Variable Structure Depth Controller for Energy Savings in an Underwater Device: Proof of Stability

Abstract

Underwater exploration is vital for advancing scientific understanding of marine ecosystems, biodiversity, and oceanic processes. Autonomous underwater vehicles and sensor platforms play a crucial role in continuous monitoring, but their operational endurance is often limited by energy constraints. Various control strategies have been proposed to enhance energy efficiency, including robust and optimal controllers, energy-optimal model predictive control, and disturbance-aware strategies. Recent work introduced a variable structure depth controller for a sensor platform with a variable buoyancy module, resulting in a 22% reduction in energy consumption. This paper extends that work by providing a formal stability proof for the proposed switching controller, ensuring safe and reliable operation in dynamic underwater environments. In contrast to the conventional approach used in controller stability proofs for switched systems—which typically relies on the existence of multiple Lyapunov functions—the method developed in this paper adopts a different strategy. Specifically, the stability proof is based on a novel analysis of the system’s trajectory in the net buoyancy force-versus-depth error plane. The findings were applied to a depth-controlled sensor platform previously developed by the authors, using a well-established system model and considering physical constraints. Despite adopting a conservative approach, the results demonstrate that the control law can be implemented while ensuring formal system stability. Moreover, the study highlights how stability regions are affected by different controller parameter choices and mission requirements, namely, by determining how these aspects affect the bounds of the switching control action. The results provide valuable guidance for selecting the appropriate controller parameters for specific mission scenarios.

1. Introduction

Underwater ocean exploration is crucial for expanding the understanding of the planet, as oceans cover more than 70% of the Earth’s surface but remain largely unexplored. These explorations may reveal vital information about marine ecosystems, biodiversity, and deep-sea environments, offering insights into how oceans regulate climate, support life, and produce resources such as food and energy.

In this context, maintaining a continuous presence underwater is essential for long-term ocean research, conservation, and resource management. Persistent monitoring allows scientists to collect real-time data on environmental changes, marine life behaviors, and ecosystem health. It also enables the detection of rapid or subtle shifts caused by climate change, pollution, and human activities like overfishing. Sensing platforms such as autonomous vehicles [1] and buoys [2] help ensure effective management of marine resources and offer early warning systems for natural disasters such as tsunamis or harmful algal blooms, ultimately safeguarding both marine ecosystems and human populations. These platforms are essential for deep-sea exploration, but their operations are often constrained by limited energy resources. Efficient control laws can optimize actuation systems to minimize energy consumption, extending mission durations and expanding exploration capabilities, particularly for long-duration missions and operations in remote or deep-sea areas. Optimizing vertical motion, which is among the most energy-intensive manoeuvres, is especially relevant in scenarios involving repeated water column profiling, sampling at multiple depths, or persistent stratified monitoring in estuarine and coastal regions. A variety of application scenarios can be identified. For example, maritime environment surveillance, namely, for adaptive robotic sampling of phytoplankton [3], automatic detection and classification of underwater mines based on images [4], leak inspections of underwater pipelines [5], and subsea archaeology [6].

In order to minimize energy consumption, in [7] a robust controller for an underwater vehicle with a 150m depth of operation was developed. The approach combines standard sliding mode control (SMC) with classical optimal control techniques. The SMC is responsible for adding robustness to the system in relation to modelling errors and environmental disturbances. The optimizer is responsible for the minimization of the net control effort. When tracking a helicoidal trajectory, an improved efficiency of up to 30% is reported, although at the expense of tracking accuracy.

In [8,9,10] an energy-optimal economic model predictive controller (EO-EMPC) was employed for energy-optimal point-to-point motion control of an AUV. The proposed controller was tested, in a simulation, on the model of a 6000 m rated AUV. Excellent results were presented in [9] for the proposed control method, reaching a reduction in energy consumption of 50% when compared with the baseline MPC. Unfortunately, no robustness analysis of the controller was provided.

The quest for energy efficiency in underwater vehicles is also reflected in [11], where a different approach, based on the capability to discern which external disturbances can be safely disregarded, was adopted. The underlying idea was to identify zero-mean periodic disturbances (such as ocean waves), as their cumulative impact on the state of the AUV over extended periods was negligible. Therefore, the AUV controller could feasibly overlook these disturbances without compromising long-term tracking accuracy on a macro scale. This allows a reduction in energy consumption on a micro scale while maintaining macro-level performance. The proposed approach was shown to lead to considerable energy reduction in a simulation applied to an AUV capable of diving up to 500 m [11].

The authors of this paper have recently proposed a variable structure depth controller for energy savings in an underwater sensor platform equipped with a variable buoyancy module [12]. The proposed controller initiates action when the system dynamics become adverse, causing the vehicle to stray from the intended reference, and deactivates when conditions become favorable, directing the vehicle back toward the desired target. The results in [12] show that the proposed controller reduces energy consumption by an average of 22% compared to the next most efficient PID-based controller for the VBM-actuated vehicle, although this comes with a trade-off in control performance.

The importance of having adequate proof of stability for control laws in underwater vehicles is paramount to ensuring safe and efficient operations in complex marine environments. Stability guarantees are critical because underwater vehicles often operate in highly dynamic and uncertain conditions, including varying currents, pressure, and obstacles. A control law that is not rigorously proven to be stable may lead to unpredictable behavior, such as oscillations, instability, or even failure during missions. Stability proofs, such as those derived from Lyapunov methods [13,14,15], provide confidence in a vehicle’s ability to safely operate autonomously, ultimately increasing mission success rates and minimizing risks. Moreover, the work developed is crucial in determining the adequate controller parameters for specific mission scenarios, offering valuable insights into how to select the most suitable settings based on the stability regions and mission requirements. It should be emphasized that the control law presented in [12] is a switched control law, making the controlled system a switched control system. For this type of system, the proof of stability is more complex than for non-switched systems, since not only the stability may have to be shown for each subsystem, but the switching between these subsystems must not cause instability [16]. Typical approaches use multiple Lyapunov functions, [16,17], but this approach requires finding suitable Lyapunov functions for each subsystem, which is a non-trivial problem. Another important aspect concerns the fact that the analysis conducted in this study does not account for the time-delay phenomenon, which has been extensively addressed in the literature over the past several decades [18,19], because the vehicle dynamics are slow when compared with possible existing time delays.

Consequently, this paper extends the work reported in [12] by providing a new formal proof of stability of the switching controller therein developed. The innovation points are the following:

(i)

A new formal proof of stability of the controller developed in [12] is developed;

(ii)

The new formal proof of stability is different from the existing ones in the literature for switched systems, which are typically based on multiple Lyapunov functions [16];

(iii)

A study on how stability regions are influenced by different controller parameter choices and mission requirements is presented, using a depth-controlled sensor platform previously developed by the authors.

This paper is organized as follows: Section 2 presents the underwater device model, Section 3 provides a description of the controller, and Section 4 offers a formal proof of stability. Section 4 is further divided into four subsections: Section 4.1 outlines the formal proof, while Section 4.2, Section 4.3, Section 4.4 delve into its three main components. Section 5 quantifies the lower and upper bounds of the control action, using a prototype previously developed by the authors as a case study. Finally, Section 6 presents the key conclusions drawn from this work.

2. Underwater Device Description and Model

This study builds on a prototype previously developed by the authors [20,21] designed to evaluate the energy efficiency of a buoyancy-driven system. The full prototype measures 1370 mm in length and has an outer radius of 200 mm and a dry weight of 34 kg. The prototype comprises three main sections: a Main Control Unit (MCU), an intermediate section for floatation foam and a variable buoyancy module (VBM). A picture of the prototype and a schematic of the electronic and electromechanical system are shown in Figure 1.

The MCU houses an Arduino Uno for signal processing, system control, and data logging onto a 32 GB Micro SDHC card. Power is supplied by a 14.8 V high-capacity Turnigy battery. A Gravity IC Digital Wattmeter monitors power consumption of the prototype actuating devices, while two DC-DC converters ensure stable voltage delivery to both the Arduino and the VBM. On the outside of the MCU, a Bar30 pressure sensor is mounted to measure depth.

Inside the VBM, a diaphragm-sealed piston is actuated by a linear electric actuator driven by a brushed DC motor. The piston position is tracked by a FESTO SDAT-MHS position transmitter, which sends data to the motor driver (ROBOCLAW 2 × 15A). The driver controls the actuator using pulse-width modulation (PWM) based on the command voltage, u. An additional DC-DC converter adjusts the MCU voltage to match the position transmitter’s requirements. The VBM supports a volume variation of ±350 cm³ and can operate at depths of up to 100 m. This module was designed to be integrated into small-sized AUVs based on modular building blocks or to be used independently as a buoy.

The prototype model comprises the combination of the VBM model and the vertical motion model developed in [20]. Considering that actuator dynamics are much faster than the remaining dynamics, the reduced-order model of the system, represented in the Laplace domain, is presented in Figure 2.

In Figure 2, $K_1$ is the actuator velocity dynamics steady-state gain, which was experimentally identified in [21], and $A$ is the area of the piston in contact with sea water. The influence of the mechanical brake of the actuator is modelled in the block “brake” of Figure 2. If $U=0$, the brake is on, so the equivalent depth voltage, $U_Z$, is zero. If $U\ne 0$, the brake is off, so $U_Z=k_z⁄k_{u}Z$ to account for the influence of pressure due to depth, $Z$, on actuator motion.

