Sliding Mode Control with Adaptive-Reaching-Law-Based Fault-Tolerant Control for AUV Sensors and Thrusters

Abstract

Ocean currents, mechanical collisions and electronic damage can cause faults in an autonomous underwater vehicle (AUV), including sensors and thrusters. For such problems, this paper designs a fault-tolerant controller that is independent of the results of the fault diagnosis. An adaptive reaching law is developed based on sliding mode control to shorten convergence times. For the chattering phenomenon, a weighted hyperbolic tangent function is adopted instead of the traditional sign function in sliding mode control. Simulations are carried out when thruster and sensor fail under the condition of ocean current disturbance, model uncertainty and sensor noise. Comparative simulation results show that the proposed method can accelerate the convergence speed of the state point and improve the trajectory tracking effect of the AUV. Consequently, the effectiveness of the proposed method is confirmed.

1. Introduction

Autonomous underwater vehicles (AUVs) are an essential tool for underwater operations, and they have found extensive applications in resource exploration, target search and terrain investigation, providing significant contributions to the maritime industry’s development [1,2]. Critical components of the AUV include sensors and thrusters. Their structures are complex and easily influenced by external factors, which can lead to their faults. The faults of sensors and thrusters can affect the AUV’s stability and increase the risk of AUV loss [3,4,5]. Therefore, the fault-tolerant control of AUV sensors and thrusters is crucial.

Due to limitations in the AUV’s space, weight, energy, and even cost, it is not feasible to enhance the system’s reliability via hardware redundancy [6]. Therefore, numerous researchers have approached the issue from a control perspective, aiming to compensate for AUV faults. Fault-tolerant control effectively mitigates system failures, enabling the system to regain its performance as much as possible and thus enhancing system stability [7,8,9].

The methods of fault-tolerant control are divided into passive fault tolerance and active fault tolerance. The uncertainties in the system impose limitations on passive fault tolerance, making active fault-tolerant control a hot research topic [10]. Active fault-tolerant control, in turn, can be further categorized into fault-diagnosis-dependent active fault tolerance and fault-diagnosis-independent adaptive fault tolerance [11,12]. Adaptive fault tolerance control possesses advantages in overcoming system disturbances and enhancing system robustness, making it a current focal point in AUV fault tolerance control research. In recent years, numerous adaptive fault-tolerant control methods for AUVs have been proposed, focusing on enhancing AUV’s tracking control performance and reducing trajectory tracking errors [13,14]. For instance, in reference [15], the issue of fault-tolerant control concerning ocean current disturbances, model uncertainties, thruster faults, and saturation constraints during AUV trajectory tracking is addressed. An online adaptive control method is employed to estimate the upper bounds of AUV model uncertainties that affect the robot. In reference [16], a similar adaptive control method is used to estimate the model uncertainties of AUV, albeit with a slightly different representation of the upper bounds compared to reference [15], resulting in improved AUV trajectory tracking performances. Reference [17] introduces an approach using radial basis function neural networks (RBFNNs) and adaptive methods to estimate dynamic uncertainties in the construction process of the controller, compensating for the impact of AUV thruster faults, thereby enabling stable trajectory tracking.

The adaptive fault-tolerant control methods discussed above for AUVs primarily focus on the investigation of actuators. These methods employ adaptive approaches to estimate AUV uncertainties and thruster faults. However, this adaptive methodology has not been applied to the fault-tolerant control of sensors. Fault-tolerant sensor control typically involves the use of observers to estimate the sensor’s state. For instance, in reference [18], an adaptive compensatory observer is designed to address the coupling problem between the fault tolerance control of actuators and sensors in nonlinear systems. In reference [19], an active fault-tolerant control method for aircraft, based on a disturbance observer, utilizes an extended state observer to estimate sensor information and incorporates the estimated sensor information into the closed-loop motion control of the aircraft, achieving sensor fault tolerance. Reference [20] proposes an adaptive fault-tolerant control method for AUV sensors based on interval observers, which can compensate for sensor faults and robot uncertainties, ensuring the stability of AUV trajectory tracking.

Currently, commonly used control methods include sliding mode control [21,22], neural network control [23], and fuzzy control [24,25]. Sliding mode control offers strong robustness, rapid response, and simplicity, making it a hot research topic in the field of control. The sliding mode control law consists of the equivalent control law and reaching control law. However, as the conventional reaching control law cannot meet the controller’s need for rapid convergence, many scholars have improved the sliding mode reaching law to enhance the controller’s precision and convergence speeds.

The focal points of research in sliding mode control have achieved finite-time rapid convergence and reduced chattering. To address the challenge of finite-time convergence in sliding mode control, many scholars have devised sliding mode reaching laws to enhance the precision and convergence speed of controllers. For instance, when dealing with uncertain boundaries, researchers employ adaptive fault-tolerant methods to compensate for the impact of sensor and thruster faults, thus achieving adaptive fault tolerance for sensors and thrusters. Traditional exponential rate laws are non-adjustable, leading to decreased system robustness due to the rapid decline of the exponential term in the power rate law. For this issue, reference [26] designed an improved variable-rate reaching law, which consists of a continuous exponential function, and the conventional constant rate reaching law. The simulation results in reference [26] demonstrated that the sliding mode manifold’s convergence time was effectively reduced. Compared with reference [27], reference [26] squared the exponential term to prevent its rapid decline and enhance the system’s stability.

AUVs are significantly affected by various factors, including oceanic conditions, sensor measurement noise, and the high-frequency switching inherent to the sign function in sliding mode control. Consequently, AUV control experiences severe chattering phenomena. Common techniques employed to mitigate this issue involve approaches such as boundary layer methods [27,28] and the hyperbolic tangent function method [29]. In our investigation conducted under the influence of ocean current disturbances, model uncertainties and sensor noise, we observed pronounced chattering phenomena in control inputs.

