Adaptive Backstepping Axial Position Tracking Control of Autonomous Undersea Vehicles with Deferred Output Constraint

Abstract

In this paper, an adaptive backstepping control scheme is proposed to solve the the surge motion tracking control problem of an autonomous undersea vehicle (AUV) with system constraint. First, an initial rectification reference signal is constructed for the subsequent implementation of deferred output constraint and making the control input smaller and smoother in the early stage of system operation. Second, a barrier Lyapunov function is adopted for developing an output-constrained state feedback adaptive controller. Then, on the basis of coordinate transformation and estimating the derivative of surge displacement by a linear and nonlinear combined differentiator, we develop an output feedback adaptive backstepping control scheme for AUVs whose velocity signals are unmeasurable. We also carried out a comparative numerical simulation with traditional adaptive control to verify the feasibility of the proposed control strategy.

1. Introduction

An autonomous underwater vehicle (AUV) is an important piece of equipment in marine scientific research, resource development and marine military forces [1]. The precise motion control of AUVs is helpful to complete the tasks of reconnaissance, underwater measurement and target tracking, and it has received considerable attention from researchers and engineers. Since the 1990s, many advanced control technologies have been adopted for this issue, such as sliding mode control [2], adaptive control [3], neural network control [4], fuzzy-PID [5], predictive formation tracking control [6], etc.

The motions of an AUV have six degrees of freedom (DOFs) including surge, sway, heave, roll, pitch and yaw motions (see Figure 1). Among them, surge, sway and heave refer to the longitudinal, sideways and vertical displacements; yaw, roll and pitch describe the rotations around the vertical, longitudinal and transverse axes. The surge motion control of AUVs mainly includes velocity and position tracking control, which plays an important role in many tasks, such as automatic docking, position keeping, precise exploration, reconnaissance and data acquisition [7,8,9]. When an AUV is in motion, due to the complex operating environment, current fluctuation and other uncertainties and disturbances, it is difficult to obtain an accurate dynamic model. In [7], Marco et al. developed a method for identifying the decoupled surge motion dynamic parameters of a small AUV. In [10], Fossen proposed PI control and PID control methods to design the controller for the accurate position tracking of the underwater vehicle’s surge motion. In [11], Smallwood et al. developed an adaptive controller while ignoring the influence of current. In [12], Riedel design a sliding mode variable structure controller for AUV without strict stability analysis. In [13], Gao studied the surge trajectory tracking problem of AUVs under the influence of current and presented an adaptive backstepping sliding mode control scheme. In [9], Herlambang et al. proposed a sliding-PID method for surge motion control and roll motion control. In [14], Yu et al. proposed an improved adaptive backstepping control algorithm to decrease the too large initial control input. In [15], Li et al. studied the backstepping control to solve the surge position tracking control problem of AUVs with input limitation. These research results promote the development of AUV control technology.

Figure 1. Six DOFs motions of AUVs.

The aforementioned control algorithms can work well while all system states can be accurately measurable. An AUV may be equipped with a compass module to measure its attitude, and angular velocities can be measured by an angular rate sensor. By using the typical positioning systems (including acoustic ultrashort baseline system, GPS, inertial navigation system, etc.), it is not difficult to obtain the positions of AUVs with very good accuracy. Usually, their translational velocities can be obtained by the Doppler velocity log (DVL). However, in some cases, due to economic and other practical limitations, AUVs may not be equipped with DVLs. Even when an AUV is equipped with a DVL, the DVL can only generate accurate velocity measurements provided that the distance to the seafloor is within a certain boundary, and the measured translational velocities by DVL may not be adequately accurate because of measurement noises [16]. During motion control design without velocity measurement or without accurate velocity measurement, it is necessary to obtain the accurate estimation of velocity for implementing an output feedback controller design. So far, there are mainly two technical means to estimate the velocity signal. A popular approach to velocity estimation entails the use of state observers [17]. Another approach is to use sliding mode differentiators [18]. For uncertain nonlinear systems, the state observer design is very complicated [19]. Even for the ideal situation, high-gain observers can be quite sensitive to measurement noises [18]. By contrast, sliding mode differentiators possess robustness to measurement noises [20]. In the context of motion control for AUVs, several output feedback control schemes have been proposed since the beginning of this century. In [21], the design and experimental results of a novel output feedback controller for slender-body underwater vehicles is proposed, with the velocity estimation achieved by using Luenberger observer. In [22,23], nonlinear observers are designed for velocity estimation. In [24], an adaptive output feedback controller based on dynamic recurrent fuzzy neural network observer is proposed. In [16], a finite-time-convergent observer is developed to solve the estimation problem of the translational velocities. Hitherto, in the context of AUV output feedback control, the results of using differentiators to estimate velocity signals are very few.

Real application systems usually operate under various constraints and restrictions arising from physical limitations or performance needs. Since this century, arising from the sustained research interest in the control field, a lot of results involving system constraints have been reported. These results promote the development of related technologies. Typical approaches of the class consist of model predictive control [25], reference governor [26] and barrier Lyapunov function [27]. In recent years, the design of state/output-constrained controllers based on barrier Lyapunov function has attracted great attention [28,29,30]. In the context of adaptive control design for AUVs, the number of relevant literature results is very limited [31]. The adaptive control design for AUVs with system constraints, as well as to meet the requirement of output feedback, is still a subject worthy of further study.