The ratio $k_z/k_u$ was determined in [22] and models the torque caused on the motor due to the force exerted by the outside pressure, as well as the motor stall torque at the applied voltage level. $F_{net}$ is the balance between the actual VBM buoyancy force, $F_{VBM}$, and the disturbances acting on the prototype, $F_{dist}$, whether internal ($F_{dist\_int}=-\psi \rho gZ$) or external ($F_{dist\_ext}$), as defined in Equation (1).

$$F_{net}=F_{VBM}+F_{dis{t}_{ext}}-\psi \rho gZ$$

(1)

$K_2$ and $T_2$ are the depth dynamics parameters identified in [20]. The parameters $\rho$ and $g$ are the water mass per unit volume at 1 atm and the acceleration of gravity, respectively. Parameter $\psi$ expresses the change in water density with depth and the loss of volume per meter depth due to hull deformation. A method to determine the loss of volume per meter depth due to hull deformation was presented in [12].

This model can be described by the following differential equations:

$$\left\{\begin{array}{c}{\dot{V}}_Z={\displaystyle \frac{K_2}{T_2\rho g}}\psi \rho gZ-{\displaystyle \frac{1}{T_2}}{V}_Z-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{VBM}-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{dist\_ext}\\ {\dot{F}}_{VBM}=-\rho g{K}_{1}A{k}_z/k_{u}Z+\rho g{K}_{1}AU\end{array}\right.$$

(2)

where $V_Z=\dot{Z}$ is the vertical speed of the prototype. It will be considered, without loss of generality, that the control purpose is to drive the depth error, $E_Z$, to $E_Z=0$. If control action, $U$, switched between two values, $U_i,i=1,2$, is used, where

$$U_i=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if i=1\\ -U_{switch}sign\left(E_Z\right)\hspace{1em}\hspace{1em}\hspace{1em}if i=2\end{array}\right.$$

(3)

then the closed-loop dynamics are described by

$$\dot{\mathbf{x}}={\mathbf{F}}_{\mathit{i}}(\mathbf{x}\left(\mathit{t}\right),F_{dist\_ext})$$

(4)

where the state $\mathbf{x}\left(\mathit{t}\right)$ is defined as $\mathbf{x}\left(\mathit{t}\right)={\left[\begin{array}{ccc}Z& V_Z& F_{VBM}\end{array}\right]}^T$ and ${\mathbf{F}}_1(\mathbf{x}\left(\mathit{t}\right),F_{dist\_ext})$ and ${\mathbf{F}}_2(\mathbf{x}\left(\mathit{t}\right),F_{dist\_ext})$ are defined as

$${\mathbf{F}}_1(\mathbf{x}\left(t\right),F_{dist\_ext})=\left\{\begin{array}{c}\dot{Z}=V_Z\\ {\dot{V}}_Z={\displaystyle \frac{K_2}{T_2\rho g}}\psi \rho gZ-{\displaystyle \frac{1}{T_2}}{V}_Z-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{VBM}-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{dist\_ext}\\ {\dot{F}}_{VBM}=0\end{array}\right.$$

(5)

$${\mathbf{F}}_2(\mathbf{x}\left(t\right),F_{dist\_ext})=\left\{\begin{array}{c}\dot{Z}=V_Z\\ {\dot{V}}_Z={\displaystyle \frac{K_2}{T_2\rho g}}\psi \rho gZ-{\displaystyle \frac{1}{T_2}}{V}_Z-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{VBM}-{\displaystyle \frac{K_2}{T_2\rho g}}{F}_{dist\_ext}\\ {\dot{F}}_{VBM}=-\rho g{K}_{1}A{k}_z/k_{u}Z-\rho g{K}_{1}A{U}_{switch}sign\left(E_Z\right)\end{array}\right.$$

(6)

3. Controller Description

To minimize the energy consumption of the VBM, a novel variable structure controller (VSC) was presented in [12]. The core concept of this VSC controller is to leverage the system’s inherent dynamics to favor low-energy depth control. This involves activating the VBM whenever the underwater device deviates from the target depth. Conversely, when the underwater device moves in a way that reduces the depth error, the VBM is deactivated. Consequently, the VBM is anticipated to be off for a significant portion of the mission, thus conserving energy. Figure 3 shows a block diagram of the controller structure. It should be noticed that although $F_{VBM}$ can be determined by measuring the volume, $V_{VBM}$, external disturbances are estimated using a disturbance observer, as described in [12]. Additional information on the controller and observer is available in [12].

In Figure 3, $Z_{ref}$ is the target depth reference. The depth control error and its derivative can be written as (7) and (8), respectively, since $Z_{ref}$ is assumed to be constant. Figure 4 presents the system model notation.

$$E_Z=Z_{ref}-Z$$

(7)

$${\dot{E}}_Z=-V_Z$$

(8)

The VSC detailed below is responsible for deciding whether it is necessary to switch off the VBM, by engaging the brake and using control law (3) with $i=1$, or to switch on the VBM, by disengaging the brake and applying control law (3) with $i=2$. Figure 5 presents a schematic to explain the VSC decision process.

In the shaded regions of the schematic in Figure 5, the VBM is switched off ($i=1$ in Equation (3)), and in the remaining regions it is switched on ($i=2$ in Equation (3)). In regions 1a, 1b, and 1c, the VBM is switched off, and in regions 2a and 2b the VBM is switched on.

If there are no disturbances present, $F_{VBM}$ is sufficient to characterize the prototype depth acceleration, since it is the only force that can change the prototype vertical motion equilibrium. In real-world conditions, there are countless possible disturbances that may occur during an underwater mission such as load lifting, buoyancy trimming errors, and hull deformation, among others. This means that it is possible for a situation to occur where $F_{VBM}$ should lead to an expected output in terms of prototype acceleration but the existence of $F_{dist}$ leads to an unexpected prototype movement. To take disturbances into account, the vertical axis of the Figure 5 schematic represents the net force, $F_{net}$ ($F_{net}=F_{VBM}+F_{dist}$). Parameter $a$ is a depth deadband ($a>0$), meaning there is an admissible depth error in which the controller is switched off ($\left|E_Z\right|<a\to U=0$). Parameter $b$ is a band in $F_{net}$ ($b>0$) which ensures that when the system is switched off its natural dynamics lead to a decrease in the depth error magnitude. Considering that the system starts in region $2_a$, the depth error, $E_Z$, and net force, $F_{net}$, are positive. In this situation, the target depth reference is deeper than the current depth ($Z_{ref}>Z$), but the positive buoyancy would lead to a decrease in depth, which would increase $E_Z$. For this reason, the VBM should be switched on in this case, so that $U=-sign\left(E_Z\right)·U_{switch}$. Since $U$ will directly act on the derivative of $F_{VBM}$, as long as certain requirements, later detailed, are fulfilled, the value of $F_{net}$ will eventually decrease until the value $-b$ is reached and the depth error will eventually decrease. A similar line of thought can be followed for region $2_b$ of Figure 5 with $E_Z<0$, while keeping in mind that in this region the depth error and the net force are both negative. In regions $1_a$ and $1_b$, the depth error and the net volume have opposite signs; therefore, despite the VBM being turned off, the natural dynamics of the system will inevitably lower the error, leading the system to region $1_c$. The system cannot remain in a steady state inside region $1_c$, so it will eventually move to the other regions. From the description above, it is possible to infer that parameter a has a direct influence on the steady-state error.

Table 1. VSC decision table.

Regions	${\mathit{E}}_{\mathit{Z}}$	${\mathit{F}}_{\mathit{n}\mathit{e}\mathit{t}}$	Decision	i	$\mathbf{Control}\mathbf{Action},\mathit{U}$
$1_c$	$-a<E_Z<a$	$F_{net}$	OFF	1	$0$
$2_a$	$E_Z>a$	$F_{net}\ge -b$	ON	2	$-U_{switch}$
$1_a$	$E_Z>a$	$F_{net}<-b$	OFF	1	$0$
$1_b$	$E_Z<-a$	$F_{net}>b$	OFF	1	$0$
$2_b$	$E_Z<-a$	$F_{net}\le b$	ON	2	$U_{switch}$

presents the different VSC decisions and control action, $U$, according to each region in the schematic of Figure 5.

4. Proof of Stability of the Closed-Loop System

4.1. Outline of the Proof of Stability

In Section 4 it will be proven that system (4) with the control action (3) and the switching decision presented in Table 1 is stable for constant depth references, $Z_{ref}$, and constant external disturbances, $F_{dist\_ext}$, as long as $U_{switch}$ remains within bounds to be determined. The proof of stability of the controller will be divided into three parts.

In part 1 it will be shown that the following condition is valid for system (4):

Condition 1: If $F_{net}$ is bounded, then ${\dot{E}}_Z=-V_Z$ is also bounded.

In part 2 it will be shown that for system (4) with the control action (3) and the switching decision presented in Table 1, the following condition is valid:

Condition 2: As long as $U_{switch}$ in (3) is higher than a lower bound to be determined, when the system (4) is in region 2a, ${\dot{F}}_{net}\le -\eta$, $\eta >0$, and when the system (4) is in region 2b, ${\dot{F}}_{net}\ge \eta$.

In part 3 it will be shown that as long as condition 2 is satisfied, for system (4) with the control action (3) and the switching decision presented in Table 1, the following conditions are valid:

Conditions 3(a) and 3(b): Using the control law presented in Section 3, and as long condition 2 is satisfied and $U_{switch}$ is lower than upper bounds to be determined, starting from an arbitrary state, $\xi$, such that $F_{net\xi }=0$, $E_{z\xi }{<E}_{Zmax}$, the system performs a cyclical sequence, S, traversing points $\alpha ,\beta ,{\gamma }_1,{\gamma }_2,\delta ,{\epsilon }_1,{\mathrm{a}\mathrm{n}\mathrm{d} \epsilon }_2$ of Figure 6, such that the following conditions are met:

3(a) $F_{net\alpha }=F_{net{\epsilon }_2}$ and $E_{Z{\epsilon }_2}\le E_{Z\alpha }$;

3(b) $\left|E_{Z\beta }\right|=\left|E_{Z\zeta }\right| \mathrm{a}\mathrm{n}\mathrm{d} \left|F_{net\zeta }\right|\le \left|F_{net\beta }\right|$.

