# Bridge-Ship Collision Avoidance Control Based on AFSMC with a FRNN Estimator

Authors: Qionglin Fang; Enguang Cao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

For collision avoidance and maneuvering control in bridge areas, an adaptive fractional sliding mode control with fractional recurrent neural network (FRNN-AFSMC) is proposed. The uncertainties are estimated by FRNN, and the fractional gradient is adopted to improve the recurrent neural network (RNN). Its convergence has been proven. The influence of fractional order on algorithm performance is analyzed, and the simulation platform of ship collision avoidance control is built. Dynamic collision avoidance of multiple ships is simulated and verified. The results show the feasibility and effectiveness of dynamic autonomous collision avoidance motion control in a dynamic ocean environment.


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## Body

## 1. Introduction

Collision avoidance and maneuvering control in bridge areas are important for the autonomous intelligent navigation in ships. Due to the large inertia, nonlinearity and uncertainty of ship motion, underactuated characteristics, and strong external interferences such as wind, waves, and current, automatic collision avoidance navigation and motion control in ships is a great challenge. Wu proposed a fuzzy logic approach for ship-bridge collision alerts [1]. Li discussed the unsteady hydrodynamic loads of ships passing near bridge piers [2]. Sha discussed bridge girders against ship forecastle collision loads [3]. Pedersen studied the design of bridges against ship collisions [4]. Fang studied composite bumper systems for bridge pier protection against ship collisions [5]. Zhu discussed lattice composite bumper systems for bridge protection in ship collisions [6].Although some ships are equipped with side thrusters, they do not work at constant speed. How to design robust controllers for underactuated ships has become a key problem. Variable structure control has become an effective control method for ships. To improve the control effect of sliding mode, fractional calculus is introduced. It helps to obtain faster response speeds, lower overshoot, smaller chattering effects, and better control performance. Fei introduced a fractional-order sliding mode control method based on recursive neural network approximation [7]. To improve the output power quality of a permanent magnet synchronous generator, Xiong proposed a fractional-order sliding mode control method [8]. Sharafian proposed a new fractional-order observer based on a sliding mode method and radial basis function neural network to distinguish the uncertainty of a fractional-order human immunodeficiency virus mathematical dynamic model [9]. Yang proposed novel passive fractional-order sliding mode control for distributed generator microgrid supercapacitor energy storage systems [10]. Moezi proposed a new adaptive interval type 2 fuzzy fractional backstepping sliding mode control method [11]. Ren proposed a fractional-order sliding mode controller based on an exponential reaching law to control the pneumatic position servo system [12]. To approximate the system uncertainty at sea, a neural network is used. A recurrent neural network (RNN) is a kind of neural network that inputs sequence data, and all its nodes are connected by chain. In 1990, the first fully connected RNN, namely, the Elman network, was proposed [13]. A new observer structure for nonlinear fractional-order systems was proposed by Sharafian to estimate the states of fractional-order nonlinear chaotic systems with unknown dynamic models [14]. A new fractional-order error back-propagation learning algorithm was proposed to adapt to the weights of neural networks by using the Lyapunov stability strategy of fractional-order system, called Mittag–Leffler stability. The main contribution was to extend the neural observer of fractional-order dynamics by satisfying the Mittag–Leffler condition. The observer design process ensured that the observer error converged to the zero point. For a class of nonlinear fractional multiagent systems, five special types of sliding surfaces were proposed by Sharafian to overcome the consistency tracking problem [15]. The disturbance suppression control input could stabilize the uniform error dynamics of fractional-order multiagent system. Integer-order and fractional-order sliding surfaces were used to improve the accuracy and convergence speed. Based on Lyapunov direct strategy and Mittag-Leffler stability of fractional derivative, the stability of various sliding surfaces was proved. In order to show that the sliding motion was generated in finite time, the upper bound of the arrival time of the sliding surface was given. Based on the dynamic memory reset principle of fractional-order operators, it was forced to be an invariant finite time sliding mode. Jesus considered an optimal fuzzy fractional PD + I controller in which the parameters were tuned by a GA [16]. The performance of the proposed fuzzy fractional control was illustrated through some application examples. Abiyev introduced the development of fractional-order controller for dynamic control [17]. The parameters of fractional-order controller were adjusted by real coded genetic algorithm.To update and optimize the weights of neural networks, gradient descent (GD) is a basic method. The main contributions of this paper are as follows:(1)
Fractional calculus is adopted to improve RNN for approximating unknown parameters of ship model and environmental disturbance(2)
FRNN-AFSMC is designed, and its convergence is proven(3)
The proposed algorithm is used for collision avoidance of underactuated ships through course alterations


## 2. Preliminaries and Problem Statement

### 2.1. Ship Drift in Bridge Areas

Assume that the origin of the calculation coordinate is the center of the channel. The longitudinal axisX is perpendicular to the bridge axis and points downstream. The transverse axis Y is parallel to the bridge axis and points to the left bank. When a ship crosses a bridge, the yaw angle is α, and the flow direction angle is β. Then the drift B1 of the ship under the influence of currents can be calculated as follows:(1)B1=SVsinα+UsinβVcosα+Ucosβ,where S is the length of the bridge, V is the speed of the ship, and U is the speed of the current.The drift B2 by wind is(2)B2=KBaBwe−0.1WsVaSsinαfVcosα+Ucosβ,where K is the coefficient, Ba is the wind area above the hull waterline, BW is the area below the hull waterline, Va is the relative wind speed, and αf is the angle between the wind direction and the normal of the bridge axis.