Motivated by the above discussion, this work investigates the output feedback adaptive control method for AUVs with deferred output constraint. The main contributions of this work can be summarized as follows:

(i)

The deferred output constraint in adaptive control design is implemented by constructing the auxiliary reference signal. Meanwhile, this construction is also beneficial to improve the smoothness of control input in the early stage of system operation.

(ii)

Both the state feedback control scheme and output feedback control scheme are developed for AUVs with deferred output constraints.

(iii)

By using the proposed control scheme, all the signals in closed loop are proved to be bounded, and better asymptotic tracking performance is achieved.

Remark 1.

PID algorithms are quite popular in industrial applications due to their simple algorithm and easy implementation. For the case that the derivative of output error is not measurable or cannot be accurately measured, PI control, as an alternative to PID control, is often adopted in the design of actual control system, which decreases the control accuracy. Moreover, the parameter tuning of PID control is not very easy. In PID control schemes, the demand of system constraint during system operation cannot be guaranteed.

The remainder of this paper is organized as follows. The problem formulation and the preliminaries are given in Section 2. The construction of the auxiliary reference signal is introduced in Section 3. State feedback control design and output feedback control design are discussed in Section 4 and Section 5, respectively. In Section 6, the numerical simulation comparison is carried out to verify the effectiveness of the proposed adaptive control scheme. Finally, Section 7 concludes this work.

2. Problem Formulation

The surge motion dynamics of AUVs can be described by the following formula as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \dot{x}=u+c,\hfill \\ & (m-X_{\dot{u}})\dot{u}=X_{u\left|u\right|}u\left|u\right|+(1-C_n)\tau +\Delta f(x,u)\hfill \end{array}\right.\end{array}$$

(1)

where x represents the surge displacement state of AUV, u is the surge forward velocity of AUV relative to ocean current, c is the velocity of ocean current, m and $X_{\dot{u}}$ are the mass of the AUV itself and the added mass, $X_{u\left|u\right|}$ is the nonlinear surge drag coefficient, $\tau$ is the thrust force and $\Delta f(x,u)$ is the unmodeled dynamics. In (1), m, $X_{\dot{u}}$, $X_{u\left|u\right|}$ and $C_n$ are unknown constants, and $m>X_{\dot{u}}$ holds. The reference trajectory is $x_d$, which satisfies Assumption 1.

The control objective of this work is to develop a controller such that x can track the given reference trajectory $x_d$, with $e=x-x_d$ being constrained during the operation of the AUV system.

Assumption 1.

$x_d$ is continuously differentiable, i.e., $x_d$ and $\frac{d{x}_d}{dt}$ are continuous with respect to time t.

Assumption 2.

c is time-varying but bounded, i.e., $\frac{dc}{dt}\ne 0$ and $|\frac{dc}{dt}|<d_c$, with $d_c$ being an unknown positive constant.

Lemma 1

([32]). For $h\in R$ and $\beta >0$,

$$\begin{array}{c}\hfill 0\le \left|h\right|-\frac{h^2}{\sqrt{h^2+{\beta }^2}}<\beta .\end{array}$$

(2)

Lemma 2

([33]).

Consider a linear and nonlinear combined differentiator as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\dot{\eta }}_1={\eta }_2\hfill \\ & {\epsilon }^2{\dot{\eta }}_2=-a_0({\eta }_1-v\left(t\right))-a_1|{\eta }_1{-v\left(t\right)|}^{a_1}sign({\eta }_1-v\left(t\right))-b_0\epsilon {\eta }_2-b_1{\left|\epsilon {\eta }_2\right|}^{a_2}sign\left(\epsilon {\eta }_2\right),\hfill \end{array}\right.\end{array}$$

where $0<a_2<1$, $\frac{a_2}{2-a_2}<a_1<1$, $a_0>0,b_0>0,b_1>0$ and $\epsilon >0$ are design parameters. For the continuous second-order differentiable signal $v\left(t\right)$, there exits $\zeta >\frac{2}{a_2}$ and $\Gamma >0$, which satisfy

$$\begin{array}{c}\hfill {\eta }_1-v\left(t\right)=o\left({\epsilon }^{a_2\zeta }\right),{\eta }_2-\dot{v}\left(t\right)=o\left({\epsilon }^{a_2\zeta -1}\right),\forall t>\epsilon \Gamma .\end{array}$$

(3)

Remark 2.

According to Assumption 2, the velocity of an ocean current is bounded, and the acceleration of an ocean current is also bounded, which is consistent with the actual situation.

For simplicity, in this paper, the arguments of functions are often omitted while no confusion arises.

3. Construction of Auxiliary Reference Signal

In output-constrained control based on barrier Lyapunov functions, the constraint parameter should be set large enough. For example, one designs a controller according to the barrier Lyapunov function as follows:

$$\begin{array}{c}\hfill V=\frac{e^2}{2(b_e^2-e^2)},\end{array}$$

(4)

where $b_e$ is the design parameter for implementing system constraint. During controller design, $b_e$ must be set properly to meet $b_e>\left|e\left(0\right)\right|$. Since $\left|e\right(0\left)\right|$ is different in each operation, it is a bit troublesome to set the appropriate value of $b_e$ during controller design. To overcome this problem, an auxiliary signal is formed as follows:

$$\begin{array}{c}\hfill x_d^{*}\left(t\right)=\left\{\begin{array}{cccc}& (x\left(0\right)-x_d\left(0\right)){\cos }^3\left(\frac{\pi t}{2t_d}\right)\nu +x_d\left(t\right),\hfill & & t<t_d\hfill \\ & x_d\left(t\right),\hfill & & t\ge t_d\hfill \end{array}\right.\end{array}$$