By ensuring conditions 3(a) and 3(b) above, it can be shown that the absolute values in both $F_{net}$ and $E_Z$ are bounded. From part 1 of the control proof, if $F_{net}$ is bounded, $V_Z$ is also bounded. From Equation (7), if $E_Z$ is bounded, $Z$ is also bounded. Finally, from (1) and assuming constant external disturbances, if $F_{net}$ and $Z$ are bounded, so is $F_{VBM}$. It can therefore be shown that under specific conditions for $U_{switch}$, the state $\mathbf{x}$ of Equation (4) remains bounded. It should be noticed that $F_{dist\_ext}$ will be considered constant, since in conventional scenarios disturbances such as fouling effects or salinity changes vary at such a low pace that they might be considered essentially constant. Also, as long as the disturbances change suddenly at sparse time instants between quasi-constant values, non-zero time derivatives only occur at those sparse time instants, so the system stability will not be affected. The next sections present in full detail the proof of stability.

Figure 6. Sequence of states in the $E_Z-F_{net}$ plane.

Figure 6. Sequence of states in the $E_Z-F_{net}$ plane.

4.2. Part 1 of the Controller Proof of Stability

Consider the block diagram of Figure 2.

Theorem 1.

For the system defined by Equation (4) and by the block diagram in Figure 2, if $F_{net}$ is bounded, ${\dot{E}}_Z$ is also bounded.

Proof of Theorem 1.

For the system represented by the block diagram of Figure 2, the transfer function between $F_{net}\left(s\right)$ and the derivative of $Z$ written in the Laplace domain, $Z\left(s\right)·s=V_Z\left(s\right)$, can be written as

$$V_Z\left(s\right)=-{\displaystyle \frac{\frac{K_2}{\rho g}}{T_{2}s+1}}{F}_{net}(\mathrm{s})$$

(9)

Given Equation (8), the error derivative can be written in the Laplace domain as

$$E_Z\left(s\right)s=-V_Z\left(s\right)$$

(10)

Replacing this equation in Equation (9) leads to

$$E_Z\left(s\right)·s={\displaystyle \frac{\frac{K_2}{\rho g}}{T_{2}s+1}}{F}_{net}(s)$$

(11)

Since all constants in Equation (11) are positive, if $F_{net}\left(t\right)$ is bounded, then ${\dot{E}}_z(t)$ is also bounded because it is a first-order low-pass filtered version of $F_{net}\left(t\right)$. Condition 1 of Section 4.1 is therefore satisfied. □

4.3. Part 2 of the Controller Proof of Stability

Theorem 2.

For the system represented by (4), the variable structure control action of Equation (3) with the switching decision block described in Table 1 ensures that when system (4) is in region 2_a, ${\dot{F}}_{net}\le -\eta$, and when the system (4) is in region 2_b, ${\dot{F}}_{net}\ge \eta$, with$\eta >$ 0 and as long as Equation (12) is satisfied.

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+sign\left(E_Z\right)\left(-\left(k_z/k_u\right)Z-\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z+{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}\right)$$

(12)

Proof of Theorem 2.

Consider region $2_a$, where $E_Z>a\bigwedge F_{net}\ge -b$. In this case, to ensure that the system moves out of this region, it is necessary to impose ${\dot{F}}_{net}<0.$ In this work it will be ensured that ${\dot{F}}_{net}<-\eta$, so that a minimum $F_{net}$ time rate is guaranteed. From the block diagram of Figure 2, the value of $F_{net}$ can be written, in the Laplace domain, as

$$F_{net}(s)={\displaystyle \frac{1}{s}}\left(\rho g\left(\left(K_{1}A\right)\left(U\left(s\right)-k_z/k_{u}Z\left(s\right)\right)\right)\right)+F_{dis{t}_{ext}}(s)-\psi \rho gZ(s)$$

(13)

So, the Laplace transform of the $F_{net}$ time derivative can be written as

$$F_{net}\left(s\right)s=\rho g{K}_{1}AU\left(s\right)-\rho g{K}_{1}A{k}_z/k_{u}Z\left(s\right)+s{F}_{dis{t}_{ext}}\left(s\right)-\psi \rho g{V}_Z\left(s\right)$$

(14)

To ensure that ${\dot{F}}_{net}<-\eta$, the condition presented in Equation (15), written in the time domain, must be met:

$${\dot{F}}_{net}<-\eta \to U<-{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+\left(k_z/k_u\right)Z+\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z-{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}$$

(15)

Since at region $2_a$, according to (3), $U=-sign\left(E_z\right)U_{switch}$, with $U_{switch}>0$, and since in region $2_a$, $E_z>0$, $U=-U_{switch}$, replacing the value of $U$ in Equation (15) leads to Equation (16):

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}-\left(k_z/k_u\right)Z-\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z+{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}$$

(16)

Consider now region $2_b$, where $E_Z<-a\bigwedge F_{net}\le b$. In this case, to ensure that the system moves out of this region, ${\dot{F}}_{net}>0.$ For the same reason, regarding region 2a, it will be ensured that ${\dot{F}}_{net}>\eta$. In order to ensure that ${\dot{F}}_{net}>\eta$, Equation (14) can be used to reach Equation (17):

$${\dot{F}}_{net}>\eta \to U>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+\left(k_z/k_u\right)Z+\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z-{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}$$

(17)

Since at region $2_b$, according to (3), $U=-sign\left(E_z\right)U_{switch}$, $U_{switch}>0$ and since in region $2_b$, $E_z<0$, $U=U_{switch}$ , replacing $U$ in Equation (17) leads to the following equation:

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+\left(k_z/k_u\right)Z+\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z-{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}$$

(18)

Conditions (16) and (18) can be written, in a compact form, as in Equation (12). Therefore, it can be shown that control law (3) with $U_{switch}$ given by (12) and using the switching decision law presented in Table 1 ensures that when the system (4) is in region 2_a, ${\dot{F}}_{net}\le -\eta$, and when the system (4) is in region 2_b, ${\dot{F}}_{net}\ge \eta$. Condition 2 of Section 4.1 is therefore satisfied. □

4.4. Part 3 of the Controller Proof of Stability

Let $S$ be a switching sequence associated with the switched system (4), the set of switching times be defined by $\left\{t_j\right\}$, and the interval of completion, $\mathrm{\beth }\left(\left.S\right|i\right)$, be defined as the set of time intervals during which subsystem $i$ is active:

$$\mathrm{\beth }\left(\left.S\right|i\right)=\bigcup _k{\overline{t}}_{i,k},{\underset{\_}{t}}_{i,k}$$

(19)

where

$$\left\{{\overline{t}}_{i,k}\right\}=\left\{\left.t_j\right|\mathrm{when}\mathrm{subsystem}i\mathrm{is}\mathrm{switched}\mathrm{on}\right\}$$

(20)

and

$$\left\{{\underset{\_}{t}}_{i,k}\right\}=\left\{\left.t_j\right|\mathrm{when}\mathrm{subsystem}i\mathrm{is}\mathrm{switched}\mathrm{off}\right\}$$

(21)

Consider now Figure 7, showing a sequence of switching actions and the corresponding states in the $E_Z-F_{net}$ plane.

As detailed in Section 4.1, it will be shown in the following that starting from point $\xi \left({\overline{t}}_{2,0}\right)$, the system will evolve to point ${\epsilon }_2\left(t\right)$, crossing $\alpha \left(t\right),\beta \left(t\right)$, ${\gamma }_1\left(t\right)$, ${\gamma }_2\left(t\right)$, $\delta \left(t\right)$, and ${\epsilon }_1\left(t\right)$ in that order. The system then enters a cyclic sequence: $S=\left\{\alpha \left(t\right),\beta \left(t\right),{\gamma }_1\left(t\right),{\gamma }_2\left(t\right),\delta \left(t\right),{\epsilon }_1\left(t\right),{\epsilon }_2\left(t\right)\right\}$, such that condition 3 of Section 4.1 is met.