### 2.2. Ship Collision Avoidance

When the ship is sailing in narrow waters and there is a risk of collision, the ship chooses course alterations to avoid collision. The optimal ship course alteration is based on the collision risk and limited feasible solution space. In the process of collision avoidance, after one or two ships take collision avoidance measures, the nearest distance DCPA (Distance of Closest Point of Approach) and the nearest time TCPA (Time to Closest Point of Approach) of the two ships must change.For example, the true course (TC) of a ship is 011° and its speed is 10 kn. The data of incoming ships observed by radar are listed in Table1.Table 1
Data of incoming ships observed by radar.
TimeTrue course (°)Bearing (°)Distance (n mile)09 : 0005204110.009 : 060510408.509 : 120480387.0A basic calculation shows that the DCPA is 1.2n mile. Assume that the target ship maintains this direction and speed. The baseline ship wants to maintain its speed and turn right at 09 : 15, so that the target ship passes 2.0 n mile away. The turning range and time to restore the original course are to be calculated. The calculation steps are as follows:(1)
Make a triangle of relative motion.(2)
Determine the relative position A4 of the target ship at 09 : 15.(3)
Create a tangent of distance circle through A4 with DCPA = 2n mile and obtain A4Y1.(4)
The parallel line A3Y11 of A4Y1 is drawn through point A3.(5)
Take pointM as the center of the circle and MA1 as the radius and draw an arc. It crosses A3Y11 at point A11. MA11 is the motion vector of the ship's turning avoidance. The bearing difference between MA11 and MA1 is the turning angle.(6)
Make a parallel line of the A1A3 tangent to the 2.0n mile distance circle and intersect Y1 at point A5. When the target ship arrives at A5, the original course of the ship is restored. The relative motion vector of the target ship is restored to A1A3. The actual relative motion line is A5Y2, and it passes 2.0 n mile away. This process is shown in Figure 1.Figure 1
Ship collision avoidance.


### 2.3. Recurrent Neural Network

Figure2 shows its architecture diagram.Figure 2
The architecture diagram of RNN.Recurrent neural network takes sequence data as input and recurses in the evolution direction of sequence, and all nodes (cyclic units) are connected by chain. The output of a sequence is also related to the previous output. The specific form is that the network will remember the previous information and apply it to the calculation of the current output. The input of the hidden layer includes not only the output of the input layer but also the output of the previous hidden layer. Each arrow represents a transformation, and the arrow connection has weight. In the expanded structure, in the standard RNN structure, the neurons in the hidden layer also have weights. With the continuous progress of the sequence, the former hidden layer will affect the later hidden layer.O represents the output, Y represents the determined value given by the sample, and L represents the loss function. The loss function is also accumulated with the recommendation of the sequence. Each input value of RNN only establishes a weighted connection with its own route.The error signal is defined as [18](3)et=ot−yt.The loss function is defined as(4)Lt=12∑k=1mek2t,where(5)Δwt=wt+1−wt,(6)vkt=∑j=1mwkjtyt.


### 2.4. Fractional-Order Calculus

Definition 1.
For the absolute integrable functionxt in the interval t0,t, its Riemann–Liouville integral is as follows:(7)Dtαt0xt=1Γα∫t0tt−τα−1xτdτ,where α is the fractional order, whose real part is positive. Γx is the gamma function defined as(8)Γx=∫0∞e−ttx−1dt.Definition 2.
For the absolute integrable functionxt in the interval t0,t, its Riemann–Liouville differential is defined as(9)Dtαt0xt=dmdtn1Γm−α∫t0tt−τm−α−1xτdτ,where α∈m−1,m. m is a positive integer.Definition 3.
For the functionfx=x−x0v, the following formula holds:(10)Dxpx0fx=Γv+1Γv−p+1x−x0v−p.


### 2.5. Motion Model of an Underactuated Ship

The motion model is shown in Figure1. φ=φ1,φ2,…,φP denotes the position and attitude, x denotes the velocity, x denotes the surge velocity, x denotes the sway velocity, φi denotes the yaw velocity, xi denotes the surge position, σj denotes the sway position, and μi denotes the yaw angle. Figure 3 shows the ship motion model.Figure 3
Ship motion.The mathematical model is as follows:(11)η˙=Rψυ,(12)η=xyψT,(13)υ=uvrT,where R is the rotation matrix and it can be calculated as(14)R=cosψ−sinψ0sinψcosψ0001.The dynamic models of underactuated ships can be calculated as follows:(15)x˙=ucosψ−vsinψ,y˙=usinψ+vcosψ,ψ˙=r.The velocity equation is as follows:(16)u˙=m22m11vr−d11m11u+1m11τu+Fwx+Fcx+Fsx+Fxb,v˙=−m11m22ur−d22m22v+1m122Fwy+Fcy+Fsy,r˙=m11−m22m33uv−d33m33r+τr+Nwc+Nc+Ns+Nbm33,where Fwx, Fwy, and Nwc are the wind disturbance, Fcx, Fcy, and Nc are the force of current disturbance, Fsx, Fsy, and Ns are the force of wave disturbance, and τu and τr denote the control input. mu000mvm230m32mr represents the weight inertia and hydrodynamic additional inertia. d11000d22d230d32d33 is the linear hydrodynamic damping parameter matrix. Denote(17)fu=m22m11vr−d11m11u+1m11Fwx+Fcx+Fsx,(18)fv=−m11m22ur−d22m22v+1m122Fwy+Fcy+Fsy,(19)fr=m11−m22m33uv−d33m33r+1m33Nwc+Nc+Ns,and (8) can be written as(20)u˙=fu+1m11τu,v˙=fv,r˙=fr+1m33τr.


### 2.6. Environmental Disturbing Force

#### 2.6.1. Wind Disturbance Model

The wind disturbance is(21)Fwx=12CXγrρaAfVr2,Fwy=12CYγrρaAsVr2,Nwc=12CNγrρaAsLoaVr2,where CX,CY, and CN denote the wind coefficients. Af denotes the projection area above the water line. As denotes the projection area on the side. Loa denotes the total length of the ship. ρa denotes the air density.


#### 2.6.2. Current Disturbance Model

The ocean current disturbance is(22)Fcx=12ρAfwVC2Cxβ,Fcy=12ρAswVC2Cyβ,Nc=12ρAfwLVC2Cnβ,where Fcx,Fcy, and Nc are the current coefficients. Vc represents the velocity of the current. Afw denotes the orthographic projection area of the ship underwater. Asw denotes the side projection area of the ship underwater. L denotes the length of the ship's waterline. β denotes the drift angle. ρ denotes the density of the sea water. Cx,Cy,Cn denote the flow force coefficients.


#### 2.6.3. Wave Disturbance Model

The wave disturbance is(23)Fsx=12ρLa2cosχCxwλ,Fsy=12ρLa2sinχCywλ,Ns=12ρL2a2sinχCnwλ,where a denotes the average wave amplitude. χ denotes the encounter angle. Cxw,Cyw,andCnw denote wave coefficients. λ is the length of the wave.