(5)

where

$$\begin{array}{c}\hfill \nu =\left\{\begin{array}{cccc}& 1,\hfill & & t<t_d,\hfill \\ & 0,\hfill & & t\ge t_d,\hfill \end{array}\right.\end{array}$$

(6)

$t_d>0$ is a design parameter. According to the above construction, we can see that both $x_d^{*}\left(0\right)=x\left(0\right)$, $x_d^{*}\left(t_d\right)=x_d\left(t_d\right)$, ${\dot{x}}_d^{*}\left(t_d\right)={\dot{x}}_d\left(t_d\right)$ and ${\ddot{x}}_d^{*}\left(t_d\right)={\ddot{x}}_d\left(t_d\right)$ hold. The construction of such auxiliary signal is the basis of achieving deferred output constraint. For an AUV, when the absolute value of the initial error is large, if the controller is designed according to the conventional method, the control input in the initial stage of operation may be very large, even exceeding the maximum power that the AUV can provide. In the process of controller design, using $x_d^{*}\left(t\right)$ instead of $x_d\left(t\right)$ as the reference signal is helpful to solve this problem.

Remark 3.

The purpose of constructing an auxiliary signal lies in the following two points: (i) it brings convenience to deferred constraint control design; (ii) it can make the control input smaller and smoother in the early stage of system operation.

4. State Feedback Control Design

4.1. Backstepping Control Design

From (1), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \dot{x}=u+c,\hfill \\ & \dot{u}=\frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}u\left|u\right|+\frac{1-C_n}{m-X_{\dot{u}}}\tau +\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)\hfill \end{array}\right.\end{array}$$

(7)

Define $e=x-x_d,z_1=x-x_d^{*}$ and $z_2=u-\alpha$, where $\alpha$ is a virtual controller to be designed. From the construction of $x_d^{*}$, we can see that $z_1\left(0\right)=0$ and $e\left(t\right)=z_1\left(t\right)$ for $t\ge t_d$. From (7), we obtain

$$\begin{array}{c}\hfill {\dot{z}}_1=u+c-{\dot{x}}_d=z_2+\alpha +c-{\dot{x}}_d^{*}.\end{array}$$

(8)

Then, we define a Lyapunov function as follows:

$$\begin{array}{c}\hfill V_1=\frac{z_1^2}{2(b_z^2-z_1^2)}.\end{array}$$

(9)

According to (8) and (9), we have

$$\begin{array}{c}\hfill {\dot{V}}_1=\sigma z_1(z_2+\alpha +c-{\dot{x}}_d^{*}),\end{array}$$

(10)

where $\sigma =\frac{b_z^2}{{(b_z^2-z_1^2)}^2}$. Through designing

$$\begin{array}{cc}\hfill \alpha =& {\dot{x}}_d^{*}-\frac{{\gamma }_1}{2}(b_z^2-z_1^2)z_1/b_z^2-\frac{{\widehat{c}}_{\rho }^2\sigma z_1}{\sqrt{{\widehat{c}}_{\rho }^2{\sigma }^2{z}_1^2+{\zeta }^2}},\hfill \end{array}$$

(11)

with $\iota >1,{\gamma }_1>\frac{1}{2}$. Applying Lemma 1 and substituting (11) into (10), we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_1=& -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma z_1(z_2+c-\frac{{\widehat{c}}_{\rho }^2\sigma z_1}{\sqrt{{\widehat{c}}_{\rho }^2{\sigma }^2{z}_1^2+{\zeta }^2}})\hfill \\ \hfill =& -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma z_1{z}_2+\sigma (z_{1}c-|z_1|{\widehat{c}}_{\rho })+\sigma |z_1|{\widehat{c}}_{\rho }-\frac{{\widehat{c}}_{\rho }^2{\sigma }^2{z}_1^2}{\sqrt{{\widehat{c}}_{\rho }^2{\sigma }^2{z}_1^2+{\zeta }^2}}\hfill \\ \hfill \le & -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma z_1{z}_2+\sigma \left|z_1\right|{\tilde{c}}_{\rho }+\zeta \hfill \end{array}$$

(12)

From (7), we obtain

$$\begin{array}{cc}\hfill {\dot{z}}_2=& \dot{u}-\dot{\alpha }\hfill \\ \hfill =& \frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}u\left|u\right|+\frac{1-C_n}{m-X_{\dot{u}}}\tau +\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)-\dot{\alpha }\hfill \end{array}$$

(13)

Let $\varpi =\frac{1-C_n}{m-X_{\dot{u}}}$, $\mathit{\theta }={[{\varpi }^{-1}\frac{X_{u\left|u\right|}}{m-X_{\dot{u}}},-\frac{1-C_n}{m-X_{\dot{u}}}]}^T,\mathit{\phi }=\left[u\right|u|,\dot{\alpha }{]}^T$. Define $V_2=\frac{1}{2\varpi }{z}_2^2$.

$$\begin{array}{cc}\hfill {\dot{V}}_2=& z_2[{\varpi }^{-1}\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)+{\mathit{\theta }}^T\mathit{\phi }+\tau ]\hfill \end{array}$$

(14)

A sigmoid neural network [34] is employed to approximate ${\varpi }^{-1}\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)$ as follows:

$$\begin{array}{cc}\hfill & {\varpi }^{-1}\frac{1}{m-X_{\dot{u}}}\Delta f(x,u)={\mathit{\omega }}^{*T}\mathit{\psi }\left(\mathit{X}\right)+ϵ\left(\mathit{X}\right)\hfill \end{array}$$