For this purpose, the following results derived from the block diagram of Figure 2 will be used:

$$F_{net}=F_{VBM}-\psi \rho gZ+F_{dist\_ext}$$

(22)

and, since $F_{dist\_ext}$ is considered to be constant,

$${\dot{F}}_{net}={\dot{F}}_{VBM}-\psi \rho g{V}_Z$$

(23)

Also, from Figure 2 and Equation (3), the value of $F_{VBM}$ and its derivative are given by

$$F_{VBM}=\left\{\begin{array}{c}constant,i=1\\ \int K_{1}A\rho g\left(-U_{switch}sign\left(E_z\right)-{\displaystyle \frac{k_Z}{k_u}}Z\right)dt,i=2\end{array}\right.$$

(24)

$${\dot{F}}_{VBM}=\left\{\begin{array}{c}0,i=1\\ K_{1}A\rho g\left(-U_{switch}sign(E_z)-{\displaystyle \frac{k_Z}{k_u}}Z\right),i=2\end{array}\right.$$

(25)

At the initial point, $\xi ({\overline{t}}_{2,0}$), it is considered that the system has zero velocity, zero net buoyancy, and an initial arbitrary error $E_{Z\xi }>\mu$. The initial point can be defined as

$$\xi \left({\overline{t}}_{2,0}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{2,0}\right)=E_{Z\xi };{\dot{E}}_Z\left({\overline{t}}_{2,0}\right)={\dot{E}}_{Z\xi }=0\\ Z\left({\overline{t}}_{2,0}\right)=Z_{\xi };V_Z\left({\overline{t}}_{2,0}\right)=V_{Z\xi }=-{\dot{E}}_{Z\xi }=0\\ F_{net}\left({\overline{t}}_{2,0}\right)=F_{net\xi }=0\end{array}\right.$$

(26)

Since $F_{net}\left({\overline{t}}_{2,0}\right)=0$, from (22), it follows that

$$F_{VBM}\left({\overline{t}}_{2,0}\right)=\psi \rho g{Z}_{\xi }-F_{dist\_ext}$$

(27)

The change in $F_{VBM}$ between points $\xi \left(F_{VBM\xi }\right)$ and $\mathrm{\alpha }\left(F_{VBM\mathrm{\alpha }}\right)$ can be determined by integrating Equation (25) with $i=2$ and noticing that in this region $E_z>0$:

$$F_{VBM\mathrm{\alpha }}-F_{VBM\xi }={\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{\dot{F}}_{VBM}dt={\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{K}_{1}A\rho g\left(-U_{switch}-{\displaystyle \frac{k_Z}{k_u}}Z\right)dt$$

(28)

And the change in $F_{net}$ between points $\xi \left(F_{net\xi }\right)$ and $\alpha \left(F_{net\alpha }\right)$ can be determined by integrating Equation (23) with $i=2$:

$$F_{net\alpha }-F_{net\xi }={\int }_{{\overline{t}}_{\mathrm{2,0}}}^{{\overline{t}}_{1,k}}\left({\dot{F}}_{VBM}-\psi \rho g{V}_Z\right)dt$$

(29)

Developing Equation (29):

$$F_{net\alpha }=F_{net\xi }+{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{\dot{F}}_{VBM}dt-{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}\psi \rho g{V}_{Z}dt$$

(30)

Considering (8):

$$F_{net\alpha }=F_{net\xi }+{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{\dot{F}}_{VBM}dt+\psi \rho g{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{\dot{E}}_{Z}dt$$

(31)

From (28):

$$F_{net\alpha }=F_{net\xi }+{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{K}_{1}A\rho g\left(-U_{switch}-{\displaystyle \frac{k_Z}{k_u}}Z\right)dt+\psi \rho g{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}{\dot{E}}_{Z}dt$$

(32)

$$F_{net\alpha }=F_{net\xi }-K_{1}A\rho g{U}_{switch}\left({\overline{t}}_{1,k}-{\overline{t}}_{2,0}\right)-K_{1}A\rho g{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}Zdt+\psi \rho g\left(E_{Z\alpha }-E_{Z\xi }\right)$$

(33)

Since $F_{net\alpha }=-b$ and $F_{net\xi }=0$,

$$E_{Z\alpha }=E_{Z\xi }-{\displaystyle \frac{b}{\psi \rho g}}+{\displaystyle \frac{K_{1}A}{\psi }}{U}_{switch}\left({\overline{t}}_{1,k}-{\overline{t}}_{2,0}\right)+{\displaystyle \frac{K_{1}A}{\psi }}{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,0}}^{{\overline{t}}_{1,k}}Zdt$$

(34)

Given that ${\dot{E}}_{Z\xi }=0$ and that according to (11) the sign of ${\dot{E}}_Z$ and $F_{net}$ are the same, it follows that point $\alpha \left({\overline{t}}_{1,k}\right)$ can be defined as

$$\alpha \left({\overline{t}}_{1,k}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{1,k}\right)=E_{Z\alpha };{\dot{E}}_Z\left({\overline{t}}_{1,k}\right)={\dot{E}}_{Z\alpha }\le 0\\ Z\left({\overline{t}}_{1,k}\right)=Z_{\alpha };V_Z\left({\overline{t}}_{1,k}\right)=V_{Z\alpha }=-{\dot{E}}_{Z\alpha }\ge 0\\ F_{net}\left({\overline{t}}_{1,k}\right)=F_{net\alpha }=-b\end{array}\right.$$

(35)

The change in $F_{VBM}$ between points $\alpha$, $F_{VBM\alpha }$, and $\beta ,F_{VBM\beta }$, can be determined by integrating Equation (25) with $i=1$:

$$F_{VBM\beta }-F_{VBM\alpha }=0\to F_{VBM\beta }=F_{VBM\alpha }$$

(36)

The change in $F_{net}$ between points $\alpha$,$F_{net\alpha }$, and $\beta , F_{net\beta }$, can be determined by integrating Equation (23) with $i=1$.

$$F_{net\beta }-F_{net\alpha }={\int }_{{\overline{t}}_{1,k}}^{{\underset{\_}{t}}_{1,k}}-\psi \rho g{V}_{Z}dt$$

(37)

Since $F_{net\alpha }=-b$ and $E_{Z\beta }=-a$,

$$F_{net\beta }=-b+\psi \rho g{\int }_{E_{Z\alpha }}^{E_{Z\beta }}d{E}_Z=-b+\psi \rho g\left(E_{Z\beta }-E_{Z\alpha }\right)=-b-\psi \rho g\left(a+E_{Z\alpha }\right)$$

(38)

Since ${\dot{E}}_{Z\alpha }<0$ and between $\alpha$ and $\beta$ $F_{net}<0$, the point $\beta \left({\underset{\_}{t}}_{1,k}\right)$ can therefore be defined as

$$\beta \left({\underset{\_}{t}}_{1,k}\right):\left\{\begin{array}{c}{E}_Z\left({\underset{\_}{t}}_{1,k}\right)=E_{Z\beta };{\dot{E}}_Z\left({\underset{\_}{t}}_{1,k}\right)={\dot{E}}_{Z\beta }<0\\ Z\left({\underset{\_}{t}}_{1,k}\right)=Z_{\beta };V_Z\left({\underset{\_}{t}}_{1,k}\right)=V_{Z\beta }=-{\dot{E}}_{Z\beta }>0\\ F_{net}\left({\underset{\_}{t}}_{1,k}\right)=F_{net\beta }=-b-\psi \rho g\left(a+E_{Z\alpha }\right)\end{array}\right.$$

(39)

The change in $F_{VBM}$ between points $\beta \left(F_{VBM\beta }\right)$ and ${\gamma }_1\left(F_{VBM{\gamma }_1}\right)$ can be determined by integrating Equation (25) with $i=2$ and noticing that in this region $E_z<0$:

$$F_{VBM{\gamma }_1}-F_{VBM\beta }={\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}{\dot{F}}_{VBM}dt={\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}{K}_{1}A\rho g\left(U_{switch}-{\displaystyle \frac{k_Z}{k_u}}Z\right)dt$$

(40)

And the change in $F_{net}$ between points $\beta \left(F_{net\beta }\right)$ and ${\gamma }_1\left(F_{net{\gamma }_1}\right)$ can be determined by integrating Equation (23) with $i=2$:

$$F_{net{\gamma }_1}-F_{net\beta }={\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}\left({\dot{F}}_{VBM}-\psi \rho g{V}_Z\right)dt$$

(41)

Developing Equation (41) leads to

$$F_{net{\gamma }_1}=F_{net\beta }+K_{1}A\rho g{U}_{switch}\left({\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}\right)-K_{1}A\rho g{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt+\psi \rho g\left(E_{Z{\gamma }_1}-E_{Z\beta }\right)$$

(42)

Since $F_{net{\gamma }_1}=b$ and $E_{Z\beta }=-a$, using Equation (38),

$$E_{Z{\gamma }_1}=E_{Z\alpha }+{\displaystyle \frac{2b}{\psi \rho g}}-{\displaystyle \frac{K_{1}A}{\psi }}{U}_{switch}\left({\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}\right)+{\displaystyle \frac{K_{1}A}{\psi }}{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt$$

(43)

Between $\beta$ and ${\gamma }_1$, the system is accelerated when $F_{net}<0$ and is decelerated when $F_{net}>0$. The velocity when $F_{net}<0$ is positive and may or not become negative when $0<F_{net}<b$. The point ${\gamma }_1\left({\overline{t}}_{1,k+1}\right)$ can therefore be defined as

$${\gamma }_1\left({\overline{t}}_{1,k+1}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{1,k+1}\right)=E_{Z{\gamma }_1};{\dot{E}}_Z\left({\overline{t}}_{1,k+1}\right)={\dot{E}}_{Z{\gamma }_1}\\ Z\left({\overline{t}}_{1,k+1}\right)=Z_{{\gamma }_1};V_Z\left({\overline{t}}_{1,k+1}\right)=V_{Z{\gamma }_1}\\ F_{net}\left({\overline{t}}_{1,k+1}\right)=F_{net{\gamma }_1}=b\end{array}\right.$$

(44)

If at ${\gamma }_1$ the velocity $V_{Z{\gamma }_1}$ is positive, the system will undergo a particular behavior between points ${\gamma }_1$ and ${\gamma }_2$. When the system enters infinitesimally at $i=1$, the control action is $U=0$, which makes $F_{VBM}=constant$ according to (24). Since ${\dot{E}}_Z$ is negative, $Z$ will increase and so does the force magnitude factor due to $-\psi \rho gZ$. This leads to a decrease in the net force and the system is pushed towards $i=2$, causing the control action to turn on. This will increase the net force, causing the system to be moved again towards $i=1$. So, in this situation, the system undergoes a switching phase with $n_{\gamma }$ switching between $i=1$ and $i=2$. This behavior will occur until the vehicle is fully decelerated, making the velocity, $\dot{Z}$, change sign from positive to negative. Also, since ${\dot{E}}_Z=-V_Z$, at this point, the sign of ${\dot{E}}_Z$ changes from negative to positive. When this happens, point ${\gamma }_2\left({\overline{t}}_{1,k+1+n_{\gamma }}\right)$ is reached. Since between ${\gamma }_1$ and ${\gamma }_2$ the average value of $F_{net}$ is constant, the relation between $Z\left({\overline{t}}_{1,k+1+n_{\gamma }}\right)=Z_{{\gamma }_2}$ and $Z_{{\gamma }_1}$ can be determined, from the block diagram of Figure 2, by

$${\dot{V}}_Z+{\displaystyle \frac{1}{T_2}}{V}_Z=-{\displaystyle \frac{K_2}{T_2\rho g}}b,V_{Z\gamma 1}>0\wedge {\underset{\_}{t}}_{2,k}<t<{\underset{\_}{t}}_{2,k+n_{\gamma }}$$

