Composite learning adaptive sliding mode control for AUV target tracking

Communicated by Dr Jing Na

Accepted Manuscript

Composite Learning Adaptive Sliding Mode Control for AUV Target Tracking Yuyan Guo, Hongde Qin, Bin Xu, Yi Han, Quan-Yong Fan, Pengchao Zhang PII: DOI: Reference:

S0925-2312(19)30373-X https://doi.org/10.1016/j.neucom.2019.03.033 NEUCOM 20567

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

22 October 2018 13 March 2019 25 March 2019

Please cite this article as: Yuyan Guo, Hongde Qin, Bin Xu, Yi Han, Quan-Yong Fan, Pengchao Zhang, Composite Learning Adaptive Sliding Mode Control for AUV Target Tracking, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.03.033

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Composite Learning Adaptive Sliding Mode Control for AUV

April 17, 2019

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Target Tracking

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Yuyan Guo1 , Hongde Qin2 , Bin Xu2,1∗ , Yi Han1 , Quan-Yong Fan1 , Pengchao Zhang3

1. School of Automation, Northwestern Polytechnical University, Xi’an, China, 710000 2. Science and Technology on Underwater Vehicle Laboratory, Harbin Engineering University, Harbin, China, 150000

3. Key Laboratory of Industrial Automation of Shaanxi Province, Shaanxi University of Technology,

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Hanzhong, Shaanxi, 723000, China

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Abstract

This paper studies the controller design for an autonomous underwater vehicle (AUV) with the target tracking task. Considering the uncertainty the nonlinear longitudinal model, a sliding mode controller is

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designed. Meanwhile the neural networks (NNs) are used to approximate the unknown nonlinear function in the model. To improve the NNs learning rapidity, the prediction error which reflect the learning performance is constructed, further the updating law is designed utilizing the composite learning technique.

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The system stability is guaranteed through the Lyapunov approach. The simulation results verify that the designed method could force the AUV to track the target until rendezvous, and the model uncertainty is

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addressed better via the composite learning algorithm.

Index Terms – Autonomous Underwater Vehicle, Target Tracking, Sliding Mode Control, Composite Learning, Neural Networks

1

Introduction

Autonomous underwater vehicle has been widely studied due to its potential civil and military value. With the capability of highly autonomous action, the civil AUVs are applied for marine monitoring, seafloor mapping ∗ To

whom all correspondences should be addressed.

E-mail: [email protected]

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and oil or gas industry, while in military applications, AUVs could be used for mine countermeasures and threat target tracking. When performing path following or position control tasks, it is important to design precision AUV controller, while advanced control approaches such as intelligent control[1], dynamic surface control [2] and the finite-time tracking control [3, 4] have been studied for surface and underwater vehicles, servo systems in recent years. Some techniques on multi-agent control [5, 6] are also concerned on the multi-

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AUV control studies [7]. Based on linearized model, representative approaches including output feedback control [8], gain-scheduling control [9] and H∞ control [10] have been studied for the path-following and diving mission of AUVs, while nonlinear control methods need to be further studied to ensure the global performance [11]. The highly nonlinear dynamic behavior of the AUV and uncertainties in hydrodynamic coefficients makes it important to investigate nonlinear robust control approaches for AUVs [12]. Studies on compensation-based robust control

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have been successfully applied to the quadrotors [13], the servomechanisms [14], the manipulators [15] and the AUVs [16, 17] via estimating the uncertainty and disturbance or their upper bounds. In [18, 19, 20], the sliding mode control have been applied to AUVs and spacecrafts to make the system robust to the model uncertainty and external disturbance. Intelligent techniques including neural networks [21, 22, 23] and fuzzy systems [24] are also utilized in AUVs where the disturbances and the model uncertainties could be compensated via the

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function approximation.

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With the universal approximation ability, NNs have been widely applied to nonlinear systems control to approximate the uncertain information [25, 26]. Unlike the usual machine learning method, the NN weights updating laws in controller design are commonly constructed based on Lyapunov stability analysis, while the

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tracking error plays an important role. Some studies have concentrated on improving the dynamic and steady learning performance based on the finite time technique [27], the boundary iterative learning [28], the echo

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networks [29] and the fuzzy systems [30], etc. In [31], based on the serial-parallel estimation model (SPEM) of the system states, the prediction error is obtained and further applied in the NNs weights updating law.