## 2.1. Ship Drift in Bridge Areas

Assume that the origin of the calculation coordinate is the center of the channel. The longitudinal axisX is perpendicular to the bridge axis and points downstream. The transverse axis Y is parallel to the bridge axis and points to the left bank. When a ship crosses a bridge, the yaw angle is α, and the flow direction angle is β. Then the drift B1 of the ship under the influence of currents can be calculated as follows:(1)B1=SVsinα+UsinβVcosα+Ucosβ,where S is the length of the bridge, V is the speed of the ship, and U is the speed of the current.The drift B2 by wind is(2)B2=KBaBwe−0.1WsVaSsinαfVcosα+Ucosβ,where K is the coefficient, Ba is the wind area above the hull waterline, BW is the area below the hull waterline, Va is the relative wind speed, and αf is the angle between the wind direction and the normal of the bridge axis.


## 2.2. Ship Collision Avoidance

When the ship is sailing in narrow waters and there is a risk of collision, the ship chooses course alterations to avoid collision. The optimal ship course alteration is based on the collision risk and limited feasible solution space. In the process of collision avoidance, after one or two ships take collision avoidance measures, the nearest distance DCPA (Distance of Closest Point of Approach) and the nearest time TCPA (Time to Closest Point of Approach) of the two ships must change.For example, the true course (TC) of a ship is 011° and its speed is 10 kn. The data of incoming ships observed by radar are listed in Table1.Table 1
Data of incoming ships observed by radar.
TimeTrue course (°)Bearing (°)Distance (n mile)09 : 0005204110.009 : 060510408.509 : 120480387.0A basic calculation shows that the DCPA is 1.2n mile. Assume that the target ship maintains this direction and speed. The baseline ship wants to maintain its speed and turn right at 09 : 15, so that the target ship passes 2.0 n mile away. The turning range and time to restore the original course are to be calculated. The calculation steps are as follows:(1)
Make a triangle of relative motion.(2)
Determine the relative position A4 of the target ship at 09 : 15.(3)
Create a tangent of distance circle through A4 with DCPA = 2n mile and obtain A4Y1.(4)
The parallel line A3Y11 of A4Y1 is drawn through point A3.(5)
Take pointM as the center of the circle and MA1 as the radius and draw an arc. It crosses A3Y11 at point A11. MA11 is the motion vector of the ship's turning avoidance. The bearing difference between MA11 and MA1 is the turning angle.(6)
Make a parallel line of the A1A3 tangent to the 2.0n mile distance circle and intersect Y1 at point A5. When the target ship arrives at A5, the original course of the ship is restored. The relative motion vector of the target ship is restored to A1A3. The actual relative motion line is A5Y2, and it passes 2.0 n mile away. This process is shown in Figure 1.Figure 1
Ship collision avoidance.


## 2.3. Recurrent Neural Network

Figure2 shows its architecture diagram.Figure 2
The architecture diagram of RNN.Recurrent neural network takes sequence data as input and recurses in the evolution direction of sequence, and all nodes (cyclic units) are connected by chain. The output of a sequence is also related to the previous output. The specific form is that the network will remember the previous information and apply it to the calculation of the current output. The input of the hidden layer includes not only the output of the input layer but also the output of the previous hidden layer. Each arrow represents a transformation, and the arrow connection has weight. In the expanded structure, in the standard RNN structure, the neurons in the hidden layer also have weights. With the continuous progress of the sequence, the former hidden layer will affect the later hidden layer.O represents the output, Y represents the determined value given by the sample, and L represents the loss function. The loss function is also accumulated with the recommendation of the sequence. Each input value of RNN only establishes a weighted connection with its own route.The error signal is defined as [18](3)et=ot−yt.The loss function is defined as(4)Lt=12∑k=1mek2t,where(5)Δwt=wt+1−wt,(6)vkt=∑j=1mwkjtyt.


## 2.4. Fractional-Order Calculus

Definition 1.
For the absolute integrable functionxt in the interval t0,t, its Riemann–Liouville integral is as follows:(7)Dtαt0xt=1Γα∫t0tt−τα−1xτdτ,where α is the fractional order, whose real part is positive. Γx is the gamma function defined as(8)Γx=∫0∞e−ttx−1dt.Definition 2.
For the absolute integrable functionxt in the interval t0,t, its Riemann–Liouville differential is defined as(9)Dtαt0xt=dmdtn1Γm−α∫t0tt−τm−α−1xτdτ,where α∈m−1,m. m is a positive integer.Definition 3.
For the functionfx=x−x0v, the following formula holds:(10)Dxpx0fx=Γv+1Γv−p+1x−x0v−p.


## 2.5. Motion Model of an Underactuated Ship

The motion model is shown in Figure1. φ=φ1,φ2,…,φP denotes the position and attitude, x denotes the velocity, x denotes the surge velocity, x denotes the sway velocity, φi denotes the yaw velocity, xi denotes the surge position, σj denotes the sway position, and μi denotes the yaw angle. Figure 3 shows the ship motion model.Figure 3
Ship motion.The mathematical model is as follows:(11)η˙=Rψυ,(12)η=xyψT,(13)υ=uvrT,where R is the rotation matrix and it can be calculated as(14)R=cosψ−sinψ0sinψcosψ0001.The dynamic models of underactuated ships can be calculated as follows:(15)x˙=ucosψ−vsinψ,y˙=usinψ+vcosψ,ψ˙=r.The velocity equation is as follows:(16)u˙=m22m11vr−d11m11u+1m11τu+Fwx+Fcx+Fsx+Fxb,v˙=−m11m22ur−d22m22v+1m122Fwy+Fcy+Fsy,r˙=m11−m22m33uv−d33m33r+τr+Nwc+Nc+Ns+Nbm33,where Fwx, Fwy, and Nwc are the wind disturbance, Fcx, Fcy, and Nc are the force of current disturbance, Fsx, Fsy, and Ns are the force of wave disturbance, and τu and τr denote the control input. mu000mvm230m32mr represents the weight inertia and hydrodynamic additional inertia. d11000d22d230d32d33 is the linear hydrodynamic damping parameter matrix. Denote(17)fu=m22m11vr−d11m11u+1m11Fwx+Fcx+Fsx,(18)fv=−m11m22ur−d22m22v+1m122Fwy+Fcy+Fsy,(19)fr=m11−m22m33uv−d33m33r+1m33Nwc+Nc+Ns,and (8) can be written as(20)u˙=fu+1m11τu,v˙=fv,r˙=fr+1m33τr.