(45)

The solution for the first-order differential Equation (45) is given by

$$V_Z(t)=\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)·e^{-(t-{\underset{\_}{t}}_{2,k})/T_2}-{\displaystyle \frac{K_2}{\rho g}}b,V_{Z\gamma 1}>0\wedge {\underset{\_}{t}}_{2,k}<t<{\underset{\_}{t}}_{2,k+n_{\gamma }}$$

(46)

where $V_{Z\gamma 1}$ is the depth velocity at ${\gamma }_1$. To obtain the expression for depth when time varies between ${\underset{\_}{t}}_{2,k}$ and ${\underset{\_}{t}}_{2,k+n_{\gamma }}$, expression (46) is integrated, leading to

$$Z(t)=Z_{\gamma 1}+\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)T_2·(1-e^{-(t-{\underset{\_}{t}}_{2,k})/T_2})-{\displaystyle \frac{K_2}{\rho g}}b\left(t-{\underset{\_}{t}}_{2,k}\right),V_{Z\gamma 1}>0\wedge {\underset{\_}{t}}_{2,k}<t<{\underset{\_}{t}}_{2,k+n_{\gamma }}$$

(47)

In (47) $Z_{\gamma 1}$ is the depth at point ${\gamma }_1$. The point ${\gamma }_2$ occurs when there is change in the sign of the velocity in (46):

$$0=\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)·e^{-(t_{{\gamma }_2}-{\underset{\_}{t}}_{2,k})/T_2}-{\displaystyle \frac{K_2}{\rho g}}b$$

(48)

Solving for $t_{{\gamma }_2}$:

$$t_{{\gamma }_2}={\underset{\_}{t}}_{2,k}-T_2\ln \left({\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)$$

(49)

Replacing (49) in (47) and considering (7) leads to

$$E_{Z\gamma 2}=E_{Z\gamma 1}-\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)T_2·\left(1-{\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)-{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)$$

(50)

If at ${\gamma }_1$ the velocity $V_{Z{\gamma }_1}$ is zero, then according to (50), $E_{Z\gamma 2}=E_{Z\gamma 1}$, which means points ${\gamma }_1$ and ${\gamma }_2$ are coincident. If $V_{Z{\gamma }_1}$ is negative, then (50) is no longer valid since no switching occurs.

The depth error in expression (50) can also be expressed as

$$E_{Z\gamma 2}=E_{Z\gamma 1}+∆E_{Z\gamma }$$

(51)

where $∆E_{Z\gamma }$ is the change in depth error between ${\gamma }_1$ and ${\gamma }_2$, given by

$$∆E_{Z\gamma }=-\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)T_2·\left(1-{\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)-{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)$$

(52)

Expression (51) is valid when $V_{Z{\gamma }_1}\ge 0$ and switching occurs. On the other hand, since points ${\gamma }_1$ and ${\gamma }_2$ are coincident when $V_{Z{\gamma }_1}<0$, Equation (51) can take a more general form:

$$E_{Z\gamma 2}=E_{Z\gamma 1}+{\lambda }_{\gamma }∆E_{Z\gamma }$$

(53)

where ${\lambda }_{\gamma }$ is a parameter that expresses the occurrence of switching at state ${\gamma }_1$:

$$\left\{\begin{array}{c}{\lambda }_{\gamma }=1,\hspace{1em}\hspace{1em}{V}_{Z{\gamma }_1}\ge 0\\ {\lambda }_{\gamma }=0,\hspace{1em}\hspace{1em}{V}_{Z{\gamma }_1}<0\end{array}\right.$$

(54)

Point ${\gamma }_2$ can therefore be defined as

$${\gamma }_2\left({\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}\right)=E_{{Z\gamma }_2}=E_{Z\gamma 1}+{\lambda }_{\gamma }∆E_{Z\gamma };{\dot{E}}_{Z{\gamma }_2}={\left(1-{\lambda }_{\gamma }\right)\dot{E}}_{Z{\gamma }_1}\ge 0\\ Z\left({\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}\right)=Z_{{\gamma }_2};V_{Z\gamma 2}={\left(1-{\lambda }_{\gamma }\right)V}_{Z{\gamma }_1}\le 0\\ F_{net}\left({\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}\right)=F_{net{\gamma }_2}=F_{net{\gamma }_1}=b\end{array}\right.$$

(55)

From point ${\gamma }_2$ until point $\delta$, ${\dot{E}}_Z$ is positive, so $V_Z$ is negative (Equation (8)). Thus, the force due to the vehicle deformation decreases, since $Z$ becomes smaller, and since $F_{VBM}$ is constant, the value of $F_{net}$ increases (Equation (22)) until the system reaches point $\delta$.

The change in $F_{VBM}$ between points ${\gamma }_2$, $F_{VBM{\gamma }_2}$, and $\delta ,F_{VBM\delta }$, can be determined by integrating Equation (25) with $i=1$:

$$F_{VBM{\gamma }_2}-F_{VBM\delta }=0\to F_{VBM{\gamma }_2}=F_{VBM\delta }$$

(56)

The change in $F_{net}$ between points ${\gamma }_2$, $F_{net{\gamma }_2}$, and $\delta ,F_{net\delta }$, can be determined by integrating Equation (23) with $i=1$:

$$F_{net\delta }-F_{net{\gamma }_2}={\int }_{{\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}}^{{\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}-\psi \rho g{V}_{Z}dt\to F_{net\delta }=b+\psi \rho g{\int }_{{\overline{t}}_{1,k+1+{\lambda }_{\gamma }{n}_{\gamma }}}^{{\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}{\displaystyle \frac{d{E}_Z}{dt}}dt$$

(57)

Equation (57) can be rewritten as

$$F_{net\delta }=b+\psi \rho g(E_{Z\delta }-E_{Z\gamma 2})$$

(58)

Since $E_{Z\delta }=\mathrm{a:}$

$$F_{net\delta }=b+\psi \rho g(a-E_{Z\gamma 2})$$

(59)

The point $\delta ({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})$ can therefore be defined as

$$\delta ({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }}):\left\{\begin{array}{c}{E}_Z({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})=E_{Z\delta }=a;{\dot{E}}_Z({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})={\dot{E}}_{Z\delta }>0\\ Z({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})=Z_{\delta };V_Z({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})=V_{Z\delta }<0\\ F_{net}({\underset{\_}{t}}_{1,k+1+{{\lambda }_{\gamma }n}_{\gamma }})=F_{net\delta }=b+\psi \rho g(a-E_{Z\gamma 2})\end{array}\right.$$

(60)

From $\delta$ to ${\epsilon }_1$, the change in $F_{VBM}$ can be determined by integrating Equation (25) with $i=2$ and noticing that in this region $E_z>0$:

$$F_{VBM{\epsilon }_1}-F_{VBM\delta }={\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}{\dot{F}}_{VBM}dt={\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}{K}_{1}A\rho g\left(-U_{switch}-{\displaystyle \frac{k_Z}{k_u}}Z\right)dt$$

(61)

The change in $F_{net}$ between points $\delta \left(F_{net\delta }\right)$ and ${\epsilon }_1\left(F_{net{\epsilon }_1}\right)$ can be determined by integrating Equation (23) with $i=2$:

$$F_{net{\epsilon }_1}-F_{net\delta }={\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}\left({\dot{F}}_{VBM}-\psi \rho g{V}_Z\right)dt$$

(62)

Developing (62):

$$F_{net{\epsilon }_1}=F_{net\delta }-K_{1}A\rho g{U}_{switch}\left({\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)-K_{1}A\rho g{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}Zdt+\psi \rho g\left(E_{Z{\epsilon }_1}-E_{Z\delta }\right)$$

(63)

Considering Equation (59) and since $F_{net{\epsilon }_1}=-b$ and $E_{Z\delta }=a$:

$$E_{Z{\epsilon }_1}=E_{Z\gamma 2}-{\displaystyle \frac{2b}{\psi \rho g}}+{\displaystyle \frac{K_{1}A}{\psi }}{U}_{switch}\left({\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)+{\displaystyle \frac{K_{1}A}{\psi }}{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}Zdt$$

(64)

Between $\delta$ and ${\epsilon }_1$, the system is accelerated when $F_{net}>0$ and is decelerated when $F_{net}<0$. The velocity, $V_Z$, in region 2a when $F_{net}>0$ is negative and may or not become positive when $0>F_{net}>-b$. The point ${\epsilon }_1\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)$ can therefore be defined as

$${\epsilon }_1\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)=E_{Z{\epsilon }_1};{\dot{E}}_Z\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)={\dot{E}}_{Z{\epsilon }_1}\\ Z\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)=Z_{{\gamma }_1};V_Z\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)=V_{Z{\epsilon }_1}\\ F_{net}\left({\overline{t}}_{1,k+2+{{\lambda }_{\gamma }n}_{\gamma }}\right)=F_{net{\epsilon }_1}=-b\end{array}\right.$$

