Fault-Tolerant Trajectory Tracking Control for a Differential-Driven Unmanned Surface Vehicle with Propeller Faults

Abstract

This article investigates the problem of adaptive fault-tolerant trajectory tracking control for a differential-driven unmanned surface vehicle (USV) with propeller faults. A new USV control system considering a propeller servo loop is established, which is composed of kinematics, kinetics including unhealthy surge force and yaw moment, and propeller motor shaft speed dynamics. Firstly, the control design of the kinematic level derives the virtual surge speed and yaw rate, which can accurately guide the tracking design of the kinetic level. Secondly, by estimating the bound of the unknown propeller fault parameters, the virtual fault-tolerant control laws are constructed in the kinetic level, which can generate the desired motor angular shaft speeds with an active compensation feature. Thirdly, in the control design of the propeller servo loop, the command duty cycles are designed to force the actual motor shaft speeds to track the desired signals produced from the kinetic level. It can be proven that tracking errors are semiglobally ultimately uniformly bounded based on Lyapunov stability theory. Finally, simulations considering single propeller and twin propeller faults prove the validity of the developed method.

1. Introduction

Owing to the significant benefits in terms of low production expenses, high intelligence levels, and high mobility, unmanned surface vehicles (USVs) have been extensively employed in oceanographic data collection, surveillance, oil and pollution clean-up, and other oceanic assignments [1,2,3,4]. As a crucial prerequisite for the autonomous operation of USVs, their ability to follow a predetermined trajectory significantly impacts the effectiveness and ultimate achievement of mission goals. Consequently, the problem of trajectory tracking for USVs has garnered significant attention in recent years. By modeling the USV as a Takagi–Sugeno fuzzy system, a fuzzy output–feedback tracking controller is firstly presented in [5]. Considering modeling imperfections of the USV system, a neural network-based adaptive tracking control algorithm is proposed in [6].

It is noted that the USVs studied in [5,6] possess full actuation. Nevertheless, since most USVs are not fitted with the actuator for sway motion, this type of USV belongs to the category of underactuated systems [7]. A global robust adaptive algorithm is proposed in [8] to settle the issue of control law design of a USV. Based on the neural networks technique, the work in [9] designs a prescribed performance controller to drive an underactuated USV with model uncertainties for tracking a circular trace. In [10], an adaptive disturbance estimation control is combined with the port-Hamiltonian framework to achieve the successful tracking of a desired trajectory. By using an output redefinition technology, a finite-time robust tracking result is yielded in [11]. However, all the aforementioned works are mainly concentrated on designing controllers at the kinematic and kinetic levels, merely fulfilling the theoretical aspect of thrust design. These controllers cannot be utilized directly since we do not possess the capability to directly input thrust into the USV, and instead, we depend on propellers to produce thrust.

Differential-driven USVs, which belong to a specific category of underactuated systems, have been extensively utilized in real-world maritime tasks. The cause is attributed to the fact that the port and starboard propellers are arranged symmetrically on the USV’s tail, and each propeller produces thrust independently, facilitating the vehicle’s maneuverability and reducing manufacturing costs. The electric motors propel the propellers of the differential-driven USV, utilizing pulse width modulation (PWM) duty cycles as the actual control command signal. The authors in [12] firstly develop one-state shaft speed dynamics of a single thruster. In [13], a voltage-operated thruster shaft speed model is established by proposing a two-state shaft speed model. An output–feedback control scheme is presented for a ship system based on three-state shaft speed dynamics of the propeller [14]. In [15], a model predictive control approach is designed for the USV with a single propeller. Recently, a disturbance observer-based tracking control approach is proposed for the differential-driven USV taking into account the propeller servo loop [16]. To achieve the user-defined tracking performance, a dynamic surface technique-based adaptive prescribed-time control strategy is developed for a differential-driven USV with completely unknown parameters in [17]. During the course of oceanic tasks, thruster faults are nearly unavoidable [18]. As stated in [19], common thruster failures are categorized as hard-over failures, floating failures, loss-of-effectiveness failures, and stuck failures. For differential-driven USVs, a fault in even a single propeller can lead to a substantial decrease in maneuverability. Note that some fault-tolerant control approaches, such as the backstepping technique [20], adaptive control [21], Lyapunov matrix method [22], and robust control [23], have been studied in relation to the motion control of USVs. Unfortunately, no fault-tolerant control result is developed to guarantee the safe operation of differential-driven USV subject to propeller faults.

Following these discussions, this article intends to propose an intelligent fault-tolerant control method for a differential-driven USV with propeller faults. The contributions of this article are summarized as follows.

(1) In comparison with the employed USV control system considering the healthy actuator mechanism in [16,17], a differential-driven USV model with propeller faults is firstly proposed since the damage of the USV’s propulsion unit is not negligible in the complex marine environment.

(2) By presenting the estimation technique for propeller fault parameter bounds, an adaptive fault-tolerant control scheme is developed to guarantee the tracking performance of the differential-driven USV even if there is the occurrence of propeller faults.

(3) Compared with the fault-tolerant tracking control method in [21], the restriction of known propeller fault parameters and the process of thrust modification are eliminated in this paper. By considering the servo loop of the differential-driven USV, the designed fault-tolerant tracking controller can be translated into the desired motor shaft speed tracking signals. On this basis, the PWM duty cycles as actual control commands are designed for motor shaft speed systems, which can directly drive twin propellers of the USV.

This article is structured as follows: Section 2 proposes a differential-driven USV model with propeller faults, while Section 3 presents the control design and stability analysis. Section 4 develops simulation results to prove the effectiveness of the developed control method, and finally, this paper is concluded in Section 5.

Table 1. Nomenclature of system variables.

Symbol	Interpretation
x	Surge displacement (m)
y	Sway displacement (m)
$\psi$	Yaw angle (rad)
u	Surge speed (m/s)
v	Sway speed (m/s)
r	Yaw rate (rad/s)
$n_1,n_2$	Shaft speeds of motors (rpm)
$M_{11},M_{22},M_{33},M_{23},M_{32}$	USV inertia including added mass (kg)
$X_u,Y_v,Y_r,N_v,N_r$	Linear damping coefficients (kg/s)
$X_{\left|u\right|u},Y_{\left|v\right|v},Y_{\left|r\right|v},Y_{\left|v\right|r},N_{\left|v\right|v},N_{\left|r\right|v},N_{\left|v\right|r},N_{\left|r\right|r}$	Nonlinear damping coefficients (kg/m)
${\tau }_u$	Surge force (N)
${\tau }_r$	Yaw moment (N·m)
$V_m$	Supply voltage (V)
$T_{m1},T_{m2}$	Transfer coefficients from voltage to rotational speed (rad/(s·V))
$T_{n1},T_{n2}$	Transfer coefficients from load torque to rotational speed (rad/(s·N·m))
$\rho$	Density of water ($\mathrm{kg}/{\mathrm{m}}^3$)
$D_1,D_2$	Port and starboard propeller diameter (m)
$t_1,t_2$	Thrust reduction factors (-)
w	Wake fraction number (-)
$Q_1,Q_2$	Moment of inertia for motors (${\mathrm{kgm}}^2$)
${\delta }_1,{\delta }_2$	Duty cycles of PWM signals (-)
${\rho }_1,{\rho }_2$	Actuation effectiveness of two propellers (-)
${\sigma }_1,{\sigma }_2$	Floating faults (-)

and

Table 2. Nomenclature of control design variables.

Symbol	Interpretation
$e_1$	Position error (m)
$e_2,e_3$	Surge and sway displacement errors (m)
$e_4$	Barrier function to manage the control coefficient in sway (-)
$e_5$	Surge speed tracking error (m/s)
$e_6$	Yaw rate tracking error (rad/s)
$e_7,e_8$	Motor shaft speed errors (rpm)
${\xi }_1,{\xi }_2$	Sway and surge control gain functions of position error dynamics (-)
$u_d,{\overline{u}}_d$	Virtual and filtered surge speed control laws (m/s)
$r_d,{\overline{r}}_d$	Virtual and filtered yaw rate control laws (rad/s)
${\chi }_{\psi }$	Control gain function of orientation regulation dynamics (-)
${\upsilon }_1$	Intermediate surge force control law (N)
${\upsilon }_2$	Intermediate yaw moment control law (N·m)
${\tau }_u^{*}$	Surge force control law (N)
${\tau }_r^{*}$	Yaw moment control law (N·m)
${\delta }_1^{*},{\delta }_2^{*}$	Duty cycle control laws of two motors (-)
$k_1,k_2,k_3,k_4,k_5,k_6$	Gain design parameters of control laws (-)
${\varrho }_{g,i}(g=1,2,3,4,i=1,2,\dots ,h),d_1,d_2$	Learning rate design parameters of adaptive laws (-)
${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$	Anti-saturation auxiliary variables (-)
$\mathrm{\Xi }$	Small predefined positive constant to avoid singularity (-)

show the nomenclature of variables.

