# An Embedded Linear Model Three-Dimensional Fuzzy PID Control System for a Bionic AUV under Wave Disturbance

Authors: Da Lyu; Haoqin Su; Yun Li; Zijun Zhang; Hao Long; Junbo Zhao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study provides a new embedded linear model (ELM) three-dimensional fuzzy PID control method for a bionic autonomous underwater vehicle (AUV), which is disturbed by ocean waves. The uncertainty of the ocean waves will make it difficult for the AUV to achieve good sailing quality, especially near the sea surface. In view of this problem, an ELM fuzzy PID controller, which adopts three-dimensional fuzzy control rules, is designed. In the ELM fuzzy PID controller, a linear model of AUV is established in order to actively perceive the effect of waves on AUV. Hence, the wave disturbance can be reflected in the control signal earlier and faster. Finally, three simulations are carried out to verify the control effect of the ELM fuzzy PID controller. An ocean wave disturbance model is established in order to simulate the force and the moment acting on the AUV. The results imply that compared with the classical PID controller and normal fuzzy PID controller, the ELM fuzzy PID controller can significantly enhance the stability of AUV and make the response faster and overshoot smaller.


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## Body

## 1. Introduction

Wave disturbance to the AUV sailing near the sea surface has always been a problem that restricts its working capacity. The French developed the world’s first AUV, Epaulard, in the 1970s [1]. After more than 40 years of development, AUV has been widely used in marine environment exploration [2, 3], marine mineral exploration [4], marine data collection [5], maritime alert [6], and other aspects.One of the essential parts of AUV is the control system. The control system of AUV is equivalent to the brain of humans. The AUV system is a six-degree-of-freedom, nonlinear, strongly coupled, and model uncertain system [7]. It needs to face underwater wave disturbance and complex topography during its missions. Therefore, how to efficiently and stably control the AUV has become one of the most challenging problems. The research on the control system of an AUV is the key to solve this problem. How to improve the AUV control system under the disturbance of waves is the main issue of this study.At present, classical and modern control theory has been widely used in industrial production [8], scientific experiments [9], aerospace [10], and military fields [11]. Taking PID control as an example, Wan [12] proposed a fractional-order PID strategy, which has been successfully applied to the heading control, diving control, and path-following system of AUV on sea trial. By comparing simulations and experiments, satisfactory performance, such as overshoot, settling time, and steady-state error, has been achieved. Khodayari and Balochian [13] design a variable integral PID controller based on a disturbance observer (DO) for AUV with a torpedo exterior, and simulation shows that the proposed controller has better tracking and robustness in the presence of interference. Guerrero et al. [14] design a nonlinear PID controller based on a set of saturation functions for trajectory tracking on an underwater vehicle. The effectiveness and robustness of the controller are demonstrated through experiments. But for these control theories, they have a common premise, that is, an accurate mathematical model of the controlled object needs to be established [15–17]. However, the systems studied in various fields are becoming more and more complex. For those complex systems that are difficult to establish accurate models [18, 19], it is often inappropriate to use classical control theory or modern control theory. But using fuzzy control theory can often achieve the purpose of precise control [20].Fuzzy logic control is abbreviated as fuzzy control. This theory was proposed by Professor Zadeh in the 1970s [21]. The most important part of a fuzzy controller is the fuzzy control law [22–24]. The fuzzy control law is the control experience summarized by researchers through the operation, learning, and accumulation of the controlled object system. These control experiences include the characteristics of the controlled object, the corresponding control strategies, and the performance indicators in various situations. The fuzziness of the controller is that the description of the fuzzy control law, which is usually natural language. The key of fuzzy control is to transform natural language into algorithm language for control. Rabah et al. [25] developed a fuzzy PI controller to adjust the parameters of the PI controller using the position and change in position data as input. The obtained results indicate that the proposed controller works well for tracking a moving target under different scenarios, especially during the night. In addition, Nouri et al. [26] propose a self-tuning fuzzy (STF) controller for the three-level boost converter. For each operating point, a proportional integral (PI) controller is designed. The results verify that the proposed control strategy can yield good performance. Farajdadian and Hosseini [27] designed fuzzy logic controllers (FLCs) for maximum power point tracking (MPPT) in a photovoltaic system, and then, fuzzy membership functions are optimized using the firefly algorithm (FA) to generate the proper duty cycle. The simulation results show that asymmetric fuzzy membership functions based on FA increase the tracking speed of MPPT and improve tracking accuracy. Septyan and Agustinah [28] propose the development of a fuzzy controller and a nonlinear dynamic feedback for an automated guided vehicle (AGV) with nonholonomic constraints. The fuzzy controller is designed using parallel distributed compensation with a 3-state kinematic error model. Simulation results show that the AGV can track sudden big orientation change from the nonsmooth reference trajectory and initial position error. Raj and Mohan [29] propose four new models of Takagi-Sugeno (TS) type fuzzy controllers. The gain variations of the controller and the stability analysis of the closed-loop control system are investigated. Yan et al. [30] reported a high-order disturbance observer (HODO)-based adaptive fuzzy tracking controller for nonlinear systems with saturation nonlinearity, uncertainties, and external disturbances. A numerical simulation of a two-stage chemical reactor shows the effectiveness of the developed control strategy. Sun et al. [31] investigated the problem of adaptive fuzzy control for a class of nontriangular structural stochastic switched nonlinear systems with full-state constraints and designed an adaptive fuzzy stochastic switched control scheme based on the Barrier Lyapunov function. The effectiveness of the proposed control scheme is verified via simulation studies. Yang et al. [32] developed an adaptive fuzzy control scheme for a dual-arm robot, applied an approximate Jacobian matrix to address the uncertain kinematic control, and constructed a decentralized fuzzy logic controller to compensate for uncertain dynamics of the robotic arms and the manipulated object. The effectiveness of the proposed approach was illustrated by extensive simulation studies using a dual-arm robot. Chao et al. [33] reported an optimal fuzzy PID controller design based on conventional PID control and nonlinear factors. In this way, a fuzzy PID controller can be developed with less parameters and optimized by using the genetic algorithm (GA). The feasibility of this technique was demonstrated by simulation. Khodayari and Balochian [13] designed a new self-tuning fuzzy fractional-order PID (AFFOPID) controller based on a nonlinear MIMO structure for an AUV. Simulation results demonstrate that the proposed controller has good performance and significant robust stability in comparison to traditional tuned PID controllers.This study proposes an ELM (embedded linear model) fuzzy PID controller, which adopts three-dimensional fuzzy control rules. Its structure is different from the self-tuning fuzzy fractional-order PID controller published before [13]. A simplified linear model is embedded in the controller in this study, and the fuzzy rules are three dimensional, which can more accurately adjust parameters online. The simulation test is designed to verify the effect of the control system under the disturbance of waves. Compared with the publicly reported control method of the AUV system, this control method uses an embedded linear model to sense the external disturbance, combined with a reasonable fuzzy rule design. Thus, the control method has a faster reaction speed and stronger control strength.


## 2. Preliminary

### 2.1. Introduction of the AUV Studied in This Study

The bionic AUV shown in Figure1 adopts a dolphin-like appearance. The propulsion system consists of two propellers T1, T1 mounted on the horizontal tail. The pitching motion of the AUV is controlled by the angle of horizontal rudder δe, and the heading motion is controlled by the different T1−T2 between the two propellers. In addition, this AUV is underactuated, as there are no actuators in the roll direction. But the hydrodynamic restoring force is designed to be large enough in the roll direction.Figure 1
The bionic AUV diagram described in this study.