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In [32], the global stability under NNs-based control is investigated by constructing switching laws. It is noticed that the leaning performance is often ignored in such design. To introduce the learning performance index in updating laws, some studies concentrates on designing composite learning laws by constructing auxiliary filter [33, 34, 35] or utilizing time-interval data [36]. Considering the task of tracking a moving target, the AUV control requires a faster convergence. Due to the existence of the model uncertainty, the fast estimation of unknown nonlinear functions should be resolved. This motivate us to apply the composite learning approach in the AUV control. In this paper, a compositelearning-based sliding mode controller is constructed for an AUV longitudinal model. The main contributions are summarized as follows: 1) Considering a new AUV nonlinear dynamics model with the target tracking mission (instead of tracking a 2

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solid path or depth), the dynamics is transferred based on LOS guidance to simplify the controller design. 2) In the designed sliding mode controller, the composite learning technique is utilized to ensure the fast convergence of the uncertainty approximation, while the NN weights updating law is constructed and modified based on the sliding mode surface and the prediction error. This paper is arranged as follows. The AUV dynamics model is studied in Section 2, to simplify the design,

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the target tracking is transformed to the pitch angle command, further a strict feed back model contains the pitch angle and the pitch rate is obtained. Section 3 designs a sliding mode controller for the transformed model. In Section 4, the stability of the system is analyzed via the Lyapunov approach. The simulation results are given in Section 5 to show the effectiveness of the controller. Finally the study is summarized in Section

2 2.1

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6.

Dynamics Analysis and Problem Formulation Longitudinal Dynamics of AUV

v˙ = k11 v2 + k14 sin Θ + k18


1 k21 v2 α + k22 vωz + k23 α + k24 α sin Θ + k25 cos Θ + k26 cos θ + k27 sin θ + k28 + k29 v2 δe v

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α˙ =

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This paper considers the following AUV longitudinal model:

ω˙ z = k31 v2 α + k32 vωz + k34 α sin Θ + k35 cos Θ + k36 cos θ + k37 sin θ + k38 + k39 v2 δe

y˙m = v sin Θ

(3)

(5) (6)

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x˙m = v cos Θ

(2)

(4)

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θ˙ = ωz

(1)

Θ = θ −α

(7)

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where v is the velocity, α is the angle of attack, ωz is the pitch rate, θ is the pitch angle, Θ is the path angle, xm and ym are the coordinates of the AUV in the ground coordinate system. The control input of the system is the elevator deflection δe . The definitions of the coefficients in (1)-(7) could be found in the Appendix.

2.2 Dynamic Transformation Considering the AUV (xm , ym ) and the target (xd , yd ) in the ground coordinate system defined in Fig. 1: The line-of-sight (LOS) angle q between the AUV and the target satisfies:   yd − ym q = arctan xd − xm 3

(8)

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y x q

 xd  y d

v

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Figure 1: Frame Definition

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D

 xm  ym

To ensure that the AUV could track the target, the AUV path angle Θ should be equal to q, consequently the velocity vector of AUV will always point at the target position. Therefore, the target tracking command could

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be transformed to the pitch angle tracking command:   yd − ym θr = arctan +α xd − xm

(9)

Let θr pass the following filter to obtain the pitch angle reference signal x1d :

(10)

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2 ωt1 ωt2
2 (s + ωt1 ) s2 + 2ξ ωt1 s + ωt2

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Define x1 = θ , x2 = ωz , then (3) and (4) could be transferred as: x˙1 = x2

(11)

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x˙2 = f2 + g2 δe

where f2 = k31 v2 α + k32 vωz + k34 α sin Θ + k35 cos Θ + k36 cos θ + k38 , g2 = ωg2 θg2 , ωg2 = k39 , θg2 = v2 .