2. Problem Formulation

2.1. USV Dynamics

To show the movement of a USV, two coordinates are established, as depicted in Figure 1. Consider a three-degree-of-freedom differential-driven USV model with kinematics and kinetics established as

$$\begin{array}{cc}\hfill \dot{x}=& ucos\left(\psi \right)-vsin\left(\psi \right)\hfill \\ \hfill \dot{y}=& usin\left(\psi \right)+vcos\left(\psi \right)\hfill \\ \hfill \dot{\psi }=& r\hfill \\ \hfill \dot{u}=& f_u(u,v,r)+{\displaystyle \frac{1}{M_{11}}}{\tau }_{du}+{\displaystyle \frac{1}{M_{11}}}{\tau }_u(W_1,W_2)\hfill \\ \hfill \dot{v}=& f_v(u,v,r)+{\displaystyle \frac{1}{M_{22}}}{\tau }_{dv}\hfill \\ \hfill \dot{r}=& f_r(u,v,r)+{\displaystyle \frac{1}{M_{33}}}{\tau }_{dr}-{\displaystyle \frac{M_{32}}{M_{22}{M}_{33}}}{\tau }_{dv}+{\displaystyle \frac{1}{M_{33}}}{\tau }_r(W_1,W_2)\hfill \end{array}$$

(1)

with

$$\begin{array}{cc}\hfill f_u(u,v,r)=& {\displaystyle \frac{1}{M_{11}}}\left[\left(M_{22}v+M_{22}^{*}r\right)r+(X_u+X_{\left|u\right|u}\left|u\right|)u\right]\hfill \\ \hfill f_v(u,v,r)=& {\displaystyle \frac{1}{M_{22}}}\left[-M_{11}ur+\left(Y_r+Y_{\left|v\right|r}\left|v\right|\right)r+\left(Y_{\left|r\right|v}\left|r\right|+Y_v+Y_{\left|v\right|v}\left|v\right|\right)v\right]\hfill \\ \hfill f_r(u,v,r)=& {\displaystyle \frac{1}{M_{33}}}[-\left(M_{22}v+M_{22}^{*}r\right)u+M_{11}uv+\left(N_v+N_{\left|v\right|v}\left|v\right|+N_{\left|r\right|v}\left|r\right|\right)v\hfill \\ \hfill & +(N_r+N_{\left|v\right|r}\left|v\right|+N_{\left|r\right|r}\left|r\right|\left)r\right]+{\displaystyle \frac{M_{32}}{M_{22}{M}_{33}}}[M_{11}ur-(Y_r+Y_{\left|v\right|r}\left|v\right|)r\hfill \\ \hfill & -\left(Y_{\left|r\right|v}\left|r\right|+Y_v+Y_{\left|v\right|v}\left|v\right|\right)v]\hfill \end{array}$$

where $M_{22}^{*}=(M_{23}+M_{32})/2$; $(x,y)$ and $\psi$ are the position and heading of a USV in the earth-fixed reference frame; u, v, and r are the surge, sway, and yaw speeds of the ship in a body-fixed reference frame; $M_{11}$, $M_{22}$, $M_{23}$, $M_{32}$, and $M_{33}$ are the vehicle inertia involving added mass; $X_u$, $Y_v$, $Y_r$, $N_v$, $N_r$, $X_{\left|u\right|u}$, $Y_{\left|v\right|v}$, $Y_{\left|r\right|v}$, $Y_{\left|v\right|r}$, $N_{\left|v\right|v}$, $N_{\left|r\right|v}$, $N_{\left|v\right|r}$, and $N_{\left|r\right|r}$ denote the damping parameters; ${\tau }_{du}$, ${\tau }_{dv}$, and ${\tau }_{dr}$ are the bounded external disturbances.

By employing the differential thrust, two propellers mounted symmetrically at the rear of the vehicle can generate both a surge force ${\tau }_u(W_1,W_2)$ and a yaw moment ${\tau }_r(W_1,W_2)$. The thrust forces $W_1$ and $W_2$, generated by two motor-powered propellers, can be depicted in Figure 2a. A fundamental equivalent model is introduced, wherein the thrusts are repositioned to mass center, as depicted in Figure 2b. Therefore, the surge force and yaw moment are formulated as

$$\begin{array}{cc}\hfill {\tau }_u(W_1,W_2)=& W_1+W_2\hfill \\ \hfill {\tau }_r(W_1,W_2)=& {\displaystyle \frac{G}{2}}(W_1-W_2)\hfill \end{array}$$

(2)

where G denotes the spacing between twin propellers.

2.2. DC Motor-Powered Propeller Model

On the basis of [16], the dynamics of the propeller servo loop is formulated as:

$$\begin{array}{cc}\hfill {\dot{n}}_1=& f_{n1}(n_1,u)+{\displaystyle \frac{K_1}{Q_1}}{\delta }_1\hfill \\ \hfill {\dot{n}}_2=& f_{n2}(n_2,u)+{\displaystyle \frac{K_2}{Q_2}}{\delta }_2\hfill \end{array}$$

(3)

with

$$\begin{array}{c}\hfill f_{n1}={\displaystyle \frac{1}{Q_1}}\left(-{\overline{\sigma }}_{fm1}u|n_1|-{\overline{\sigma }}_{fn1}{n}_1\left|n_1\right|-n_1\right)\\ \hfill f_{n2}={\displaystyle \frac{1}{Q_2}}\left(-{\overline{\sigma }}_{fm2}u|n_2|-{\overline{\sigma }}_{fn2}{n}_2\left|n_2\right|-n_2\right)\end{array}$$

where $n_1$ and $n_2$ represent the shaft speeds of DC motors; $Q_1$ and $Q_2$ denote the rotor moments of inertia for DC motors; $K_1=V_m{T}_{m1}$ and $K_2=V_m{T}_{m2}$ denote control gains; $T_{m1}$ and $T_{m2}$ denote the motor transfer coefficients, and $V_m$ denotes the supply voltage; ${\overline{\sigma }}_{fm1}={\sigma }_{fm1}{T}_{n1}\rho D_1^4(1-w),{\overline{\sigma }}_{fm2}={\sigma }_{fm2}{T}_{n2}\rho D_2^4(1-w)$, ${\overline{\sigma }}_{fn1}={\sigma }_{fn1}{T}_{n1}\rho D_1^5$, and ${\overline{\sigma }}_{fn2}={\sigma }_{fn2}{T}_{n2}\rho D_2^5$; $0<w<1$ is the wake fraction number, ${\sigma }_{fm1}$, ${\sigma }_{fm2}$, ${\sigma }_{fn1}$, and ${\sigma }_{fn2}$ denote nondimensional constants, $\rho$ represents the density of water, and $D_1$ and $D_2$ are propeller diameters; ${\delta }_1$ and ${\delta }_2$ are the duty cycles of PWM signals, which can be described as

$$\begin{array}{c}\hfill {\delta }_a=\left\{\begin{array}{c}-1,\mathrm{if}{\delta }_a^{*}\le -1\hfill \\ {\delta }_a^{*},\mathrm{if}-1<{\delta }_a^{*}<1,a=1,2\hfill \\ 1,\mathrm{if}{\delta }_a^{*}\ge 1\hfill \end{array}\right.\end{array}$$

(4)

where ${\delta }_a^{*}$ are the designed command duty cycles.

From [17], the thrust forces $W_1$ and $W_2$ in (2) can be further written as

$$\begin{array}{c}\hfill W_1(u,n_1)={\overline{\sigma }}_{Wm1}u|n_1|+{\overline{\sigma }}_{Wn1}{n}_1\left|n_1\right|\\ \hfill W_2(u,n_2)={\overline{\sigma }}_{Wm2}u|n_2|+{\overline{\sigma }}_{Wn2}{n}_2\left|n_2\right|\end{array}$$

(5)

where ${\overline{\sigma }}_{Wm1}=(1-t_1){\sigma }_{Wm1}\rho D_1^3(1-w)$, ${\overline{\sigma }}_{Wm2}=(1-t_2){\sigma }_{Wm2}\rho D_2^3(1-w)$, ${\overline{\sigma }}_{Wn1}=(1-t_1){\sigma }_{Wn1}\rho D_1^4$, and ${\overline{\sigma }}_{Wn2}=(1-t_2){\sigma }_{Wn2}\rho D_2^4$; ${\sigma }_{Wma}$ and ${\sigma }_{Wna}$ represent nondimensional constants, and $t_a$ denotes thrust deduction numbers.

2.3. Propeller Faults

It is noted that twin propellers are responsible for generating actual force and moment of the differential-driven USV. During real-world operations, twin propellers may encounter faults due to physical damage, motor malfunction, control system failure, power issues, environmental factors, and improper maintenance, etc. The thrust forces $W_1^f$ and $W_2^f$, generated by two motor-powered propellers with faults, can be depicted in Figure 3a. A fundamental equivalent model is used, where the unhealthy thrust forces are repositioned to the mass center, as depicted in Figure 3b. To depict propeller faults, the unhealthy thrust forces produced by twin propellers can be mathematically formulated as

$$\begin{array}{cc}\hfill W_1^f=& {\rho }_1{W}_1+{\sigma }_1\hfill \\ \hfill W_2^f=& {\rho }_2{W}_2+{\sigma }_2\hfill \end{array}$$

(6)

where ${\rho }_1$ and ${\rho }_2$ are the actuation effectiveness of two propellers; ${\sigma }_1$ and ${\sigma }_2$ are the unanticipated forces arising from floating faults, which are time-varying.

From (2), the actual force and moment can be written as

$$\begin{array}{cc}\hfill {\tau }_u^f(W_1^f,W_2^f)=& B{\tau }_u(W_1,W_2)+{\sigma }_1+{\sigma }_2\hfill \\ \hfill {\tau }_r^f(W_1^f,W_2^f)=& B{\tau }_r(W_1,W_2)+{\displaystyle \frac{G}{2}}({\sigma }_1-{\sigma }_2)\hfill \end{array}$$

(7)

with $B=min\{{\rho }_1,{\rho }_2\}$. Because ${\sigma }_1$ and ${\sigma }_2$ are two uncertain terms, they are considered as bounded disturbances. Thus, the differential-driven USV with propeller faults is established as

$$\begin{array}{cc}\hfill \dot{x}=& ucos\left(\psi \right)-vsin\left(\psi \right)\hfill \\ \hfill \dot{y}=& usin\left(\psi \right)+vcos\left(\psi \right)\hfill \\ \hfill \dot{\psi }=& r\hfill \\ \hfill \dot{u}=& f_u(u,v,r)+{\overline{\tau }}_{du}+{\displaystyle \frac{B}{M_{11}}}{\tau }_u(W_1,W_2)\hfill \\ \hfill \dot{v}=& f_v(u,v,r)+{\displaystyle \frac{1}{M_{22}}}{\tau }_{dv}\hfill \\ \hfill \dot{r}=& f_r(u,v,r)+{\overline{\tau }}_{dr}+{\displaystyle \frac{B}{M_{33}}}{\tau }_r(W_1,W_2)\hfill \\ \hfill {\dot{n}}_1=& f_{n1}(n_1,u)+{\displaystyle \frac{K_1}{Q_1}}{\delta }_1\hfill \\ \hfill {\dot{n}}_2=& f_{n2}(n_2,u)+{\displaystyle \frac{K_2}{Q_2}}{\delta }_2\hfill \end{array}$$

(8)

where ${\overline{\tau }}_{du}={\displaystyle \frac{1}{M_{11}}}({\tau }_{du}+{\sigma }_1+{\sigma }_2)$ and ${\overline{\tau }}_{dr}={\displaystyle \frac{1}{M_{33}}}[{\tau }_{dr}+{\displaystyle \frac{G}{2}}({\sigma }_1-{\sigma }_2)]-{\displaystyle \frac{M_{32}}{M_{22}{M}_{33}}}{\tau }_{dv}.$

2.4. Control Objective

Definition 1.

