# Adaptive Learning Based Tracking Control of Marine Vessels with Prescribed Performance

Authors: Zhao Xu; Shuzhi Sam Ge; Changhua Hu; Jinwen Hu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A novel adaptive tracking controller of fully actuated marine vessels is proposed with completely unknown dynamics and external disturbances. The model of dominant dynamic behaviors and unknown disturbances of the vessel are learned by a neural network in real time. The controller is designed and it unifies backstepping and adaptive neural network techniques with predefined tracking performance constraints on the tracking convergence rate and the transient and steady-state tracking error. The stability of the proposed adaptive tracking controller of the vessel is proven with a uniformly bounded tracking error. The proposed adaptive tracking controller is shown to be effective in the tracking control of marine vessels by simulations.


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## Body

## 1. Introduction

Tracking control of marine vessels has been widely used in civil and military applications such as shipping transportation, sea investigation, search and rescue, and security surveillance. Previous works on tracking control can be briefly classified into model-based and approximation-based approaches. Model-based methods require accurate dynamic models of the systems, which makes them difficult to be applied in reality due to the following reasons:(1) the dynamics of a vessel are generally time-varying and highly nonlinear and thus their actual mathematical model is hard to be obtained precisely using current modeling techniques; (2) vessels always suffer from unknown stochastic environmental disturbances.Recently, adaptive NN control design techniques were presented for controlling marine vehicles with model uncertainties [1, 2], by which the dynamics of the vessels can be partially known or even completely unknown. In [3], a simple yet computationally efficient NN tracking control approach is presented for control of the fully actuated vessels. Reference [4] extends the control strategy in [3] to the underactuated vessels. Other approximation methods such as fuzzy tracking control schemes are also proposed for marine vehicles. In [5], a fuzzy tracking controller is proposed against the model uncertainty of the vessel dynamics which can obtain with a uniformly bounded tracking error. However, one problem with the existing approximation-based control schemes is that it is of great challenge to guarantee the transient response of the vessel dynamics; for example, unsuitable initialization of the weight of NN may lead to poor transient tracking performance.For the purpose of solving this problem, an efficient prescribed performance control method was proposed in [6, 7] to regularize the response of strict-feedback systems in advance. The main idea is to formulate the tracking control as a constrained optimization problem with predefined constraints on the transient and steady-state tracking errors. It is then transformed into a relatively easier unconstrained optimization problem by defining a transformation function that incorporates the errors and their constraints into a single function analytically. This technique has been applied in several applications [8–11]. In [12], an adaptive NN control method was proposed with guaranteed transient and steady-state tracking performances for marine surface vessels with model uncertainties. Radial basis function (RBF) NNs are applied to approximate unknown vessel dynamics before control action applies with the assumptions of persistent excitation and recurrent orbits of the desired trajectory. However, the vessels always suffer from unknown environmental disturbances, and the estimated and stored model in memory may be violated during operating and the control performance cannot be guaranteed. Moreover, the persistent excitation condition was very restrictive making the method difficult to be applied in practice.This paper addresses the tracking control of surface vessels with unknown system dynamics and stochastic disturbances. An adaptive NN controller is proposed to obtain desired tracking performances. First, the method can work with predefined convergence rate, maximum overshoot, and steady-state error as the desired tracking performance indices. Second, the adaptive NN control combined with backstepping technique is applied to generate control commands for tracking control with unknown model dynamics and external disturbances. Third, the stability of the control method is analyzed and the tracking error is proven to be uniformly bounded.


## 2. Problem Formulation

### 2.1. Dynamics of Surface Vessels

We consider the surge, sway, and yaw motions of the marine vessel. Letη=[ηx,ηy,ηψ]T represent position (ηx,ηy) and heading (ηψ) of the vessel. The vector v=[vx,vy,rψ]T represents the alongship, athwartship, and rotational velocity, respectively, in the body-fixed frame. The vessel dynamics is considered as follows:(1)η˙=Jηv(2)Mv˙+Cvv+Dvv+gη=τ+τdη,v,t,where(3)Jη=cos⁡ψn-sin⁡ψn0sin⁡ψncos⁡ψn0001.M is the diagonal inertia matrix, C(v) is the matrix of Coriolis and centripetal, D(v) is the damping matrix, g(η) is the restoring force, τ is the vector of input signals, and τd is the time-varying unknown external disturbance and unmodeled dynamics. In this paper, we assume that M, C(v), and D(v) are completely unknown.