The limitations of fixed boundary layer thickness and the effectiveness of the hyperbolic tangent function in reducing chattering have been observed. This is due to the fact that the system’s trajectory cannot asymptotically converge to the designated switching surface, resulting in the absence of sliding mode behavior on the switching surface. This, in turn, reduces the system’s robustness on the switching surface. To address this issue, we propose a chattering reduction method based on a weighted hyperbolic tangent function that is aimed at reducing chattering in the propulsion control input, thereby lowering energy consumption and extending the AUV’s operational lifespan.

In summary, building upon the methods discussed above and the preliminary research in this paper [15,26,27], we have further delved into the fault-tolerant control problem in the presence of AUV model uncertainties, ocean current disturbances, sensor measurement noise and sensor and thruster faults. Our objective is to ensure the finite-time convergence of tracking errors and relatively smooth control signals. In comparison to existing outcomes, this paper makes the following significant contributions:

(1)

To achieve the fault tolerance control of AUV sensors and actuators, this paper employs an adaptive approach to compensate for the effects of faults and uncertainties. The approach in this paper differs from traditional fault tolerance control methods. Most traditional methods primarily focus on actuator faults, with limited research on fault tolerance when both sensors and actuators experience failures. Additionally, traditional sensor fault tolerance control is predominantly centered around observer design, heavily relying on the effectiveness of observers and fault diagnosis. In contrast, this paper incorporates parameter adaptation techniques into terminal sliding mode control, ensuring that AUV tracking errors converge to equilibrium within a finite time.

(2)

To achieve rapid error convergence during AUV trajectory tracking, this paper introduces an adaptive sliding mode reaching law. The subject of this paper exhibits nonlinear and coupled characteristics and operates within complex marine environments over extended periods. When designing a nonsingular terminal sliding mode controller using conventional reaching laws, as seen in [26,27], it was observed that the control performance was relatively better when the state points were closer to the sliding surface. However, when state points were far from the sliding surface, the fault tolerance performance was less effective, leading to increased convergence times and reduced convergence speeds. As a solution to this issue, this paper introduces a power reaching law into the variable reaching law, creating an adaptive reaching law. In this approach, when state points are far from the sliding surface, the power term plays a dominant role, while the variable term predominantly comes into play when state points are relatively closer to the sliding surface. This ensures that state points rapidly converge within a finite time, thus enhancing the effectiveness of trajectory tracking control.

(3)

To reduce the vibration in AUV actuators, this paper introduces a weight-based hyperbolic tangent function as a replacement for the conventional sign function in sliding mode control. Differing from traditional anti-vibration methods, this approach addresses the limitations associated with maintaining a fixed boundary layer thickness and using the hyperbolic tangent function. It results in a relatively smoother output of control signals for AUVs.

This paper’s structure of organization is as follows: Section 2 describes the AUV dynamics model in an ocean environment, along with modeling uncertainty, sensor noise, sensor faults and thruster faults. In Section 3, the sliding mode fault tolerance control method based on the adaptive reaching law is studied. In Section 4, simulations are carried out to verify the designed controller. Finally, conclusions are given in Section 5.

2. Mathematical Models and Problem Description

2.1. AUV Dynamic Model under the Influence of Ocean Currents

The AUV dynamic model under the effect of ocean currents is as follows, citing reference [30]:

$$\begin{gathered} \dot{\eta }=J\left(\eta \right)v \\ M\left(v\right)\dot{v}+C_{RB}\left(v\right)v+g\left(\eta \right)+C_A\left(v_r\right)v_r+D\left(v_r\right)v_r=\tau \end{gathered}$$

(1)

where $J\left(\eta \right)$ denotes the transformation matrix from the inertial coordinate frame to the body-fixed coordinate frame; $M$ denotes the inertia matrix involving the added mass; $C_{RB}$ denotes the rigid body’s centripetal and Coriolis force matrix, $C_A$ denotes the hydrodynamic centripetal and Coriolis force matrix; $D$ is the hydrodynamic resistance matrix; $g$ denotes the restoring force matrix; and $\tau$ denotes the control force and moment acting on the center of gravity of the AUV. $\upsilon ={\left[u v w p q r\right]}^T$ denotes the velocity vector relative to the body-fixed coordinate system; $\eta ={\left[x y z \varphi \theta \psi \right]}^T$ denotes the velocity relative to the position and attitude vector of the vehicle under the inertial coordinate frame; ${\upsilon }_r=\upsilon -{\upsilon }_c$, ${\upsilon }_c$ denotes the current velocity, and it is relative to the body-fixed coordinate frame. The coordinate frames are shown in Figure 1.

The paper transforms Equation (1) in the inertial coordinate frame and designs an AUV fault-tolerant controller. Equation (2) is defined as follows:

$$M_{\eta }\left(\eta \right)\ddot{\eta }+C_{RB\eta }\left(\eta ,\dot{\eta }\right)\dot{\eta }+C_{A\eta }\left({\eta }_r,\dot{{\eta }_r}\right)\dot{{\eta }_r}+D_{\eta }\left({\eta }_r,\dot{{\eta }_r}\right)\dot{{\eta }_r}+g_{\eta }\left(\eta \right)=J^{-T}\tau$$

(2)

where $M_{\eta }\left(\eta \right)=J^{-T}M{J}^{-1}$; $C_{RB\eta }\left(\eta ,\dot{\eta }\right)=J^{-T}\left[C_{RB}\left(q\right)-M{J}^{-1}\dot{J}\right]J^{-1}$;$C_{A\eta }\left(\eta ,\dot{\eta }\right)=J^{-T}{C}_A\left(q_r\right)J^{-1}$; $D_{\eta }\left({\eta }_r,{\dot{\eta }}_r\right)=J^{-T}D\left(q_r\right)J^{-1}$; and $g_{\eta }\left(\eta \right)=J^{-T}g$; ${\eta }_r=\eta -V_c$. $V_c$ is the vector of ocean currents in the inertial coordinate frame.