Consider a nonlinear system $\dot{l}\left(t\right)=f(l,t)$, if there are the positive constants J and $T(J,l(0\left)\right)$, such that the state solution $l\left(t\right)\in {\Re }^n$ meets $\left|\right|l\left(t\right)\left|\right|\le J$ for $t>t_0+T(J,l\left(0\right))$ and $l\left(0\right)\in {\mathrm{\Omega }}_l$, where ${\mathrm{\Omega }}_l$ denotes a compact set, then $l\left(t\right)$ is said to be semiglobally ultimately uniformly bounded.

The control objective of this work is to find the backstepping-based fault-tolerant trajectory tracking control for the differential-driven USV with propeller faults (8), such that (1) all closed-loop signals of the USV system are semiglobally ultimately uniformly bounded and (2) the USV position $(x,y)$ can track the desired reference trajectory $(x_d,y_d)$, in which $(x_d,y_d)$ and its derivative $({\dot{x}}_d,{\dot{y}}_d)$ are bounded.

3. Main Results

3.1. Control Laws

Define the coordinate transformations in kinematic, kinetic, and actuated levels of the USV as

$$\begin{array}{cc}\hfill & \underset{\mathrm{Kinematic}\mathrm{level}}{\underbrace{e_1=\sqrt{e_2^2+e_3^2},e_4=ln\left({\displaystyle \frac{{\epsilon }_{\psi }+{\xi }_1}{{\epsilon }_{\psi }-{\xi }_1}}\right)}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \underset{\mathrm{Kinetic}\mathrm{level}}{\underbrace{e_5={\overline{u}}_d-u,e_6={\overline{r}}_d-r}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \underset{\mathrm{Actuated}\mathrm{level}}{\underbrace{e_7=n_1-n_{f1},e_8=n_2-n_{f2}}}\hfill \end{array}$$

(11)

where $e_1$ denotes the position error; $e_2=x_d-x$ and $e_3=y_d-y$ represent the surge and sway displacement errors; $e_4$ denotes a barrier function to manage the control coefficient in sway ${\xi }_1={\displaystyle \frac{1}{e_1}}[e_{2}sin\left(\psi \right)-e_{3}cos\left(\psi \right)]$; $e_5$ and $e_6$ are the surge and sway velocity errors; $e_7$ and $e_8$ are the DC motor shaft speed errors; ${\epsilon }_{\psi }=\sqrt{1-{{\rm Y}}^2}$ is a design constant with $0<{\rm Y}<1$; ${\overline{u}}_d$, ${\overline{r}}_d$, $n_{f1}$, and $n_{f2}$ denote the outputs of the first-order command filters with respect to the virtual speed control laws $u_d$, $r_d$ and reference motor shaft speed commands $n_1^{*}$, $n_2^{*}$, respectively, which can be defined as follows:

$$\begin{array}{cc}\hfill & {\mu }_1{\dot{\overline{u}}}_d+{\overline{u}}_d=u_d,{\overline{u}}_d\left(0\right)=u_d\left(0\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & {\mu }_2{\dot{\overline{r}}}_d+{\overline{r}}_d=r_d,{\overline{r}}_d\left(0\right)=r_d\left(0\right)\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\mu }_3{\dot{n}}_{f1}+n_{f1}=n_1^{*},n_{f1}\left(0\right)=n_1^{*}\left(0\right)\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\mu }_4{\dot{n}}_{f2}+n_{f2}=n_2^{*},n_{f2}\left(0\right)=n_2^{*}\left(0\right)\hfill \end{array}$$

(15)

where ${\mu }_g>0$$(g=1,2,3,4)$ are design parameters.

From (8)–(11), the following error dynamics are constructed as

$$\begin{array}{cc}\hfill {\dot{e}}_1=& {\xi }_{1}v-{\xi }_{2}u+{\varpi }_1\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\dot{e}}_4=& 2{\epsilon }_{\psi }{\chi }_{\psi }\left[{\xi }_{2}r+{\displaystyle \frac{1}{e_1}}\left(u{\xi }_1{\xi }_2+v{\xi }_2^2\right)+{\varpi }_3\right]\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\dot{e}}_5=& {\overline{f}}_1\left(X_1\right)-{\overline{\tau }}_{du}-{\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}+{\upsilon }_1-{\upsilon }_1+{\Delta }_1\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\dot{e}}_6=& {\overline{f}}_2\left(X_2\right)-{\overline{\tau }}_{dr}-{\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}+{\upsilon }_2-{\upsilon }_2+{\Delta }_2\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\dot{e}}_7=& {\overline{f}}_3\left(X_3\right)+{\displaystyle \frac{K_1}{Q_1}}({\delta }_1^{*}+{\Delta }_3)\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\dot{e}}_8=& {\overline{f}}_4\left(X_4\right)+{\displaystyle \frac{K_2}{Q_2}}({\delta }_2^{*}+{\Delta }_4)\hfill \end{array}$$

(21)

where ${\chi }_{\psi }={\displaystyle \frac{1}{{\epsilon }_{\psi }^2-{\xi }_1^2}}$, ${\xi }_2={\displaystyle \frac{1}{e_1}}(e_{2}cos\left(\psi \right)+e_{3}sin\left(\psi \right))$ is the control coefficient in surge; ${\varpi }_1={\displaystyle \frac{1}{e_1}}(e_2{\dot{x}}_d+e_3{\dot{y}}_d)$, ${\varpi }_2={\displaystyle \frac{1}{e_1}}({\dot{x}}_{d}sin\left(\psi \right)-{\dot{y}}_{d}cos\left(\psi \right))$, and ${\varpi }_3={\varpi }_2-{\displaystyle \frac{1}{e_1}}{\xi }_1{\varpi }_1$ are the residual terms; ${\overline{f}}_1\left(X_1\right)={\dot{\overline{u}}}_d-{\displaystyle \frac{1}{M_{11}}}[(M_{22}v+M_{22s}r)r+(X_u+X_{\left|u\right|u}\left|u\right|\left)u\right]$, ${\overline{f}}_2\left(X_2\right)={\dot{\overline{r}}}_d+{\displaystyle \frac{1}{M_{33}}}[-(M_{22}v+M_{22s}r)u+M_{11}uv+(N_v+N_{\left|v\right|v}\left|v\right|+N_{\left|r\right|v}\left|r\right|)v+(N_r+N_{\left|v\right|r}\left|v\right|+N_{\left|r\right|r}\left|r\right|\left)r\right]-{\displaystyle \frac{M_{32}}{M_{22}{M}_{33}}}[M_{11}ur-(Y_r+Y_{\left|v\right|r}\left|v\right|)r-(Y_{\left|r\right|v}\left|r\right|+Y_v+Y_{\left|v\right|v}\left|v\right|\left)v\right]$, ${\overline{f}}_3\left(X_3\right)=f_{n1}-{\dot{n}}_{f1}+{\displaystyle \frac{s^2}{2M_{11}^2{\zeta }_5}}{(n_1^{*}-n_1)}^2$, and ${\overline{f}}_4\left(X_4\right)=f_{n2}-{\dot{n}}_{f2}+{\displaystyle \frac{s^2}{2M_{33}^2{\zeta }_6}}{(n_1^{*}-n_1)}^2$ are unknown nonlinear terms; ${\Delta }_1={\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}-{\displaystyle \frac{B}{M_{11}}}{\tau }_u(W_1,W_2)$, ${\Delta }_2={\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}-{\displaystyle \frac{B}{M_{33}}}{\tau }_r(W_1,W_2)$, ${\Delta }_3={\delta }_1-{\delta }_1^{*}$, and ${\Delta }_4={\delta }_2-{\delta }_2^{*}$ are error terms; ${\tau }_u^{*}$ and ${\tau }_r^{*}$ are the virtual surge force and moment with intermediate control laws ${\upsilon }_1$ and ${\upsilon }_2$;

Establish the following compensating signal dynamics as

$$\begin{array}{cc}\hfill {\dot{\lambda }}_1=& -p_1{\lambda }_1-{\xi }_2{\overline{u}}_d+{\xi }_2{u}_d+{\xi }_2{\lambda }_3-q_1\mathrm{sign}\left({\lambda }_1\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_2=& -2{\epsilon }_{\psi }{\chi }_{\psi }{p}_2{\lambda }_2+2{\epsilon }_{\psi }{\chi }_{\psi }{\xi }_2{\overline{r}}_d-2{\epsilon }_{\psi }{\chi }_{\psi }{\xi }_2{r}_d\hfill \\ \hfill & -2{\epsilon }_{\psi }{\chi }_{\psi }{\xi }_2{\lambda }_4-2{\chi }_{\psi }{\epsilon }_{\psi }{q}_2\mathrm{sign}\left({\lambda }_2\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_3=& -p_3{\lambda }_3-{\xi }_2{\lambda }_1-q_3\mathrm{sign}\left({\lambda }_3\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_4=& -p_4{\lambda }_4+{\chi }_{\psi }{\xi }_2{\lambda }_2-q_4\mathrm{sign}\left({\lambda }_4\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_5=& -p_5{\lambda }_5-q_5\mathrm{sign}\left({\lambda }_5\right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_6=& -p_6{\lambda }_6-q_6\mathrm{sign}\left({\lambda }_6\right)\hfill \end{array}$$

(27)

where $p_c>0$ and $q_c>0$$(c=1,2,3,4,5,6)$ are design parameters; ${\lambda }_c$ is the compensating signals.