### 2.2. Mathematical Model of the AUV

In order to study the effect of the control method described in this study, the mathematical model of the AUV is needed for the following simulation. In this study, the earth-fixed coordinate and the body-fixed coordinate, which are shown in Figure2 are selected to describe the motion of AUV. The origin of the earth-fixed coordinate with axis xe,ye,ze is selected on the ground, the xe axis points to the north direction in the horizontal plane, the ye axis is straight up, and the ze axis is perpendicular to the xe and ye axis, forming a right-handed system with them. The origin of the body-fixed coordinate with axis xb,yb,zb is selected at the center of buoyance of AUV. xb axis points forward along the longitudinal axis of AUV, yb axis points upward perpendicular to X axis, and zb axis is perpendicular to xb axis and yb axis, forming a right-handed system with them.Figure 2
The earth-fixed coordinate and the body-fixed coordinate.In order to establish the mathematical model of the AUV, the following assumptions were made: the AUV is a rigid body; the AUV is completely immersed in the sea; and the influence of the curvature and rotation of the Earth are not considered.According to the theorem of momentum, the kinetic equations of the AUV are established in the body coordinate system of the AUV as in equation (1). In the formula, F is the projection of the resultant force of all external forces received by the AUV on the body coordinate system, and M is the projection of the resultant moment of force received by the AUV on the body coordinate system. Q is the momentum of AUV. K is the moment of momentum of AUV.(1)dQdt+ω×Q=F,dKdt+ω×K+v0×Q=M.The momentumQ and the moment of momentum K are as in equation (2), in which, m is the mass of AUV, vc is the velocity vector of mass center relative to earth-fixed coordinate, rc is the vector of mass center to buoyance center, v0 is the velocity vector of buoyance center, ω is the angular velocity of AUV, and J0 is the moment of inertia matrix of AUV relative to the buoyance center.(2)Q=mvc,K=J0⋅ω+rc×mv0.In equation (1), the external forces on the AUV include ideal fluid inertial force, viscous position force, viscous damping force, buoyancy, gravity, propeller thrust, and propeller unbalanced moment. [34].All the external forces and moments acting on the AUV are substituted into equation (1), and the higher-order small quantities of AUV motion parameters are ignored. Then, the kinetic equation of the AUV can be obtained as equations (3)–(6).(3)m+λ11v˙x+mzcω˙y−mycω˙z+mvzωy−vyωz+ycωxωy+zcωxωz−xcωy2+ωz2=T1+T2−0.5⋅CxSρv2S−ΔGsinθm+λ22v˙y+mxc+λ26ω˙z−mzcω˙x+mvxωz−vzωx+zcωyωz+xcωxωy−ycωx2+ωz2=0.5⋅Cyαα+Cyδeδe+Cyω¯zω¯zρv2S−ΔGcosθcosφm+λ33v˙z−mxc−λ35ω˙y+mycω˙x+mvyωx−vxωy+xcωzωx+ycωyωz−zcωx2+ωy2=0.5⋅Czββ+CzΔTT1−T2+Czω¯yω¯yρv2S+ΔGcosθsinφ,(4)Jxx+λ44ω˙x+mycv˙z−mzcv˙y+mycvyωx−vxωy+mzcvzωx−vxωz+Jzz−Jyyωyωz=0.5⋅ρv2SLmxββ+mxΔTT1−T2+mxω¯xω¯x+mxω¯yω¯y+Gcosθycsinφ+zccosφJyy+λ55ω˙y+mzcv˙x−mxc−λ35v˙z+mzcvzωy−vyωz+mxcvxωy−vyωx+Jxx−Jzzωzωx=0.5⋅ρv2SLmyββ+myΔTT1−T2+myω¯xω¯x+myω¯yω¯y−Gxccosθsinφ+zcsinθJzz+λ66ω˙z−mycv˙x+mxc+λ26v˙y+mxcvxωz−vzωx+mycvyωz−vzωy+Jyy−Jxxωxωy=0.5⋅ρv2SLmzαα+mzδeδe+mzω¯zω¯z+Gycsinθ−xccosθcosφ,(5)θ˙=ωysinφ+ωzcosφ,ψ˙=ωysecθcosφ−ωzsecθsinφ,φ˙=ωx−ωytanθcosφ+ωztanθsinφ,(6)x˙0=vxcosθcosψ+vysinψsinφ−sinθcosψcosφ+vzsinψcosφ+sinθcosψsinφ,y˙0=vxsinθ+vycosθcosφ−vzcosθsinφ,z˙0=−vxcosθsinψ+vycosψsinφ−sinθsinψcosφ+vzcosψcosφ−sinθsinψsinφ.Among them, equation (3) is the force equation in the directions of the three-body coordinate axis, and equation (4) is the moment equation in the directions of the three-body coordinate axis. Equations (5) and (6) are the coordinate transformation equations of angular velocity and velocity between two coordinate systems, respectively. In the equations, α is the attack angle (head up is positive). β is the sideslip angle (right deviation is positive). δe is the horizontal rudder angle (upward deflection is positive). V is the volume of AUV. θ is the pitch angle of AUV (along the positive direction of zb axis, clockwise is positive). φ is the roll angle of AUV (along the positive direction of xb axis, clockwise is positive). ψ is the yaw angle of AUV (along the positive direction of yb axis, clockwise is positive). T1 is the thrust of the propellor 1. T2 is the thrust of the propellor 2. λij is the added mass of the AUV. ΔG is the resultant force of gravity and buoyancy. The other mainly used parameters in the equation are shown in Table 1.Table 1
The mainly used parameters of the AUV in the equation.
Variable namesData valueUnitRemarksM100KgMassB981NBuoyancyJxx28.9Kg⋅m2Moment of inertia aboutx axisJyy677Kg⋅m2Moment of inertia abouty axisJzz677Kg⋅m2Moment of inertia aboutz axisL2.6mLength of the AUVCx0.141Coefficient of dragLd0.9mDistance of the two propellerCzω¯y1.171Derivative ofCz to ω¯yCzα2.321Derivative ofCz to αCzδe0.511Derivative ofCz to δeMyα0.691Derivative ofMy to αMyδe−0.281Derivative ofMy to δeCyβ−2.321Derivative ofCy to βMzβ0.691Derivative ofMz to βxc0.12mDistance between mass center and buoyancy center onxb axisyc0.03mDistance between mass center and buoyancy center onyb axiszc0mDistance between mass center and buoyancy center onzb axis


## 2.1. Introduction of the AUV Studied in This Study

The bionic AUV shown in Figure1 adopts a dolphin-like appearance. The propulsion system consists of two propellers T1, T1 mounted on the horizontal tail. The pitching motion of the AUV is controlled by the angle of horizontal rudder δe, and the heading motion is controlled by the different T1−T2 between the two propellers. In addition, this AUV is underactuated, as there are no actuators in the roll direction. But the hydrodynamic restoring force is designed to be large enough in the roll direction.Figure 1
The bionic AUV diagram described in this study.


## 2.2. Mathematical Model of the AUV

In order to study the effect of the control method described in this study, the mathematical model of the AUV is needed for the following simulation. In this study, the earth-fixed coordinate and the body-fixed coordinate, which are shown in Figure2 are selected to describe the motion of AUV. The origin of the earth-fixed coordinate with axis xe,ye,ze is selected on the ground, the xe axis points to the north direction in the horizontal plane, the ye axis is straight up, and the ze axis is perpendicular to the xe and ye axis, forming a right-handed system with them. The origin of the body-fixed coordinate with axis xb,yb,zb is selected at the center of buoyance of AUV. xb axis points forward along the longitudinal axis of AUV, yb axis points upward perpendicular to X axis, and zb axis is perpendicular to xb axis and yb axis, forming a right-handed system with them.Figure 2
The earth-fixed coordinate and the body-fixed coordinate.In order to establish the mathematical model of the AUV, the following assumptions were made: the AUV is a rigid body; the AUV is completely immersed in the sea; and the influence of the curvature and rotation of the Earth are not considered.According to the theorem of momentum, the kinetic equations of the AUV are established in the body coordinate system of the AUV as in equation (1). In the formula, F is the projection of the resultant force of all external forces received by the AUV on the body coordinate system, and M is the projection of the resultant moment of force received by the AUV on the body coordinate system. Q is the momentum of AUV. K is the moment of momentum of AUV.(1)dQdt+ω×Q=F,dKdt+ω×K+v0×Q=M.The momentumQ and the moment of momentum K are as in equation (2), in which, m is the mass of AUV, vc is the velocity vector of mass center relative to earth-fixed coordinate, rc is the vector of mass center to buoyance center, v0 is the velocity vector of buoyance center, ω is the angular velocity of AUV, and J0 is the moment of inertia matrix of AUV relative to the buoyance center.(2)Q=mvc,K=J0⋅ω+rc×mv0.In equation (1), the external forces on the AUV include ideal fluid inertial force, viscous position force, viscous damping force, buoyancy, gravity, propeller thrust, and propeller unbalanced moment. [34].All the external forces and moments acting on the AUV are substituted into equation (1), and the higher-order small quantities of AUV motion parameters are ignored. Then, the kinetic equation of the AUV can be obtained as equations (3)–(6).(3)m+λ11v˙x+mzcω˙y−mycω˙z+mvzωy−vyωz+ycωxωy+zcωxωz−xcωy2+ωz2=T1+T2−0.5⋅CxSρv2S−ΔGsinθm+λ22v˙y+mxc+λ26ω˙z−mzcω˙x+mvxωz−vzωx+zcωyωz+xcωxωy−ycωx2+ωz2=0.5⋅Cyαα+Cyδeδe+Cyω¯zω¯zρv2S−ΔGcosθcosφm+λ33v˙z−mxc−λ35ω˙y+mycω˙x+mvyωx−vxωy+xcωzωx+ycωyωz−zcωx2+ωy2=0.5⋅Czββ+CzΔTT1−T2+Czω¯yω¯yρv2S+ΔGcosθsinφ,(4)Jxx+λ44ω˙x+mycv˙z−mzcv˙y+mycvyωx−vxωy+mzcvzωx−vxωz+Jzz−Jyyωyωz=0.5⋅ρv2SLmxββ+mxΔTT1−T2+mxω¯xω¯x+mxω¯yω¯y+Gcosθycsinφ+zccosφJyy+λ55ω˙y+mzcv˙x−mxc−λ35v˙z+mzcvzωy−vyωz+mxcvxωy−vyωx+Jxx−Jzzωzωx=0.5⋅ρv2SLmyββ+myΔTT1−T2+myω¯xω¯x+myω¯yω¯y−Gxccosθsinφ+zcsinθJzz+λ66ω˙z−mycv˙x+mxc+λ26v˙y+mxcvxωz−vzωx+mycvyωz−vzωy+Jyy−Jxxωxωy=0.5⋅ρv2SLmzαα+mzδeδe+mzω¯zω¯z+Gycsinθ−xccosθcosφ,(5)θ˙=ωysinφ+ωzcosφ,ψ˙=ωysecθcosφ−ωzsecθsinφ,φ˙=ωx−ωytanθcosφ+ωztanθsinφ,(6)x˙0=vxcosθcosψ+vysinψsinφ−sinθcosψcosφ+vzsinψcosφ+sinθcosψsinφ,y˙0=vxsinθ+vycosθcosφ−vzcosθsinφ,z˙0=−vxcosθsinψ+vycosψsinφ−sinθsinψcosφ+vzcosψcosφ−sinθsinψsinφ.Among them, equation (3) is the force equation in the directions of the three-body coordinate axis, and equation (4) is the moment equation in the directions of the three-body coordinate axis. Equations (5) and (6) are the coordinate transformation equations of angular velocity and velocity between two coordinate systems, respectively. In the equations, α is the attack angle (head up is positive). β is the sideslip angle (right deviation is positive). δe is the horizontal rudder angle (upward deflection is positive). V is the volume of AUV. θ is the pitch angle of AUV (along the positive direction of zb axis, clockwise is positive). φ is the roll angle of AUV (along the positive direction of xb axis, clockwise is positive). ψ is the yaw angle of AUV (along the positive direction of yb axis, clockwise is positive). T1 is the thrust of the propellor 1. T2 is the thrust of the propellor 2. λij is the added mass of the AUV. ΔG is the resultant force of gravity and buoyancy. The other mainly used parameters in the equation are shown in Table 1.Table 1
The mainly used parameters of the AUV in the equation.
Variable namesData valueUnitRemarksM100KgMassB981NBuoyancyJxx28.9Kg⋅m2Moment of inertia aboutx axisJyy677Kg⋅m2Moment of inertia abouty axisJzz677Kg⋅m2Moment of inertia aboutz axisL2.6mLength of the AUVCx0.141Coefficient of dragLd0.9mDistance of the two propellerCzω¯y1.171Derivative ofCz to ω¯yCzα2.321Derivative ofCz to αCzδe0.511Derivative ofCz to δeMyα0.691Derivative ofMy to αMyδe−0.281Derivative ofMy to δeCyβ−2.321Derivative ofCy to βMzβ0.691Derivative ofMz to βxc0.12mDistance between mass center and buoyancy center onxb axisyc0.03mDistance between mass center and buoyancy center onyb axiszc0mDistance between mass center and buoyancy center onzb axis