Because measuring the dynamic model of the AUV is difficult and the AUV works in a challenging maritime environment, the effect of ocean currents and model errors impairs the precision of AUV fault tolerance control. The modeling uncertainties are represented by the equations below [31]:

$$\begin{gathered} M_{\eta }={\widehat{M}}_{\eta }+\Delta M_{\eta }, C_{A\eta }={\widehat{C}}_{A\eta }+\Delta C_{A\eta },C_{RB\eta }={\widehat{C}}_{RB\eta }+\Delta C_{RB\eta }, \\ {D}_{\eta }={\widehat{D}}_{\eta }+\Delta D_{\eta },g_{\eta }={\widehat{g}}_{\eta }+\Delta g_{\eta } \end{gathered}$$

(3)

where${\widehat{M}}_{\eta },{\widehat{C}}_{\eta },{\widehat{D}}_{\eta }, \mathrm{a}\mathrm{n}\mathrm{d} {\widehat{g}}_{\eta }$ represent the nominal values; $M_{\eta },C_{\eta },D_{\eta }, \mathrm{a}\mathrm{n}\mathrm{d} g_{\eta }$ represent the real values; ${\Delta M}_{\eta },{\Delta C}_{A\eta },{\Delta D}_{\eta }, \mathrm{a}\mathrm{n}\mathrm{d} {\Delta g}_{\eta }$ represent modeling uncertainties. Therefore, ocean current disturbances and AUV model uncertainties can be expressed as follows:

$${F_1=\Delta M}_{\eta }\ddot{\eta }+\Delta C_{RB\eta }\left(\dot{\eta }\right)\dot{\eta }+{\Delta C}_{A\eta }\left(\dot{{\eta }_r}\right)\dot{{\eta }_r}+\Delta D_{\eta }\left(\dot{{\eta }_r}\right)\dot{{\eta }_r}+\Delta g_{\eta }\left(\eta \right)$$

(4)

Assuming that ocean current disturbances and model uncertainty $F_1$ are bounded, the uncertain terms of the AUV model are related: $\eta ,\dot{\eta },{\dot{\eta }}_r$ and $\ddot{\eta }$. $\dot{\eta }$ is a variate related to $\eta$. Inspired by references [15,16], it is assumed that the uncertainties of the system are bounded: that is, $F_1\le M_{\eta }\left({\rho }_0+{\rho }_1‖\eta ‖+{\rho }_2{‖\dot{\eta }‖}^2\right)$.

One of the AUV’s most important components is its sensors. Serving as the AUV’s observation unit, sensors are frequently employed in underwater environments. This paper mainly simulates the sensor’s faults in Doppler velocity log (DVL) and inertial measurement units (IMUs). DVL is used to measure the AUV’s velocity, while the IMU measures the AUV’s angular velocity. The position and altitude of the AUV are measured via an inertial navigation system (INS), and the position and altitude are continuously differentiable in a first-order manner.

They are susceptible to the influences of many factors, such as ocean currents, mechanical collisions and electronic damage, which can result in sensor faults, including invalidation and bias faults.

When invalidation faults occur, the sensor’s measured value is constant. In the case of bias faults, a constant error exists between the sensor’s measured value and the actual value. In brief, the sensor’s measured value can be described in the following expression while simulating sensor faults [32]:

$${\dot{\eta }}^{\prime }=\dot{\eta }-\Delta \dot{\eta }=k\dot{\eta }+f\left(t\right)+\rho \left(t\right)$$

(5)

where ${\dot{\eta }}^{\prime }$ represents the measured value of the sensor and $\dot{\eta }$ represents the actual value of the sensor, $\dot{\eta }={\left[u,v,w,p,q,r\right]}^{\mathrm{T}}$; $k\in \left[0,1\right]$; this means that the sensor is working properly without faults when $k=1$. When $0<k<1$, the sensor is susceptible to the loss of accuracy faults. $f\left(t\right)$ represents the error between the measured value and the actual value of the sensor when the sensor suffers from faults.

ρ(t) is sensor noise, and it consists of white noise $n\left(t\right)$ and random walk noise $b\left(t\right)$. White noise n(t) is external noise caused by factors like the AD converter and outside disturbances during the process of measurement. Random walk noise $b\left(t\right)$ is the integrated parameter of internal errors generated by internal factors such as sensor internal machinery and temperature. Therefore, the noise, $\rho \left(t\right)$, in this paper can be expressed as follows [33]:

$$\rho \left(t\right)=n\left(t\right)+b\left(t\right)$$

(6)

Combining Equation (5) and the AUV dynamic mode shown in Equation (2), expression $\ddot{\eta }$ is described as follows:

$$\ddot{\eta }={\widehat{M}}_{\eta }^{-1}\left[J^{-T}Bu-{\widehat{C}}_{\eta }\left(\dot{\eta }-\Delta \dot{\eta }\right)-{\widehat{D}}_{\eta }\left(\dot{\eta }-\Delta \dot{\eta }\right)-{\widehat{g}}_{\eta }\right]$$

(7)

It can be observed in Equation (7) that the uncertain terms of the AUV model caused by sensors can be expressed as follows:

$$F_2=-\left({\widehat{C}}_{\eta }+{\widehat{D}}_{\eta }\right)\Delta \dot{\eta }$$

(8)

We assume that the uncertain term, F₂, caused by sensor faults is bounded: that is, $F_2\le {\rho }_3‖{\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }‖$.