Construct the following compensated tracking errors as

$$\begin{array}{cc}\hfill & \underset{\mathrm{Kinematic}\mathrm{level}}{\underbrace{{\zeta }_1=e_1-{\lambda }_1,{\zeta }_2=e_4-{\lambda }_2}}\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \underset{\mathrm{Kinetic}\mathrm{level}}{\underbrace{{\zeta }_3=e_5-{\lambda }_3,{\zeta }_4=e_6-{\lambda }_4}}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \underset{\mathrm{Actuated}\mathrm{level}}{\underbrace{{\zeta }_5=e_7-{\lambda }_5,{\zeta }_6=e_8-{\lambda }_6.}}\hfill \end{array}$$

(30)

Invoking (16)–(27) into (28)–(30) yields

$$\begin{array}{cc}\hfill {\dot{\zeta }}_1=& {\xi }_{1}v-{\xi }_2{u}_d+{\xi }_2{\zeta }_3+{\varpi }_1+p_1{\lambda }_1+q_1\mathrm{sign}\left({\lambda }_1\right)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\dot{\zeta }}_2=& 2{\epsilon }_{\psi }{\chi }_{\psi }\left({\xi }_2{r}_d-{\xi }_2{\lambda }_4+{\displaystyle \frac{1}{e_1}}u{\xi }_1{\xi }_2+{\displaystyle \frac{1}{e_1}}v{\xi }_2^2+{\varpi }_3\right)\hfill \\ \hfill & +2p_2{\chi }_{\psi }{\epsilon }_{\psi }{\lambda }_2+2{\chi }_{\psi }{\epsilon }_{\psi }{q}_2\mathrm{sign}\left({\lambda }_2\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\dot{\zeta }}_3=& {\overline{f}}_1\left(X_1\right)-{\overline{\tau }}_{du}-{\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}+{\upsilon }_1-{\upsilon }_1+{\Delta }_1+p_3{\lambda }_3\hfill \\ \hfill & +{\xi }_2{\lambda }_1+q_3\mathrm{sign}\left({\lambda }_3\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\dot{\zeta }}_4=& {\overline{f}}_2\left(X_2\right)-{\overline{\tau }}_{dr}-{\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}+{\upsilon }_2-{\upsilon }_2+{\Delta }_2+p_4{\lambda }_4\hfill \\ \hfill & -{\chi }_{\psi }{\xi }_2{\lambda }_2+q_4\mathrm{sign}\left({\lambda }_4\right)\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\dot{\zeta }}_5=& {\overline{f}}_3\left(X_3\right)+{\displaystyle \frac{K_1}{Q_1}}({\delta }_1^{*}+{\Delta }_3)+p_5{\lambda }_5+q_5\mathrm{sign}\left({\lambda }_5\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\dot{\zeta }}_6=& {\overline{f}}_4\left(X_4\right)+{\displaystyle \frac{K_2}{Q_2}}({\delta }_2^{*}+{\Delta }_4)+p_6{\lambda }_6+q_6\mathrm{sign}\left({\lambda }_6\right).\hfill \end{array}$$

(36)

Construct the control laws in kinematic, kinetic, and actuated levels as

$$\begin{array}{cc}\hfill u_d=& {\displaystyle \frac{1}{{\xi }_2}}\left(k_1{\zeta }_1+p_1{e}_1+{\xi }_{1}v+{\varpi }_1\right)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill r_d=& -{\displaystyle \frac{k_2}{{\xi }_2}}{\zeta }_4{\chi }_{\psi }-{\displaystyle \frac{p_2}{{\xi }_2}}{e}_4-{\displaystyle \frac{1}{e_1}}{\xi }_{1}u-{\displaystyle \frac{1}{e_1}}v{\xi }_2-{\displaystyle \frac{{\varpi }_3}{{\xi }_2}}\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\upsilon }_1=& k_3{\zeta }_5+{\xi }_2{e}_1+p_3{e}_5+{\widehat{\theta }}_1^T{\phi }_1\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\upsilon }_2=& k_4{\zeta }_6-{\chi }_{\psi }{\xi }_2{e}_4+p_4{z}_6+{\widehat{\theta }}_2^T{\phi }_2\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\tau }_u^{*}=& \mathrm{sign}\left({\displaystyle \frac{B}{M_{11}}}\right){\displaystyle \frac{{\zeta }_3{\widehat{s}}_1^2{\upsilon }_1^2}{\sqrt{{\zeta }_3^2{\widehat{s}}_1^2{\upsilon }_1^2+o_1^2}}}\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\tau }_r^{*}=& \mathrm{sign}\left({\displaystyle \frac{B}{M_{33}}}\right){\displaystyle \frac{{\zeta }_4{\widehat{s}}_2^2{\upsilon }_2^2}{\sqrt{{\zeta }_4^2{\widehat{s}}_2^2{\upsilon }_2^2+o_2^2}}}\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\delta }_1^{*}=& -{\displaystyle \frac{Q_1}{K_1}}\left(k_5{\zeta }_5+p_5{e}_7+{\widehat{\theta }}_3^T{\phi }_3+K_{z1}{\mathrm{\Lambda }}_1\right)\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\delta }_2^{*}=& -{\displaystyle \frac{Q_2}{K_2}}\left(k_6{\zeta }_6+p_6{e}_8+{\widehat{\theta }}_4^T{\phi }_4+K_{z2}{\mathrm{\Lambda }}_2\right)\hfill \end{array}$$

(44)

where $k_c>0$, $o_a>0$, and $K_{za}>0$ are design parameters; ${\phi }_g={[{\phi }_{g,1},{\phi }_{g,2},\dots ,{\phi }_{g,h}]}^T$$(g=1,2,3,4)$ denote fuzzy basis function vectors, ${\widehat{\theta }}_g={[{\widehat{\theta }}_{g,1},{\widehat{\theta }}_{g,2},\dots ,{\widehat{\theta }}_{g,h}]}^T$ are weight estimation vectors, and h is the number of fuzzy rules; ${\widehat{s}}_1$ and ${\widehat{s}}_2$ are the estimates of $s_1={\displaystyle \frac{1}{s_1^{*}}}$ and $s_2={\displaystyle \frac{1}{s_2^{*}}}$, where $s_1^{*}=inf\left|{\displaystyle \frac{B}{M_{11}}}\right|$ and $s_2^{*}=inf\left|{\displaystyle \frac{B}{M_{33}}}\right|$.

Design the adaptive updating laws as

$$\begin{array}{cc}\hfill {\dot{\widehat{\theta }}}_{1,i}=& {\varphi }_{1,i}{\phi }_{1,i}{\zeta }_3-{\varrho }_{1,i}{\widehat{\theta }}_{1,i}\hfill \\ \hfill {\dot{\widehat{\theta }}}_{2,i}=& {\varphi }_{2,i}{\phi }_{2,i}{\zeta }_4-{\varrho }_{2,i}{\widehat{\theta }}_{2,i}\hfill \\ \hfill {\dot{\widehat{\theta }}}_{3,i}=& {\varphi }_{3,i}{\phi }_{3,i}{\zeta }_5-{\varrho }_{3,i}{\widehat{\theta }}_{3,i}\hfill \\ \hfill {\dot{\widehat{\theta }}}_{4,i}=& {\varphi }_{4,i}{\phi }_{4,i}{\zeta }_6-{\varrho }_{4,i}{\widehat{\theta }}_{4,i}\hfill \\ \hfill {\dot{\widehat{s}}}_1=& {\kappa }_1{\zeta }_3{\upsilon }_1-d_1{\widehat{s}}_1\hfill \\ \hfill {\dot{\widehat{s}}}_2=& {\kappa }_2{\zeta }_4{\upsilon }_2-d_2{\widehat{s}}_2\hfill \end{array}$$

(45)

where, for $g=1,2,3,4, i=1,2,\dots ,h, a=1,2$, ${\varphi }_{g,i}$, ${\varrho }_{g,i}$, ${\kappa }_a$, and $d_a$ are positive constants to be designed.

In order to solve the input saturation problem, we design the auxiliary systems as

$$\begin{array}{c}\hfill {\dot{\mathrm{\Lambda }}}_1=\left\{\begin{array}{c}-{ϵ}_1{\mathrm{\Lambda }}_1-{\displaystyle \frac{1}{{\mathrm{\Lambda }}_1}}\left({\displaystyle \frac{K_1}{Q_1}}\left|{\zeta }_5{\Delta }_3\right|+{\Delta }_3^2/2\right)+{\Delta }_3,\left|{\mathrm{\Lambda }}_1\right|\ge \mathrm{\Xi }\hfill \\ 0,|{\mathrm{\Lambda }}_1|<\mathrm{\Xi }\hfill \end{array}\right.\\ \hfill {\dot{\mathrm{\Lambda }}}_2=\left\{\begin{array}{c}-{ϵ}_2{\mathrm{\Lambda }}_2-{\displaystyle \frac{1}{{\mathrm{\Lambda }}_2}}\left({\displaystyle \frac{K_2}{Q_2}}\left|{\zeta }_6{\Delta }_4\right|+{\Delta }_4^2/2\right)+{\Delta }_4,\left|{\mathrm{\Lambda }}_1\right|\ge \mathrm{\Xi }\hfill \\ 0,|{\mathrm{\Lambda }}_2|<\mathrm{\Xi }\hfill \end{array}\right.\end{array}$$

(46)

where $\mathrm{\Xi }$ and ${ϵ}_a$ denote positive constants to be determined.