(65)

If at ${\epsilon }_1$ the velocity $V_{Z{\epsilon }_1}$ is positive, the system will undergo a particular behavior between points ${\epsilon }_1$ and ${\epsilon }_2$. When the system enters infinitesimally at $i=1$, the control action is $U=0$, which makes $F_{VBM}=constant$ according to (24). Since ${\dot{E}}_Z$ is positive, $Z$ will decrease and so will the force magnitude factor due to $-\psi \rho gZ$. This will lead to an increase in the net force, and the system will be pushed towards $i=2$, causing the control action to turn on. This will increase the net force, causing the system to be moved again towards $i=1$. So, in this situation, the system undergoes a switching phase with $n_{\epsilon }$ switchings between $i=1$ and $i=2$. This behavior will occur until the vehicle is fully decelerated, making the velocity, $\dot{Z}$, change sign from negative to positive. Also, since ${\dot{E}}_Z=-V_Z$, at this point, the sign of ${\dot{E}}_Z$ changes from positive to negative. When this happens, point ${\epsilon }_2\left({\overline{t}}_{1,k+2+{\lambda n}_{\gamma }+n_2}\right)$ is reached. Since between ${\epsilon }_1$ and ${\epsilon }_2$ the average value of $F_{net}$ is constant and equal to $-b$, the relation between $Z\left({\overline{t}}_{1,k+1+n_{\gamma }}\right)=Z_{{\gamma }_2}$ and $Z_{{\gamma }_1}$ can be determined, from the block diagram of Figure 2, as

$${\dot{V}}_Z+{\displaystyle \frac{1}{T_2}}{V}_Z={\displaystyle \frac{1}{T_2}}{\displaystyle \frac{K_2}{\rho g}}b,V_{Z\epsilon 1}<0\wedge {\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}<t<{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }+n_{\epsilon }}$$

(66)

The solution for the first-order differential Equation (66) is given by

$$V_Z=\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)·e^{-\left(t-{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)/T_2}+{\displaystyle \frac{K_2}{\rho g}}b,V_{Z\epsilon 1}<0\wedge {\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}<t<{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }+n_{\epsilon }}$$

(67)

Integrating (67):

$$\begin{gathered} Z=Z_{{\epsilon }_1}+\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)T_2\left(1-e^{-\left(t-{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)/T_2}\right)+{\displaystyle \frac{K_2}{\rho g}}b\left(t-{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right), \\ {V}_{Z\epsilon 1}<0\wedge {\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}<t<{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }+n_{\epsilon }} \end{gathered}$$

(68)

In (68) $Z_{{\epsilon }_1}$ is the depth at point ${\epsilon }_1$. The point ${\epsilon }_2$ occurs when the velocity in (67) changes its sign:

$$0=\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)·e^{-\left(t_{{\epsilon }_2}-{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)/T_2}+{\displaystyle \frac{K_2}{\rho g}}b$$

(69)

Solving for $t_{{\epsilon }_2}$:

$$t_{{\epsilon }_2}={\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-T_2\ln \left({\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)$$

(70)

Plugging (70) into (68) and considering (7):

$$E_{Z{\epsilon }_2}=E_{Z{\epsilon }_1}-\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)T_2\left(1-{\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)+{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)$$

(71)

If at ${\epsilon }_1$ the velocity $V_{Z{\epsilon }_1}$ is zero, then, according to (71), $E_{Z\epsilon 2}=E_{Z\epsilon 1}$, which means points ${\epsilon }_1$ and ${\epsilon }_2$ are coincident. If $V_{Z{\epsilon }_1}$ is positive, then (71) is no longer valid since no switching occurs.

The depth error in (71) can also be expressed as

$$E_{Z{\epsilon }_2}=E_{Z{\epsilon }_1}+∆E_{Z\epsilon }$$

(72)

where $∆E_{Z\epsilon }$ is the change in depth error between ${\epsilon }_1$ and ${\epsilon }_2$, given by

$$∆E_{Z\epsilon }=-\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)T_2\left(1-{\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)+{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)$$

(73)

Expression (72) is valid when $V_{Z{\epsilon }_1}\le 0$ and switching occurs. On the other hand, since points ${\epsilon }_1$ and ${\epsilon }_2$ are coincident when $V_{Z{\gamma }_1}>0$, Equation (72) can take a more general form:

$$E_{Z{\epsilon }_2}=E_{Z{\epsilon }_1}+{\lambda }_{\epsilon }∆E_{Z\epsilon }$$

(74)

where ${\lambda }_{\epsilon }$ is a parameter that expresses the occurrence of switching at state ${\epsilon }_1$:

$$\left\{\begin{array}{c}{\lambda }_{\epsilon }=1,\hspace{1em}\hspace{1em}{V}_{Z{\epsilon }_1}\le 0\\ {\lambda }_{\epsilon }=0,\hspace{1em}\hspace{1em}{V}_{Z{\epsilon }_1}>0\end{array}\right.$$

(75)

Point ${\epsilon }_2$ can therefore be defined as

$${\epsilon }_2\left({\overline{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right):\left\{\begin{array}{c}{E}_Z\left({\overline{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=E_{{Z\epsilon }_2}=E_{Z\epsilon 1}+{\lambda }_{\epsilon }∆E_{Z\epsilon };{\dot{E}}_{Z{\epsilon }_2}={\left(1-{\lambda }_{\epsilon }\right)\dot{E}}_{Z{\epsilon }_1}\le 0\\ Z\left({\overline{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=Z_{{\epsilon }_2};V_{Z\epsilon 2}={\left(1-{\lambda }_{\epsilon }\right)V}_{Z{\epsilon }_1}\ge 0\\ F_{net}\left({\overline{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=F_{net{\epsilon }_2}=F_{net{\epsilon }_1}=-b\end{array}\right.$$

(76)

The change in $F_{VBM}$ between points ${\epsilon }_2$, $F_{VBM{\epsilon }_2}$, and $\zeta ,F_{VBM\zeta }$, can be determined by integrating Equation (25) with $i=1$:

$$F_{VBM\zeta }-F_{VBM{\epsilon }_2}=0\to F_{VBM\zeta }=F_{VBM{\epsilon }_2}$$

(77)

The change in $F_{net}$ between points ${\epsilon }_2$,$F_{VBM{\epsilon }_2}$, and $\zeta ,F_{VBM\zeta }$, can be determined by integrating Equation (23) with $i=1$.

$$F_{net\zeta }-F_{net{\epsilon }_2}={\int }_{{\overline{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}}^{{\underset{\_}{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}}-\psi \rho g{V}_{Z}dt$$

(78)

Since $F_{net{\epsilon }_2}=-b$ and $E_{Z\zeta }=-a$:

$$F_{net\zeta }=-b-\psi \rho g\left(a+E_{Z{\epsilon }_2}\right)$$

(79)

Point $\zeta$ can therefore be defined as

$$\zeta \left({\underset{\_}{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right):\left\{\begin{array}{c}{E}_Z\left({\underset{\_}{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=E_{Z\zeta };{\dot{E}}_Z\left({\underset{\_}{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)={\dot{E}}_{Z\zeta }<0\\ Z\left({\underset{\_}{t}}_{,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=Z_{\zeta };V_Z\left({\underset{\_}{t}}_{1,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=V_{Z\zeta }=-{\dot{E}}_{Z\zeta }>0\\ F_{net}\left({\underset{\_}{t}}_{,k+2+{\lambda }_{\gamma }{n}_{\gamma }+{\lambda }_{\epsilon }{n}_{\epsilon }}\right)=F_{net\zeta }=-b-\psi \rho g\left(a+E_{Z{\epsilon }_2}\right)\end{array}\right.$$

(80)

With all the points in the diagram defined, condition 3 of Section 4.1 will be demonstrated in the following. Condition 3(a) requires that $F_{net\alpha }=F_{net{\epsilon }_2}$ and $E_{Z{\epsilon }_2}<E_{Z\alpha }$. From Figure 7, $F_{net\alpha }=F_{net{\epsilon }_2}$ by definition. The condition $E_{Z{\epsilon }_2}<E_{Z\alpha }$ can be satisfied by ensuring the following conditions:

(i) $\left|E_{Z{\gamma }_2}\right|\le E_{Z\alpha }$;

(ii) $E_{Z{\epsilon }_2}\le \left|E_{Z{\gamma }_2}\right|$.

The condition for part (i) is

$$\left|E_{Z{\gamma }_2}\right|\le \left|E_{Z\alpha }\right|$$

(81)

Replacing Equations (43), (52), and (53) in (81) leads to

$$U_{switch}\le {\displaystyle \frac{1}{\left({\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}\right)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2\left|E_{Z\alpha }\right|+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\gamma }\left|∆E_{Z\gamma }\right|\right)+{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt\right]$$

(82)

The condition for part (ii) is

$$\left|E_{{Z\epsilon }_2}\right|\le \left|E_{{Z\gamma }_2}\right|$$

(83)

Replacing Equations (64), (73), and (74) in (83) leads to

$$U_{switch}\le {\displaystyle \frac{1}{\left({\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2\left|E_{{Z\gamma }_2}\right|+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\epsilon }\left|∆E_{Z\epsilon }\right|\right)-{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}Zdt\right]$$

(84)

Inequalities (82) and (84) specify upper bounds that $U_{switch}$ must comply with for condition 3(a) to be satisfied.