The thruster provides the power source for AUV, which works in a challenging maritime environment. Therefore, it is prone to faults, including insufficient output, invalidation, and so on. The force and torque output of the thruster under fault conditions can be simulated as follows:

$${\tau }^{\prime }=B^{\prime }u$$

(9)

where ${\tau }^{\prime }$ represents the real output force and torque when the thruster fails; $B^{\prime }=B-\Delta B=B\left(I-L\right)$, and $B^{\prime }$ represents the thrust allocation matrix under the condition of thruster faults, where $L=diag\left(l_{ii}\right)\left(i=1,2,\dots ,8\right)$ and $l_{ii}\in \left[0,1\right]$. When the thruster is working properly, $l_{ii}=0$; when the thruster fails, $l_{ii}\ne 0$.

The uncertain term, $F_3$, is caused by the AUV thruster fault; thus, it can be described as follows:

$$F_3=-J^{-T}\Delta Bu$$

(10)

We assume that the uncertain term, $F_3$, caused by thrust faults is bounded: that is, $F_3\le {\rho }_4‖{\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta }+{\widehat{M}}_{\eta }\left({\dot{\alpha }}_1+{\ddot{\eta }}_d\right)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1}‖$.

Therefore, the total uncertain terms, $F$, for AUV ocean current interference, model uncertainty, sensor noise and sensor and thruster faults can be expressed as follows:

$$\begin{array}{cc}\hfill F& =F_1+F_2+F_3\hfill \\ \hfill & =M_{\eta }\ddot{\eta }+\Delta C_{RB\eta }\left(\dot{\eta }\right)\dot{\eta }+\Delta C_{A\eta }\left({\dot{\eta }}_r\right){\dot{\eta }}_r+\Delta D_{\eta }\left({\dot{\eta }}_r\right){\dot{\eta }}_r+\Delta g\left(\eta \right)\hfill \\ \hfill & +\left(C_{\eta }+D_{\eta }\right)\Delta \dot{\eta }-J^{-T}\Delta Bu\hfill \\ \hfill & \le M_{\eta }({\rho }_0+{\rho }_1\Vert \eta \Vert +{\rho }_2{‖\dot{\eta }‖}^2)+{\rho }_3\Vert {\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }\Vert +{\rho }_4\Vert {\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta }\hfill \\ \hfill & +{\widehat{M}}_{\eta }({\dot{\alpha }}_1+{\ddot{\eta }}_d)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1}\Vert \hfill \end{array}$$

(11)

where ${\rho }_0,{\rho }_1,{\rho }_2,{\rho }_3$, and ${\rho }_4$ are unknown and positive vectors; $‖·‖$represents the 2-norm of the vector. Reference [16] only considers the $\dot{\eta }$ signal and reports that the uncertain terms are only related to the velocity signal. Reference [15] not only considers the $\dot{\eta }$ signal in the AUV model’s uncertain terms but also considers the $\eta$ signal. The uncertain terms directly affect the position and velocity outputs of AUV; thus, acceleration signal $\ddot{\eta }$ is not considered in the paper. The research content of this paper is based on the fault tolerance control of sensors and thrusters. The sensors are affected by the marine environment, measurement noise, and the internal random walk noise of the sensors. Although the acceleration signal is bounded, its amplitude changes relatively greatly. Therefore, this paper adopts the representation form of the uncertain terms in reference [15] and considers the influence of ocean current disturbance, model uncertainty, sensor noise and sensor faults and thruster faults in the uncertain terms.

Therefore, under the influence of ocean current disturbance, model uncertainty, sensor noise, sensor faults and thruster faults, the state space equation of AUVs in the inertial coordinate system is expressed as follows.

$$\ddot{\eta }={\widehat{M}}_{\eta }^{-1}\left(J^{-T}Bu-{\widehat{C}}_{\eta }\dot{\eta }-{\widehat{D}}_{\eta }\dot{\eta }-{\widehat{g}}_{\eta }-F\right)$$

(12)

2.2. Problem Description

Under the influence of ocean current disturbance, model uncertainty, sensor noise, sensor faults and thruster faults, this paper designs a class of nonlinear adaptive fault-tolerant controllers. Without relying on fault diagnosis results, the controllers can ensure the stability of the system and enable the state points to converge rapidly to the equilibrium point within a finite time period, achieving the adaptive fault tolerance control of AUV sensors and thrusters. The adaptive fault-tolerant controller designed in this paper can estimate the upper bound of the model’s uncertain terms in an online manner and design an adaptive sliding mode fault-tolerant controller based on the estimation. Relative to the chattering problem of sliding mode control, a hyperbolic tangent function is adopted for reducing chattering.

The controller does not depend on fault diagnosis results. It can compensate for various uncertain factors online based on state errors when ensuring stability, enabling the system to converge rapidly to the desired state and achieving the adaptive fault-tolerant control of AUV sensors and thrusters. Simulations verify that the control method can effectively compensate for the influence of uncertain factors, such as ocean current disturbance, model uncertainty, sensor noise, sensor faults and thruster faults, ensuring the stable operation and high tracking accuracy of the system.