The desired motor shaft speeds are obtained as

$$\begin{array}{cc}\hfill n_1^{*}=& {\displaystyle \frac{-{\overline{\sigma }}_{Wm1}u+\sqrt{{\overline{\sigma }}_{Wm1}^2{u}^2+4\left(\frac{{\tau }_u^{*}}{2}+\frac{{\tau }_r^{*}}{G}\right){\overline{\sigma }}_{Wn1}}}{2{\overline{\sigma }}_{Wn1}}}\hfill \\ \hfill n_2^{*}=& {\displaystyle \frac{-{\overline{\sigma }}_{Wm2}u+\sqrt{{\overline{\sigma }}_{Wm2}^2{u}^2+4\left(\frac{{\tau }_u^{*}}{2}-\frac{{\tau }_r^{*}}{G}\right){\overline{\sigma }}_{Wn2}}}{2{\overline{\sigma }}_{Wn2}}}.\hfill \end{array}$$

(47)

3.2. Stability Analysis

The control block diagram is shown in Figure 4. The following theorem is presented to prove the system stability.

Theorem 1.

Consider the differential-driven USV with propeller faults (8), control laws (37)–(44), adaptive laws (45), and the auxiliary systems (46), with the appropriate design parameters. Then, the position tracking error $e_1$ converges within the compact set ${\mathrm{\Omega }}_{e_1}=\left\{\right|e_1\left(t\right)|\le {\iota }_{e_1}>\sqrt{2\mathrm{\Theta }/\mathrm{\Psi }}\}$, and all closed-loop signals are ensured to be semiglobally ultimately uniformly bounded.

Proof. 

(1) Kinematic Level: Design the first Lyapunov candidate function as

$$\begin{array}{c}\hfill V_1={\displaystyle \frac{1}{2}}{\zeta }_1^2+{\displaystyle \frac{1}{4{\epsilon }_{\psi }}}{\zeta }_2^2.\end{array}$$

(48)

Using (31), (32), (37), (38), and the Young’s inequality, we have

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & -2\left(k_1+p_1-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_1^2\right)-4{\epsilon }_{\psi }\left(k_2{\chi }_{\psi }^2+p_2{\chi }_{\psi }-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{4{\epsilon }_{\psi }}}{\zeta }_2^2\right)\hfill \\ \hfill & +{\xi }_2{\zeta }_1{\zeta }_3-{\chi }_{\psi }{\xi }_2{\zeta }_2{\zeta }_4+{\displaystyle \frac{1}{2}}{q}_1^2+{\displaystyle \frac{1}{2}}{\chi }_{\psi }^2{q}_2^2.\hfill \end{array}$$

(49)

(2) Kinetic Level: Design the second Lyapunov candidate function as

$$\begin{array}{cc}\hfill V_2=& {\displaystyle \frac{1}{2}}\left({\zeta }_3^2+{\zeta }_4^2+\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{1}{{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}^2+\sum _{a=1}^2{\displaystyle \frac{s_a^{*}}{{\kappa }_a}}{\tilde{s}}_a^2\right)\hfill \end{array}$$

(50)

where ${\tilde{\theta }}_{a,i}={\theta }_{a,i}-{\widehat{\theta }}_{a,i}$ and ${\tilde{s}}_a=s_a-{\widehat{s}}_a$; ${\varphi }_{a,i}>0$ and ${\kappa }_a>0$ are design parameters.

Based on (33) and (34), the time derivative of $V_2$ is

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\zeta }_3\left({\overline{f}}_1\left(X_1\right)-{\overline{\tau }}_{du}-{\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}+{\upsilon }_1-{\upsilon }_1+{\Delta }_1+p_3{\lambda }_3+{\xi }_2{\lambda }_1+q_3\mathrm{sign}\left({\lambda }_3\right)\right)\hfill \\ & +{\zeta }_4\left({\overline{f}}_2\left(X_2\right)-{\overline{\tau }}_{dr}-{\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}+{\upsilon }_2-{\upsilon }_2+{\Delta }_2+p_4{\lambda }_4-{\chi }_{\psi }{\xi }_2{\lambda }_2+q_4\mathrm{sign}\left({\lambda }_4\right)\right)\hfill \\ \hfill & -\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{1}{{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}{\dot{\widehat{\theta }}}_{a,i}-\sum _{a=1}^2{\displaystyle \frac{s_a^{*}}{{\kappa }_a}}{\tilde{s}}_a{\dot{\widehat{s}}}_a.\hfill \end{array}$$

(51)

Since ${\overline{f}}_a\left(X_a\right)(a=1,2)$ are unknown, they can be approximated via fuzzy logic systems in Lemma 1 of [24,25] as

$$\begin{array}{cc}\hfill {\overline{f}}_a\left(X_a\right)=& {\theta }_a^T{\phi }_a\left(X_a\right)+{\pi }_a\left(X_a\right),\left|\right|{\pi }_a\left(X_a\right)\left|\right|\le {\pi }_a^{*}\hfill \end{array}$$

(52)

where ${\theta }_a={[{\theta }_{a,1},{\theta }_{a,2},\dots ,{\theta }_{a,h}]}^T$ are the ideal weight vectors, and ${\pi }_a\left(X_a\right)$ are the approximation errors with ${\delta }_a^{*}>0$.

Employing the Young’s inequality in [26,27,28], it yields that

$$\begin{array}{cc}\hfill {\zeta }_b{q}_b\mathrm{sign}\left({\lambda }_b\right)\le & {\displaystyle \frac{1}{2}}{\zeta }_b^2+{\displaystyle \frac{1}{2}}{q}_b^2\hfill \\ \hfill -{\zeta }_3{\overline{\tau }}_{du}\le & {\displaystyle \frac{1}{2}}{\zeta }_3^2+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}\hfill \\ \hfill -{\zeta }_4{\overline{\tau }}_{dr}\le & {\displaystyle \frac{1}{2}}{\zeta }_4^2+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}\hfill \end{array}$$

(53)

where ${\overline{\tau }}_{du}^{*}$ and ${\overline{\tau }}_{dr}^{*}$ are the positive constants because of the boundedness of disturbances.

Substituting (52) and (53) into (51) gives

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & {\zeta }_3\left({\tilde{\theta }}_1^T{\phi }_1+{\pi }_1+{\Delta }_1+{\zeta }_3-{\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}-k_3{\zeta }_3-{\xi }_2{\zeta }_1-p_3{\zeta }_3+{\upsilon }_1\right)\hfill \\ & +{\zeta }_4\left({\tilde{\theta }}_2^T{\phi }_2+{\pi }_2+{\Delta }_2+{\zeta }_4-{\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}-k_4{\zeta }_4+{\chi }_{\psi }{\xi }_2{\zeta }_2-p_4{\zeta }_4+{\upsilon }_2\right)\hfill \\ \hfill & +\sum _{b=3}^4{\displaystyle \frac{1}{2}}{q}_b^2+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}-\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{1}{{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}{\dot{\widehat{\theta }}}_{a,i}-\sum _{a=1}^2{\displaystyle \frac{s_a^{*}}{{\kappa }_a}}{\tilde{s}}_a{\dot{\widehat{s}}}_a+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}.\hfill \end{array}$$

(54)

On the basis of Lemma 5 in [29] and the Young’s inequality, we obtain

$$\begin{array}{cc}\hfill -{\displaystyle \frac{B}{M_{11}}}{\tau }_1^{*}{\zeta }_3\le & s_1^{*}{o}_1-s_1^{*}{\zeta }_3{\widehat{s}}_1{\upsilon }_1\hfill \\ \hfill -{\displaystyle \frac{B}{M_{33}}}{\tau }_2^{*}{\zeta }_4\le & s_2^{*}{o}_2-s_2^{*}{\zeta }_4{\widehat{s}}_2{\upsilon }_2\hfill \\ \hfill {\zeta }_3{\pi }_1\le & {\displaystyle \frac{1}{2}}{\zeta }_3^2+{\displaystyle \frac{1}{2}}{\pi }_1^{*2}\hfill \\ \hfill {\zeta }_4{\pi }_2\le & {\displaystyle \frac{1}{2}}{\zeta }_4^2+{\displaystyle \frac{1}{2}}{\pi }_2^{*2}.\hfill \end{array}$$

(55)

From Assumption 2 in [16], it follows that

$$\begin{array}{c}\hfill {\zeta }_3{\Delta }_1\le {\displaystyle \frac{s}{M_{11}}}{\zeta }_3\left|n_1^{*}-n_1\right|\le {\displaystyle \frac{1}{2}}{\zeta }_3^2+{\displaystyle \frac{s^2}{2M_{11}^2}}{(n_1^{*}-n_1)}^2\\ \hfill {\zeta }_4{\Delta }_2\le {\displaystyle \frac{s}{M_{33}}}{\zeta }_4\left|n_2^{*}-n_2\right|\le {\displaystyle \frac{1}{2}}{\zeta }_4^2+{\displaystyle \frac{s^2}{2M_{33}^2}}{(n_2^{*}-n_2)}^2\end{array}$$

(56)

where s represents the Lipschitz constant.