(15)

where $\left|ϵ\left(\mathit{X}\right)\right|\le {ϵ}_{x,N}$, with ${ϵ}_{x,N}$ being an unknown constant,

$$\begin{array}{c}\hfill \mathit{\psi }\left(\mathit{X}\right)=\frac{{\xi }_{x1}}{{\xi }_{x2}+\exp (-\mathit{X}/{\xi }_{x3})}+{\xi }_{x4},\end{array}$$

(16)

$\mathit{X}=[x,u,x_d,{\dot{x}}_d,{\ddot{x}}_d]$, ${\xi }_{x1}-{\xi }_{x4}$ are the design parameters, and ${ϵ}_N$ is an unknown bounded constant. Combining (15) with (14) yields

$$\begin{array}{cc}\hfill {\dot{V}}_2=& z_2[{\mathit{\omega }}^{*T}\mathit{\psi }\left(\mathit{X}\right)+ϵ\left(\mathit{X}\right)+{\mathit{\theta }}^T\mathit{\phi }+\tau ],\hfill \end{array}$$

(17)

Based on (12) and (17), the following control law is designed as

$$\begin{array}{cc}\hfill \tau =& -\sigma z_1-{\gamma }_2{z}_2-{\widehat{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{X}\right)-{\widehat{\mathit{\theta }}}^T\mathit{\phi }\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\omega }}}=& {\gamma }_3{z}_2\mathit{\psi }\left(\mathit{X}\right)-k_1{\gamma }_3\widehat{\mathit{\omega }},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\theta }}}=& {\gamma }_4{z}_2\mathit{\phi }-k_2{\gamma }_4\widehat{\mathit{\theta }},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\dot{\widehat{c}}}_{\rho }=& {\gamma }_5\sigma \left|z_1\right|-k_3{\gamma }_5{\widehat{c}}_{\rho },\hfill \end{array}$$

(21)

where ${\gamma }_2>1$, ${\gamma }_3>0,{\gamma }_4>0,{\gamma }_5>0,k_1>0$, $k_2>0,k_3>0$, and $\widehat{\mathit{\omega }}$ and $\widehat{\mathit{\theta }}$ are the estimations of ${\mathit{\omega }}^{*}$ and $\mathit{\theta }$, respectively.

4.2. Stability Analysis

The system stability and constraint property of the AUV surge motion is established by the following theorem.

Theorem 1.

Consider the closed-loop surge motion AUV system consisting of (1), (18)–(21). The boundedness of all signals in the closed-loop system and $|z_1|<b_z$ are guaranteed. The rectification error $z_1$ converges to a tunable set as follows:

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|z_1\right|\le \sqrt{\frac{{\rho }_1}{{\lambda }_1}}.\end{array}$$

(22)

The definitions of ${\rho }_1$ and ${\lambda }_1$ in (22) are available in (30).

Proof. 

Let us define a Lyapunov function as

$$\begin{array}{c}\hfill V=V_1+V_2+\frac{1}{2{\gamma }_3}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{1}{2{\gamma }_4}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+\frac{1}{2{\gamma }_5}{\tilde{c}}_{\rho }^2,\end{array}$$

(23)

where $\tilde{\mathit{\omega }}=\mathit{\omega }-\widehat{\mathit{\omega }},\tilde{\mathit{\theta }}=\mathit{\theta }-\widehat{\mathit{\theta }},{\tilde{c}}_{\rho }=c_{\rho }-{\widehat{c}}_{\rho }$.

Applying (18), we have

$$\begin{array}{cc}\hfill \dot{V}\le & -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma \left|z_1\right|{\tilde{c}}_{\rho }+\zeta -{\gamma }_1{z}_1^2-{\gamma }_2{z}_2^2+z_2[{\tilde{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{X}\right)+ϵ\left(\mathit{X}\right)+{\tilde{\mathit{\theta }}}^T\mathit{\phi }]\hfill \\ \hfill & +\frac{1}{{\gamma }_3}{\tilde{\mathit{\omega }}}^T\dot{\tilde{\mathit{\omega }}}+\frac{1}{{\gamma }_4}{\tilde{\mathit{\theta }}}^T\dot{\tilde{\mathit{\theta }}}+\frac{1}{{\gamma }_5}{\tilde{c}}_{\rho }{\dot{\tilde{c}}}_{\rho }.\hfill \end{array}$$

(24)

By employing (19), we obtain

$$\begin{array}{cc}\hfill & z_2{\tilde{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{X}\right)+\frac{1}{{\gamma }_3}{\tilde{\mathit{\omega }}}^T\dot{\tilde{\mathit{\omega }}}\hfill \\ \hfill =& z_2{\tilde{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{X}\right)-\frac{1}{{\gamma }_3}{\tilde{\mathit{\omega }}}^T({\gamma }_3{z}_2\mathit{\psi }\left(\mathit{X}\right)-k_1{\gamma }_3\widehat{\mathit{\omega }})\hfill \\ \hfill =& -k_1{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+k_1{\tilde{\mathit{\omega }}}^T\mathit{\omega }\hfill \\ \hfill \le & -k_1{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }\hfill \\ \hfill =& -\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }.\hfill \end{array}$$

(25)