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Assumption 1 The predefined pitch angle reference signal x1d and its derivatives are continuous and availn o 2 + x˙2 + x¨2 ≤ B able, and [x1d , x˙1d , x¨1d ]T ∈ Ωd with known compact set Ωd = [x1d , x˙1d , x¨1d ]T : x1d ⊂ R3 , 0 1d 1d

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whose size B0 is a known positive constant.

Remark 1 The mission of the AUV in this paper is to track the target until reach the minimum distance. Utilizing the LOS guidance law, the position tracking of system (1)-(7) is transferred to the pitch angle tracking command, and further a simple second-order system is obtained to simplify the controller design.

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Control Goal

Considering the dynamic uncertainty in system (11), the control goal is to design a composite learning sliding mode controller δe such that the pitch angle x1 could track the reference signal x1d , further the AUV could reach the position of the target. 4

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3 3.1

Controller Design RBF Neural Networks

In this paper, the RBF Neural Networks (RBFNNs) framework is used to approximate the unknown nonlinear

1, 2, · · · , l are obtained via Gaussian function:

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function f2 . The inputs are chosen as Xin = [Θ, θ , ωz ]T . There are l hidden layer nodes, where θ f 2 j , j =

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Xin − c j 1 θ f 2 j = √ exp − , j = 1, 2, · · · l 2σ 2j 2π

where c j denotes the center vector of the jth hidden neuron.

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With the universal approximation property, it is known that there exist optimal weights vector ω ∗f 2 such that:

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f2 = ω ∗T f 2 θ f 2 + ε2

(13)

where θ f 2 = [θ f 21 , θ f 22 , · · · θ f 2m ]T , ε2 is the inevitable bounded reconstruction error which satisfies |ε2 | ≤ ε2m .  T The NN weights vector ωˆ f 2 = ωˆ f 21 , ωˆ f 22 , · · · ωˆ f 2l in this paper is updated via the composite learning-based

Sliding Mode Control

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3.2

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adaptive law (20), which will be further explained.

In this section, the sliding mode controller is designed for system (11) based on RBFNNs. Define the pitch

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angle tracking error e1 = x1 − x1d , then its derivative is obtained as: e˙1 = x˙1 − x˙1d = x2 − x˙1d

(14)

s = ae1 + e˙1

(15)

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Design the sliding mode surface s as:

where a > 0 is a user-defined parameter. Utilizing the RBF NNs to estimate f2 , the derivative of s is calculated as s˙ = ae˙1 + e¨1 = a (x2 − x˙1d ) + ω ∗T f 2 θ f 2 + ε2 + g2 δe − x¨1d

(16)

The control input δe is designed as gˆ2 δe = −ωˆ Tf2 θ f 2 − ae˙1 − k1 s − k2 |s|r sgn (s) + x¨1d 5

(17)

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where k1 and k2 are the positive design parameters, 0 < r < 1 is a user-defined parameter, ωˆ 2 is designed in to estimate ω2∗ , gˆ2 = ωˆ g2 θg2 with ωˆ g2 as the estimation of ωg2 . Define the prediction error: zn2 = x2 − xˆ2

(18)

x˙ˆ2 = ωˆ Tf2 θ f 2 + gˆ2 δe + β2 zn2 where β2 is a user-defined positive parameter. The adaptive law of ωˆ f 2 is designed as

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ω˙ˆ f 2 = γ f 2 [θ f 2 (s + γz2 zn2 ) − δ f 2 ωˆ f 2 ]

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where xˆ2 is obtained via the following SPEM [31]: (19)

(20)

with γ f 2 > 0, γz2 > 0 and δ f 2 > 0 as the user-defined parameters. The adaptive law of ωˆ g2 is designed as

ω˙ˆ g2 = Proj {γg2 [θg2 δe (s + γz2 zn2 ) − δg2 ωˆ g2 ]}

(21)

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with γg2 > 0 and δg2 > 0 as the user-defined parameters. The parameter projection function Proj (τ) has the   τ      

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following form:

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Proj (τ) =

      

0

if ω g2 ≤ ωˆ g2 ≤ ω¯ g2




or (ωˆ g2 = ω¯ g2 , τ < 0)
or ωˆ g2 = ω g2 , τ > 0

(22)

otherwise

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where ω g2 and ω¯ g2 are the known lower and upper bound of ωg2 .