## 3. Design Scheme of ELM Fuzzy PID Controller

### 3.1. The Embedded Model and the Fuzzy Logic

In this study, we embedded a simplified linear model of the AUV into the controller. In order to obtain an accurate linear model, the mathematical model of the AUV established above has been simplified and divided into the vertical motion model (7) and the horizontal motion model (9) according to the small disturbance theory [35].In order to ensure the stability of the control system, the accuracy of the embedded linear model needs to be verified. Figure3 shows the comparison between the simulation results of the embedded linear model and the results of the real AUV lake test data with a 6-degree yaw rudder in the lateral direction. Figure 4 shows the comparison between the simulation results of the embedded linear model and the real AUV lake test data with a 6-degree elevator in the pitch direction. It can be seen from the results that the error of the embedded linear model and the real experimental data is small in the yaw and pitch directions. The error of the yaw direction and pitch direction is both less than 5%, so it can be used to reflect the motion of the actual AUV without disturbance.(7)α˙ω˙zθ.d˙=a21a34−a31a24a23a31−a21a33a21a32−a31a22a23a31−a21a3300a33a24−a23a34a23a31−a21a33a33a22−a23a32a23a31−a21a33000100−v00v00,αωzθd+a31a25−a21a35a23a31−a21a33a23a35−a33a25a23a31−a21a3300⋅δe.Figure 3
Simulation results of the embedded linear model and the results of the real AUV lake test data with a 6-degree yaw rudder.Figure 4
Simulation results of the embedded linear model and the real AUV lake test data with a 6-degree elevator.d is the depth of the AUV. The coefficients in equation (7) are defined as follows:(8)a21=mxc+λ261/2ρv02S,a22=μcosα0−Cyω¯zLv0,a23−m+λ22cosα01/2ρv0S,a24=−Cyα,a25=Cyδe,a26=−ΔGsinθ01/2ρv02S,a31=Jzz+λ661/2ρv02SL,a32=μx¯ccosα0−mzω¯zLv0,a33=−mxc+λ26cosα01/2ρv02SL,a34=−mzα,a35=mzδe,a36=−Gxcsinθ01/2ρv02SL,(9)ω˙yβ˙ψ˙CTD.=a43a52−a53a42a23a31−a21a33a43a54−a53a44a23a31−a21a3300a42a51−a52a41a23a31−a21a33a44a51−a54a41a23a31−a21a330010000v0−v00⋅ωyβψCTD+a53a49−a43a59a23a31−a21a33a59a41−a49a51a23a31−a21a3300⋅ΔT.CTD is the cross-track distance of the AUV. The coefficients in equation (9) are defined as follows:(10)a41=λ35−mxc1/2ρv02S,a42=−μ+Czω¯yLv0,a43=m+λ331/2ρv0S,a44=−Czβ,a49=Czδr,a59=myδr,a51=Jyy+λ551/2ρv02SL,a52=−myω¯y−μx¯cLv0,a53=λ35−mxc1/2ρv0SL,a54=−myβ.In this study, the basic design idea of the ELM fuzzy PID controller is shown in Figure5. The feedback state error between the linear model and the actual AUV is obtained, and then, it is inputted into the controller together with the error of the control quantity and the error derivative of the control quantity.Figure 5
Design of the embedded linear model (ELM) fuzzy PID control method.It is assumed that the feedback state vector of the linear model is as follows:(11)X˜=α˜ω˜zθ˜d˜β˜ω˜yψ˜CTD˜T=LδeYΔTT,where the function Lδe is the solution of equation (7) when the horizontal rudder angle is δe. The YΔT is the solution of equation (9) when the propeller differential quantity is ΔT. Lδe and YΔT can be derived from the solution of nonhomogeneous state equation of linear time-invariant system as follows:(12)LδeYΔT=eA1t−t0x1t0+∫t0teA1t−τB1δeτdτeA2t−t0x2t0+∫t0teA2t−τB2ΔTτdτ,where A1 is the state matrix of AUV vertical motion system. B1 is the control matrix of AUV vertical motion system. A2 is the state matrix of AUV horizontal motion system. B2 is the control matrix of AUV horizontal motion system. The feedback state vector of the actual AUV is as follows:(13)X=αωzθdβωyψCTDT.The expected state vector of the AUV is as follows:(14)Xc=αcωzcθcdcβcωycψcCTDcT.Hence, the feedback state error between the linear model and the actual AUV is as follows:(15)ΔXi=X˜i−Xi.The error between the expected and actual control quantity is as follows:(16)ei=Xci−Xi.The derivative of error between the expected and actual control quantity is as follows:(17)e′i=Xc′i−X′i.The controller adopts three-dimensional fuzzy control rules, in which the input variables of the controller consist of the feedback state errorΔX, the control quantity error e, and the derivative of control quantity error e′. The output includes the three control parameters of the controller Kp, Ki, and Kd, and the control compensation Q. The four parameters are adaptively adjusted according to the changes in the external environment and working conditions. The control law can be expressed as follows:(18)C=Kp⋅e+Ki⋅∫e⋅dt+Kd⋅e′+Q,where(19)KpKiKdQ=Fee′ΔX.RelationshipF is fuzzy logic, which includes fuzzy rules and membership functions. In this study, the fuzzy controller adopted a three-input and four-output structure. The inputs included the feedback state error ΔX between the linear model and the actual AUV, the control quantity error e, and the derivative of the control quantity error e′. The output includes the three PID control parameters Kp, Ki, and Kd, and the control compensation Q. The fuzzy control set of ΔX, e, e′, and Q is NLNMZPMPL, respectively, representing negative large, negative middle, zero, positive middle, and positive large. The fuzzy control set of Kp, Ki, and Kd is ZSML, which means zero, small, medium, and large, respectively.The fuzzy rules about the three control parametersKp, Ki, and Kd of PID controller can be design as the fuzzy rules by Chao et al. [33]. The fuzzy rules about Q mainly depend on ΔXi. As an example, for the depth control loop in vertical motion, ΔXi is d˜−d. According to the engineering experience, there are the following relationships: if d˜−d is a large/small positive number, it proves that AUV is more disturbed by waves in positive direction. Then, the compensation Qdepth should be negatively large/small. It is analyzed in a similar way, if d˜−d is a large/small negative number, and the compensation Qdepth should be positively large/small. Then, the fuzzy rules are formulated in Table 2.Table 2
Fuzzy control rule table of ELM fuzzy PID controller.
KpKiKdQe′e′e′e′NLNMZPMPLNLNMZPMPLNLNMZPMPLNLNMZPMPLΔXNLeNLMLLLMZZZZZSSZSSPLPLPMNLNLNMSSMSSSMMMSMMSMMPLPMPMNMNLZSMLMSMLLLMSMLMSPMPMPMNMNMPMSSMSSSMMMSMMSMMPLPMPMNMNLPLMLLLMZZZZZSSZSSPLPLPMNLNLNMeNLMLLLMZZZZZSSZSSPLPMPMNMNLNMSSMSSSMMMSMMSMMPMPMZNMNMZSMLMSMLLLMSMLMSPMPMZNMNMPMSSMSSSMMMSMMSMMPMPMZNMNMPLMLLLMZZZZZSSZSSPLPMPMNMNLZeNLMLLLMZZZZZSSZSSZZZZZNMSSMSSSMMMSMMSMMZZZZZZSMLMSMLLLMSMLMSZZZZZPMSSMSSSMMMSMMSMMZZZZZPLMLLLMZZZZZSSZSSZZZZZPMeNLMLLLMZZZZZSSZSSNLPMPMNMPLNMSSMSSSMMMSMMSMMNMPMZNMPMZSMLMSMLLLMSMLMSNMPMZNMPMPMSSMSSSMMMSMMSMMNMPMZNMPMPLMLLLMZZZZZSSZSSNLPMPMNMPLPLeNLMLLLMZZZZZSSZSSNLPLPMNLPLNMSSMSSSMMMSMMSMMNLPMPMNMPLZSMLMSMLLLMSMLMSNMPMPMNMPMPMSSMSSSMMMSMMSMMNLPMPMNMPLPLMLLLMZZZZZSSZSSNLPLPMNLPLThe membership function of the fuzzy control rules adopted the triangular membership function, as shown in Figure6. The membership function is used to describe the membership degree of variables. The closer the membership function value is to 1, the more certain the input variable belongs to the set. The closer the membership function is to 0, the more uncertain the input variable belongs to the set.Figure 6
Membership functionBix corresponding to the fuzzy set NLNMZPMPL and membership function B^ix corresponding to the fuzzy set. ZSML.