By using (20), we obtain

$$\begin{array}{cc}\hfill & z_2{\tilde{\mathit{\theta }}}^T\mathit{\phi }+\frac{1}{{\gamma }_4}{\tilde{\mathit{\theta }}}^T\dot{\tilde{\mathit{\theta }}}\hfill \\ \hfill =& z_2{\tilde{\mathit{\theta }}}^T\mathit{\phi }-\frac{1}{{\gamma }_4}{\tilde{\mathit{\theta }}}^T({\gamma }_4{z}_2\mathit{\phi }-k_2{\gamma }_4\widehat{\mathit{\theta }})\hfill \\ \hfill =& -k_2{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+k_2{\tilde{\mathit{\theta }}}^T\mathit{\theta }\hfill \\ \hfill \le & -k_2{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+\frac{k_2}{2}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+\frac{k_2}{2}{\mathit{\theta }}^T\mathit{\theta }\hfill \\ \hfill =& -\frac{k_2}{2}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}+\frac{k_2}{2}{\mathit{\theta }}^T\mathit{\theta }.\hfill \end{array}$$

(26)

According to (24)–(26), we obtain

$$\begin{array}{cc}\hfill \dot{V}\le & -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma \left|z_1\right|{\tilde{c}}_{\rho }+\zeta -{\gamma }_1{z}_1^2-{\gamma }_2{z}_2^2+z_2ϵ\left(\mathit{X}\right)+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }+\frac{k_2}{2}{\mathit{\theta }}^T\mathit{\theta }\hfill \\ \hfill & +\frac{1}{{\gamma }_5}{\tilde{c}}_{\rho }{\dot{\tilde{c}}}_{\rho }-\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}-\frac{k_2}{2}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}.\hfill \end{array}$$

(27)

Similar to (25), from (21), we can derive

$$\begin{array}{c}\hfill \sigma |z_1|{\tilde{c}}_{\rho }+\frac{1}{{\gamma }_5}{\tilde{c}}_{\rho }{\dot{\tilde{c}}}_{\rho }\le -\frac{k_3}{2}{\tilde{c}}_{\rho }^2+\frac{k_3}{2}{c}_{\rho }^2\end{array}$$

(28)

Combining (28) with (27), we obtain

$$\begin{array}{cc}\hfill \dot{V}\le & -\frac{{\gamma }_1}{2}\frac{z_1^2}{b_z^2-z_1^2}+\zeta -{\gamma }_1{z}_1^2-({\gamma }_2-\frac{1}{2})z_2^2+\frac{1}{2}{ϵ}_{x,N}^2+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }\hfill \\ \hfill & +\frac{k_2}{2}{\mathit{\theta }}^T\mathit{\theta }+\frac{k_3}{2}{c}_{\rho }^2-\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}-\frac{k_2}{2}{\tilde{\mathit{\theta }}}^T\tilde{\mathit{\theta }}-\frac{k_3}{2}{\tilde{c}}_{\rho }^2\hfill \\ \hfill \le & {\lambda }_{1}V+{\rho }_1,\hfill \end{array}$$

(29)

in which ${\lambda }_1=min({\gamma }_1,2\varpi ({\gamma }_2-\frac{1}{2}),k_1{\gamma }_3,k_2{\gamma }_4,k_3\mu )$, ${\rho }_1=\zeta +\frac{1}{2}{ϵ}_N^2+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }+\frac{k_2}{2}{\mathit{\theta }}^T\mathit{\theta }+\frac{k_3}{2}{c}_{\rho }^2$. Then, from (28), we obtain

$$\begin{array}{c}\hfill V\left(t\right)\le {\mathrm{e}}^{-\lambda t}(V\left(0\right)-\frac{{\rho }_1}{{\lambda }_1})+\frac{{\rho }_1}{{\lambda }_1}.\end{array}$$

(30)

Since $V\left(0\right)$ is bounded, from (30), we can draw a conclusion that $V\left(t\right)$ is bounded for $t>0$. According to the definition of V, we can see that $\tilde{\mathit{\omega }}$, $\widehat{\mathit{\omega }}$ and $z_2$ are bounded, and

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|z_1\right|\le \sqrt{\frac{{\rho }_1}{{\lambda }_1}}.\end{array}$$

(31)

Hence, by setting proper parameters $\zeta ,\mu$, $k_1-k_3$ and ${\gamma }_1-{\gamma }_4$, we can achieve the desired steady-state accuracy. On the other hand, since $b_z>\left|z_1\left(0\right)\right|$ holds, from (30), we can see that $b_z^2-z_1^2>0$ holds for $t>0$, which implies

$$\begin{array}{c}\hfill |z_1\left(t\right)|<b_z,\forall t>0.\end{array}$$

(32)

According to the construction of $x_d^{*}$, it follows from (32) that $|x_1\left(t\right)-x_d\left(t\right)|<b_z$ holds for $t\ge t_d$. Furthermore, we can see that $|x_1\left(t\right)|<b_z+\left|x_d\right|$ holds for $t\ge t_d$. □

5. Output Feedback Control Design

In order to derive the adaptive control scheme based on output feedback, let us first consider the adaptive control design problem when u and c are known.

Define $\varsigma =u+c$. We perform the following coordinate transformation for system (1) as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \dot{x}=\varsigma ,\hfill \\ & \dot{\varsigma }=\frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}(\varsigma -c)|\varsigma -c|+\frac{1}{m-X_{\dot{u}}}\Delta f(\varsigma -c)-\dot{c}+\frac{1-C_n}{m-X_{\dot{u}}}\tau .\hfill \end{array}\right.\end{array}$$

(33)

Let $z_1=x-x_d^{*}$ and $z_2=\varsigma -\alpha$, where $\alpha ={\dot{x}}_d^{*}-\frac{\iota }{2}(b_z^2-z_1^2)z_1/b_z^2$.