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Defining ω˜ f 2 = ω ∗f 2 − ωˆ f 2 and ω˜ g2 = ωg2 − ωˆ g2 yields: s˙ = −k1 s − k2 |s|r sgn (s) + ω˜ Tf2 θ f 2 + ε2 + ω˜ g2 θg2 δe

(23)

z˙n2 = ω˜ Tf2 θ f 2 + ε2 + ω˜ g2 θg2 δe − β2 zn2

(24)

and

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Stability Analysis

Theorem 1 Considering the closed-loop system (11) under Assumption 1, together with the designed control law (17) and the adaptive laws (19), (20), (21), the error signals in (25) are uniformly ultimately bounded (UUB). 6

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Proof. Choose the Lyapunov function as 1 1 1 −1 2 ˜ f 2 + γg2 LA = s2 + ω˜ Tf2 γ −1 ω˜ g2 f2 ω 2 2 2

(25)

The derivatives of (25) is calculated as

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˙ˆ f 2 − γ −1 ω˜ g2 ω˙ˆ g2 + γz2 zn2 z˙n2 L˙ A = ss˙ − ω˜ Tf2 γ −1 g2 f2 ω

(26)

Substituting the error dynamics (23), (24) and adaptive laws (20), (21), into (26), we have:

L˙ A = −k1 s2 − k2 |s|r+1 − γz2 β2 z2n2 + sε2 + δ f 2 ω˜ Tf2 ωˆ f 2 + δg2 ω˜ g2 ωˆ g2 + γz2 zn2 ε2 Considering the following inequalities:

(28) (29) (30) (31)

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1 2 1 sε2 ≤ s2 + ε2m 2 2 1 2 1 2 γz2 zn2 ε2 ≤ γz2 zn2 + ε2m 2 2

1 1 2 δ f 2 ω˜ Tf2 ωˆ f 2 ≤ δ f 2 ω ∗f 2 − δ f 2 ω˜ Tf2 ω˜ f 2 2 2 1 1 2 2 δg2 ω˜ g2 ωˆ g2 ≤ δg2 ωg2 − δg2 ω˜ g2 2 2

(27)

(32)

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≤ −σ LA + Pa

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Then we know L˙ A satisfies     2 2 2 ˙LA ≤ − k1 − 1 s2 − δ f 2 ω˜ Tf2 ω˜ f 2 − γz2 β2 − 1 z2n2 − δg2 ω˜ g2 + ε2m + δ f 2 kω2∗ k2 + δg2 ωg2 2 2

 σ = min (2k1 − 1) , δ f 2 γ f 2 , (2β2 − 1) , δg2 γg2

2 2 Pa = ε2m + δ f 2 kω2∗ k2 + δg2 ωg2

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where

Furthermore, the following inequality can be obtained   Pa −σt Pa 0 ≤ LA ≤ + LA (0) − e σ σ

(33)

Then it is concluded that all the signals in the Lyapunov function (25) are bounded. Define N = ω˜ Tf2 θ f 2 + ε2 + ω˜ g2 θg2 δe , it is known that N is a bounded signal satisfying |N| ≤ Nm . The derivative of the sliding mode surface is obtained as: s˙ = −k1 s − k2 |s|r sgn (s) + N

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(34)

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Similar to the analysis in [37], if s is not zero, k1 and k2 could be chosen as Nm + η1 |s| Nm k2 = r + η2 |s|

(35)

k1 =

(36)

s˙ = −k1 0 s − k2 |s|r sgn (s) s˙ = −k1 s − k2 0 |s|r sgn (s) where k1 0 = k1 −

N s




≥ η1 , k2 0 = k2 −



N |s| sgn(s) r



≥ η2 .

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where η1 and η2 are positive constants. Then (34) could be written as:

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Then it is concluded that the system trajectory could converge to the regions Ω = min {Ω1 , Ω2 } where   Nm Ω1 = |s| ≤ k1 − η1 (   1r ) Nm Ω2 = |s| ≤ k2 − η2

(38)

(39) (40)

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This completes the proof.