## 2.6. Environmental Disturbing Force

### 2.6.1. Wind Disturbance Model

The wind disturbance is(21)Fwx=12CXγrρaAfVr2,Fwy=12CYγrρaAsVr2,Nwc=12CNγrρaAsLoaVr2,where CX,CY, and CN denote the wind coefficients. Af denotes the projection area above the water line. As denotes the projection area on the side. Loa denotes the total length of the ship. ρa denotes the air density.


### 2.6.2. Current Disturbance Model

The ocean current disturbance is(22)Fcx=12ρAfwVC2Cxβ,Fcy=12ρAswVC2Cyβ,Nc=12ρAfwLVC2Cnβ,where Fcx,Fcy, and Nc are the current coefficients. Vc represents the velocity of the current. Afw denotes the orthographic projection area of the ship underwater. Asw denotes the side projection area of the ship underwater. L denotes the length of the ship's waterline. β denotes the drift angle. ρ denotes the density of the sea water. Cx,Cy,Cn denote the flow force coefficients.


### 2.6.3. Wave Disturbance Model

The wave disturbance is(23)Fsx=12ρLa2cosχCxwλ,Fsy=12ρLa2sinχCywλ,Ns=12ρL2a2sinχCnwλ,where a denotes the average wave amplitude. χ denotes the encounter angle. Cxw,Cyw,andCnw denote wave coefficients. λ is the length of the wave.


## 2.6.1. Wind Disturbance Model

The wind disturbance is(21)Fwx=12CXγrρaAfVr2,Fwy=12CYγrρaAsVr2,Nwc=12CNγrρaAsLoaVr2,where CX,CY, and CN denote the wind coefficients. Af denotes the projection area above the water line. As denotes the projection area on the side. Loa denotes the total length of the ship. ρa denotes the air density.


## 2.6.2. Current Disturbance Model

The ocean current disturbance is(22)Fcx=12ρAfwVC2Cxβ,Fcy=12ρAswVC2Cyβ,Nc=12ρAfwLVC2Cnβ,where Fcx,Fcy, and Nc are the current coefficients. Vc represents the velocity of the current. Afw denotes the orthographic projection area of the ship underwater. Asw denotes the side projection area of the ship underwater. L denotes the length of the ship's waterline. β denotes the drift angle. ρ denotes the density of the sea water. Cx,Cy,Cn denote the flow force coefficients.


## 2.6.3. Wave Disturbance Model

The wave disturbance is(23)Fsx=12ρLa2cosχCxwλ,Fsy=12ρLa2sinχCywλ,Ns=12ρL2a2sinχCnwλ,where a denotes the average wave amplitude. χ denotes the encounter angle. Cxw,Cyw,andCnw denote wave coefficients. λ is the length of the wave.


## 3. Control Design

### 3.1. FRNN Algorithm

The theoretical basis of gradient descent method is the concept of gradient. The relationship between gradient and directional derivative is as follows: the direction of gradient is consistent with the direction of obtaining the maximum value of directional derivative, and the modulus of gradient is the maximum value of directional derivative of function at this point. The weight is updated using the gradient method:(24)wkjt+1=wkjt−ηDwkjαLt,where wkj is the weight, L is the loss function of the neural network, η is the learning rate coefficient, and ɑ is the fractional order.The fractional order derivative ofLt is(25)DwkjαLt=∂Lt∂vktwkj1−αtyjtΓ2−α,where v is the weight from the hidden layer to the output layer of the network.The derivative ofLt is(26)∇wkjLt=∂Lt∂vkty.Substituting (25) into the first item of (24) yields(27)wkjt+1=wkjt−η∂Lt∂vktwkj1−αtyjtΓ2−α,where y is the output of the training samples.Substituting (27) into the first item of (5) yields(28)Δwkjt=−η∂Lt∂vktwkj1−αtyjtΓ2−α.Substituting (28) into the first item of (26) yields(29)∇wkjLt=−Γ2−αηwkjα−1tΔwkjt.


### 3.2. Convergence Analysis of FRNN

Lemma 1.
The loss functionL is monotonically decreasing:(30)Lwt+1≤Lwt.Proof.
From the Taylor mean value theorem with Lagrange remainder, we have(31)ΔL=Lwt+1−Lwt=L′Δwt+12L″Δwt2.
Applying (29) to (31), we have(32)ΔL<−Γ2−αηwα−1t+cΔwt2,where c>1/2L″. From (32), one can obtain that (30) will hold if the following is satisfied:(33)η≤Γ2−αwt|α−1c.
The proof of Lemma1 is completed.Lemma 2.
The set {L(t)} is convergent:(34)limt⟶∞Lwt=L∗.Proof.
From (4), we have Lwt≥0.
Every bounded monotonic sequence of real numbers converges. From Lemma1, Lwt+1≤Lwt. Therefore, there exists L∗≥0 satisfying(35)limt⟶∞Lwt=L∗.
The proof of Lemma2 is completed.Theorem 1.
Assume that (33) is valid. Then, the set DwtαLwt converges to zero.(36)limt⟶+∞DwtαLwt=0Proof.
Denote(37)λ=Γ2−αηwα−1t−c≥0.
From (32), one can obtain(38)Lwt+1≤Lwt−Γ2−αηwα−1t−cΔwt2=Lwt−λΔwt2=Lw0−λ∑t=0kΔwt2.
SinceLwt≥0, we have(39)λ∑t=0kΔwt2≤Lw0.
Whent⟶+∞, it holds that(40)∑t=0∞Δwt2≤∞.
Thus, we have(41)limt⟶∞Δwt=0.
Substituting (24) into (5) yields(42)Δwt=−ηDwtαLt.
Substituting (42) into (41) yields(43)limt⟶∞−ηDwtαLt=0.
Thus, we have(44)limt⟶+∞DwtαLwt=0.
From Theorem1, we conclude that FRNN converges.