### 3.2. Design of AUV Vertical and Horizontal Motion Controller

For the vertical motion of the AUV, in order to realize the simultaneous control of AUV position and attitude, a two-stage controller is adopted. The diagram of the two-stage ELM fuzzy PID controller for the vertical motion of the AUV is shown in Figure7. The error of the depth dc−d is used as the input of the first-stage controller. The output of the first-stage controller is used as the expected pitch angle θc which is the input of the second-stage controller. The output value of the second stage controller is taken as the horizontal rudder angle δe of the AUV.Figure 7
Diagram of the two-stage ELM fuzzy PID controller for the vertical motion of the AUV.For the horizontal motion of the AUV, the two-stage ELM fuzzy PID controller is also adopted, as shown in Figure8. The error of the cross-track distance CTDc−CTD is used as the input of the first-stage controller. The output of the first-stage controller is used as the expected yaw angle ψc, which is the input of the second-stage controller. Thus, the output value of the second-stage controller is used as the difference of the two thrusters T1−T2 of the AUV.Figure 8
Diagram of the two-stage ELM fuzzy PID controller for the horizontal motion of the AUV.


## 3.1. The Embedded Model and the Fuzzy Logic

In this study, we embedded a simplified linear model of the AUV into the controller. In order to obtain an accurate linear model, the mathematical model of the AUV established above has been simplified and divided into the vertical motion model (7) and the horizontal motion model (9) according to the small disturbance theory [35].In order to ensure the stability of the control system, the accuracy of the embedded linear model needs to be verified. Figure3 shows the comparison between the simulation results of the embedded linear model and the results of the real AUV lake test data with a 6-degree yaw rudder in the lateral direction. Figure 4 shows the comparison between the simulation results of the embedded linear model and the real AUV lake test data with a 6-degree elevator in the pitch direction. It can be seen from the results that the error of the embedded linear model and the real experimental data is small in the yaw and pitch directions. The error of the yaw direction and pitch direction is both less than 5%, so it can be used to reflect the motion of the actual AUV without disturbance.(7)α˙ω˙zθ.d˙=a21a34−a31a24a23a31−a21a33a21a32−a31a22a23a31−a21a3300a33a24−a23a34a23a31−a21a33a33a22−a23a32a23a31−a21a33000100−v00v00,αωzθd+a31a25−a21a35a23a31−a21a33a23a35−a33a25a23a31−a21a3300⋅δe.Figure 3
Simulation results of the embedded linear model and the results of the real AUV lake test data with a 6-degree yaw rudder.Figure 4
Simulation results of the embedded linear model and the real AUV lake test data with a 6-degree elevator.d is the depth of the AUV. The coefficients in equation (7) are defined as follows:(8)a21=mxc+λ261/2ρv02S,a22=μcosα0−Cyω¯zLv0,a23−m+λ22cosα01/2ρv0S,a24=−Cyα,a25=Cyδe,a26=−ΔGsinθ01/2ρv02S,a31=Jzz+λ661/2ρv02SL,a32=μx¯ccosα0−mzω¯zLv0,a33=−mxc+λ26cosα01/2ρv02SL,a34=−mzα,a35=mzδe,a36=−Gxcsinθ01/2ρv02SL,(9)ω˙yβ˙ψ˙CTD.=a43a52−a53a42a23a31−a21a33a43a54−a53a44a23a31−a21a3300a42a51−a52a41a23a31−a21a33a44a51−a54a41a23a31−a21a330010000v0−v00⋅ωyβψCTD+a53a49−a43a59a23a31−a21a33a59a41−a49a51a23a31−a21a3300⋅ΔT.CTD is the cross-track distance of the AUV. The coefficients in equation (9) are defined as follows:(10)a41=λ35−mxc1/2ρv02S,a42=−μ+Czω¯yLv0,a43=m+λ331/2ρv0S,a44=−Czβ,a49=Czδr,a59=myδr,a51=Jyy+λ551/2ρv02SL,a52=−myω¯y−μx¯cLv0,a53=λ35−mxc1/2ρv0SL,a54=−myβ.In this study, the basic design idea of the ELM fuzzy PID controller is shown in Figure5. The feedback state error between the linear model and the actual AUV is obtained, and then, it is inputted into the controller together with the error of the control quantity and the error derivative of the control quantity.Figure 5
Design of the embedded linear model (ELM) fuzzy PID control method.It is assumed that the feedback state vector of the linear model is as follows:(11)X˜=α˜ω˜zθ˜d˜β˜ω˜yψ˜CTD˜T=LδeYΔTT,where the function Lδe is the solution of equation (7) when the horizontal rudder angle is δe. The YΔT is the solution of equation (9) when the propeller differential quantity is ΔT. Lδe and YΔT can be derived from the solution of nonhomogeneous state equation of linear time-invariant system as follows:(12)LδeYΔT=eA1t−t0x1t0+∫t0teA1t−τB1δeτdτeA2t−t0x2t0+∫t0teA2t−τB2ΔTτdτ,where A1 is the state matrix of AUV vertical motion system. B1 is the control matrix of AUV vertical motion system. A2 is the state matrix of AUV horizontal motion system. B2 is the control matrix of AUV horizontal motion system. The feedback state vector of the actual AUV is as follows:(13)X=αωzθdβωyψCTDT.The expected state vector of the AUV is as follows:(14)Xc=αcωzcθcdcβcωycψcCTDcT.Hence, the feedback state error between the linear model and the actual AUV is as follows:(15)ΔXi=X˜i−Xi.The error between the expected and actual control quantity is as follows:(16)ei=Xci−Xi.The derivative of error between the expected and actual control quantity is as follows:(17)e′i=Xc′i−X′i.The controller adopts three-dimensional fuzzy control rules, in which the input variables of the controller consist of the feedback state errorΔX, the control quantity error e, and the derivative of control quantity error e′. The output includes the three control parameters of the controller Kp, Ki, and Kd, and the control compensation Q. The four parameters are adaptively adjusted according to the changes in the external environment and working conditions. The control law can be expressed as follows:(18)C=Kp⋅e+Ki⋅∫e⋅dt+Kd⋅e′+Q,where(19)KpKiKdQ=Fee′ΔX.RelationshipF is fuzzy logic, which includes fuzzy rules and membership functions. In this study, the fuzzy controller adopted a three-input and four-output structure. The inputs included the feedback state error ΔX between the linear model and the actual AUV, the control quantity error e, and the derivative of the control quantity error e′. The output includes the three PID control parameters Kp, Ki, and Kd, and the control compensation Q. The fuzzy control set of ΔX, e, e′, and Q is NLNMZPMPL, respectively, representing negative large, negative middle, zero, positive middle, and positive large. The fuzzy control set of Kp, Ki, and Kd is ZSML, which means zero, small, medium, and large, respectively.The fuzzy rules about the three control parametersKp, Ki, and Kd of PID controller can be design as the fuzzy rules by Chao et al. [33]. The fuzzy rules about Q mainly depend on ΔXi. As an example, for the depth control loop in vertical motion, ΔXi is d˜−d. According to the engineering experience, there are the following relationships: if d˜−d is a large/small positive number, it proves that AUV is more disturbed by waves in positive direction. Then, the compensation Qdepth should be negatively large/small. It is analyzed in a similar way, if d˜−d is a large/small negative number, and the compensation Qdepth should be positively large/small. Then, the fuzzy rules are formulated in Table 2.Table 2
Fuzzy control rule table of ELM fuzzy PID controller.
KpKiKdQe′e′e′e′NLNMZPMPLNLNMZPMPLNLNMZPMPLNLNMZPMPLΔXNLeNLMLLLMZZZZZSSZSSPLPLPMNLNLNMSSMSSSMMMSMMSMMPLPMPMNMNLZSMLMSMLLLMSMLMSPMPMPMNMNMPMSSMSSSMMMSMMSMMPLPMPMNMNLPLMLLLMZZZZZSSZSSPLPLPMNLNLNMeNLMLLLMZZZZZSSZSSPLPMPMNMNLNMSSMSSSMMMSMMSMMPMPMZNMNMZSMLMSMLLLMSMLMSPMPMZNMNMPMSSMSSSMMMSMMSMMPMPMZNMNMPLMLLLMZZZZZSSZSSPLPMPMNMNLZeNLMLLLMZZZZZSSZSSZZZZZNMSSMSSSMMMSMMSMMZZZZZZSMLMSMLLLMSMLMSZZZZZPMSSMSSSMMMSMMSMMZZZZZPLMLLLMZZZZZSSZSSZZZZZPMeNLMLLLMZZZZZSSZSSNLPMPMNMPLNMSSMSSSMMMSMMSMMNMPMZNMPMZSMLMSMLLLMSMLMSNMPMZNMPMPMSSMSSSMMMSMMSMMNMPMZNMPMPLMLLLMZZZZZSSZSSNLPMPMNMPLPLeNLMLLLMZZZZZSSZSSNLPLPMNLPLNMSSMSSSMMMSMMSMMNLPMPMNMPLZSMLMSMLLLMSMLMSNMPMPMNMPMPMSSMSSSMMMSMMSMMNLPMPMNMPLPLMLLLMZZZZZSSZSSNLPLPMNLPLThe membership function of the fuzzy control rules adopted the triangular membership function, as shown in Figure6. The membership function is used to describe the membership degree of variables. The closer the membership function value is to 1, the more certain the input variable belongs to the set. The closer the membership function is to 0, the more uncertain the input variable belongs to the set.Figure 6
Membership functionBix corresponding to the fuzzy set NLNMZPMPL and membership function B^ix corresponding to the fuzzy set. ZSML.