### 2.2. Prescribed Performance Control Preliminaries

The aim of this paper is to let the vessel position and heading inη track a smooth, bounded desired trajectory ηr with a uniformly bounded tracking error ε(t)=η(t)-ηr(t). The predefined constraints of the tracking error ε(t)=[ε1(t),ε2(t),ε3(t)]T∀t≥0 are given as follows:(4)-κϱit<εit<ϱitifεi0≥0-ϱit<εit<κϱitifεi0≤0,where 0≤κ≤1 and ϱi(t),i=1,2,3, are called performance functions [13] which are smooth, bounded, and strictly positive and decreasing.Thus, the constraints on the tracking errorεi(t) are defined by ϱi(t) and κ. Given the following exponential performance function(5)ϱit=ϱi0-ϱi∞ε-lit+ϱi∞,i=1,2,3,where ϱi0=ϱi(0) and ϱi∞=limt→∞ϱi(t), the constant ϱi0 in (5) should be set such that (4) is satisfied at t=0. li determines how fast ϱi(t) decreases. Moreover, it is implied that the overshoot is required to be no larger than κϱi(0).The idea in [6, 7] of transforming the tracking error into a new function is borrowed here, by which the error constraints can be incorporated into the function analytically so that the controller optimization can be solved more efficiently. The error transformation is defined by(6)ξi=Tiεiϱi,i=1,2,3,where Ti(·),i=1,2,3 are smooth, strictly increasing functions which define objective mappings:(7)Ti:-κ,1⟶-∞,∞forεi0≥0Ti:-1,κ⟶-∞,∞forεi0≤0for i=1,2,3. Defining ξ(t)=[ξ1(t),ξ2(t),ξ3(t)]T, after differentiation we have(8)ξ˙=T∂ε˙-ϱ˙εϱ,where T∂=diag([dT1/dε1/ϱ11/ϱ1⋯dT3/dε3/ϱ31/ϱ3]) and ϱ˙ε/ϱ=ϱ˙1ε1/ϱ1⋯ϱ˙3ε3/ϱ3T.According to [6], if ξi(t) is able to be kept bounded ∀t≥0 with the transformation function Ti(·),i=1,2,3, then (4) is guaranteed.


## 2.1. Dynamics of Surface Vessels

We consider the surge, sway, and yaw motions of the marine vessel. Letη=[ηx,ηy,ηψ]T represent position (ηx,ηy) and heading (ηψ) of the vessel. The vector v=[vx,vy,rψ]T represents the alongship, athwartship, and rotational velocity, respectively, in the body-fixed frame. The vessel dynamics is considered as follows:(1)η˙=Jηv(2)Mv˙+Cvv+Dvv+gη=τ+τdη,v,t,where(3)Jη=cos⁡ψn-sin⁡ψn0sin⁡ψncos⁡ψn0001.M is the diagonal inertia matrix, C(v) is the matrix of Coriolis and centripetal, D(v) is the damping matrix, g(η) is the restoring force, τ is the vector of input signals, and τd is the time-varying unknown external disturbance and unmodeled dynamics. In this paper, we assume that M, C(v), and D(v) are completely unknown.