For condition 3(b) to be satisfied, $\left|E_{Z\beta }\right|=\left|E_{Z\zeta }\right|\mathrm{and}\left|F_{net\zeta }\right|<\left|F_{net\beta }\right|$. The first condition is ensured by the definition of points $\beta$ and $\zeta$. For the second condition to be met, (38) and (79) can be inserted into (85):

$$\left|F_{net\zeta }\right|\le \left|F_{net\beta }\right|$$

(85)

Leading to Equation (86):

$$E_{Z{\epsilon }_2}\le E_{Z\alpha }$$

(86)

The result from expression (86) means that as long as both upper bounds for $U_{switch}$ are met, then $\left|F_{net\zeta }\right|\le \left|F_{net\beta }\right|$ and $F_{net}$ is bounded, meeting the requirements of condition 3(b).

4.5. Quantifying the Controller Action Limits

4.5.1. Quantifying the Upper Limits of $U_{switch}$

As was shown in the previous sections, the controller is stable if it satisfies conditions (12), (82), and (84). Expression (82) is repeated below.

$$U_{switch}<{\displaystyle \frac{1}{\left({\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}\right)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2E_{Z\alpha }+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\gamma }\left|∆E_{Z\gamma }\right|\right)+{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt\right]$$

(87)

And $∆E_{Z\gamma }$ is repeated below.

$$∆E_{Z\gamma }=-\left(V_{Z\gamma 1}+{\displaystyle \frac{K_2}{\rho g}}b\right)T_2·\left(1-{\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)-{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{\frac{V_{Z\gamma 1}\rho g}{b{K}_2}+1}}\right)$$

(88)

Since (87) defines an upper bound for $U_{switch}$, it is interesting to find the minimum value for its right side. In this situation $\left|∆E_{Z\gamma }\right|$ should take the highest possible value and ${\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt$ should take the lowest, contributing to a conservative estimation of $U_{switch}$. Since (88) is monotonic and always negative and since $E_{Z\gamma 2}\le E_{Z\gamma 1}$, the maximum value of (88) occurs when $V_{Z\gamma 1}$ is maximal. To estimate the maximum value of $U_{switch}$ according to expression (87), a few assumptions will be made:

1.

From $\alpha$ to ${\gamma }_1$, $F_{net}=F_{net\beta }=constant$ for the purposes of estimating $V_{Z\gamma 1}$. Since $F_{net\beta }$ is the maximum value $F_{net}$ takes between $\alpha$ and ${\gamma }_1$, this will contribute to an overestimation of $V_{Z\gamma 1}$, leading to a conservative estimation of $U_{switch}$.

2.

The integral factor ${\int }_{{\overline{t}}_{2,k}}^{{\underset{\_}{t}}_{2,k}}Zdt=0$. Since $Z$ is always positive, assuming that its integral is zero contributes to a conservative estimation.

3.

From $\beta$ to ${\gamma }_1$, ${\dot{F}}_{net}=\eta$. The difference ${\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}$ corresponds to the required time for $F_{net}$ to increase from $F_{net\beta }$ to $F_{net\gamma 1}=b$. Since from ${\underset{\_}{t}}_{2,k}$ to ${\overline{t}}_{2,k}$, ${\dot{F}}_{net}>\eta$, it will be considered, in a conservative scenario, that ${\dot{F}}_{net}=\eta$.

Considering the block diagram of Figure 2:

$$s·Z\left(s\right)={\displaystyle \frac{-K_2}{\rho g}}{\displaystyle \frac{1}{T_{2}s+1}}{F}_{net}(s)$$

(89)

This means that maximum depth velocity occurs in a steady state, in which

$$V_Z={\displaystyle \frac{-K_2}{\rho g}}{F}_{net}$$

(90)

From assumption 1,

$$V_{Z\gamma 1\_max}={\displaystyle \frac{-K_2}{\rho g}}{F}_{net\beta }$$

(91)

From (38):

$$V_{Z\gamma 1\_max}={\displaystyle \frac{-K_2}{\rho g}}\left(-b-\psi \rho g\left(a+E_{Z\alpha }\right)\right)=K_2\psi \left(a+E_{Z\alpha }\right)+{\displaystyle \frac{K_2}{\rho g}}b$$

(92)

From assumption 3,

$${\left.{\dot{F}}_{net}\right|}_{{\underset{\_}{t}}_{2,k}\to {\overline{t}}_{2,k}}={\displaystyle \frac{F_{net\gamma 1}-F_{net\beta }}{{\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}}}={\displaystyle \frac{0-(-b-\psi \rho g\left(a+E_{Z\alpha }\right))}{{\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}}}=\eta$$

(93)

Rearranging:

$${\underset{\_}{t}}_{2,k}-{\overline{t}}_{2,k}={\displaystyle \frac{b+\psi \rho g\left(a+E_{Z\alpha }\right)}{\eta }}$$

(94)

Considering assumption 2 and expressions (91) and (94), expression (87) can be rewritten as

$$U_{switch}<{\displaystyle \frac{\eta }{b+\psi \rho g\left(a+E_{Z\alpha }\right)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2E_{Z\alpha }+{\displaystyle \frac{2b}{\psi \rho g}}+{\lambda }_{\gamma }∆E_{Z\gamma }\right)\right]$$

(95)

Expression (95) with ${\lambda }_{\gamma }=1$ and $V_{Z\gamma 1}=V_{Z\gamma 1\_max}$ represents a conservative estimation of the upper bound for $U_{switch}$ given by expression (87). A similar process can be followed for estimating (84), repeated below.

$$U_{switch}<{\displaystyle \frac{1}{\left({\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2\left|E_{{Z\gamma }_2}\right|+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\epsilon }∆E_{Z\epsilon }\right)-{\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}Zdt\right]$$

(96)

where

$$∆E_{Z\epsilon }=-\left(V_{Z{\epsilon }_1}-{\displaystyle \frac{K_2}{\rho g}}b\right)T_2\left(1-{\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)+{\displaystyle \frac{K_2}{\rho g}}b{T}_2\ln \left({\displaystyle \frac{1}{-\frac{V_{Z{\epsilon }_1}\rho g}{K_{2}b}+1}}\right)$$

(97)

Since (96) defines another upper bound for $U_{switch}$, finding the minimum value for its right side leads to a conservative estimation. In this situation $∆E_{Z\epsilon }$ and ${\displaystyle \frac{k_Z}{k_u}}{\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}Zdt$ should take their highest values, contributing to a conservative estimation of $U_{switch}$. Since (97) is monotonic and always positive and since $E_{Z\epsilon 2}\ge E_{Z\epsilon 1}$, the maximum value of (96) occurs when $\left|V_{Z{\epsilon }_1}\right|$ is maximal. To estimate the maximum value of $U_{switch}$ according to expression (96), a few new assumptions will be made:

4.

From ${\gamma }_2$ to ${\epsilon }_1$, $F_{net}=F_{net\delta }=constant$ for the purposes of estimating $\left|V_{Z{\epsilon }_1}\right|$. Since $F_{net\delta }$ is the maximum value $F_{net}$ takes between ${\gamma }_2$ and ${\epsilon }_1$, this will contribute to an overestimation of $\left|V_{Z{\epsilon }_1}\right|$, leading to a conservative estimation of $U_{switch}$.

5.

To calculate the integral factor, $Z$ will be assumed to take its maximum value, $Z_{max}$, in a given mission, thus considering a conservative estimation.

6.

From $\delta$ to ${\epsilon }_1$, ${\dot{F}}_{net}=-\eta$. The difference ${\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}$ corresponds to the required time for $F_{net}$ to decrease from $F_{net\delta }$ to $F_{net{\epsilon }_1}=0$. Since from ${\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}$ to ${\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}$, ${\dot{F}}_{net}<-\eta$, it will be considered, in a conservative scenario, that ${\dot{F}}_{net}=-\eta$.

From (90) and assumption 4:

$$V_{Z{\epsilon }_1\_max}={\displaystyle \frac{-K_2}{\rho g}}{F}_{net\delta }$$

(98)

Since $F_{net\delta }>0$ and $V_{Z{\epsilon }_1}<0$:

$$\left|V_{Z{\epsilon }_1\_max}\right|={\displaystyle \frac{K_2}{\rho g}}{F}_{net\delta }$$

(99)

From (59):

$$\left|V_{Z{\epsilon }_1\_max}\right|={\displaystyle \frac{K_2}{\rho g}}\left(b+\psi \rho g(a-E_{Z\gamma 2})\right)=K_2\psi \left(a-E_{Z\gamma 2}\right)+{\displaystyle \frac{K_2}{\rho g}}b$$

(100)

From assumption 5,

$${\int }_{{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}^{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}{Z}_{max}dt=Z_{max}\left({\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}\right)$$

(101)

From assumption 6,

$${\left.{\dot{F}}_{net}\right|}_{{\overline{t}}_{2,k+1+n_1}\to {\underset{\_}{t}}_{2,k+1+n_1}}={\displaystyle \frac{F_{net{\epsilon }_1}-F_{net\delta }}{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}}={\displaystyle \frac{0-b-\psi \rho g(a-E_{Z\gamma 2})}{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}}=-\eta$$

(102)

$$\eta ={\displaystyle \frac{b+\psi \rho g(a-E_{Z\gamma 2})}{{\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}}}$$

(103)

$${\underset{\_}{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}-{\overline{t}}_{2,k+1+{{\lambda }_{\gamma }n}_{\gamma }}={\displaystyle \frac{b+\psi \rho g(a-E_{Z\gamma 2})}{\eta }}$$

(104)

Considering Equations (100), (101), and (104), expression (96) becomes

$$U_{switch}<{\displaystyle \frac{\eta }{b+\psi \rho g(a-E_{Z\gamma 2})}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2\left|E_{{Z\gamma }_2}\right|+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\epsilon }∆E_{Z\epsilon }\right)-{\displaystyle \frac{k_Z}{k_u}}{Z}_{max}{\displaystyle \frac{b+\psi \rho g(a-E_{Z\gamma 2})}{\eta }}\right]$$

(105)

$$U_{switch}<{\displaystyle \frac{\eta }{b+\psi \rho g(a+\left|E_{Z\gamma 2}\right|)}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2\left|E_{{Z\gamma }_2}\right|+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\epsilon }∆E_{Z\epsilon }\right)\right]-{\displaystyle \frac{k_Z}{k_u}}{Z}_{max}$$

(106)

Since one of the goals of this analysis is to show that $\left|E_{Z{\gamma }_2}\right|\le E_{Z\alpha }$, in a worst-case scenario, $\left|E_{Z{\gamma }_2}\right|=E_{Z\alpha }$. In that case, expression (106) becomes

$$U_{switch}<{\displaystyle \frac{\eta }{b+\psi \rho g(a+E_{Z\alpha })}}\left[{\displaystyle \frac{\psi }{K_{1}A}}\left(2E_{Z\alpha }+{\displaystyle \frac{2b}{\psi \rho g}}-{\lambda }_{\epsilon }∆E_{Z\epsilon }\right)\right]-{\displaystyle \frac{k_Z}{k_u}}{Z}_{max}$$

(107)

Expression (107) with ${\lambda }_{\epsilon }=1$ and $\left|V_{Z\epsilon 1}\right|=\left|V_{Z\epsilon 1\_max}\right|$ represents a conservative estimation of the upper bound for $U_{switch}$ given by expression (96).