Overall, the proposed adaptive sliding mode fault-tolerant control method provides a fault-tolerant control scheme for the AUV that does not rely on fault diagnosis results. The scheme can compensate for the influence of uncertain factors and disturbances online, ensuring the stability and high task-tracking performance of the system and enhancing its environmental adaptability and robustness.

3. Research on Designing Sliding Mode Fault-Tolerant Controllers Based on an Adaptive Reaching Law

3.1. Research on Fault-Tolerant Controller

The effect of traditional fault tolerance control based on fault diagnosis is determined by the outcome of fault diagnoses. Fault-tolerant control based on incorrect fault diagnosis results will directly affect AUV fault-tolerant control effects and even damage the AUV’s stability. Compared with the traditional fault-tolerant control method, this paper investigates a fault-tolerant control system that is independent of diagnostic fault results and offers an adaptive sliding mode fault-tolerant control method based on an adaptive reaching law.

In contrast to the conventional reaching law, the proposed adaptive reaching law can reduce the state point’s convergence time to the sliding mode switching manifold. Simulations verify the effectiveness of the proposed method.

3.1.1. Sliding Mode Manifold

The sliding mode manifold is an essential component of sliding mode control, which can have an impact on control precision.

Reference [34] adopted the nonsingular terminal sliding mode manifold from the perspective of the stability proof of the Lyapunov function, which can make the sliding mode function converge to the switching manifold.

To reduce the convergence time of position errors and error rates, inspired by reference [15], based on the idea of backstepping control, the position errors and error change rates are combined to design a new nonlinear sliding manifold with the following specific form:

$$s=z_1+\beta z_2^{\mathsf{\gamma }}$$

(13)

where $z_1=\eta -{\eta }_d$ and $z_2=\dot{\eta }-{\dot{\eta }}_d-{\alpha }_1$, ${\alpha }_1=-{\lambda }_1{z}_1$; ${\eta }_d$ is the desired position vector; ${\dot{\eta }}_d$ is the desired velocity vector; $\beta ,{\mathsf{\lambda }}_1,\mathrm{a}\mathrm{n}\mathrm{d}\gamma$ are constant, and $\beta$ > 0, ${\mathsf{\lambda }}_1>0$, and $1<\gamma <2$. The range of $\gamma$ is dependent on the sliding mode control law, avoiding the singular phenomenon.

By calculating the first-order derivative of the sliding manifold and combining it with Equation (12), we can obtain the following:

$$\begin{array}{cc}\hfill \dot{s}& ={\dot{z}}_1+\beta \gamma z_2^{\gamma -1}{\dot{z}}_2={\dot{z}}_1+\beta \gamma z_2^{\gamma -1}\left(\ddot{\eta }-{\dot{\alpha }}_1-{\ddot{\eta }}_d\right)\hfill \\ \hfill & ={\dot{z}}_1+\beta \gamma z_2^{\gamma -1}\left\{{\widehat{M}}_{\eta }^{-1}\left[J^{-T}Bu-{\widehat{C}}_{\eta }\dot{\eta }-{\widehat{D}}_{\eta }\dot{\eta }-{\widehat{g}}_{\eta }-F\right]-{\dot{\alpha }}_1-{\ddot{\eta }}_d\right\}\hfill \end{array}$$

(14)

3.1.2. Adaptive Reaching Law

There are two stages in the sliding mode motion, including the approaching motion and sliding motion. The approaching motion is the shifting of a state point from its starting condition to the switching surface until it reaches the switching surface. A reaching law can also fulfill the requirement in which the motion point in any position of the state space reaches the switching surface within a specified period. It is used to improve the dynamic quality of the approaching motion and increase the control precision of the sliding mode controller.

Once chosen, the parameters of the conventional reaching law are fixed. If the parameter values of the settings used are too large, the system’s chattering will be aggravated. If the parameter values used are too small, the system’s convergence time will be increased, and the controller’s precision will be lowered. Inspired by reference [27], this paper proposes an adaptive reaching law, which is combined with the power reaching law and the variable-rate reaching law specifically as follows:

$$\dot{s}=-k_1{\left|s\right|}^{\alpha }sgn\left(s\right)-\frac{k_2}{{\mu +\left(1-\mu \right)}^{-\delta {\left|s\right|}^2}}sgn\left(s\right)$$

(15)

where $k_1>0$; $k_2ϵ\left(0,1\right)$; $\delta >0$; $\alpha ϵ\left(0,1\right)$; and $\mu ϵ\left(0,1\right)$. According to the Lyapunov stability theory, we need to guarantee $\dot{V}$ < 0; thus, $k_1$ and $k_2$ are positive.

According to the distance between the state point and the sliding manifold, the reaching speed of the state point is adaptively adjusted. When the state point is far from the sliding manifold, the power item plays a major role, which can cause quick convergence from the state point to the sliding manifold. When the state point is close to the sliding manifold, the variable-rate item plays a major role, which can effectively reduce the convergence time from the state point to the sliding mode manifold.

3.1.3. Control Law

The sliding mode motion has two stages: that is, the approaching motion stage and sliding motion stage. The control law of the approaching motion stage needs the reaching control law $u_{sw}$, and the sliding motion stage needs the equivalent control law $u_{eq}$ [35]. Therefore, the sliding mode control law consists of the reaching control law and the equivalent control law. The expression of the sliding mode control law is shown in Equation (16):

$$u=u_{eq}+u_{sw}$$

(16)

where $u$ is the total control law; $u_{eq}$ is the equivalent control law; $u_{sw}$ is the reaching control law.