Similar to the derivation in (10)–(12), by calculating the time derivative of $V_1=\frac{z_1^2}{2(b_z^2-z_1^2)}$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_1=& \sigma z_1(z_2+\alpha -{\dot{x}}_d^{*})=-\frac{\iota }{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma z_1{z}_2,\hfill \end{array}$$

(34)

where $\sigma =\frac{b_z^2}{{(b_z^2-z_1^2)}^2}$. On the basis of (33) and the definition of $z_2$, we obtain

$$\begin{array}{cc}\hfill {\dot{z}}_2=& \dot{\varsigma }-\dot{\alpha }\hfill \\ \hfill =& \frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}(\varsigma -c)|\varsigma -c|+\frac{1}{m-X_{\dot{u}}}\Delta f(\varsigma -c)-\dot{c}+\frac{1-C_n}{m-X_{\dot{u}}}\tau -\dot{\alpha }\hfill \end{array}$$

(35)

Define $V_2=\frac{1}{2\varpi }{z}_2^2$ with $\varpi =\frac{1-C_n}{m-X_{\dot{u}}}$. According to (35), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2=& z_2[{\varpi }^{-1}(\frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}(\varsigma -c)|\varsigma -c|+\frac{1}{m-X_{\dot{u}}}\Delta f(\varsigma -c)-\dot{c})-{\varpi }^{-1}\dot{\alpha }+\tau ]\hfill \end{array}$$

(36)

Similar to (15), the following equation holds as

$$\begin{array}{cc}\hfill & {\varpi }^{-1}(\frac{X_{u\left|u\right|}}{m-X_{\dot{u}}}(\varsigma -c)|\varsigma -c|+\frac{1}{m-X_{\dot{u}}}\Delta f(\varsigma -c)-\dot{c})={\mathit{\omega }}^{*T}\mathit{\psi }\left(\mathit{Y}\right)+ϵ\left(\mathit{Y}\right),\hfill \end{array}$$

(37)

where $\left|ϵ\left(\mathit{Y}\right)\right|\le {ϵ}_{y,N}$, $\mathit{\psi }\left(\mathit{Y}\right)=\frac{{\xi }_{y1}}{{\xi }_{y2}+\exp (-\mathit{Y}/{\xi }_{y3})}+{\xi }_{y4}$, $\mathit{Y}=[x,\varsigma ,x_d^{*},{\dot{x}}_d^{*},{\ddot{x}}_d^{*}]$, ${\xi }_{y1}-{\xi }_{y4}$ are the design parameters and ${ϵ}_{y,N}$ is an unknown bounded constant.

Combining (36) with (37) yields

$$\begin{array}{cc}\hfill {\dot{V}}_2=& z_2[{\mathit{\omega }}^{*T}\mathit{\psi }\left(\mathit{Y}\right)+ϵ\left(\mathit{Y}\right)+p\dot{\alpha }+\tau ],\hfill \end{array}$$

(38)

where p denotes $-\varpi$. Based on (38), the following control law is designed as

$$\begin{array}{cc}\hfill \tau =& -\sigma z_1-{\mu }_1{z}_2-{\widehat{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{Y}\right)-\widehat{p}\dot{\alpha }\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\omega }}}=& {\mu }_2{z}_2\mathit{\psi }\left(\mathit{Y}\right)-k_1{\mu }_3\widehat{\mathit{\omega }},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \dot{\widehat{p}}=& {\mu }_3{z}_2\dot{\alpha }-k_2{\mu }_4\widehat{p},\hfill \end{array}$$

(41)

where ${\mu }_1>1$, ${\mu }_2>0,{\mu }_3>0,k_1>0$, $k_2>0$, and $\widehat{\mathit{\omega }}$ and $\widehat{\mathit{\theta }}$ are the estimations of ${\mathit{\omega }}^{*}$ and $\mathit{\theta }$, respectively.

Theorem 2.

Consider the closed-loop surge motion AUV system consisting of (1), (39)–(41), with u and c assumed to be known. The boundedness of all signals in the closed-loop system and $|z_1|<b_z$ are guaranteed. The rectification error $z_1$ converges to a tunable set as follows:

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|z_1\right|\le \sqrt{\frac{{\rho }_2}{{\lambda }_2}}.\end{array}$$

(42)

The definitions of ${\rho }_2$ and ${\lambda }_2$ in (42) are available in (45).

Proof. 

Let us define a Lyapunov function as

$$\begin{array}{c}\hfill V=V_1+V_2+\frac{1}{2{\mu }_2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{1}{2{\mu }_3}{\tilde{p}}^2,\end{array}$$

(43)

where $\tilde{\mathit{\omega }}=\mathit{\omega }-\widehat{\mathit{\omega }},\tilde{p}=p-\widehat{p}$.

