum error after the third inflection point of the vehicle for each new reference.
is the time it takes for the prototype to reach the ±
band of its own control structure, while
is the time it takes for the vehicle to reach the ±
band of the control structure that led to the highest
in its respective test.
is the maximum error after a new disturbance.
Table 8 presents the performance obtained for the previously selected controllers, as well as the energy consumption repeated from
Table 7, for an easy assessment of the trade-offs between performance and energy consumption.
The results presented in
Table 8 show that structure 7b is the fastest in every test by either settling time definition. The settling times obtained (between 80 and 130 s) are within the ones obtained for this type of vehicle [
15]. Additionally, it leads to very little overshoot and presents the best results regarding robustness to the error caused by step disturbances. In fact, when a disturbance appears, structure 7b reacts very quickly and the maximum error after disturbance is only slightly bigger than the maximum error. Structure 11b leads to the smallest depth error but it is much slower than either 3b or 7b and has significantly less robustness to step disturbances, leading to a very large error. Overall, controller 7b, an I-PD controller with a deadband in the target volume, significantly outperforms the best PM controller regarding energy consumption, response time, and robustness to disturbances. The maximum steady-state error is around 0.5 m, which, although being significantly worse than the one obtained by the 11b PM controller (steady-state error below 0.02 m), is an acceptable value for hovering control, even for shallow waters, where depth control should be tighter [
17].
5. Conclusions
This paper focused on the closed-loop depth control of a submersible platform, which can be used to monitor different ocean parameters through data collection from various sensors. Specifically, the energy consumption of two distinct actuation mechanisms for such platforms was examined: variable buoyancy and propeller actuated devices. Employing a prototype previously developed by the authors, this paper developed an intricate model of the platform utilizing both actuation solutions. Despite its linear foundation, the proposed model accommodates various nonlinearities such as saturations, sensor quantization, and actuator brake models. Additionally, it enables a straightforward estimation of the energy consumption associated with each actuation solution.
Several PID-based controllers were formulated and tested through simulation using the developed model. These controllers were employed to evaluate the dynamic response and energy demands of variable buoyancy and propeller actuated devices across diverse operational scenarios. The findings indicate that variable buoyancy systems can significantly reduce the energy required for hovering operations in the presence of buoyancy disturbances. In the scenarios analyzed in this study, adopting a variable buoyancy actuation system led to an energy consumption approximately 60% lower than that of an identical prototype powered by propellers. However, this reduction comes with a trade-off, as the variable buoyancy system exhibits a greater depth control error compared to the propeller-driven solution. Nonetheless, this compromise results in significantly faster settling times, with the variable buoyancy system outperforming the propeller-driven one.
Subsequent efforts will concentrate on refining control laws to potentially decrease the energy consumption of variable buoyancy systems. Specifically, the authors plan to explore the benefits of developing buoyancy disturbance observers toward this objective.