### 3.3. Neural Fractional-Order Sliding Mode Control

Denoteηd as the desired position and attitude. Denote υd as the desired velocity.(45)ηd=xdydψd,(46)υd=udvdrd.The tracking error of position and attitude is(47)x¯=x−xd,y¯=y−yd,ψ¯=ψ−ψd.The tracking error of velocity is(48)u¯=u−ud,v¯=v−vd,r¯=r−rd.The following fractional sliding surface functions are constructed:(49)s1t=u¯+c1D−αu¯,where c1>0. α is the fractional order.The derivative of (49) is obtained:(50)s˙1t=u˙−u˙d+c1D1−αu¯.Substituting (20) into (50), one can obtain(51)s˙1t=fu+1m11τu−u˙d+c1D1−αu¯.The sliding mode control law is constructed:(52)τu=m11u˙d−c1D1−αu¯−η1sgns1−f^u,where η1>0. f^u is the approximation of fu:(53)f^u=W^uTϕ,(54)fu=WuTϕ+ε1,where ε1 is the approximation error.Subtracting (53) from (54),(55)fu−f^u=WuTϕ+ε1−W^uTh=WuT−W^uTϕ+ε1.Denote(56)W˜u=Wu−W^u.(55) can be written as(57)fu−f^u=W˜uϕ+ε1.The adaptive control law is constructed:(58)W^˙u=1λ1s1ϕ.The fractional sliding surface is constructed:(59)s2t=c2v¯+Dβv¯,where c2>0. β is the fractional order.The derivative of (59) is obtained:(60)s˙2t=c2v¯˙+D1+βv¯=c2v¯˙+Dβ−1v¯¨.Substituting (48) into (60), one can obtain(61)s˙2t=c2v¯˙+Dβ−1v¨−v¨d,and vd is constructed as follows:(62)vd=−sinψx˙d+cosψy˙d+η2x¯sinψ−η2y¯cosψ.The derivative of (62) is obtained:(63)v˙d=−sinψx¨d−cosψrx˙d+cosψy¨d−sinψry˙d+η2x¯˙sinψ+η2cosψrx¯−η2y¯˙cosψ+η2y¯sinψr,and ud is constructed as follows:(64)ud=cosψx˙d+sinψy˙d−η2x¯cosψ−η2y¯sinψ.The derivative of (64) is obtained:(65)u˙d=cosψx¨d−sinψrx˙d+sinψy¨d+cosψry˙d−η2x¯˙cosψ+η2sinψrx¯−η2y¯˙sinψ−η2y¯cosψr.It can be obtained by calculation that(66)−2ru˙d+r2vd=−2rcosψx¨d−sinψrx˙d+sinψy¨d+cosψry˙d−η2x¯˙cosψ+η2sinψrx¯−η2y¯˙sinψ−η2y¯cosψr+r2−sinψx˙d+cosψy˙d+η2x¯sinψ−η2y¯cosψ=r2sinψx˙d−r2cosψy˙d−η2r2sinψx¯+η2r2cosψy¯−2rcosψx¨d−2rsinψy¨d+2η2rcosψx¯˙+2η2rsinψy¯˙.The derivative of (63) can obtain(67)v¨d=−sinψx¨d−cosψrx¨d−cosψrx¨d+sinψr2x˙d−cosψr˙x˙d+cosψy¨d−sinψry¨d−sinψry¨d−sinψr˙y˙d+η2x¯¨sinψ+η2x¯˙cosψr+η2cosψrx¯˙+η2cosψr˙x¯−η2sinψr2x¯−η2y¯¨cosψ+η2y¯˙rsinψ+η2y¯˙sinψr+η2y¯cosψr2+η2y¯r˙sinψ.Substituting (66) into (67), one can obtain(68)v¨d=y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−r˙ud.The control rate is designed as follows:(69)τr=m33ud−v¨−D1−βk2sgns2+η2s2+c2v¯˙+y⃛dcosψ−sinψx⃛d+η2x¯¨sinψ−η2y¯¨cosψ−2ru˙d+r2vd−udf^r.where k2>0,η2>0. f^r is the approximation of fr:(70)f^r=1udD1−βW^rTφ,(71)fr=1udD1−βWrTφ+ε2,where ε2 is the approximation error.Subtracting (70) from (71),(72)fr−f^r=1udD1−βWrTϕ+ε2−W^rTϕ=1udD1−βWrT−W^rTϕ+ε2.Denote(73)W˜r=Wr−W^r.(72) can be written as(74)fr−f^r=1udD1−βW˜rϕ+ε2.The adaptive control rate is constructed:(75)W^˙r=1λ2s2ϕ.