## 2.2. Prescribed Performance Control Preliminaries

The aim of this paper is to let the vessel position and heading inη track a smooth, bounded desired trajectory ηr with a uniformly bounded tracking error ε(t)=η(t)-ηr(t). The predefined constraints of the tracking error ε(t)=[ε1(t),ε2(t),ε3(t)]T∀t≥0 are given as follows:(4)-κϱit<εit<ϱitifεi0≥0-ϱit<εit<κϱitifεi0≤0,where 0≤κ≤1 and ϱi(t),i=1,2,3, are called performance functions [13] which are smooth, bounded, and strictly positive and decreasing.Thus, the constraints on the tracking errorεi(t) are defined by ϱi(t) and κ. Given the following exponential performance function(5)ϱit=ϱi0-ϱi∞ε-lit+ϱi∞,i=1,2,3,where ϱi0=ϱi(0) and ϱi∞=limt→∞ϱi(t), the constant ϱi0 in (5) should be set such that (4) is satisfied at t=0. li determines how fast ϱi(t) decreases. Moreover, it is implied that the overshoot is required to be no larger than κϱi(0).The idea in [6, 7] of transforming the tracking error into a new function is borrowed here, by which the error constraints can be incorporated into the function analytically so that the controller optimization can be solved more efficiently. The error transformation is defined by(6)ξi=Tiεiϱi,i=1,2,3,where Ti(·),i=1,2,3 are smooth, strictly increasing functions which define objective mappings:(7)Ti:-κ,1⟶-∞,∞forεi0≥0Ti:-1,κ⟶-∞,∞forεi0≤0for i=1,2,3. Defining ξ(t)=[ξ1(t),ξ2(t),ξ3(t)]T, after differentiation we have(8)ξ˙=T∂ε˙-ϱ˙εϱ,where T∂=diag([dT1/dε1/ϱ11/ϱ1⋯dT3/dε3/ϱ31/ϱ3]) and ϱ˙ε/ϱ=ϱ˙1ε1/ϱ1⋯ϱ˙3ε3/ϱ3T.According to [6], if ξi(t) is able to be kept bounded ∀t≥0 with the transformation function Ti(·),i=1,2,3, then (4) is guaranteed.


## 3. Adaptive Tracking Control Design

In this section, we will develop an adaptive tracking controller by combining the backstepping design technique and a RBF NN. The main advantage of RBF is local approximation and fast learning, which has been widely used in researches [14, 15].Assume that full state informationv and η of the vessel is available. Given the reference trajectory ηr=[ηxr,ηyr,ηψr]T and the tracking error(9)εt=ηt-ηrt, we can get(10)ε˙t=Jηv-η˙rt. The error is transformed based on (6) and (8) as follows:(11)ξ˙=T∂ε˙-ϱ˙εϱ=T∂η˙t-η˙rt-ϱ˙εϱ=T∂η˙t-T∂r=T∂Jv-T∂r,where r=η˙r(t)+ϱ˙ε/ϱ.According to [6, 13], the boundedness of the solution of (11) guarantees the predefined tracking performance of ε(t) introduced via (4) for all t≥0. We introduce a virtual control based on the transformed error(12)ϕ1=JTr-K1T∂JTξwith K1=K1T>0 and ϕ˙1=∂ϕ1/∂ηη˙+∂ϕ1/∂η˙rη¨r+∂ϕ1/∂ξξ˙1. Define a second error variable in the transformed space as(13)ξa=vt-ϕ1t.Assume that exogenous effects and uncertainties have finite energy; we haveτdiη,v,t≤miη,v+ni(t),i=1,2,3, where miη,v:R3×R3→R+ is a positive, smooth, and nondecreasing function and ni(t) is a positive scalar. Consider the following desired control law:(14)τ¯=-T∂JTξ-K2ξa+Cvv+Dvv+gη+Mϕ˙1-SGNξamη,v+ni,where SGNξa is the diagonal matrix of signum functions sgnξa,i,i=1,2,3, K2=K2T>0, and K2=diag(kii)∈R3×3.Since the dynamics and disturbance of the vessel are completely unknown, a RBF NNW^TS(ξ) is used to approximate the vessel model and effect of the disturbance given by(15)W^TSξ=Mϕ˙1+Cvv+Dvv+gη-SGNξamη,v+ni-ϵξ,where W^≔blockdiagW^1T,W^2T,W^3T are the estimation of the desired weight W∗, S(ξ)=[S1T(ξ),S2T(ξ),S3T(ξ)]T are the basis functions, ξ=[ηT,vT,ϕ1T,ϕ˙1T] are the inputs of the NN, and ϵ(ξ) is the approximation error. Thus, the control law and weight update law are defined as follows:(16)τ=-T∂JTξ-K2ξa+W^TSZ(17)W^˙i=-ΥiSiξξa,i+μiW^i,where Υi are constant matrix and μi>0,i=1,2,3, and Z=ηT,vT,ξT,ξ˙T are the inputs of NN.Theorem 1.
Consider the system consisting of vessel model (2), desired bounded trajectory ηd, the control law (16), and NN weight adaptation law (17). Given the performance functions ϱi(t),i=1,2,3, and constant κ satisfying 0≤κ≤1,i=1,2,3, which incorporate desired performance bounds on tracking errors εi(t),i=1,2,3, if initial conditions satisfy(18)εi0<ϱi0ifεi0≠0κ≠0,ifεi0=0 with transformation functions(19)Tiεitϱit=ailn⁡κ+εit/ϱit1-εit/ϱit,forεi0≥0ailn⁡1+εit/ϱitκ-εit/ϱit,forεi0≤0,where ai>0, i=1,2,3, are positive design constants, then the following properties hold:(1)
The tracking error of the vessel is uniformly bounded.(2)
The predefined tracking performance of the vessel in the sense of (4) and (5) is guaranteed.Proof.
See the appendix.