Substituting (55) and (56) into (54) results in

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -\left(k_3+p_3-2\right){\zeta }_3^2-{\xi }_2{\zeta }_1{\zeta }_3+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}-(k_4+p_4-2){\zeta }_4^2+{\chi }_{\psi }{\xi }_2{\zeta }_2{\zeta }_4\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+\sum _{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{{\kappa }_a}}{\tilde{s}}_a{\widehat{s}}_a+\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{a,i}}{{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}{\widehat{\theta }}_{a,i}+\sum _{a=1}^2{s}_a^{*}{o}_a+\sum _{a=1}^2{\displaystyle \frac{1}{2}}{\pi }_a^{*2}\hfill \\ \hfill & +{\displaystyle \frac{s^2}{2M_{11}^2}}{(n_1^{*}-n_1)}^2+{\displaystyle \frac{s^2}{2M_{33}^2}}{(n_2^{*}-n_2)}^2.\hfill \end{array}$$

(57)

Based on ${\tilde{\theta }}_{a,i}={\theta }_{a,i}-{\widehat{\theta }}_{a,i}$ and ${\tilde{s}}_a=s_a-{\widehat{s}}_a$, one has

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\varrho }_{a,i}}{{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}{\widehat{\theta }}_{a,i}\le & -{\displaystyle \frac{{\varrho }_{a,i}}{2{\varphi }_{a,i}}}{\tilde{\theta }}_{a,i}^2+{\displaystyle \frac{{\varrho }_{a,i}}{2{\varphi }_{a,i}}}{\theta }_{a,i}^2\hfill \\ \hfill {\displaystyle \frac{s_a^{*}{d}_a}{{\kappa }_a}}{\tilde{s}}_a{\widehat{s}}_a\le & -{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{\tilde{s}}_a^2+{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2.\hfill \end{array}$$

(58)

Substituting (58) into (57) obtains that

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -2\left(k_3+p_3-2\right)\left({\displaystyle \frac{1}{2}}{\zeta }_3^2\right)-{\xi }_2{\zeta }_1{\zeta }_3+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}-(k_4+p_4-2)\left({\displaystyle \frac{1}{2}}{\zeta }_4^2\right)+{\chi }_{\psi }{\xi }_2{\zeta }_2{\zeta }_4\hfill \\ & -\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{a,i}}{2}}\left({\displaystyle \frac{{\tilde{\theta }}_{a,i}^2}{2{\varphi }_{a,i}}}\right)-\sum _{a=1}^2{\displaystyle \frac{d_a}{2}}\left({\displaystyle \frac{s_a^{*}}{2{\kappa }_a}}{\tilde{s}}_a^2\right)+\sum _{b=3}^4{\displaystyle \frac{1}{2}}{q}_b^2+\sum _{a=1}^2{s}_a^{*}{o}_a+\sum _{a=1}^2{\displaystyle \frac{1}{2}}{\pi }_a^{*2}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+\sum _{a=1}^2\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{a,i}}{2{\varphi }_{a,i}}}{\theta }_{a,i}^2+\sum _{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2+{\displaystyle \frac{s^2}{2M_{11}^2}}{(n_1^{*}-n_1)}^2\hfill \\ \hfill & +{\displaystyle \frac{s^2}{2M_{33}^2}}{(n_2^{*}-n_2)}^2.\hfill \end{array}$$

(59)

Remark 1.

In (51), the unknown fault parameters $B=min\{{\rho }_1,{\rho }_2\}$ will hinder the virtual surge force ${\tau }_u^{*}$ and yaw moment ${\tau }_r^{*}$ design. Therefore, intermediate control terms ${\upsilon }_1-{\upsilon }_1$ and ${\upsilon }_2-{\upsilon }_2$ are introduced. Based on (39) and (40), the key terms $-k_3{\zeta }_3^2$ and $-k_4{\zeta }_4^2$ are obtained in (59) to ensure the stability of the closed-loop system. By designing ${\tau }_u^{*}$ (41) and ${\tau }_r^{*}$ (42), and using Lemma 5 in [29], the residual terms ${\zeta }_3{v}_1$ and ${\zeta }_4{v}_2$ can be counteracted via inequalities (55).

Remark 2.

Since the propeller servo loop design is considered in this paper, the error terms ${\Delta }_1={\displaystyle \frac{B}{M_{11}}}{\tau }_u^{*}-{\displaystyle \frac{B}{M_{11}}}{\tau }_u(W_1,W_2)$ and ${\Delta }_2={\displaystyle \frac{B}{M_{33}}}{\tau }_r^{*}-{\displaystyle \frac{B}{M_{33}}}{\tau }_r(W_1,W_2)$ are produced, which must be eliminated to guarantee stability analysis. Based on the relationship between the thrust and motor speed through the linear growth condition in [16], the inequalities in (57) are obtained to produce ${\displaystyle \frac{1}{2}}{\zeta }_3^2$ and ${\displaystyle \frac{1}{2}}{\zeta }_4^2$, which can be addressed by adjusting the control gain parameters $k_3$ and $k_4$ in (59). Note that the terms ${\displaystyle \frac{s^2}{2M_{11}^2}}{(n_1^{*}-n_1)}^2$ and ${\displaystyle \frac{s^2}{2M_{33}^2}}{(n_2^{*}-n_2)}^2$ will be addressed in the actuated level since the $n_1$ and $n_2$ are the states of the propeller servo loop.

Remark 3.

In (59), the terms $-{\xi }_2{\zeta }_1{\zeta }_3$ and ${\chi }_{\psi }{\xi }_2{\zeta }_2{\zeta }_4$ are to counteract the items with opposite symbols to them in (49) to guarantee stability analysis. (54) and (55) provide the design criteria for adaptive laws and fault compensation laws in (45), which can obtain the negative definite terms $-{\sum }_{a=1}^2{\sum }_{i=1}^h{\displaystyle \frac{{\varrho }_{a,i}}{2}}\left({\displaystyle \frac{{\tilde{\theta }}_{a,i}^2}{2{\varphi }_{a,i}}}\right)$ and $-{\sum }_{a=1}^2{\displaystyle \frac{d_a}{2}}\left({\displaystyle \frac{s_a^{*}}{2{\kappa }_a}}{\tilde{s}}_a^2\right)$ to ensure the stability of the closed-loop system.

(3) Actuated Level: Design the third Lyapunov candidate function as

$$\begin{array}{cc}\hfill V_3=& {\displaystyle \frac{1}{2}}\left({\zeta }_5^2+{\zeta }_6^2+\sum _{b=3}^4\sum _{i=1}^h{\displaystyle \frac{1}{{\varphi }_{b,i}}}{\tilde{\theta }}_{b,i}^2+\sum _{a=1}^2{\mathrm{\Lambda }}_a^2\right)\hfill \end{array}$$

(60)

where ${\tilde{\theta }}_{b,i}={\theta }_{b,i}-{\widehat{\theta }}_{b,i}$ and ${\varphi }_{b,i}>0$$(b=3,4)$ are design parameters.

Akin to the process in (51)–(59), the derivative of $V_3$ can be obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_3\le & -2\left(k_5+p_5-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_5^2\right)+\sum _{c=5}^6{\displaystyle \frac{1}{2}}{q}_c^2-2\left(k_6+p_6-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_6^2\right)+\sum _{b=3}^4{\displaystyle \frac{1}{2}}{\pi }_b^{*2}\hfill \\ \hfill & -\sum _{b=3}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{b,i}}{2}}\left({\displaystyle \frac{{\tilde{\theta }}_{b,i}^2}{2{\varphi }_{b,i}}}\right)+\sum _{a=1}^2{\mathrm{\Lambda }}_a{\dot{\mathrm{\Lambda }}}_a+\sum _{b=3}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{b,i}}{2{\varphi }_{b,i}}}{\theta }_{b,i}^2+\sum _{a=1}^2{\displaystyle \frac{1}{2}}{K}_{za}^2{\mathrm{\Lambda }}_a^2\hfill \\ \hfill & +{\displaystyle \frac{K_1}{Q_1}}{\zeta }_5{\Delta }_3+{\displaystyle \frac{K_2}{Q_2}}{\zeta }_6{\Delta }_4-{\displaystyle \frac{s^2}{2M_{11}^2}}{(n_1^{*}-n_1)}^2-{\displaystyle \frac{s^2}{2M_{33}^2}}{(n_2^{*}-n_2)}^2.\hfill \end{array}$$

(61)

(4) Overall Closed-Loop System: Design the total Lyapunov function as

$$\begin{array}{c}\hfill V=V_1+V_2+V_3.\end{array}$$

(62)

Then, we consider the subsequent two scenarios.

(1) If $|{\mathrm{\Lambda }}_a|\ge \mathrm{\Xi }$, adopting (46), and the Young’s inequality, one gets

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_1{\dot{\mathrm{\Lambda }}}_1\le -{ϵ}_1{\mathrm{\Lambda }}_1^2-{\displaystyle \frac{K_1}{Q_1}}\left|{\zeta }_5{\Delta }_3\right|+{\displaystyle \frac{1}{2}}{\mathrm{\Lambda }}_1^2\\ \hfill {\mathrm{\Lambda }}_2{\dot{\mathrm{\Lambda }}}_2\le -{ϵ}_2{\mathrm{\Lambda }}_2^2-{\displaystyle \frac{K_2}{Q_2}}\left|{\zeta }_6{\Delta }_5\right|+{\displaystyle \frac{1}{2}}{\mathrm{\Lambda }}_2^2.\end{array}$$

(63)

Based on (49), (59), (61), and (63), we have

$$\begin{array}{cc}\hfill \dot{V}\le & -2\left(k_1+p_1-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_1^2\right)-4{\epsilon }_{\psi }\left(k_2{\chi }_{\psi }^2+p_2{\chi }_{\psi }-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{4{\epsilon }_{\psi }}}{\zeta }_2^2\right)\hfill \\ \hfill & -2\left(k_3+p_3-2\right)\left({\displaystyle \frac{1}{2}}{\zeta }_3^2\right)-(k_4+p_4-2)\left({\displaystyle \frac{1}{2}}{\zeta }_4^2\right)-2\left(k_5+p_5-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_5^2\right)\hfill \\ \hfill & -2\left(k_6+p_6-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_6^2\right)+\sum _{c=1}^6{\displaystyle \frac{1}{2}}{q}_c^2-\sum _{g=1}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2}}\left({\displaystyle \frac{{\tilde{\theta }}_{g,i}^2}{2{\varphi }_{g,i}}}\right)+\sum _{g=1}^4{\displaystyle \frac{1}{2}}{\pi }_g^{*2}\hfill \\ \hfill & -\sum _{a=1}^2{\displaystyle \frac{d_a}{2}}\left({\displaystyle \frac{s_a^{*}}{2{\kappa }_a}}{\tilde{s}}_a^2\right)-\left(2{ϵ}_1-K_{z1}^2-1\right)\left({\displaystyle \frac{1}{2}}{\mathrm{\Lambda }}_1^2\right)-\left(2{ϵ}_2-K_{z2}^2-1\right)\left({\displaystyle \frac{1}{2}}{\mathrm{\Lambda }}_2^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+\sum _{g=1}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2{\varphi }_{g,i}}}{\theta }_{g,i}^2+\sum _{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2+\sum _{a=1}^2{s}_a^{*}{o}_a\hfill \\ \hfill \le & -{\mathrm{\Psi }}_{1}V+{\mathrm{\Theta }}_1\hfill \end{array}$$