### 3.4. Stability Analysis

Theorem 2.
With fractional-order sliding mode controllers (52) and (69) and adaptive control laws (58) and (75), the system converges in an asymptotically stable way.Proof.
Define the Lyapunov function:(76)V=12s12+12λ1W˜uTW˜u+12s22+12λ2W˜rTW˜r.
The derivative of (76) can obtain(77)V˙=s1s˙1+λ1W˜uTW˜˙u+s2s˙2+λ2W˜rTW˜˙r.
Substituting (31) into (30), one can obtain(78)s˙1t=fu+u˙d−c1D1−αu¯−η1sgns1−f^u−u˙d+c1D1−αu¯=fu−f^u−η1sgns1.
Substituting (36) into (78), one can obtain(79)s˙1t=W˜uTφ+ε1−η1sgns1.
Substituting (79) into (77), one can obtain(80)V˙=s1W˜uTh+ε1−η1sgns1+λ1W˜uTW˜˙u+s2s˙2+λ2W˜rTW˜˙r=ε1s1−η1s1+W˜uTs1h+λ1W˜˙u+s2s˙2+λ2W˜rTW˜˙r.
The derivative of (35) can obtain(81)W˜˙u=W˙u−W^˙u=−W^˙u.
Substituting (37) into (81), one can obtain(82)W˜˙u=−1λ1s1h.
Substituting (82) into (80), one can obtain(83)V˙=ε1s1−η1s1+s2s˙2+λ2W˜rTW˜˙r.
Substituting (47) into (40), one can obtain(84)s˙2t=c2v¯˙+Dβ−1v¨−y⃛dcosψ−sinψx⃛d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−r˙ud.
Substituting (12) into (84), one can obtain(85)s˙2t=c2v¯˙+Dβ−1v¨−y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud−1mrτrud=c2v¯˙+Dβ−1v¨−Dβ−1y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud−1mrτrud.
Substituting (69) into (85), one can obtain(86)s˙2t=c2v¯˙+Dβ−1v¨−Dβ−1y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud+Dβ−1−v¨−D1−βk2sgns2+η2s2+c2v¯˙+y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ−2ru˙d+r2vd−udf^r=−k2sgns2−η2s2+Dβ−1frud−udf^r.
Substituting (86) into (83), one can obtain(87)V˙=ε1s1−η1s1+s2−k2sgns2−η2s2+s2Dβ−1frud−udf^r+λ2W˜rTW˜˙r=ε1s1−η1s1−k2s2−η2s22+s2Dβ−1frud−udf^r+λ2W˜rTW˜˙r.
Substituting (84) into (87), one can obtain(88)V˙=ε1s1−η1s1−k2s2−η2s22+s2Dβ−11udD1−βW˜rh+ε2ud+λ2W˜rTW˜˙r=ε1s1−η1s1−k2s2−η2s22+s2W˜rh+ε2+λ2W˜rTW˜˙r.
The derivative of (83) can obtain(89)W˜˙r=W˙r−W^˙r=−W^˙r.
Substituting (75) into (89), one can obtain(90)W˜˙r=−1λ2s2φ.
Substituting (86) into (84), one can obtain(91)V˙=ε1s1−η1s1−k2s2−η2s22+s2W˜rh+ε2−W˜rTs2φ=ε1s1−η1s1−k2s2−η2s22+ε2s2.
The following conditions are established:(92)ε1≤η1,(93)ε2≤k2.
We have(94)V˙≤0.
The system is asymptotically stable, given that the stability of the presented method cannot be sustained due to the dynamic memory of the fractional operators.


## 3.1. FRNN Algorithm

The theoretical basis of gradient descent method is the concept of gradient. The relationship between gradient and directional derivative is as follows: the direction of gradient is consistent with the direction of obtaining the maximum value of directional derivative, and the modulus of gradient is the maximum value of directional derivative of function at this point. The weight is updated using the gradient method:(24)wkjt+1=wkjt−ηDwkjαLt,where wkj is the weight, L is the loss function of the neural network, η is the learning rate coefficient, and ɑ is the fractional order.The fractional order derivative ofLt is(25)DwkjαLt=∂Lt∂vktwkj1−αtyjtΓ2−α,where v is the weight from the hidden layer to the output layer of the network.The derivative ofLt is(26)∇wkjLt=∂Lt∂vkty.Substituting (25) into the first item of (24) yields(27)wkjt+1=wkjt−η∂Lt∂vktwkj1−αtyjtΓ2−α,where y is the output of the training samples.Substituting (27) into the first item of (5) yields(28)Δwkjt=−η∂Lt∂vktwkj1−αtyjtΓ2−α.Substituting (28) into the first item of (26) yields(29)∇wkjLt=−Γ2−αηwkjα−1tΔwkjt.


## 3.2. Convergence Analysis of FRNN

Lemma 1.
The loss functionL is monotonically decreasing:(30)Lwt+1≤Lwt.Proof.
From the Taylor mean value theorem with Lagrange remainder, we have(31)ΔL=Lwt+1−Lwt=L′Δwt+12L″Δwt2.
Applying (29) to (31), we have(32)ΔL<−Γ2−αηwα−1t+cΔwt2,where c>1/2L″. From (32), one can obtain that (30) will hold if the following is satisfied:(33)η≤Γ2−αwt|α−1c.
The proof of Lemma1 is completed.Lemma 2.
The set {L(t)} is convergent:(34)limt⟶∞Lwt=L∗.Proof.
From (4), we have Lwt≥0.
Every bounded monotonic sequence of real numbers converges. From Lemma1, Lwt+1≤Lwt. Therefore, there exists L∗≥0 satisfying(35)limt⟶∞Lwt=L∗.
The proof of Lemma2 is completed.Theorem 1.
Assume that (33) is valid. Then, the set DwtαLwt converges to zero.(36)limt⟶+∞DwtαLwt=0Proof.
Denote(37)λ=Γ2−αηwα−1t−c≥0.
From (32), one can obtain(38)Lwt+1≤Lwt−Γ2−αηwα−1t−cΔwt2=Lwt−λΔwt2=Lw0−λ∑t=0kΔwt2.
SinceLwt≥0, we have(39)λ∑t=0kΔwt2≤Lw0.
Whent⟶+∞, it holds that(40)∑t=0∞Δwt2≤∞.
Thus, we have(41)limt⟶∞Δwt=0.
Substituting (24) into (5) yields(42)Δwt=−ηDwtαLt.
Substituting (42) into (41) yields(43)limt⟶∞−ηDwtαLt=0.
Thus, we have(44)limt⟶+∞DwtαLwt=0.
From Theorem1, we conclude that FRNN converges.