Based on the nonsingular terminal sliding manifold and the composition of the sliding mode control law, combined with the Lyapunov stability theory, the AUV control law is finally proposed as follows. Proof of the Lyapunov stability is in Appendix A.

$$\begin{gathered} u=u_1+u_2+u_3+u_4 \\ {u}_1=B^(-1) J^T [\left({\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta }\right)+{\widehat{M}}_{\eta }\left({\dot{\alpha }}_1+{\ddot{\eta }}_d\right)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1} \\ +{\widehat{M}}_{\eta }{z}_2^{1-\gamma }{\gamma }^{-1}{\beta }^{-1}(-k_1{|s|}^{\alpha }sgn(s)-\frac{k_2}{\mu +(1-\mu )e^{-\delta {|s|}^2}}sgn(s))] \\ {u}_2=B^{-1}{J}^T{\widehat{M}}_{\eta }\left({\widehat{\rho }}_0+{\widehat{\rho }}_1‖\eta ‖+{\widehat{\rho }}_2{‖\dot{\eta }‖}^2\right)sgn\left(s\right) \\ {u}_3=B^{-1}{J}^T{\widehat{M}}_{\eta }{\widehat{\rho }}_3‖{\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }‖sgn\left(s\right) \\ {u}_4=B^{-1}{J}^T{\widehat{M}}_{\eta }{\widehat{\rho }}_4\Vert {\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta }+{\widehat{M}}_{\eta }\left({\dot{\alpha }}_1+{\ddot{\eta }}_d\right) -{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1}\Vert sgn(s) \end{gathered}$$

(17)

where $u_1$ represents the sum of the equivalent control law and the approaching control law without considering the uncertainties of the AUV; $u_2$ represents the adaptive control law considering the uncertainties of the AUV; $u_3$ represents the adaptive control law to compensate for AUV sensor noise and faults; $u_4$ represents the adaptive control law to compensate for AUV thruster faults.

The estimate of uncertainty ${\widehat{\rho }}_i\left(\mathrm{i}=0,1,2,3,4\right)$ is updated using the adaptive law as follows:

$$\begin{gathered} {\dot{\widehat{\rho }}}_0=q_0\beta \gamma z_2^{\gamma -1}\left|s\right| \\ {\dot{\widehat{\rho }}}_1=q_1\beta \gamma z_2^{\gamma -1}‖\eta ‖\left|s\right| \\ {\dot{\widehat{\rho }}}_2=q_2\beta \gamma z_2^{\gamma -1}{‖\dot{\eta }‖}^2\left|s\right| \\ {\dot{\widehat{\rho }}}_3=q_3\beta \gamma z_2^{\gamma -1}‖{\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }‖\left|s\right| \\ {\dot{\widehat{\rho }}}_4=q_4\beta \gamma z_2^{\gamma -1}‖{\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+\widehat{g}+{\widehat{M}}_{\eta }\left({\dot{\alpha }}_1+{\ddot{\eta }}_d\right)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1}‖\left|s\right| \end{gathered}$$

(18)

The block diagram of the adaptive reaching law sliding mode fault-tolerant control method proposed in this paper is shown in Figure 2.

3.1.4. Chattering Reduction Method Based on the Weighted Hyperbolic Tangent Function

This paper investigates sensor noise and considers its participation in closed control for AUVs. We restrain sensor noise by using a filter. When restrained sensor information participates in the AUV’s control, the controller’s chattering can increase.

The hyperbolic tangent function is one of the most effective methods for solving the chattering of sliding mode control inputs. The function is shown in Equation (19), where $\sigma$ represents the weighting factor. The selection of parameters directly affects the robustness and stability of the system. When the weighting factor parameter is too large, the chattering of the control input is aggravated. When the weighting factor parameter is too small, the robustness of the system is reduced [36]. Therefore, in this paper, the weighting factor is appropriately selected according to the state of the system.

$$\tanh \left(\sigma s\right)=\frac{e^{\sigma x}-e^{-\sigma x}}{e^{\sigma x}+e^{-\sigma x}}$$

(19)

Combining Equations (17) and (19), the final sliding mode control law can be transformed into the following form:

$$\begin{gathered} u=u_1+u_2+u_3+u_4 \\ {u}_1=B^{-1}{J}^T[({\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta })+{\widehat{M}}_{\eta }({\dot{\alpha }}_1+{\ddot{\eta }}_d)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1} \\ +{\widehat{M}}_{\eta }{z}_2^{1-\gamma }{\gamma }^{-1}{\beta }^{-1}(-k_1{\left|s\right|}^{\alpha }tanh({\sigma }_{1}s)-\frac{k_2}{\mu -\left(1-\mu \right)e^{-\delta {\left|s\right|}^2}}tanh({\sigma }_{2}s))] \\ {u}_2=B^{-1}{J}^T{\widehat{M}}_{\eta }({\widehat{\rho }}_0+{\widehat{\rho }}_1\Vert \eta \Vert +{\widehat{\rho }}_2{\Vert \dot{\eta }\Vert }^2)tanh({\sigma }_{3}s) \\ {u}_3=B^{-1}{J}^T{\widehat{M}}_{\eta }{\widehat{\rho }}_3\Vert {\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }\Vert tanh({\sigma }_{4}s) \\ {u}_4=B^{-1}{J}^T{\widehat{M}}_{\eta }{\widehat{\rho }}_4\Vert {\widehat{C}}_{\eta }\dot{\eta }+{\widehat{D}}_{\eta }\dot{\eta }+{\widehat{g}}_{\eta }+{\widehat{M}}_{\eta }({\dot{\alpha }}_1+{\ddot{\eta }}_d)-{\widehat{M}}_{\eta }{z}_2^{2-\gamma }{\gamma }^{-1}{\beta }^{-1}\Vert tanh({\sigma }_{5}s) \end{gathered}$$

(20)

Via the above analysis, it can be known that the hyperbolic tangent function based on weighting factors can effectively reduce the chattering of sliding mode control, enabling the system to be more stable.