(64)

where ${\mathrm{\Psi }}_1=min\{2(k_1+p_1-{\displaystyle \frac{1}{2}}),4{\epsilon }_{\psi }(k_2{\chi }_{\psi }^2+p_2{\varpi }_{\psi }-{\displaystyle \frac{1}{2}}),2(k_3+p_3-2),$$2(k_4+p_4-2),$$2(k_5+p_5-{\displaystyle \frac{3}{2}}),2(k_6,p_6-{\displaystyle \frac{3}{2}}),{\displaystyle \frac{{\varrho }_{1,i}}{2}},{\displaystyle \frac{{\varrho }_{2,i}}{2}},{\displaystyle \frac{{\varrho }_{3,i}}{2}},{\displaystyle \frac{{\varrho }_{4,i}}{2}},{\displaystyle \frac{d_1}{2}},{\displaystyle \frac{d_2}{2}},2{ϵ}_1-K_{z1}^2-1,2{ϵ}_2-K_{z2}^2-1\}$, and ${\mathrm{\Theta }}_1={\sum }_{c=1}^6{\displaystyle \frac{1}{2}}{q}_c^2+{\sum }_{g=1}^4{\displaystyle \frac{1}{2}}{\pi }_g^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+{\sum }_{g=1}^4{\sum }_{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2{\varphi }_{g,i}}}{\theta }_{g,i}^2+{\sum }_{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2+{\sum }_{a=1}^2{s}_a^{*}{o}_a$.

(2) When $|{\mathrm{\Lambda }}_a|<\mathrm{\Xi }$, utilizing the Young’s inequality, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{K}_{za}^2{\mathrm{\Lambda }}_a^2\le -{\displaystyle \frac{1}{2}}{K}_{za}^2{\mathrm{\Lambda }}_a^2+K_{za}^2{\mathrm{\Xi }}^2\\ \hfill {\displaystyle \frac{K_a}{Q_a}}{\zeta }_{a+4}{\Delta }_{a+2}\le {\displaystyle \frac{1}{2}}{\zeta }_{a+4}^2+{\displaystyle \frac{K_a^2}{2Q_a^2}}{\Delta }_{a+2}^2.\end{array}$$

(65)

Then, substituting (49), (59), (61), and (65) yields

$$\begin{array}{cc}\hfill \dot{V}\le & -2\left(k_1+p_1-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_1^2\right)-4{\epsilon }_{\psi }\left(k_2{\chi }_{\psi }^2+p_2{\chi }_{\psi }-{\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{1}{4{\epsilon }_{\psi }}}{\zeta }_2^2\right)\hfill \\ \hfill & -2\left(k_3+p_3-2\right)\left({\displaystyle \frac{1}{2}}{\zeta }_3^2\right)-(k_4+p_4-2)\left({\displaystyle \frac{1}{2}}{\zeta }_4^2\right)-2\left(k_5+p_5-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_5^2\right)\hfill \\ \hfill & -2\left(k_6+p_6-{\displaystyle \frac{3}{2}}\right)\left({\displaystyle \frac{1}{2}}{\zeta }_6^2\right)+\sum _{c=1}^6{\displaystyle \frac{1}{2}}{q}_c^2-\sum _{g=1}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2}}\left({\displaystyle \frac{{\tilde{\theta }}_{g,i}^2}{2{\varphi }_{g,i}}}\right)\hfill \\ \hfill & -\sum _{a=1}^2{\displaystyle \frac{d_a}{2}}\left({\displaystyle \frac{s_a^{*}}{2{\kappa }_a}}{\tilde{s}}_a^2\right)+\sum _{a=1}^2{K}_{za}^2{\mathrm{\Xi }}^2+\sum _{a=1}^2{\displaystyle \frac{K_a^2}{2Q_a^2}}{\Delta }_{a+2}^2-\sum _{a=1}^2{K}_{za}^2\left({\displaystyle \frac{1}{2}}{\mathrm{\Lambda }}_a^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+\sum _{g=1}^4\sum _{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2{\varphi }_{g,i}}}{\theta }_{g,i}^2+\sum _{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2+\sum _{a=1}^2{s}_a^{*}{o}_a+\sum _{g=1}^4{\displaystyle \frac{1}{2}}{\pi }_g^{*2}\hfill \\ \hfill \le & -{\mathrm{\Psi }}_{2}V+{\mathrm{\Theta }}_2\hfill \end{array}$$

(66)

where ${\mathrm{\Psi }}_2=min\{2(k_1+p_1-{\displaystyle \frac{1}{2}}),4{\epsilon }_{\psi }(k_2{\chi }_{\psi }^2+p_2{\varpi }_{\psi }-{\displaystyle \frac{1}{2}}),2(k_3+p_3-2),$$2(k_4+p_4-2),$$2(k_5+p_5-{\displaystyle \frac{3}{2}}),2(k_6,p_6-{\displaystyle \frac{3}{2}}),{\displaystyle \frac{{\varrho }_{1,i}}{2}},{\displaystyle \frac{{\varrho }_{2,i}}{2}},{\displaystyle \frac{{\varrho }_{3,i}}{2}},{\displaystyle \frac{{\varrho }_{4,i}}{2}},{\displaystyle \frac{d_1}{2}},{\displaystyle \frac{d_2}{2}},K_{z1}^2,K_{z2}^2\}$ and ${\mathrm{\Theta }}_2={\sum }_{c=1}^6{\displaystyle \frac{1}{2}}{q}_c^2+{\sum }_{g=1}^4{\displaystyle \frac{1}{2}}{\pi }_g^{*2}+{\sum }_{c=1}^6{\displaystyle \frac{1}{2}}{q}_c^2+{\sum }_{a=1}^2{K}_{za}^2{\mathrm{\Xi }}^2+{\sum }_{a=1}^2{\displaystyle \frac{G_a^2}{2K_a^2}}{\Delta }_{a+2}^2+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{du}^{*2}+{\displaystyle \frac{1}{2}}{\overline{\tau }}_{dr}^{*2}+{\sum }_{g=1}^4{\sum }_{i=1}^h{\displaystyle \frac{{\varrho }_{g,i}}{2{\varphi }_{g,i}}}{\theta }_{g,i}^2+{\sum }_{a=1}^2{\displaystyle \frac{s_a^{*}{d}_a}{2{\kappa }_a}}{s}_a^2+{\sum }_{a=1}^2{s}_a^{*}{o}_a$.

Summarizing (64) and (66) results in

$$\begin{array}{c}\hfill \dot{V}\le -\mathrm{\Psi }V+\mathrm{\Theta }\end{array}$$

(67)

where $\mathrm{\Psi }=min\{{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2\}$ and $\mathrm{\Theta }=max\{{\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2\}$. And the design parameters meet $k_1>{\displaystyle \frac{1}{2}}-p_1$, $k_2>{\displaystyle \frac{1}{{\chi }_{\psi }^2}}({\displaystyle \frac{1}{2}}-p_2{\varpi }_{\psi })$, $k_3>2-p_3$, $k_4>2-p_4$, $k_5>{\displaystyle \frac{3}{2}}-p_5$, $k_6>{\displaystyle \frac{3}{2}}-p_6$, ${ϵ}_1>{\displaystyle \frac{1}{2}}{K}_{z1}^2+{\displaystyle \frac{1}{2}}$, and ${ϵ}_2>{\displaystyle \frac{1}{2}}{K}_{z2}^2+{\displaystyle \frac{1}{2}}$.

By solving (67), one has

$$\begin{array}{c}\hfill 0\le V\left(t\right)\le {\displaystyle \frac{\mathrm{\Theta }}{\mathrm{\Psi }}}+\left(V\left(0\right)-{\displaystyle \frac{\mathrm{\Theta }}{\mathrm{\Psi }}}\right)e^{-\mathrm{\Psi }t}.\end{array}$$

(68)

Based on (62), we have $e_1$, $e_2$ $e_3$, $e_4$, $e_5$, $e_6$, ${\widehat{\theta }}_1$, ${\widehat{\theta }}_2$, ${\widehat{\theta }}_3$, ${\widehat{\theta }}_4$, ${\widehat{s}}_1$, ${\widehat{s}}_2$, ${\mathrm{\Lambda }}_1$, and ${\mathrm{\Lambda }}_2$ are semiglobally ultimately uniformly bounded. Furthermore, it means that all the closed-loop signals are guaranteed to be semiglobally ultimately uniformly bounded.

From (62) and (68), it follows that

$$\begin{array}{c}\hfill |e_1\left(t\right)|\le \sqrt{2V\left(t\right)}=\sqrt{{\displaystyle \frac{2\mathrm{\Theta }}{\mathrm{\Psi }}}+\left(2V\left(0\right)-{\displaystyle \frac{2\mathrm{\Theta }}{\mathrm{\Psi }}}\right){exp}^{(-\mathrm{\Psi }t)}}.\end{array}$$

(69)

Thus, for any ${\iota }_{e_1}>\sqrt{2\mathrm{\Theta }/\mathrm{\Psi }}$, there is a time $t_{e_1}$ satisfying $\forall t>t_{e_1},\left|e_1\right|\le {\iota }_{e_1}$. Therefore, Theorem 1 is demonstrated. □

4. Validation

Simulation results are exhibited to prove the validity of the developed method. A desired trajectory is chosen as

$$\begin{array}{cc}\hfill x_d=& 10sin\left(0.1t\right)\hfill \\ \hfill y_d=& -10cos\left(0.1t\right)+10.\hfill \end{array}$$

(70)

The hydrodynamic analysis-derived nominal parameters of the USV are listed as [8] $M_{11}=25.8$ kg, $M_{22}=33.8$ kg, $M_{33}=2.76$ kg, $M_{23}=M_{32}=6.2$ kg, $X_u+X_{\left|u\right|u}\left|u\right|=-12\mathrm{kg}/\mathrm{s}-2.5\mathrm{kg}/\mathrm{m}\left|u\right|$, $Y_{\left|r\right|v}\left|r\right|+Y_v+Y_{\left|v\right|v}\left|v\right|=-8\mathrm{kg}/\mathrm{m}\left|r\right|-17\mathrm{kg}/\mathrm{s}-4.5\mathrm{kg}/\mathrm{m}\left|v\right|$, $N_r+N_{\left|v\right|r}\left|v\right|+N_{\left|r\right|r}\left|r\right|=-0.5\mathrm{kg}/\mathrm{s}-0.4\mathrm{kg}/\mathrm{m}\left|v\right|-0.1\mathrm{kg}/\mathrm{m}\left|r\right|$, $Q_a=1{\mathrm{kgm}}^2$, $V_m=24$ V, $T_{ma}=300\mathrm{rad}/(\mathrm{s}·\mathrm{V})$, ${\sigma }_{Wma}=-0.001$, ${\sigma }_{Wna}=0.00006$, ${\sigma }_{fma}=-0.1$, ${\sigma }_{fna}=0.05$, $\rho =1000\mathrm{kg}/{\mathrm{m}}^3$, $\gamma =1$, $D_a=0.1$, $t_a=0.2$, and $ϵ=0.2$.