## 3.3. Neural Fractional-Order Sliding Mode Control

Denoteηd as the desired position and attitude. Denote υd as the desired velocity.(45)ηd=xdydψd,(46)υd=udvdrd.The tracking error of position and attitude is(47)x¯=x−xd,y¯=y−yd,ψ¯=ψ−ψd.The tracking error of velocity is(48)u¯=u−ud,v¯=v−vd,r¯=r−rd.The following fractional sliding surface functions are constructed:(49)s1t=u¯+c1D−αu¯,where c1>0. α is the fractional order.The derivative of (49) is obtained:(50)s˙1t=u˙−u˙d+c1D1−αu¯.Substituting (20) into (50), one can obtain(51)s˙1t=fu+1m11τu−u˙d+c1D1−αu¯.The sliding mode control law is constructed:(52)τu=m11u˙d−c1D1−αu¯−η1sgns1−f^u,where η1>0. f^u is the approximation of fu:(53)f^u=W^uTϕ,(54)fu=WuTϕ+ε1,where ε1 is the approximation error.Subtracting (53) from (54),(55)fu−f^u=WuTϕ+ε1−W^uTh=WuT−W^uTϕ+ε1.Denote(56)W˜u=Wu−W^u.(55) can be written as(57)fu−f^u=W˜uϕ+ε1.The adaptive control law is constructed:(58)W^˙u=1λ1s1ϕ.The fractional sliding surface is constructed:(59)s2t=c2v¯+Dβv¯,where c2>0. β is the fractional order.The derivative of (59) is obtained:(60)s˙2t=c2v¯˙+D1+βv¯=c2v¯˙+Dβ−1v¯¨.Substituting (48) into (60), one can obtain(61)s˙2t=c2v¯˙+Dβ−1v¨−v¨d,and vd is constructed as follows:(62)vd=−sinψx˙d+cosψy˙d+η2x¯sinψ−η2y¯cosψ.The derivative of (62) is obtained:(63)v˙d=−sinψx¨d−cosψrx˙d+cosψy¨d−sinψry˙d+η2x¯˙sinψ+η2cosψrx¯−η2y¯˙cosψ+η2y¯sinψr,and ud is constructed as follows:(64)ud=cosψx˙d+sinψy˙d−η2x¯cosψ−η2y¯sinψ.The derivative of (64) is obtained:(65)u˙d=cosψx¨d−sinψrx˙d+sinψy¨d+cosψry˙d−η2x¯˙cosψ+η2sinψrx¯−η2y¯˙sinψ−η2y¯cosψr.It can be obtained by calculation that(66)−2ru˙d+r2vd=−2rcosψx¨d−sinψrx˙d+sinψy¨d+cosψry˙d−η2x¯˙cosψ+η2sinψrx¯−η2y¯˙sinψ−η2y¯cosψr+r2−sinψx˙d+cosψy˙d+η2x¯sinψ−η2y¯cosψ=r2sinψx˙d−r2cosψy˙d−η2r2sinψx¯+η2r2cosψy¯−2rcosψx¨d−2rsinψy¨d+2η2rcosψx¯˙+2η2rsinψy¯˙.The derivative of (63) can obtain(67)v¨d=−sinψx¨d−cosψrx¨d−cosψrx¨d+sinψr2x˙d−cosψr˙x˙d+cosψy¨d−sinψry¨d−sinψry¨d−sinψr˙y˙d+η2x¯¨sinψ+η2x¯˙cosψr+η2cosψrx¯˙+η2cosψr˙x¯−η2sinψr2x¯−η2y¯¨cosψ+η2y¯˙rsinψ+η2y¯˙sinψr+η2y¯cosψr2+η2y¯r˙sinψ.Substituting (66) into (67), one can obtain(68)v¨d=y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−r˙ud.The control rate is designed as follows:(69)τr=m33ud−v¨−D1−βk2sgns2+η2s2+c2v¯˙+y⃛dcosψ−sinψx⃛d+η2x¯¨sinψ−η2y¯¨cosψ−2ru˙d+r2vd−udf^r.where k2>0,η2>0. f^r is the approximation of fr:(70)f^r=1udD1−βW^rTφ,(71)fr=1udD1−βWrTφ+ε2,where ε2 is the approximation error.Subtracting (70) from (71),(72)fr−f^r=1udD1−βWrTϕ+ε2−W^rTϕ=1udD1−βWrT−W^rTϕ+ε2.Denote(73)W˜r=Wr−W^r.(72) can be written as(74)fr−f^r=1udD1−βW˜rϕ+ε2.The adaptive control rate is constructed:(75)W^˙r=1λ2s2ϕ.


## 3.4. Stability Analysis

Theorem 2.
With fractional-order sliding mode controllers (52) and (69) and adaptive control laws (58) and (75), the system converges in an asymptotically stable way.Proof.
Define the Lyapunov function:(76)V=12s12+12λ1W˜uTW˜u+12s22+12λ2W˜rTW˜r.
The derivative of (76) can obtain(77)V˙=s1s˙1+λ1W˜uTW˜˙u+s2s˙2+λ2W˜rTW˜˙r.
Substituting (31) into (30), one can obtain(78)s˙1t=fu+u˙d−c1D1−αu¯−η1sgns1−f^u−u˙d+c1D1−αu¯=fu−f^u−η1sgns1.
Substituting (36) into (78), one can obtain(79)s˙1t=W˜uTφ+ε1−η1sgns1.
Substituting (79) into (77), one can obtain(80)V˙=s1W˜uTh+ε1−η1sgns1+λ1W˜uTW˜˙u+s2s˙2+λ2W˜rTW˜˙r=ε1s1−η1s1+W˜uTs1h+λ1W˜˙u+s2s˙2+λ2W˜rTW˜˙r.
The derivative of (35) can obtain(81)W˜˙u=W˙u−W^˙u=−W^˙u.
Substituting (37) into (81), one can obtain(82)W˜˙u=−1λ1s1h.
Substituting (82) into (80), one can obtain(83)V˙=ε1s1−η1s1+s2s˙2+λ2W˜rTW˜˙r.
Substituting (47) into (40), one can obtain(84)s˙2t=c2v¯˙+Dβ−1v¨−y⃛dcosψ−sinψx⃛d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−r˙ud.
Substituting (12) into (84), one can obtain(85)s˙2t=c2v¯˙+Dβ−1v¨−y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud−1mrτrud=c2v¯˙+Dβ−1v¨−Dβ−1y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud−1mrτrud.
Substituting (69) into (85), one can obtain(86)s˙2t=c2v¯˙+Dβ−1v¨−Dβ−1y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ+r2vd−2ru˙d−frud+Dβ−1−v¨−D1−βk2sgns2+η2s2+c2v¯˙+y¨dcosψ−sinψx¨d+η2x¯¨sinψ−η2y¯¨cosψ−2ru˙d+r2vd−udf^r=−k2sgns2−η2s2+Dβ−1frud−udf^r.
Substituting (86) into (83), one can obtain(87)V˙=ε1s1−η1s1+s2−k2sgns2−η2s2+s2Dβ−1frud−udf^r+λ2W˜rTW˜˙r=ε1s1−η1s1−k2s2−η2s22+s2Dβ−1frud−udf^r+λ2W˜rTW˜˙r.
Substituting (84) into (87), one can obtain(88)V˙=ε1s1−η1s1−k2s2−η2s22+s2Dβ−11udD1−βW˜rh+ε2ud+λ2W˜rTW˜˙r=ε1s1−η1s1−k2s2−η2s22+s2W˜rh+ε2+λ2W˜rTW˜˙r.
The derivative of (83) can obtain(89)W˜˙r=W˙r−W^˙r=−W^˙r.
Substituting (75) into (89), one can obtain(90)W˜˙r=−1λ2s2φ.
Substituting (86) into (84), one can obtain(91)V˙=ε1s1−η1s1−k2s2−η2s22+s2W˜rh+ε2−W˜rTs2φ=ε1s1−η1s1−k2s2−η2s22+ε2s2.
The following conditions are established:(92)ε1≤η1,(93)ε2≤k2.
We have(94)V˙≤0.
The system is asymptotically stable, given that the stability of the presented method cannot be sustained due to the dynamic memory of the fractional operators.