## 3.2. Design of AUV Vertical and Horizontal Motion Controller

For the vertical motion of the AUV, in order to realize the simultaneous control of AUV position and attitude, a two-stage controller is adopted. The diagram of the two-stage ELM fuzzy PID controller for the vertical motion of the AUV is shown in Figure7. The error of the depth dc−d is used as the input of the first-stage controller. The output of the first-stage controller is used as the expected pitch angle θc which is the input of the second-stage controller. The output value of the second stage controller is taken as the horizontal rudder angle δe of the AUV.Figure 7
Diagram of the two-stage ELM fuzzy PID controller for the vertical motion of the AUV.For the horizontal motion of the AUV, the two-stage ELM fuzzy PID controller is also adopted, as shown in Figure8. The error of the cross-track distance CTDc−CTD is used as the input of the first-stage controller. The output of the first-stage controller is used as the expected yaw angle ψc, which is the input of the second-stage controller. Thus, the output value of the second-stage controller is used as the difference of the two thrusters T1−T2 of the AUV.Figure 8
Diagram of the two-stage ELM fuzzy PID controller for the horizontal motion of the AUV.


## 4. Simulation of ELM Fuzzy PID Control for AUV

### 4.1. Wave Disturbance Model for AUV

In order to verify whether the control system can effectively suppress the disturbance of waves to the AUV, this study proposes a wave disturbance model to simulate the influence of waves to the AUV.It is assumed that the wave disturbances to AUV are regular waves. The fluid studied satisfies the ideal fluid hypothesis of uniform, incompressible, nonviscous, and nonrotational. According to the Froude-Krylov hypothesis [36], the pressure distributions in regular waves are as follows:(20)p=ρgH2⋅coshk⋅z+hcoshk⋅h⋅coskxcosθ−kysinθ−ωt,where ρ is the density of water, g is the acceleration of gravity, k is the wave number, ω is the wave frequency, γ is the angle between the direction of AUV motion and the direction of wave travel, h is the water depth, and H is the wave height. For the six-degree-of-freedom movement of the AUV, the wave interference force is expressed as follows:(21)F=−∬p⋅ndS,M=−∬p⋅n×rdS,where S is the surface area of AUV, and n is the normal vector outside of AUV surface. By using Gauss theorem, the integral along the surface of the AUV can be transformed into volume integral, as shown in(22)Fx=−∫∫∫∂p∂xdV,Fy=−∫∫∫∂p∂ydV,Fz=−∫∫∫∂p∂zdV,Mx=−∫∫∫∂p∂zy−∂p∂yzdV,My=−∫∫∫∂p∂xz−∂p∂zxdV,Mz=−∫∫∫∂p∂yx−∂p∂xydV,where V is the volume of the AUV. As shown in Figure 9, the AUV can be approximately simplified to a cylinder of equal diameter. By substituting equations (20)–(22), and considering the ocean depth as infinite, the interference force and moment of the waves on the AUV can be simplified as follows:(23)Fx=Ex⋅2ρgπHk2sinγ⋅coshk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sinkRsinγ2coskLcosγ2,Fy=Ey⋅2ρgπHk2cosγ⋅coshk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sin−kRsinγ2sinkLcosγ2,Fz=Ez⋅2ρgπHk2cosγsinγ⋅sinhk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sinkRsinγ2sinklcosγ2,Mx=ρgπHDxekD+kR−ekDk3cosγ⋅kD−1−1sin2γ⋅2sinωt−kLcosγ−kRsinγ2sinkRsinγ2coskLcosγ2+kRsinγsinkLcosγ2sinωt+kRsinγ−kLcosγ2,My=ρgπHDyekD+kR−ekDk3sinγ⋅−kD+1+1cos2γ⋅2sinωt−kLcosγ−kRsinγ2sinkRsinγ2sinkLcosγ2+LkcosγsinkRsinγ2sinωt+kLcosγ−kRsinγ2,Mz=ρgπHDzekD+kR−ekDk2sin2γ⋅Rsinγcosωt+kRsinγ−kLcosγ2sinkLcosγ2+LcosγsinkRsinγ2cosωt−kLcosγ+kRsinγ2,wherein Ex, Ey, Ez, Dx, Dy, and Dz are diffraction coefficients of waves in the directions of six-degrees-of-freedom of the AUV, respectively.Figure 9
Diagram of AUV under wave disturbance.The wave scales are shown in Table3. According to the international wave scales, waves are divided into ten levels. During the actual process, AUVs are generally affected by waves below the SLIGHT, which are usually the CALM-RIPPLED and the SMOOTH-WAVELET. Generally, the AUV missions are not performed in the SLIGHT level and above.Table 3
International wave scale. Source:https://wenku.baidu.com/view/98aac42e25284b73f242336c1eb91a37f1113286.html.
Wave scale nameWave height rangeDescription of sea surfaceCalm-glassy0 mThe sea surface is as smooth as a mirror, or there are only swellsCalm-rippled0–0.1 mRipple and swell exist at the same timeSmooth-wavelet0.1–0.5 mThe waves are very small, the crest begins to break, and the spray is not white but glassySlight0.5–1.25 mThe crests of the waves are broken, and some of them form white sprayModerate1.25–2.50 mWaves have distinct shapes and form white waves everywhereRough2.50–4.0 mThere were big wave crests, the spray occupied a large area on the crest, and the wind began to cut off the spray on the crestVery rough4–6 mThe wave crests sometimes appear the long wave shape of storm waveHigh6–9 mThe crests of the waves were covered with sprayVery high9–14 mDense spray covered the slope of the waves, and the sea turned white except in certain places within the troughPhenomenal>14 mThe whole sea surface is covered with dense spray layer, the air is full of water droplets, and the visibility is significantly reducedAccording to the wave disturbance model established above, the wave force and moment on the AUV decrease with the depth. AUV is most vulnerable to instability in shallow water due to interference. Therefore, the simulation is performed when the initial depth of AUV is 5 meters.


### 4.2. Design of Simulation Test

The model of the AUV and its ELM fuzzy PID controller is established in Matlab Simulink as shown in Figure10. The dynamic equation of the AUV is realized by using the 6DOF module in Simulink, and four modules are established to simulate the real force and moment acting on the AUV, which are hydrodynamic force module, gravity module, control force module, and wave disturbance force module, respectively. The controller includes two channels, which are vertical motion control channel and horizontal motion control channel. The structure of the two control channels is shown in Chapter 3.2, and the embedded linear model of the AUV is established by S-function.Figure 10
AUV model and its controller in Simulink under wave disturbance.In order to verify the control effect of the ELM fuzzy PID controller when AUV is sailing in a straight line, simulation test I is designed as follows: the AUV maintains a fixed depth of 5 meters and sails to the north direction at a speed of 4 knots. During the simulation, the wave disturbance model established above is adopted to disturb the AUV. Under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves, as a comparison, the classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated, respectively. The structure of the classical PID controller is similar to the controller proposed by Hammad et al. [37] The structure of the fuzzy PID controller is similar to the controller proposed by Zhang et al. [38] The basic information setting in the simulation process is shown in Table 4.Table 4
Controller tuning parameters and basic information used in the simulation.
Simulation informationClassical PIDNormal fuzzy PIDELM fuzzy PIDDuration of simulation/s40 s (test I)/50 s (test II, III)40 s (test I)/50 s (test II, III)40 s (test I)/50 s (test II, III)Step size/s0.01 s0.01 s0.01 sIterative methodRunge-KuttaRunge-KuttaRunge-KuttaKp_depth0.02[0, 0.05][0, 0.05]Ki_depth0.005[0, 0.01][0, 0.01]Kd_depth0.02[0, 0.05][0, 0.05]Qdepth——[−0.1, 0.1]Kp_θ1[0, 2][0, 2]Ki_θ0[0, 0.01][0, 0.01]Kd_θ1[0, 2][0, 2]Qθ——[−0.01, 0.01]Kp_CTD0.005[0, 0.01][0, 0.01]Ki_CTD0[0, 0.01][0, 0.01]Kd_CTD0.000 5[0, 0.001][0, 0.001]QCTD——[−0.1, 0.1]Kp_ψ1[0, 2][0, 2]Ki_ψ0[0, 0.01][0, 0.01]Kd_ψ3[0, 5][0, 0.01]Qψ——[−0.01, 0.01]It is necessary to verify the control ability of the ELM fuzzy PID controller in the process of AUV depth changing or horizontal transformation. Thus, simulation test II is designed as follows: the AUV sails to the north direction at a speed of 4 knots under the disturbance of the CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves, respectively. At the time of 20s, the depth transfers from 5 m to 10 m. Similar to test I, the classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated for comparison. The simulation information is shown in Table4.Simulation test III is similar to test II. The AUV sails to the north direction at a speed of 4 knots, and the cross-track distance of the AUV transfers from 0 m to 5 m at 20 s. During the simulation, the AUV is disturbed by the three kinds of waves. The classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated for comparison. The simulation information is shown in Table4.Since the actuator has a time delay when executing the control signal, it is considered to set the reaction time of the actuator to 0.2 seconds in the simulation process. Considering the influence of rudder surface saturation, the saturation values of the elevator and yaw rudder are set to 10 degrees.