The initial values of the USV system are chosen as $x\left(0\right)=0$, $y\left(0\right)=3$, $\psi \left(0\right)=-1.1$ in case 1, $x\left(0\right)=0.1$, $y\left(0\right)=2.5$, $\psi \left(0\right)=-1.1$ in case 2, and $x\left(0\right)=0$, $y\left(0\right)=4$, $\psi \left(0\right)=-1.1$ in case 3. The other initial states in three cases are $u\left(0\right)=0$, $v\left(0\right)=0$, $r\left(0\right)=0$, $n_1\left(0\right)=0$ and $n_2\left(0\right)=0$.

The fuzzy sets of fuzzy logic systems ${\theta }_g^T{\phi }_g\left(X_g\right)(g=1,2,3,4)$ are chosen over the interval $[-{\displaystyle \frac{11}{2}},{\displaystyle \frac{11}{2}}]$. For $h=1,\dots ,12$, define $X_1={[u,v,r]}^T$, $X_2={[u,v,r]}^T$, $X_3={[u,n_1]}^T$, $X_4={[u,n_2]}^T$ and

$$\begin{array}{cc}\hfill X_1^0=& {\underset{3}{\underbrace{\left[{\left[-13/2+h,-13/2+h\right]}^T,\dots ,{\left[-13/2+h,-13/2+h\right]}^T\right]}}}^T\hfill \\ \hfill X_2^0=& {\underset{3}{\underbrace{\left[{\left[-13/2+h,-13/2+h\right]}^T,\dots ,{\left[-13/2+h,-13/2+h\right]}^T\right]}}}^T\hfill \\ \hfill X_3^0=& {\underset{2}{\underbrace{\left[{\left[-13/2+h,-13/2+h\right]}^T,{\left[-13/2+h,-13/2+h\right]}^T\right]}}}^T\hfill \\ \hfill X_4^0=& {\underset{2}{\underbrace{\left[{\left[-13/2+h,-13/2+h\right]}^T,{\left[-13/2+h,-13/2+h\right]}^T\right]}}}^T.\hfill \end{array}$$

Thus, the fuzzy membership functions of fuzzy logic systems ${\theta }_g^T{\phi }_g\left(X_g\right)$ are selected as ${\mu }_{g^h}\left(X_g\right)=exp(-{(X_g-X_g^0)}^T(X_g-X_g^0)/2)$. Next, the fuzzy basis function vectors are obtained as ${\phi }_g\left(X_g\right)=[{\phi }_{g,1}\left(X_g\right),{\phi }_{g,2}\left(X_g\right),\dots ,{\phi }_{g,12}\left(X_g\right)]$ with ${\phi }_{g,h}\left(X_g\right)={\displaystyle \frac{{\mu }_{g^h}}{{\sum }_{h=1}^{12}{\mu }_g^h}}$.

The design parameters are selected as $k_1=2$, $k_2=3$, $k_3=3$, $k_4=5$, $k_5=210$, $k_6=198$, $p_c=1(c=1,2,3,4,5,6)$, $q_c=0.01(c=1,2,3,4,5,6)$, $o_a=1(a=1,2)$, ${\varphi }_{g,i}=1(g=1,2,3,4,i=1,2,\dots ,11)$, ${\varrho }_{g,i}=1$, ${\kappa }_a=0.01$, and $d_a=0.02$. In this simulation, three kinds of propeller fault cases are taken into account, as shown in

Table 3. Three kinds of propeller fault cases.

Case	Unhealthy Propellers	Fault Occurrence
1	Port propeller	Lose 80% effectiveness; ${\sigma }_1=sin\left(t\right)cos\left(t\right)$
2	Starboard propeller	40 s, Lose 70% effectiveness; ${\sigma }_2=2-exp(-t-1)$
3	Twin propellers	20 s, Lose 20% effectiveness (port); 30 s, Lose 60% effectiveness (starboard); ${\sigma }_1=0.5exp(-t-1)+0.5$; ${\sigma }_2=1-exp(-t-1)$

. In the first case, it is supposed that the port propeller suffers a loss-of-effectiveness fault at 10 s. Thus, 80% effectiveness of the propeller is lost. While, in the second case, assume that the starboard propeller loses 70% effectiveness from 40 s. Next, in the third case, suppose that the port propeller loses 20% effectiveness from 20 s and the starboard propeller loses 60% effectiveness from 30 s.

From Figure 5, under the developed fault-tolerant control method, the reference trajectory can be followed even if the port propeller loses 80% effectiveness, the starboard propeller loses 70% effectiveness, or the twin propellers lose 20% and 60% effectiveness. The position errors $e_1$ under three cases are exhibited in Figure 6. We see that under case 1, case 2, and case 3, the position error can converge within the compact set ${\mathrm{\Omega }}_{e_1}=\left\{\right|e_1\left(t\right)|\le 1.2\mathrm{m}\}$, ${\mathrm{\Omega }}_{e_1}=\left\{\right|e_1\left(t\right)|\le 1.05\mathrm{m}\}$, ${\mathrm{\Omega }}_{e_1}=\left\{\right|e_1\left(t\right)|\le 1.1\mathrm{m}\}$, respectively. Figure 7 shows the healthy thrust forces $W_a$ and the thrust forces from suffered faults $W_a^f$. Correspondingly, the surge force ${\tau }_u^f$ and yaw moment ${\tau }_r^f$ are exhibited in Figure 8. Based on Figure 9, when the loss-of-effectiveness faults happen, the left-side DC motor increases the output power reaching 3274 rpm under case 1, the right side DC motor increases the output power reaching 2212 rpm under case 2, and the DC motors increase the output power reaching 1662 rpm and 2025 rpm under case 3. Thus, it is clear that from Figure 8, the surge force and yaw moment can be increased actively to maintain the tracking performance. Note that the increase in DC motor shaft speeds are dual to the improvement in duty cycles of PWM signals ${\delta }_a$ for increasing the armature voltages of motors, which are shown in Figure 10. Based on Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, we can demonstrate that the differential-driven USV achieves fault-tolerant trajectory tracking performance.

To verify the fault-tolerant control advantage of the developed method, a comparison on tracking results between the fault-tolerant control and non-fault-tolerant control was conducted. The non-fault-tolerant control means that the fault-tolerant control laws ${\tau }_u^{*}$ (41) and ${\tau }_r^{*}$ (42) are not used. Thus, the desired motor shaft speeds are produced by the non-fault-tolerant control laws (39) and (40), which cannot actively compensate for faults. In this comparison, we consider that the case 3 fault type and all the control design parameters remain unchanged. Figure 11 and Figure 12 show the tracking control results and the position error convergence performance using the fault-tolerant control and non-fault-tolerant control, respectively. It can be seen that the presented control scheme results in good fault-tolerant tracking performance. The performance metrics are shown in

Table 4. Performance metrics under propeller faults.

Metric	Under Fault-Tolerant Control	Under Non-Fault-Tolerant Control
Average position error $e_1$	0.9915/m	3.095/m
Performance recovery capability	Yes	No

.

To demonstrate the robustness of the proposed control strategy against real-world marine disturbances, the effects of waves, winds, and currents are simulated. In the numerical simulations, the environmental disturbances are modeled as Gaussian random noise. Based on [30], high-frequency wave motions are simulated using a second-order bandstop filter, while low-frequency disturbances caused by wave drift, ocean currents, and winds in the yaw direction are represented by a first-order transfer function. Therefore, the disturbances are considered as ${\tau }_{du}=sin\left(\psi \right)\overline{G}\left(s\right)$, ${\tau }_{dv}=cos\left(\psi \right)\overline{G}\left(s\right)$, and ${\tau }_{dr}=G^{\prime }\left(s\right)$, where $\overline{G}\left(s\right)$ and $G^{\prime }\left(s\right)$ are the high-frequency wave motion and the slow-varying environmental disturbances, respectively. We chose two reference trajectories: (1) ${[x_d,y_d]}^T={[4sin\left(t\right),3t]}^T$ and (2) ${[x_d,y_d]}^T=[10sin\left((0.05t+1/360)\pi \right),10sin{\left(\left(0.02t\right)\pi \right]}^T$. The design parameters are the same as previous simulation, and the case 3 fault type is considered. Figure 13 shows that two tracking missions are accomplished under real-world marine disturbances based on our method.

5. Conclusions

In this paper, a fault-tolerant trajectory tracking control method has been presented for the differential-driven USV with propeller faults. The designed PWM duty cycles of propeller motors have been considered as the actual control command that can be employed on a differential-driven USV in practice. The USV system with the presented controller remains stable even if there are propeller faults, model uncertainties, and ocean disturbances. Finally, simulation verification has been proposed to prove the validity of the proposed method.