## 4. Simulation Studies

A container ship [19] is used for testing. The length of the ship is 175 m. Table 2 lists sea weather condition data.Table 2
Sea weather condition data.
Time of observationSignificant wave height (m)Maximum wave height (m)Wind speed (m/s)Maximum wind speed (m/s)Wind directionWind force07.50 PM1.402.209.912.6North-northeast508.00 PM1.602.4010.4012.60North5.0008.10 PM1.602.4010.8014.10North-northeast6.0008.20 PM1.602.4011.5013.80North-northeast6.0008.30 PM1.702.2010.6013.70North-northeast5.0008.40 PM1.702.2010.6013.30North5.00Table3 lists the tidal height on Dec. 30, 2019.Table 3
Tidal height on Dec. 30, 2019.
Tide timeTidal height12.00 AM44701.00 AM51602.00 AM54902.04 AM54903.00 AM52404.00 AM43405.00 AM319Table4 lists the ship collision avoidance parameters between two ships.Table 4
Ship collision avoidance parameters.
Before collision avoidanceAfter collision avoidanceShip no.Course (°)Speed (kn)Course113710.5118.32314.67.5298.3


### 4.1. Results

Figure4 shows the speed response curve of ship 1.Figure 4
Speed response curve of ship 1.
(a)(b)(c)Figure5 shows the position and attitude response curve of ship 1.Figure 5
Position and attitude response of ship 1.
(a)(b)(c)Figure6 shows the roll response curve of ship 1.Figure 6
Roll response curve of ship 1.
(a)(b)Figure7 shows the input rudder angle of ship 1.Figure 7
Input rudder angle curve of ship 1.Figure8 shows the speed response curve of ship 2.Figure 8
Speed response curve of ship 2.
(a)(b)(c)Figure9 shows the position and attitude response curve of ship 1.Figure 9
Position and attitude response curve of ship 2.
(a)(b)(c)Figure10 shows the roll response curve of ship 2.Figure 10
Roll response curve of ship 2.
(a)(b)Figure11 shows the input rudder angle of ship 2.Figure 11
Input rudder angle curve of ship 2.Figure12 shows the resultant positions of ships 1 and 2 to avoid collisions.Figure 12
Position of two ships.Figures4–12 show that FRNN-AFSMC can realize the collision avoidance between two ships.


### 4.2. Performance Comparison

Table4 shows the performance comparison of different algorithms, including SMC, ASMC with RNN (RNN-ASMC), and FRNN-AFSMC.Table5 shows that FRNN-AFSMC has less overshoot and a shorter system regulation time with a wider parameter selection range.Table 5
Comparison of the control performances of different algorithms.
SMCRNN-ASMCFRNN-AFOSMCOvershoot19%13%9%Adjustment time (s)192176156Figure13 shows the yawing response curve of the closed-loop control system, where the horizontal axis represents the time in seconds and the longitudinal axis represents the yaw angle in unit of rad.Figure 13
Performance comparison of different algorithms.Figure13 shows that, compared with basic sliding mode control algorithm, the overshoot of FRNN-AFOSMC is smaller and the regulation time of the system is shorter.


## 4.1. Results

Figure4 shows the speed response curve of ship 1.Figure 4
Speed response curve of ship 1.
(a)(b)(c)Figure5 shows the position and attitude response curve of ship 1.Figure 5
Position and attitude response of ship 1.
(a)(b)(c)Figure6 shows the roll response curve of ship 1.Figure 6
Roll response curve of ship 1.
(a)(b)Figure7 shows the input rudder angle of ship 1.Figure 7
Input rudder angle curve of ship 1.Figure8 shows the speed response curve of ship 2.Figure 8
Speed response curve of ship 2.
(a)(b)(c)Figure9 shows the position and attitude response curve of ship 1.Figure 9
Position and attitude response curve of ship 2.
(a)(b)(c)Figure10 shows the roll response curve of ship 2.Figure 10
Roll response curve of ship 2.
(a)(b)Figure11 shows the input rudder angle of ship 2.Figure 11
Input rudder angle curve of ship 2.Figure12 shows the resultant positions of ships 1 and 2 to avoid collisions.Figure 12
Position of two ships.Figures4–12 show that FRNN-AFSMC can realize the collision avoidance between two ships.


## 4.2. Performance Comparison

Table4 shows the performance comparison of different algorithms, including SMC, ASMC with RNN (RNN-ASMC), and FRNN-AFSMC.Table5 shows that FRNN-AFSMC has less overshoot and a shorter system regulation time with a wider parameter selection range.Table 5
Comparison of the control performances of different algorithms.
SMCRNN-ASMCFRNN-AFOSMCOvershoot19%13%9%Adjustment time (s)192176156Figure13 shows the yawing response curve of the closed-loop control system, where the horizontal axis represents the time in seconds and the longitudinal axis represents the yaw angle in unit of rad.Figure 13
Performance comparison of different algorithms.Figure13 shows that, compared with basic sliding mode control algorithm, the overshoot of FRNN-AFOSMC is smaller and the regulation time of the system is shorter.


## 5. Conclusions

In this paper, FRNN-AFSMC is proposed and shown to converge for collision avoidance in bridge areas. The uncertainties are estimated by FRNN. The influence of fractional order on the algorithm performance is analyzed. Multiple ships dynamic collision avoidance is simulated and verified. The results show the performance comparison of different algorithms, including SMC, ASMC with RNN (RNN-ASMC), and FRNN-AFSMC. The accuracy of the control algorithm will be improved in future research.



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