Applying (34), (38) and (39), we have

$$\begin{array}{cc}\hfill \dot{V}\le & -\frac{\iota }{2}\frac{z_1^2}{b_z^2-z_1^2}+\sigma z_1{z}_2-\sigma z_1{z}_2-{\mu }_1{z}_2^2+z_2[{\tilde{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{Y}\right)+ϵ\left(\mathit{Y}\right)+\tilde{p}\dot{\alpha }]\hfill \\ \hfill & +\frac{1}{{\mu }_2}{\tilde{\mathit{\omega }}}^T\dot{\tilde{\mathit{\omega }}}+\frac{1}{{\mu }_3}\tilde{p}\dot{\tilde{p}}.\hfill \end{array}$$

(44)

Similar to (25), from (40) and (41), we can obtain $z_2{\tilde{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{Y}\right)+\frac{1}{{\mu }_2}{\tilde{\mathit{\omega }}}^T\dot{\tilde{\mathit{\omega }}}\le -\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }$ and $z_2\tilde{p}\dot{\alpha }+\frac{1}{{\mu }_3}\tilde{p}\dot{\tilde{p}}\le -\frac{k_2}{2}{\tilde{p}}^2+\frac{k_2}{2}{p}^2$. From the above three inequalities, we can obtain

$$\begin{array}{cc}\hfill \dot{V}\le & -\frac{\iota }{2}\frac{z_1^2}{b_z^2-z_1^2}-({\mu }_1-\frac{1}{2})z_2^2-\frac{k_1}{2}{\tilde{\mathit{\omega }}}^T\tilde{\mathit{\omega }}-\frac{k_2}{2}{\tilde{p}}^2+\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }+\frac{k_2}{2}{p}^2+\frac{1}{2}{ϵ}_{y,N}^2\hfill \\ \hfill \le & {\lambda }_{2}V+{\rho }_2,\hfill \end{array}$$

(45)

in which ${\lambda }_2=min(\iota ,2\varpi ({\mu }_1-\frac{1}{2}),k_1{\mu }_2,k_2{\mu }_4)$, ${\rho }_2=\frac{k_1}{2}{\mathit{\omega }}^T\mathit{\omega }+\frac{k_2}{2}{p}^2+\frac{1}{2}{ϵ}_{y,N}^2$. Up to this point, similar to the corresponding analysis in Section 4, we can conclude that $|z_1\left(t\right)|<b_z$ holds, all signals in the closed AUV system are bounded and

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|z_1\right|\le \sqrt{\frac{{\rho }_2}{{\lambda }_2}}.\end{array}$$

(46)

□

It should be noted that the control algorithm (39)–(41) is applicable only to the case that both u and c are available. To be specific, when u is not measurable, $\varsigma$ and $z_2$ become uncertain variables, which causes the adaptive control algorithm (39)–(41) to be invalid. On the other hand, although the adaptive control algorithm (39)–(41) is invalid when u or c is unknown, it can still provide reference and help for controller design in the cases that only x is available.

To proceed with the controller design, we construct the following combined differentiator to estimate $\varsigma \left(t\right)$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \dot{\chi }=\widehat{\varsigma }\hfill \\ & {\epsilon }^2\dot{\widehat{\varsigma }}=-a_0(\chi -x\left(t\right))-a_1{|\chi -x\left(t\right)|}^{a_1}sign(\chi -x\left(t\right))-b_0\epsilon \widehat{\varsigma }-b_1{\left|\epsilon \widehat{\varsigma }\right|}^{a_2}sign\left(\epsilon \widehat{\varsigma }\right),\hfill \end{array}\right.\end{array}$$

(47)

According to Lemma 2, by reasonably setting parameters $\epsilon$, $a_0,a_1,a_2$, $b_0$ and $b_1$, $\widehat{\varsigma }$ can accurately track the signal $\varsigma \left(t\right)$, such that ${\widehat{z}}_2=\widehat{\varsigma }-\alpha$ is an accurate estimation of $z_2$.

On the basis of the above signal reconstruction, we can develop the output feedback barrier adaptive control law as follows:

$$\begin{array}{cc}\hfill \tau =& -\sigma z_1-{\mu }_1{\widehat{z}}_2-{\widehat{\mathit{\omega }}}^T\mathit{\psi }\left(\widehat{\mathit{Y}}\right)-\widehat{p}\dot{\alpha }\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\omega }}}=& {\mu }_2{\widehat{z}}_2\mathit{\psi }\left(\widehat{\mathit{Y}}\right)-k_1{\mu }_3\widehat{\mathit{\omega }},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \dot{\widehat{p}}=& {\mu }_3{\widehat{z}}_2\dot{\alpha }-k_2{\mu }_4\widehat{p},\hfill \end{array}$$

(50)

where

$$\begin{array}{c}\hfill \mathit{\psi }\left(\widehat{\mathit{Y}}\right)=\frac{{\xi }_{\widehat{y}1}}{{\xi }_{\widehat{y}2}+\exp (-\widehat{\mathit{Y}}/{\xi }_{\widehat{y}3})}+{\xi }_{\widehat{y}4},\end{array}$$

(51)

$\widehat{\mathit{Y}}=[x,\widehat{\varsigma },x_d^{*},{\dot{x}}_d^{*},{\ddot{x}}_d^{*}]$, ${\xi }_{\widehat{y}1}-{\xi }_{\widehat{y}4}$ are design parameters of the neural network, and the other variables and parameters are similar to the corresponding ones in (39)–(41).

Remark 4.

Compared with the similar research results given in [13,14,15], the output feedback barrier adaptive control scheme (48)–(50) possesses the following characteristics:

(1) 

The proposed control scheme may be used in the cases that velocity signals are unmeasurable. Therefore, the applicable scope is extended.

(2) 

By incorporating the initial rectification technique into the barrier backstepping design, deferred output constraint adaptive control scheme is developed. So the robustness and safety of AUV systems have been improved.