4. Simulation

This paper designs an adaptive fault-tolerant controller, which is independent of the fault diagnosis results of sensors and thrusters. Firstly, this paper performs a three-layer wavelet reconstruction of the actual sensor output signal based on the Daubechies wavelet function to achieve noise reduction with respect to the sensor signal. Then, the paper designs a simulation function of sensor and thruster faults. Finally, the signal is fed back to the fault-tolerant controller, which is used to compensate for the effects of sensor and thruster faults to achieve the fault-tolerant trajectory tracking of AUVs.

To verify the fault-tolerant control effect of the proposed method, the Canadian Scientific Submersible Facility ROPOS underwater vehicle is adopted, along with ocean current disturbances, model uncertainty, sensor noise and unknown sensor faults and thruster faults. The parameters of POROS come from reference [37]. It is equipped with eight thrusters, and the thruster’s configuration is shown in Figure 3. The saturation limit of the thruster is ±900 N.

Model uncertainty is considered at 30%. The angle between the ocean current’s direction and the lateral direction of the AUV is 30°, and the angle between the ocean current’s direction and the vertical direction of the AUV is 60°. The ocean current is simulated via a first-order Gaussian–Markov process to simulate the spatial ocean current. The spatial ocean current can be described as follows:

$$\begin{gathered} V_c=\left[u_c,v_c,w_c,0,0,0\right] \\ {V}_c^n=\left[\begin{array}{c}{V}_c\cos \left({\alpha }_c\right)\cos \left({\beta }_c\right)\\ V_c\sin \left({\beta }_c\right)\\ V_c\sin \left({\alpha }_c\right)\cos \left({\beta }_c\right)\end{array}\right] \end{gathered}$$

(21)

The current amplitude ${\dot{V}}_c$ can be described as follows:

$${\dot{V}}_c+\mu V_c=\omega$$

(22)

where $\omega$ denotes Gaussian white noise with a variance of 0.5 and a mean of −0.8. Its power spectral density conforms to a uniform distribution, and the instantaneous statistical amplitude distribution characteristics follow a Gaussian distribution: $\mu =3,{\alpha }_c=\frac{pi}{4}$, and ${\beta }_c=\frac{pi}{6}$.

The initial pose of the AUV is $\eta \left(0\right)={\left[\mathrm{0.4,0.5},-0.4,\frac{pi}{18},\frac{pi}{9},\frac{pi}{9}\right]}^T$, and the initial velocity vector is $\dot{\eta }\left(0\right)={\left[0.02,0.02,0.02,0.01,0.01,0.01\right]}^T$. The expected trajectories, ${\mathrm{y}}_{\mathrm{d}}={\left[{\eta }_1{,\eta }_2,{\eta }_{3,0,0,0}\right]}^T,$of this paper are, respectively, ${\eta }_1=0.$4t, ${\eta }_2=0.4+0.4\sin \left(0.2t\right)$, and ${\eta }_3=0.1t$. The design parameters of the sliding mode fault-tolerant controller based on the adaptive reaching law are shown in

Table 1. Parameters of the controller.

Sliding Mode Manifold	$\beta =2,\gamma =1.01,\lambda =15$
Reaching law	$k_1=8,k_2=0.8,\alpha =0.45,\mu =0.5,\delta =0.9$
Adaptive parameter	$q_0=0.1,q_1=0.1,q_2=0.01,q_3=0.0001,q_0={10}^{-5}$
Chattering	${\sigma }_1=\left[0.05,0.05,0.05,0.5,0.5,0.15\right],{\sigma }_2=4,{\sigma }_3=0.01,$${\sigma }_4={\sigma }_5={10}^{-6}$

.

4.1. Performance of the Proposed Method

The noise of the sensor is simulated via Gaussian white noise. The mean value of Gaussian white noise is set to 0, the variance is 0.01, the angular random walk noise of the gyroscope output is 0.1°/h, and the accumulated velocity deviation of the Doppler velocity log (DVL) is 0.5 m/s in an hour.

In this subsection, simulations are conducted with respect to ocean current disturbance, AUV model uncertainty and sensor noise.

The paper simulates two fault types, including abrupt and slow faults, where A% fault denotes that an amplitude loss of (100-A)% is needed.

This paper expresses abrupt faults, as shown in Equation (23); slow faults are shown in Equation (24):

$$\begin{gathered} \dot{\eta }-\Delta \dot{\eta }=p_{11}\dot{\eta } \\ {p}_{11}=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{cc}1& \end{array} t\end{array}<t_0\\ A\% t\ge t_0\end{array}\right. \end{gathered}$$

(23)

$$p_{11}=\left\{\begin{array}{cc}0& t<t_0\\ a\left(t-t_0\right)/\left(t_1-t_0\right)+bsin\left(w_1\left(t-t_0\right)\right)& t_0\le t<t_1\\ a+bsin\left(w_2\left(t-t_1\right)\right)& t\ge t_1\end{array}\right.$$

(24)

where ${\mathrm{t}}_0=50$, ${\mathrm{t}}_1=70$, ${\mathrm{w}}_1=\frac{pi}{5}$, ${\mathrm{w}}_2=\frac{pi}{10}$, $\mathrm{a}=\mathrm{A}\%/({\mathrm{t}}_1-{\mathrm{t}}_0)$, and $\mathrm{b}=0.01$.

Measured values are constant when sensors have invalidation faults. Measured values are greater than actual values when sensors have bias faults.