## 4.1. Wave Disturbance Model for AUV

In order to verify whether the control system can effectively suppress the disturbance of waves to the AUV, this study proposes a wave disturbance model to simulate the influence of waves to the AUV.It is assumed that the wave disturbances to AUV are regular waves. The fluid studied satisfies the ideal fluid hypothesis of uniform, incompressible, nonviscous, and nonrotational. According to the Froude-Krylov hypothesis [36], the pressure distributions in regular waves are as follows:(20)p=ρgH2⋅coshk⋅z+hcoshk⋅h⋅coskxcosθ−kysinθ−ωt,where ρ is the density of water, g is the acceleration of gravity, k is the wave number, ω is the wave frequency, γ is the angle between the direction of AUV motion and the direction of wave travel, h is the water depth, and H is the wave height. For the six-degree-of-freedom movement of the AUV, the wave interference force is expressed as follows:(21)F=−∬p⋅ndS,M=−∬p⋅n×rdS,where S is the surface area of AUV, and n is the normal vector outside of AUV surface. By using Gauss theorem, the integral along the surface of the AUV can be transformed into volume integral, as shown in(22)Fx=−∫∫∫∂p∂xdV,Fy=−∫∫∫∂p∂ydV,Fz=−∫∫∫∂p∂zdV,Mx=−∫∫∫∂p∂zy−∂p∂yzdV,My=−∫∫∫∂p∂xz−∂p∂zxdV,Mz=−∫∫∫∂p∂yx−∂p∂xydV,where V is the volume of the AUV. As shown in Figure 9, the AUV can be approximately simplified to a cylinder of equal diameter. By substituting equations (20)–(22), and considering the ocean depth as infinite, the interference force and moment of the waves on the AUV can be simplified as follows:(23)Fx=Ex⋅2ρgπHk2sinγ⋅coshk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sinkRsinγ2coskLcosγ2,Fy=Ey⋅2ρgπHk2cosγ⋅coshk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sin−kRsinγ2sinkLcosγ2,Fz=Ez⋅2ρgπHk2cosγsinγ⋅sinhk⋅−D+h+kR/2⋅sinhkR/2coshkh⋅cosωt−kLcosγ−kRsinγ2sinkRsinγ2sinklcosγ2,Mx=ρgπHDxekD+kR−ekDk3cosγ⋅kD−1−1sin2γ⋅2sinωt−kLcosγ−kRsinγ2sinkRsinγ2coskLcosγ2+kRsinγsinkLcosγ2sinωt+kRsinγ−kLcosγ2,My=ρgπHDyekD+kR−ekDk3sinγ⋅−kD+1+1cos2γ⋅2sinωt−kLcosγ−kRsinγ2sinkRsinγ2sinkLcosγ2+LkcosγsinkRsinγ2sinωt+kLcosγ−kRsinγ2,Mz=ρgπHDzekD+kR−ekDk2sin2γ⋅Rsinγcosωt+kRsinγ−kLcosγ2sinkLcosγ2+LcosγsinkRsinγ2cosωt−kLcosγ+kRsinγ2,wherein Ex, Ey, Ez, Dx, Dy, and Dz are diffraction coefficients of waves in the directions of six-degrees-of-freedom of the AUV, respectively.Figure 9
Diagram of AUV under wave disturbance.The wave scales are shown in Table3. According to the international wave scales, waves are divided into ten levels. During the actual process, AUVs are generally affected by waves below the SLIGHT, which are usually the CALM-RIPPLED and the SMOOTH-WAVELET. Generally, the AUV missions are not performed in the SLIGHT level and above.Table 3
International wave scale. Source:https://wenku.baidu.com/view/98aac42e25284b73f242336c1eb91a37f1113286.html.
Wave scale nameWave height rangeDescription of sea surfaceCalm-glassy0 mThe sea surface is as smooth as a mirror, or there are only swellsCalm-rippled0–0.1 mRipple and swell exist at the same timeSmooth-wavelet0.1–0.5 mThe waves are very small, the crest begins to break, and the spray is not white but glassySlight0.5–1.25 mThe crests of the waves are broken, and some of them form white sprayModerate1.25–2.50 mWaves have distinct shapes and form white waves everywhereRough2.50–4.0 mThere were big wave crests, the spray occupied a large area on the crest, and the wind began to cut off the spray on the crestVery rough4–6 mThe wave crests sometimes appear the long wave shape of storm waveHigh6–9 mThe crests of the waves were covered with sprayVery high9–14 mDense spray covered the slope of the waves, and the sea turned white except in certain places within the troughPhenomenal>14 mThe whole sea surface is covered with dense spray layer, the air is full of water droplets, and the visibility is significantly reducedAccording to the wave disturbance model established above, the wave force and moment on the AUV decrease with the depth. AUV is most vulnerable to instability in shallow water due to interference. Therefore, the simulation is performed when the initial depth of AUV is 5 meters.


## 4.2. Design of Simulation Test

The model of the AUV and its ELM fuzzy PID controller is established in Matlab Simulink as shown in Figure10. The dynamic equation of the AUV is realized by using the 6DOF module in Simulink, and four modules are established to simulate the real force and moment acting on the AUV, which are hydrodynamic force module, gravity module, control force module, and wave disturbance force module, respectively. The controller includes two channels, which are vertical motion control channel and horizontal motion control channel. The structure of the two control channels is shown in Chapter 3.2, and the embedded linear model of the AUV is established by S-function.Figure 10
AUV model and its controller in Simulink under wave disturbance.In order to verify the control effect of the ELM fuzzy PID controller when AUV is sailing in a straight line, simulation test I is designed as follows: the AUV maintains a fixed depth of 5 meters and sails to the north direction at a speed of 4 knots. During the simulation, the wave disturbance model established above is adopted to disturb the AUV. Under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves, as a comparison, the classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated, respectively. The structure of the classical PID controller is similar to the controller proposed by Hammad et al. [37] The structure of the fuzzy PID controller is similar to the controller proposed by Zhang et al. [38] The basic information setting in the simulation process is shown in Table 4.Table 4
Controller tuning parameters and basic information used in the simulation.
Simulation informationClassical PIDNormal fuzzy PIDELM fuzzy PIDDuration of simulation/s40 s (test I)/50 s (test II, III)40 s (test I)/50 s (test II, III)40 s (test I)/50 s (test II, III)Step size/s0.01 s0.01 s0.01 sIterative methodRunge-KuttaRunge-KuttaRunge-KuttaKp_depth0.02[0, 0.05][0, 0.05]Ki_depth0.005[0, 0.01][0, 0.01]Kd_depth0.02[0, 0.05][0, 0.05]Qdepth——[−0.1, 0.1]Kp_θ1[0, 2][0, 2]Ki_θ0[0, 0.01][0, 0.01]Kd_θ1[0, 2][0, 2]Qθ——[−0.01, 0.01]Kp_CTD0.005[0, 0.01][0, 0.01]Ki_CTD0[0, 0.01][0, 0.01]Kd_CTD0.000 5[0, 0.001][0, 0.001]QCTD——[−0.1, 0.1]Kp_ψ1[0, 2][0, 2]Ki_ψ0[0, 0.01][0, 0.01]Kd_ψ3[0, 5][0, 0.01]Qψ——[−0.01, 0.01]It is necessary to verify the control ability of the ELM fuzzy PID controller in the process of AUV depth changing or horizontal transformation. Thus, simulation test II is designed as follows: the AUV sails to the north direction at a speed of 4 knots under the disturbance of the CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves, respectively. At the time of 20s, the depth transfers from 5 m to 10 m. Similar to test I, the classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated for comparison. The simulation information is shown in Table4.Simulation test III is similar to test II. The AUV sails to the north direction at a speed of 4 knots, and the cross-track distance of the AUV transfers from 0 m to 5 m at 20 s. During the simulation, the AUV is disturbed by the three kinds of waves. The classical PID, normal fuzzy PID, and ELM fuzzy PID are simulated for comparison. The simulation information is shown in Table4.Since the actuator has a time delay when executing the control signal, it is considered to set the reaction time of the actuator to 0.2 seconds in the simulation process. Considering the influence of rudder surface saturation, the saturation values of the elevator and yaw rudder are set to 10 degrees.