(37)

Remark 2 From the stability analysis, it is concluded that the controller parameters should satisfy k1 >

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0.5, β2 > 0.5 to ensure the stability. Further, by increasing k1 , γ f 2 , β2 and γg2 ,

Pa σ

could be arbitrarily small.

It should be noted that when tuning the parameters, the restriction on actuator and the system dynamic per-

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formance should also be considered based on the simulation results. Remark 3 The analysis of the sliding mode surface s follows the design in [37]. If the Lyapunov function

Simulation

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5

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L = 12 s2 , similar conclusion can be obtained.

The initial states are set as xm0 = 0m , ym0 = −100m, v0 = 25m/s, α0 = 0deg, ωz0 = 0deg/s, θ0 = 90deg.

The parameters in LOS guidance law are set as ky = kx = 1, ωt1 = 0.5, ωt2 = 0.1, ξ = 0.7. The sliding mode

controller parameters are chosen as k1 = 0.8, k2 = 0.1, a = 0.8, r = 0.5. For the NNs design, the updating law parameters are chosen as γ f 2 = 0.8, γz2 = 2, δ f 2 = 0.001, while the parameter of the SPEM is chosen as β2 = 3. The number of the hidden layer nodes is set as l = 43 , while the neuron centers are evenly spaced in [−1.6; 1.6] × [−1.6; 1.6] × [−0.5; 0.5]. In the adaptive laws of ωg2 , parameters are chosen as γg2 = 0.00005, δg2 = 0.001.

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The target path is set as: xd (t) = xd0 + 10t yd (t) = yd0 − 0.3t + 0.2 sin (t) where xd0 = 300m, yd0 = −30m. In the simulation, when the distance between the AUV and the target is less

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than 0.1m, it is considered that they rendezvous successfully and the simulation would be stopped. The simulation results are as follows. To make a comparison, the results under the composite learning method are marked as “CL”, while the results under the so-called σ -modification method without the prediction error item γz2 zn2 in the weights updating law are marked as “NN”.

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AUV Path Target Path Rendezvous Point

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-60 -70

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100

200

300

400

500

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Figure 2: Path of the AUV and the Target

Define the minimum distance between the AUV and the target as rmin

q  2 2 = min (xm − xd ) + (ym − yd ) ,

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the results under the CL and NN methods are 0.0475m and 0.7057m, respectively.

Fig.2 shows that the AUV could successfully track the target until rendezvous with the designed approach. From Fig.3 and Fig.4, it can been seen that the designed controller could ensure the tracking of the pitch angle command. Meanwhile, utilizing the composite learning technique, the tracking error could converge faster. Fig.5 shows the response of the AOA, the path angle and the pitch rate, which are all bounded. The elevator deflection is shown in Fig.6. Moreover, the response of the NN weights norm is shown in Fig.7 and Fig. 8, which presents that the convergence speed of weights in the composite learning approach is faster. The minimum distance indicates that the designed approach could obtain higher accuracy.

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90 80 70 60

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Figure 3: Pitch Angle

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Figure 4: Pitch Angle Tracking Error

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Figure 5: AOA, Path Angle and Pitch Rate

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Figure 6: Elevator Deflection

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Figure 7: NN Weights under CL method

Figure 8: NN Weights under NN method

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Conclusion

This paper focus on the control of an AUV longitudinal model. Through the dynamic transformation, the position command is transferred to the pitch angle command and a second-order system is then obtained. A sliding mode controller is designed for the transformed system to ensure the pitch angle tracking and system robustness. Utilizing the SPEM, the prediction error is obtained and applied in the NNs weights updating law

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to improve the learning and tracking performance. Simulation results show the effectiveness of the proposed approach on an AUV.

Acknowledgments

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This work was supported by National Natural Science Foundation of China (61622308, 61873206), National Ten Thousand Talent Program for Young Top-notch Talents (W03070131), Fok Ying-Tong Education Foundation (161058), and the Stable Supporting Fund of Science and Technology on Underwater Vehicle Technology,

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Qinling-Bashan Mountains Bioresources Comprehensive Development C.I.C. (QBXT-17-7).