To verify the control effect of the fault-tolerant controller in this paper, sensor failure faults and sensor bias faults are simulated.

Sensor invalidation faults are represented by a constant, and sensor bias faults are represented by a difference of 0.1 between the real value and output value:

(1)

Sensors and thrusters are faultless in the proposed method shown in Figure 4.

(2)

At 50 s, an abrupt invalidation fault occurred in the DVL measuring longitudinal velocity. DVL’s measured value is 0.2. A slow bias fault occurred in the IMU measuring the yaw angular velocity. IMU’s measured value is 0.1 less than the actual angular yaw velocity. Moreover, Th1 failed abruptly by 75%, and the experimental result is shown in Figure 5.

4.2. Comparison Studies

(1) Simulation Comparison experiment of the proposed method and the traditional control method under 30% model uncertainty and ocean current disturbance

To verify the control effect of the controller designed in this paper, the proposed method and the traditional method were compared under the conditions of no fault in sensors and thrusters. The simulation result of the method in this paper is shown in Figure 4. The simulation result of the traditional method is shown in Figure 6.

The results in Figure 4 and Figure 6 reveal that, in the absence of sensor and thruster faults, the proposed and traditional methods were subjected to simulation and comparison within the condition of AUV ocean current disturbance, model uncertainty and sensor noise.

The results show that the convergence time is 6.5 s in the proposed method when the position error is less than 0.01 in 6 DOF, while the convergence time is 14.72 s in the traditional method.

Table 2. Comparison of the proposed fault-tolerant controller and traditional controller.

		MATE a						Conv Time b (s)	Consumption c (×109 N2)
		X (m)	Y (m)	Z (m)	Roll (rad)	Pitch (rad)	Yaw (rad)
No fault	Proposed method	0.005835	0.005216	0.008697	0.002535	0.002318	0.008297	6.50	4.46
	Traditional method	0.012322	0.009210	0.008769	0.009238	0.023862	0.029805	14.72	10.1
Faults	Proposed method	0.007403	0.005158	0.008662	0.002522	0.002288	0.011250	6.50	4.45
	Traditional method	0.014043	0.009290	0.008509	0.008744	0.021107	0.032924	14.64	8.82

^a MATE is the mean absolute tracking error from $t=0$ to the end. ^b Conv time denotes convergence time $T$ for $t \ge T$ with $|e_i|\le 0.01,i=1,2,\dots ,6.$ ^c Consumption denotes the integral of the square of the control input for thrusters.

, where MATE denotes the mean of the absolute tracking error, shows the proposed method and traditional method. It confirms that the proposed method achieves higher tracking accuracies when the system reaches a steady state. Also, the proposed method has shorter convergence times and lower chattering.

(2) Simulation comparison experiment of the proposed method and the traditional control method under the sensor and thruster faults

To verify the effectiveness of the fault-tolerant sliding mode control method based on the adaptive reaching law proposed in this paper, when an abrupt invalidation fault occurred in the DVL measuring longitudinal velocity, a slow bias fault occurred in the IMU measuring the yaw angle velocity, and Th1 failed abruptly by 75%; the simulation is carried out based on the traditional method in reference [18]. The simulation result is shown in Figure 7.

It can be observed in Figure 5 and Figure 7 that when the concurrent faults occurred, the fault-tolerant controller designed in this paper and the traditional fault-tolerant controller were simulated and compared.

The simulation results show that the tracking errors of the method in this paper converge in about 6.5 s, while the tracking errors of the traditional method converge in 14.64 s. Compared with the traditional method, the mean absolute tracking errors of the method in this paper are slightly smaller than the traditional method, which are 0.007403, 0.005158, 0.008662, 0.002522, 0.002288, and 0.011250.

For the convenience of describing the control effects of the controllers designed using the method discussed in this paper and the traditional method, this paper compares them from the perspectives of average tracking errors and average tracking variances. The comparison results are shown in Table 2.

In summary, this section has given results for the two different conditions under the proposed method and the traditional method. Simulation results show that the proposed method has a relatively better control effect. Compared with the fault-tolerant controller designed via the traditional method, it is superior to the fault-tolerant controller designed via the traditional method with respect to mean absolute tracking error, convergence time and control consumption.

5. Conclusions

An adaptive fault-tolerant control method independent of fault diagnosis results and observers was studied under the conditions of AUV ocean current disturbance, model uncertainty, sensor noise and sensor and thruster faults. The sensor’s signal is filtered through wavelet denoising to reduce sensor noise. The fault-tolerant controller is designed by the residuals between the position information and denoised sensor information, and it is based on the nonsingular terminal sliding mode control method. The sliding mode manifold design incorporates the concept of backstepping. To address the influence of model uncertainty, external disturbance, sensor faults and thruster faults, an adaptive control method is adopted to compensate for the influence of uncertainty, disturbance and faults. To solve the chattering problem of the sliding mode control, a hyperbolic tangent function based on the weight factor is used instead of the traditional sign function in this paper. Simulation results show that the adaptive AUV fault-tolerant control method can effectively compensate for the influence of faults and has a good control effect. Compared with the traditional fault-tolerant control method, the proposed method can effectively improve the convergence speed of tracking errors.

6. Future Directions

Looking ahead, several promising research directions emerge from our work, which could further advance the field. We propose the following areas for future investigation:

• Our work is focused on the method of sliding mode fault controllers in order to compensate for sensor and thruster faults. We will think about the practical relationship between thruster output and sensor frequency.
• This paper concentrates on the method of sliding mode control with an adaptive reaching law; we attained shorter convergence times. In subsequent research studies, we will reduce chattering under the condition of shorter convergence times.