6. Numerical Simulation

The REMUS UAV model is used to verify the performance of the controller, and the simulation parameters are selected according to [35] as follows: $m=30.5$, $X_{u\left|u\right|}=-1.62$, $X_{\dot{u}}=-0.93$, $c=0.5+0.02\sin \left(0.01\pi t\right),C_u=0.2$. $\Delta f(x,u)$ is set as $2+\sin \left(u\right)\cos \left(x\right)$. The reference trajectory is set as $x_d=0.5+0.5\cos \left(\frac{\pi t}{6}\right)$. The initial state is $x\left(0\right)=2$ m and $u\left(0\right)=0.2$ m/s.

The output feedback barrier adaptive control scheme (48)–(50) is adopted for simulation, with the parameters and gains in the proposed adaptive control scheme (48)–(50) set as follows: $t_d=2,b_z=0.3,k_1=0.05,k_2=0.05,k_3=0.05,{\mu }_1=10,{\mu }_2=10,{\mu }_3=10$, ${\xi }_{\widehat{y}1}=2,{\xi }_{\widehat{y}2}=10,{\xi }_{\widehat{y}3}=1,{\xi }_{\widehat{y}4}=-1$, $\epsilon =0.004,a_0=5,a_1=0.5,b_0=2,b_1=0.5$. The simulation results are shown in Figure 2, Figure 3, Figure 4 and Figure 5. Figure 2 shows the tracking curve of surge displacement, with the corresponding tracking error plotted in Figure 3. From Figure 2 and Figure 3, we can see that a favorable tracking performance was obtained. From Figure 4, we can see that $|z_1|$ is constrained within $[0,b_z]$. Figure 5 shows the profile of the control input.

Figure 2. Surge displacement(output feedback barrier adaptive control).

Figure 3. Surge tracking error (output feedback barrier adaptive control).

Figure 4. Deferred output error constraint (output feedback barrier adaptive control).

Figure 5. Control input (output feedback barrier adaptive control).

For comparison, a traditional adaptive control algorithm is adopted for simulation as follows:

$$\begin{array}{cc}\hfill \tau =& -z_1^c-{\gamma }_2{z}_2^c-{\widehat{\mathit{\omega }}}^T\mathit{\psi }\left(\mathit{X}\right)-{\widehat{\mathit{\theta }}}^T\mathit{\phi }\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\omega }}}=& {\gamma }_3{z}_2^c\mathit{\psi }\left(\mathit{X}\right)-k_1{\gamma }_3\widehat{\mathit{\omega }},\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{\theta }}}=& {\gamma }_4{z}_2^c\mathit{\phi }-k_2{\gamma }_4\widehat{\mathit{\theta }},\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\dot{\widehat{c}}}_{\rho }=& {\gamma }_5\left|z_1^c\right|-k_3{\gamma }_5{\widehat{c}}_{\rho },\hfill \end{array}$$

(55)

where $z_1^c=x-x_d$, $z_2^c=u-{\dot{\alpha }}_c$, ${\alpha }_c={\dot{x}}_d-{\gamma }_1{z}_1^c-\frac{{\widehat{c}}_{\rho }^2{z}_1^c}{\sqrt{{\widehat{c}}_{\rho }^2{z}_1^{c2}+{\zeta }^2}}$.

$$\begin{array}{c}\hfill \mathit{\psi }\left(\mathit{X}\right)=\frac{{\xi }_{x1}}{{\xi }_{x2}+\exp (-\mathit{X}/{\xi }_{x3})}+{\xi }_{x4},\end{array}$$

(56)

$\mathit{X}=[x,u,x_d,{\dot{x}}_d,{\ddot{x}}_d]$. The design parameters and gains are set follows: $\zeta =0.1$, ${\gamma }_1=10,{\gamma }_2=10,{\gamma }_3=10,{\gamma }_4=10,{\gamma }_5=0.1,k_1=0.05,k_2=0.05,k_3=0.05$, ${\xi }_{x1}=2,{\xi }_{x2}=10,{\xi }_{x3}=1,{\xi }_{x4}=-1$. Comparing Figure 2 and Figure 3 with Figure 6 and Figure 7, we can see that the tracking precision in the output feedback barrier adaptive control scheme is better than that in the traditional adaptive control scheme. Comparing Figure 4 with Figure 8, we can see that the deferred output error constraint is achieved in the output feedback barrier adaptive control scheme, whereas no system constraint is achieved in the traditional adaptive control scheme. Comparing Figure 5 with Figure 9, we can see that the control input of the output feedback barrier adaptive control is smoother and smaller than that of the traditional adaptive control scheme in the early stage of operation.

Figure 6. Surge displacement (traditional adaptive control).

Figure 7. Surge tracking error (traditional adaptive control).

Figure 8. Absolute value of surge position tracking error (traditional adaptive control).

Figure 9. Control input (traditional adaptive control).

7. Conclusions

The surge position tracking control problem of AUVs is studied in this work. On the basis of constructing an auxiliary trajectory signal and adopting the barrier Lyapunov function, we develop a state-feedback deferred constraint adaptive backstepping control scheme at first. After that, by using coordinate transformation to transform the original system into a chained form system, and by utilizing a linear and nonlinear combined differentiator to accurately estimate the derivative of surge displacement, we develop the output feedback deferred output constraint adaptive controller for AUVs whose velocity signals are not measurable. Finally, the simulation results verify the effectiveness of the proposed adaptive control schemes.