4.5.2. Quantifying the Lower Limit of $U_{swit\mathrm{c}\mathrm{h}}$

The $U_{switch}$ lower limit can be determined by (12), repeated below.

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+sign\left(E_Z\right)\left(-\left(k_z/k_u\right)Z-\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_Z+{\displaystyle \frac{1}{\rho g{K}_{1}A}}{\dot{F}}_{dis{t}_{ext}}\right)$$

(108)

To estimate the minimum value of $U_{switch}$ according to expression (108), it will be assumed that external disturbances are constant ($F_{dis{t}_{ext}}=const\to {\dot{F}}_{dis{t}_{ext}}=0$). In a conservative scenario, the right side of expression (108) is maximal for region 2b, in which case:

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+{\displaystyle \frac{k_z}{k_u}}{Z}_{max}+\left({\displaystyle \frac{\psi }{K_{1}A}}\right)V_{Zmax}$$

(109)

where $Z_{max}$ and $V_{Zmax}$ are the maximum absolute values of $Z$ and $V_Z,$ respectively. $Z_{max}$ can be overestimated as the maximum depth of a given mission. $V_{Zmax}$ can be overestimated from assumption 1 of Section 4.5.1 as the steady-state velocity at $F_{net\beta }$:

$$V_{Zmax}={\displaystyle \frac{-K_2}{\rho g}}{F}_{net\beta }$$

(110)

Considering these estimations, expression (109) becomes

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+{\displaystyle \frac{k_z}{k_u}}{Z}_{max}+\left({\displaystyle \frac{\psi }{K_{1}A}}\right){\displaystyle \frac{-K_2}{\rho g}}{F}_{net\beta }$$

(111)

Considering $F_{net\beta }$ from expression (38):

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+{\displaystyle \frac{k_z}{k_u}}{Z}_{max}+\left({\displaystyle \frac{\psi }{K_{1}A}}\right){\displaystyle \frac{-K_2}{\rho g}}\left(-b-\psi \rho g\left(a+E_{Z\alpha }\right)\right)$$

(112)

Finally, the lower bound for $U_{switch}$ can be conservatively estimated by

$$U_{switch}>{\displaystyle \frac{\eta }{\rho g{K}_{1}A}}+{\displaystyle \frac{k_z}{k_u}}{Z}_{max}+{\displaystyle \frac{K_2\psi }{{\rho gK}_{1}A}}b+{\displaystyle \frac{K_2{\psi }^2}{K_{1}A}}\left(a+E_{Z\alpha }\right)$$

(113)

5. Closed-Loop System Stability: Case Study

In order to present a case study of the application of the results obtained in this work to a prototype, in this section, expressions (95), (107), and (113) are used to determine the bounds of $U_{switch}$ according to the rate of change in net force, $\eta$, and the depth error at point $\alpha$, $E_{Z\alpha }$. The system model parameters for these expressions are unique to the real system being modelled. For the purposes of this work, the prototype developed in [21] with the parameters fully determined in [12] was considered. The prototype, whose model can be described by the block diagram of Figure 2, uses a diaphragm-sealed piston, driven by a DC motor, to change buoyancy (the reader is referred to [12,21] for further details).

The system model parameters are presented in

Table 2. System model parameters.

Parameter	Value	Unit
$A$	7.7 × 10−3	[m2]
$g$	9.81	[ms−2]
$K_1$	4.44 × 10−4	[ms−1V−1]
$K_2$	7.9355 × 103	[ms−1m−3]
$k_z/k_u$	3.31 × 10−2	[Vm−1]
$T_2$	36.3	[s]
$a$	0.5	[m]
$\rho$	1 × 103	[kgm−3]
$\psi$	3.4 × 10−7	[m3m−1]
$Z_{max}$	100	[m]

. The choice of parameter $a$ is related to the minimum depth error that is attainable with the proposed controller, which depends on the particular mission goals. In the particular example presented in the paper, it was considered that an error around ±0.5% of the maximum rated depth of the prototype ($Z_{max}$) would be a reasonable choice. Figure 8, Figure 9 and Figure 10 represent the limits for the values of $U_{switch}$, according to expressions (95), (107), and (103), that are necessary to obtain a given change in net force, $\eta$. The difference between these figures is the value of $b$: Figure 8 presents the results for $b$ = 0.0175 N (0.5% of the available buoyancy change of the prototype, which is ±350 cm³), Figure 9 the results for $b$ = 0.035 N (1% of the available buoyancy change), and Figure 10 the results for $b$ = 0.07 N (2% of the available buoyancy change).

In Figure 8a, Figure 9a, and Figure 10a, the two curved surfaces on top represent the upper bounds for $U_{switch}$ and the plane on the bottom represents the lower bound. As long as the lower bound for $U_{switch}$ is lower than the upper bound, there exist multiple values of $U_{switch}$ that guarantee the stability of the controller. An interesting result appears when the value of $b$ is very small: the upper-limit surfaces curve and intersect the lower-limit surface at low values of $E_{Z\alpha }$, as is particularly noticeable in Figure 8a. This indicates that very near the origin, with low $b$ values, the system may become locally unstable, leading to an increase in $E_{Z\alpha }$ because conditions (95), (107), and (103) are not simultaneously satisfied. However, when this occurs, the value of $E_{Z\alpha }$ of the next switching sequence will be higher, eventually leading to a bounded state, as described in Section 4.3. Figure 8b, Figure 9b, and Figure 10b represent 2D cross-sections of Figure 8a, Figure 9a, and Figure 10a, respectively. In Figure 8b, Figure 9b, and Figure 10b, the dash-dotted lines correspond to the upper bound for $U_{switch}$ according to expression (95) for each of the selected $E_{Z\alpha }$ values. The continuous lines represent the upper bounds according to expression (107), and the lower bound is represented by the dashed lines. The color of each line represents a particular value of $E_{Z\alpha }$. The horizontal magenta line corresponds to the physical limit of the battery voltage. In each of Figure 8b, Figure 9b, and Figure 10b, the regions below the upper bounds and above the lower bounds are possible $U_{switch}$ solutions, as long as they are lower than the battery voltage. These regions are shaded in green for the particular case of $E_{Z\alpha }$ = 1 m. As an example, let us consider Figure 9b. If $U_{switch}=6\mathrm{V}$, the lower limit is higher than the upper limits for each of the presented values of $E_{Z\alpha }$. This means that this value of $U_{switch}$ is not an admissible solution. On the other hand, if $U_{switch}=14\mathrm{V}$, the lower bound is lower than both upper bounds for each of the presented values of $E_{Z\alpha }$ and the controller is therefore stable. The values obtained in Figure 8, Figure 9 and Figure 10 show that the controller can be implemented since there are admissible values of $U_{switch}$ below the battery voltage level. Also, it should be noted that the upper and lower bounds were obtained with very conservative estimations and that the real bounds would broaden the range of admissible values for $U_{switch}$. In this manner, the practical utility of the controller is shown.

6. Conclusions

This paper presents a formal proof of stability for an energy-efficient, variable structure depth controller recently developed by the authors. This controller is meant to be used in underwater vehicles and platforms using variable buoyancy actuation. It is demonstrated that if the control action remains within specified upper and lower bounds, the system state remains bounded regardless of its initial rest conditions. These control limits can be quantified for a given system, providing valuable insight into the selection of controller parameters. The study’s findings were applied to a depth-controlled sensor platform previously developed by the authors, using an established system model while accounting for physical constraints. Despite following a highly conservative approach, the results indicate that the control law can be implemented while ensuring formal system stability. Additionally, the study illustrates how stability regions are influenced by different controller parameter choices and mission requirements. The results presented here serve as a guideline for selecting appropriate controller parameters for specific mission scenarios. Future work will focus on several points. First, an experimental validation of the controller through sea trials will be pursued. Second, a comparison of the controller developed in [12] with robust controllers, model predictive control, disturbance-aware strategies, and others will be performed. Third, the stability proof will be extended to scenarios involving non-constant external disturbances, for instance, the mouths of rivers, where salinity might cause sudden changes in the vehicle buoyancy, or dynamic waters, where vertical currents might cause sudden disturbances to the vehicle buoyancy abilities. Fourth, the impact of the controller parameters on the stability of the system and on energy savings will be analyzed.