## 5. Results and Analysis

### 5.1. Results of Test I

For test I, Figure11 shows the vertical control simulation results of classical PID, normal fuzzy PID, and ELM fuzzy PID in test I when AUV is straightly sailing in the three waves. The comparison of the vertical motion control results of the three methods is shown in Table 5. Figure 12 shows the horizontal control simulation results of the three control methods in test I. As it can be indicated from the figures, the comparison of the horizontal motion control results of the three methods is shown in Table 6.Figure 11
Variation of AUV depthd under the disturbance of the CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (c), and SLIGHT waves (e), and variation of AUV pitch angle θ under the disturbance of the CALM-RIPPLED waves (b), SMOOTH-WAVELET waves (d), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 5
Comparison of the vertical motion control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test I.
Simulation resultsClassical PIDFuzzy PIDELM fuzzy PIDAmplitude of depth (calm-rippled)0.3 m0.1 m0.01 mAverage error of depth (calm-rippled)0.132 m0.0 265m0.0 077mAmplitude of pitch angle (calm-rippled)1°0.5°0.4°Average error of pitch angle (calm-rippled)0.289 6°0.186 2°0.145 7°Amplitude of depth (smooth-wavelet)1.6 m0.3 m0.05 mAverage error of depth (smooth-wavelet)0.512 m0.0 737 m0.0 188 mAmplitude of pitch angle (smooth-wavelet)4.5°2.5°2°Average error of pitch angle (smooth-wavelet)1.408 1°0.874 8°0.719 1°Amplitude of depth (slight)3 m0.6 m0.1 mAverage error of depth (slight)1.0 154 m0.1 418 m0.0 367 mAmplitude of pitch angle (slight)9°5.2°4.2°Average error of pitch angle (slight)2.822 3°1.728 2°1.437 8°Figure 12
Variation of AUV cross-track distancez under the disturbance of the CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (c), and SLIGHT waves (e), and variation of AUV yaw angle ψ under the disturbance of the CALM-RIPPLED waves (b), SMOOTH-WAVELET waves (d), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 6
Comparison of the horizontal motion control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test I.
Simulation resultsClassical PIDFuzzy PIDELM fuzzy PIDAmplitude of cross-track distance (calm-rippled)0.5 m0.1 m0.04 mAverage error of cross-track distance (calm-rippled)0.1 610 m0.0 173 m0.0 061 mAmplitude of yaw angle (calm-rippled)1.1°0.38°0.25°Average error of yaw angle (calm-rippled)0.345 3°0.133 9°0.087 7°Amplitude of cross-track distance (smooth-wavelet)1 m0.2 m0.1 mAverage error of cross-track distance (smooth-wavelet)0.3 215 m0.0 355 m0.0 129 mAmplitude of yaw angle (smooth-wavelet)2.3°0.7°0.5°Average error of yaw angle (smooth-wavelet)0.690 6°0.270 1°0.181 0°Amplitude of cross-track distance (slight)2 m0.4 m0.2 mAverage error of cross-track distance (slight)0.6 426 m0.0 736 m0.0 301 mAmplitude of yaw angle (slight)4.5°1.5°1.2°Average error of yaw angle (slight)1.381 7°0.547 6°0.375 0°Figure13 shows the control signal of the elevator in the pitch direction. It can be seen from the figure that the control signal of the ELM fuzzy PID control method is stronger than the other two control methods. The rudder saturation occurs several times at the beginning of the simulation. But the control strength decreases after the first wave disturbance cycle, and the rudder saturation does not appear. But the control strength is still the largest of the three methods. The ELM fuzzy PID control method has a stronger ability to sense disturbance, because the slope of its control signal is the largest and response speed is the fastest.Figure 13
The control signal of the elevator in the pitch direction.Based on these, the following conclusions can be drawn: first, the classical PID controller can ensure the stability of AUV that keeps a direct route in CALM-RIPPLED waves. However, in SMOOTH-WAVELET waves, the motion stability of AUV is seriously affected. Furthermore, in the SLIGHT waves, the oscillation of the AUV with the classical PID controller makes it to basically lose the working ability. The AUV with a normal fuzzy PID controller has the ability to keep straightly sailing under the disturbance of CALM-RIPPLED and SMOOTH-WAVELET waves. But its stability begins to be affected under the disturbance of SLIGHT waves. However, the ELM fuzzy PID controller is able to keep the AUV stable under the disturbance of all the three waves.Figure14 shows the control signal of the yaw rudder in the yaw direction. It can be seen from the figure that, similar to the pitch direction, the control signal of the ELM fuzzy PID control method is stronger than the other two control methods. The control force is the largest in each wave disturbance cycle, and there is no rudder saturation in the simulation process. Moreover, the ELM fuzzy PID control method has a stronger ability to sense a disturbance in the yaw direction and it has a faster response speed.Figure 14
The control signal of left and right propulsive force difference in the yaw direction.


### 5.2. Results of Test II

For test II, Figure15 shows the trajectory and attitude angles of the AUV with the three control methods under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. The trajectories of PID and trajectory of normal fuzzy PID are translated in the negative direction of Z axis in order to explicitly express the three trajectories. Table 7 shows the critical parameters of the control process of depth transformation. As it can be summarized from the figures, the AUV with the classical PID controller is able to maintain stability in the process of depth transformation under the disturbance of CALM-RIPPLED waves. But in SMOOTH-WAVELET and SLIGHT waves, the depth transformation process is unable to maintain stability, while the overshoot is over 40%. Furthermore, the variation of attitude angles and position is also unacceptable. For the normal fuzzy PID controller, it makes the overshoot less and variation smaller. It can maintain the stability of AUV in the process of depth transformation under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. For the ELM fuzzy PID controller, it is able to maintain the stability of AUV in the process under the three different wave scales. Its variation of attitude angles and position is the smallest of the three methods. In addition, the response speed is the fastest and the overshoot is small enough to keep stable under the three wave scales.Figure 15
Trajectory of AUV with the three control methods when the depthd changes from 5 m to 10 m under the disturbance of CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (b), and SLIGHT waves (c), and variation of attitudes and position of the AUV under the disturbance of CALM-RIPPLED waves (d), SMOOTH-WAVELET waves (e), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 7
Comparison of the control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test II.
Simulation resultsClassical PIDNormal fuzzy PIDELM fuzzy PIDResponse time of depth in test II (calm-rippled)3.45 s4.73 s2.87 sOvershoot of depth in test II (calm-rippled)1.82 m0.47 m0.77 mResponse time of depth in test II (smooth-wavelet)3.92 s5.29 s3.08 sOvershoot of depth in test II (smooth-wavelet)2.57 m0.48 m0.88 mResponse time of depth in test II (slight)2.34 s5.36 s3.01 sOvershoot of depth in test II (slight)3.18 m0.47 m0.97 m


### 5.3. Results of Test III

For test III, Figure16 shows the trajectory and attitude angles of the AUV with the three control methods in test III under the disturbance of the three waves. Similar to test II, the trajectory of PID and trajectory of normal fuzzy PID are translated in the negative direction of Z axis in order to explicitly express the three trajectories. Table 8 shows the critical parameters of the control process of horizontal transformation. The results are consistent with test II. For the PID controller, the AUV with the classical PID controller is able to maintain stability in the process of horizontal transformation under the disturbance of CALM-RIPPLED waves. But in SMOOTH-WAVELET and SLIGHT waves, the variation of the attitude angles and position is violent and unacceptable. Furthermore, the response speed is the slowest of the three methods. For the normal fuzzy PID controller, there is an improvement in the response speed and variation range. The overshoot of the cross-track distance is less than 20%; consequently, it can maintain the stability of AUV in the process of horizontal transformation under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. However, the horizontal variation is tended to be a little bit large in the SLIGHT waves. For the ELM fuzzy PID controller, it is able to maintain the stability of the AUV in the process under all the three different wave scales. Its variation of attitudes and position is the smallest of the three methods. The overshoot is less than the normal fuzzy PID controller. In addition, the response speed is almost as fast as the fuzzy PID controller.Figure 16
Trajectory of AUV with the three control methods when the cross-track distanceCTD changes from 0 m to 5 m (left) under the disturbance of CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (b), and SLIGHT waves (c), and variation of attitudes and position of the AUV under the disturbance of CALM-RIPPLED waves (d), SMOOTH-WAVELET waves (e), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 8
Comparison of the control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test III.
Simulation resultsClassical PIDNormal fuzzy PIDELM fuzzy PIDResponse time of CTD in test III (calm-rippled)6.95 s5.12 s5.56 sOvershoot of CTD in test III (calm-rippled)0.20 m0.46 m0.13 mResponse time of CTD in test III (smooth-wavelet)7.23 s4.87 s5.26 sOvershoot of CTD in test III (smooth-wavelet)0.58 m0.65 m0.12 mResponse time of CTD in test III (slight)11.82 s4.91 s4.92 sOvershoot of CTD in test III (slight)1.12 m0.67 m0.31 m