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Appendix The coefficients ki j , i = 1, 2, 3; j = 1, 2, · · · , 9 in the AUV model (1)-(7) are with the following expressions:

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k11 = −Ax /m, k14 = −P/m, k18 = T /m    1  2 k21 = λ26 Aαmz − (Jz1 + λ66 ) Aαy + (Jz1 + λ66 ) λ22 − λ26 Ax /m D26
 1  ω k22 = (Jz1 + λ66 ) m − Aω y − λ26 Amz1 D26  1  2 λ26 − (Jz1 + λ66 ) (m + λ22 ) k23 = mD26  P  2 k24 = −λ26 + (Jz1 + λ66 ) λ22 mD26 P (Jz1 + λ66 ) k25 = D26 k26 = −λ26 Bxc /D26 , k27 = λ26 Bh/D26 , k28 = −λ26 T h/D26

Aδy [− (Jz1 + λ66 ) − λ26 xe ] D26

 1  1  α ω k31 = Amz (m + λ22 ) − λ26 Ax + Aαy , k32 = λ26 m − Aω y − Amz1 (m + λ22 ) D26 D26

k29 =

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k34 = −λ26 P/D26 , k35 = λ26 P/D26 , k36 = − (m + λ22 ) Bxc /D26 k37 = (m + λ22 ) Bh/D26 , k38 = − (m + λ22 ) T h/D26 , k39 =

Aδy [−λ26 − xe (m + λ22 )] D26

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2 D26 = (Jz1 + λ66 ) (m + λ22 ) − λ26

Ax = 0.5ρSCxS , Aαy = 0.5ρSCyα , Aδy = 0.5ρSCyδe ω¯

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y ω¯ z α α ω Aω y = 0.5ρSCy , Amz = 0.5ρSmz , Amz1 = 0.5ρSCz 4

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λ22 = K22 ρV, λ26 = K26 ρV 3 , λ66 = K66 ρV 3

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Yuyan Guo received the B.S. degree in measurement and control from Northwestern Polytechnical University, China, 2015 and the M.S. degree in Control Theory and Control Engineering from Northwestern Polytechnical University, China, 2018. He is currently studying for a Ph.D. degree in Control Science and Engineering at

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Northwestern Polytechnical University. His research interests include intelligent guidance and control.

Hongde Qin received the B.S. degree in Naval Architecture and Ocean Engineering from Harbin Engineering University, China, in 1995 and the Ph.D. degree in Design and Construction of Naval Architecture and Ocean Structure from Harbin Engineering University, China in 2003. He is currently a full Professor in the College

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of Ship Building Engineering, Harbin Engineering University, China. He is also the director of Science and Technology on Underwater Vehicle Laboratory. His research interests include overall design of unmanned

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underwater vehicles, architecture and intelligent control, underwater navigation, wave load and structural

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strength analysis of ocean structures.

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Bin Xu received the B.S. degree in measurement and control from Northwestern Polytechnical University, China, 2006 and the Ph.D. degree in Computer Science from Tsinghua University, China, 2012. He is currently professor with School of Automation, Northwestern Polytechnical University. His research interests include intelligent control and adaptive control with applications.

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Yi Han received the B.S. degree in Automation from Southwest Jiaotong University, China, 2016. He is currently studying for a M.S. degree in Control Engineering at Northwestern Polytechnical University. His

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research interests include intelligent control and robot control.

Quan-Yong Fan received the B.S. degree in automation from Henan University, Henan, China, in 2011, the

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M.S. degree in control theory and control engineering from Northeastern University, Liaoning, China, in 2013, and the Ph.D. degree in navigation guidance and control from Northeastern University in 2017. He is currently

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associate professor with School of Automation, Northwestern Polytechnical University. His current research

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interests include fault-tolerant control, intelligent control and event-triggered control.

Pengchao Zhang received his B.S. degrees from Shaanxi University of technology in 1999 and M.S. degree from Northwestern Polytechnical University in Xi?an, China. Now, he is a professor in Shaanxi University of Technology and also a doctoral student in Northwestern Polytechnical University. His research interest is mainly in the area of industrial robot and power electronics.

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