## 5.1. Results of Test I

For test I, Figure11 shows the vertical control simulation results of classical PID, normal fuzzy PID, and ELM fuzzy PID in test I when AUV is straightly sailing in the three waves. The comparison of the vertical motion control results of the three methods is shown in Table 5. Figure 12 shows the horizontal control simulation results of the three control methods in test I. As it can be indicated from the figures, the comparison of the horizontal motion control results of the three methods is shown in Table 6.Figure 11
Variation of AUV depthd under the disturbance of the CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (c), and SLIGHT waves (e), and variation of AUV pitch angle θ under the disturbance of the CALM-RIPPLED waves (b), SMOOTH-WAVELET waves (d), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 5
Comparison of the vertical motion control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test I.
Simulation resultsClassical PIDFuzzy PIDELM fuzzy PIDAmplitude of depth (calm-rippled)0.3 m0.1 m0.01 mAverage error of depth (calm-rippled)0.132 m0.0 265m0.0 077mAmplitude of pitch angle (calm-rippled)1°0.5°0.4°Average error of pitch angle (calm-rippled)0.289 6°0.186 2°0.145 7°Amplitude of depth (smooth-wavelet)1.6 m0.3 m0.05 mAverage error of depth (smooth-wavelet)0.512 m0.0 737 m0.0 188 mAmplitude of pitch angle (smooth-wavelet)4.5°2.5°2°Average error of pitch angle (smooth-wavelet)1.408 1°0.874 8°0.719 1°Amplitude of depth (slight)3 m0.6 m0.1 mAverage error of depth (slight)1.0 154 m0.1 418 m0.0 367 mAmplitude of pitch angle (slight)9°5.2°4.2°Average error of pitch angle (slight)2.822 3°1.728 2°1.437 8°Figure 12
Variation of AUV cross-track distancez under the disturbance of the CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (c), and SLIGHT waves (e), and variation of AUV yaw angle ψ under the disturbance of the CALM-RIPPLED waves (b), SMOOTH-WAVELET waves (d), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 6
Comparison of the horizontal motion control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test I.
Simulation resultsClassical PIDFuzzy PIDELM fuzzy PIDAmplitude of cross-track distance (calm-rippled)0.5 m0.1 m0.04 mAverage error of cross-track distance (calm-rippled)0.1 610 m0.0 173 m0.0 061 mAmplitude of yaw angle (calm-rippled)1.1°0.38°0.25°Average error of yaw angle (calm-rippled)0.345 3°0.133 9°0.087 7°Amplitude of cross-track distance (smooth-wavelet)1 m0.2 m0.1 mAverage error of cross-track distance (smooth-wavelet)0.3 215 m0.0 355 m0.0 129 mAmplitude of yaw angle (smooth-wavelet)2.3°0.7°0.5°Average error of yaw angle (smooth-wavelet)0.690 6°0.270 1°0.181 0°Amplitude of cross-track distance (slight)2 m0.4 m0.2 mAverage error of cross-track distance (slight)0.6 426 m0.0 736 m0.0 301 mAmplitude of yaw angle (slight)4.5°1.5°1.2°Average error of yaw angle (slight)1.381 7°0.547 6°0.375 0°Figure13 shows the control signal of the elevator in the pitch direction. It can be seen from the figure that the control signal of the ELM fuzzy PID control method is stronger than the other two control methods. The rudder saturation occurs several times at the beginning of the simulation. But the control strength decreases after the first wave disturbance cycle, and the rudder saturation does not appear. But the control strength is still the largest of the three methods. The ELM fuzzy PID control method has a stronger ability to sense disturbance, because the slope of its control signal is the largest and response speed is the fastest.Figure 13
The control signal of the elevator in the pitch direction.Based on these, the following conclusions can be drawn: first, the classical PID controller can ensure the stability of AUV that keeps a direct route in CALM-RIPPLED waves. However, in SMOOTH-WAVELET waves, the motion stability of AUV is seriously affected. Furthermore, in the SLIGHT waves, the oscillation of the AUV with the classical PID controller makes it to basically lose the working ability. The AUV with a normal fuzzy PID controller has the ability to keep straightly sailing under the disturbance of CALM-RIPPLED and SMOOTH-WAVELET waves. But its stability begins to be affected under the disturbance of SLIGHT waves. However, the ELM fuzzy PID controller is able to keep the AUV stable under the disturbance of all the three waves.Figure14 shows the control signal of the yaw rudder in the yaw direction. It can be seen from the figure that, similar to the pitch direction, the control signal of the ELM fuzzy PID control method is stronger than the other two control methods. The control force is the largest in each wave disturbance cycle, and there is no rudder saturation in the simulation process. Moreover, the ELM fuzzy PID control method has a stronger ability to sense a disturbance in the yaw direction and it has a faster response speed.Figure 14
The control signal of left and right propulsive force difference in the yaw direction.


## 5.2. Results of Test II

For test II, Figure15 shows the trajectory and attitude angles of the AUV with the three control methods under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. The trajectories of PID and trajectory of normal fuzzy PID are translated in the negative direction of Z axis in order to explicitly express the three trajectories. Table 7 shows the critical parameters of the control process of depth transformation. As it can be summarized from the figures, the AUV with the classical PID controller is able to maintain stability in the process of depth transformation under the disturbance of CALM-RIPPLED waves. But in SMOOTH-WAVELET and SLIGHT waves, the depth transformation process is unable to maintain stability, while the overshoot is over 40%. Furthermore, the variation of attitude angles and position is also unacceptable. For the normal fuzzy PID controller, it makes the overshoot less and variation smaller. It can maintain the stability of AUV in the process of depth transformation under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. For the ELM fuzzy PID controller, it is able to maintain the stability of AUV in the process under the three different wave scales. Its variation of attitude angles and position is the smallest of the three methods. In addition, the response speed is the fastest and the overshoot is small enough to keep stable under the three wave scales.Figure 15
Trajectory of AUV with the three control methods when the depthd changes from 5 m to 10 m under the disturbance of CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (b), and SLIGHT waves (c), and variation of attitudes and position of the AUV under the disturbance of CALM-RIPPLED waves (d), SMOOTH-WAVELET waves (e), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 7
Comparison of the control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test II.
Simulation resultsClassical PIDNormal fuzzy PIDELM fuzzy PIDResponse time of depth in test II (calm-rippled)3.45 s4.73 s2.87 sOvershoot of depth in test II (calm-rippled)1.82 m0.47 m0.77 mResponse time of depth in test II (smooth-wavelet)3.92 s5.29 s3.08 sOvershoot of depth in test II (smooth-wavelet)2.57 m0.48 m0.88 mResponse time of depth in test II (slight)2.34 s5.36 s3.01 sOvershoot of depth in test II (slight)3.18 m0.47 m0.97 m


## 5.3. Results of Test III

For test III, Figure16 shows the trajectory and attitude angles of the AUV with the three control methods in test III under the disturbance of the three waves. Similar to test II, the trajectory of PID and trajectory of normal fuzzy PID are translated in the negative direction of Z axis in order to explicitly express the three trajectories. Table 8 shows the critical parameters of the control process of horizontal transformation. The results are consistent with test II. For the PID controller, the AUV with the classical PID controller is able to maintain stability in the process of horizontal transformation under the disturbance of CALM-RIPPLED waves. But in SMOOTH-WAVELET and SLIGHT waves, the variation of the attitude angles and position is violent and unacceptable. Furthermore, the response speed is the slowest of the three methods. For the normal fuzzy PID controller, there is an improvement in the response speed and variation range. The overshoot of the cross-track distance is less than 20%; consequently, it can maintain the stability of AUV in the process of horizontal transformation under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves. However, the horizontal variation is tended to be a little bit large in the SLIGHT waves. For the ELM fuzzy PID controller, it is able to maintain the stability of the AUV in the process under all the three different wave scales. Its variation of attitudes and position is the smallest of the three methods. The overshoot is less than the normal fuzzy PID controller. In addition, the response speed is almost as fast as the fuzzy PID controller.Figure 16
Trajectory of AUV with the three control methods when the cross-track distanceCTD changes from 0 m to 5 m (left) under the disturbance of CALM-RIPPLED waves (a), SMOOTH-WAVELET waves (b), and SLIGHT waves (c), and variation of attitudes and position of the AUV under the disturbance of CALM-RIPPLED waves (d), SMOOTH-WAVELET waves (e), and SLIGHT waves (f).
(a)(b)(c)(d)(e)(f)Table 8
Comparison of the control results of classical PID, normal fuzzy PID, and ELM PID controller under the disturbance of three waves in test III.
Simulation resultsClassical PIDNormal fuzzy PIDELM fuzzy PIDResponse time of CTD in test III (calm-rippled)6.95 s5.12 s5.56 sOvershoot of CTD in test III (calm-rippled)0.20 m0.46 m0.13 mResponse time of CTD in test III (smooth-wavelet)7.23 s4.87 s5.26 sOvershoot of CTD in test III (smooth-wavelet)0.58 m0.65 m0.12 mResponse time of CTD in test III (slight)11.82 s4.91 s4.92 sOvershoot of CTD in test III (slight)1.12 m0.67 m0.31 m


## 6. Conclusions

In this study, a new ELM fuzzy PID controller is introduced to the AUV under the disturbance of ocean waves. An ocean wave disturbance model is established in order to simulate the force acting on the AUV. Finally, the control effect of the ELM fuzzy PID control system under the disturbance of CALM-RIPPLED, SMOOTH-WAVELET, and SLIGHT waves is simulated by using the established wave disturbance model to AUV.From the results of simulations, the conclusion can be drawn as below: compared with the classical PID controller and the normal fuzzy PID controller, the ELM fuzzy PID controller can achieve a better control effect when the AUV disturbed by waves, which mainly reflected in three aspects: stability, response speed, and overshoot. In the first aspect, the ELM fuzzy PID controller can significantly reduce the wave disturbance to AUV and enhance the stability, especially in large wave disturbance. In the second aspect, the simulation results show that the AUV with the ELM fuzzy PID controller achieves a comprehensive optimization of response speed in all the three wave scales. In the last aspect, the overshoot of the AUV with the ELM fuzzy PID controller is small enough to keep the stability after the transformation in all the three wave scales.The research results can be applied to various types of AUV, which can reduce their own posture oscillations when AUV takes photographs or detects sonar signals, thereby reducing the overlap of detection images or reducing the difficulty of camera focusing. Moreover, it can obviously improve the maneuverability and robustness of the AUV and then improve the ability to perform missions in large wave conditions.



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*Source: 4126595-2022-03-04.xml*