## 4. Simulations

In this section, the parameters of a supply vessel from Norwegian University of Science and Technology [16] are adopted for simulation with a 1 : 70 scaling to test the performance of the proposed adaptive NN controller. The vessel parameters used are given in Table 1.Table 1
Parameters of the target vessel.
Parameter Description Value m Total mass 23.8 kg L v Length of vessel 1.255 m B v Breadth of vessel 0.290 m z g Position alongZ of the center of gravity 0.046 m I v z Moments of inertia alongZ 1.76 kg⋅m2 ρ a i r Density of air 1.29 kg/m3 ρ w a t e r Density of water 1025 kg/m3 g v Gravitational acceleration 9.8 m/s2The objective of this work is to make the vessel track the given smooth, bounded reference trajectory with predefined bounds for tracking errors. In this section, two simulation scenarios with different desired trajectories are tested. The tracking performance of the proposed controller is also evaluated under two conditions: with the predefined performance bounds and without the predefined performance bounds.


### 4.1. Scenario  I

In this scenario, the reference trajectory is given as follows:(20)ηrt,tr=η0+-2.0t3tr3+3.0t2tr2ηf-η0,where η0=4.2,0.8,1.2T is the initial of the vessel, ηf=0,0,0T is the final position, and tr=50 s represents the stopping time.The parametersκ,i=1,2,3 appearing in (4) are set to be 0.8. The parameters in exponential performance functions (5) are ϱi0=0.1, ϱi∞=0.01, and li=5,i=1,2,3. Furthermore, the parameters in error transformations function (19) are chosen as ai=0.008,i=1,2,3.The inputs of NNs areZ=ηT,vT,ξT,ξ˙T. The number of nodes is l=512. The centers of the nodes are evenly distributed in -2,2,-2,2,-π,π,-0.1,0.1,-3,3,[-3,3]. The variance of center of Gaussian radial basis function is 5. The initial weights are zeros. The control gain matrices in (17) are Υ=diag[0.001,0.001,0.0001] and μ=diag[0.001,0.001,0.0001]. The Lyapunov gain matrices are tuned as K1=K2=diag[20,20,20].For comparative studies, we compare our method with the adaptive NN control without prescribed performance bound. Figure1 shows the positions and heading evolution. The tracking errors are illustrated in Figure 2. It is observed that the proposed controller has much better transient and steady-state performance while the tracking errors are within the predefined bounds as compared to the one without considering the predefined tracking error bounds. The norm of the tracking error is shown in Figure 3(a). From Figure 3(b), we can see that, during transient stage, the control input of the controller without predefined error bounds oscillates seriously and requires a larger control effort, which results in larger overshoots and oscillations of the tracking performances as seen in Figure 2, whereas the control input of our proposed method is smooth. In addition, the proposed controller with predefined error bounds converges faster and the tracking errors stay strictly within the predefined bounds, while the one without considering the performance bounds cannot guarantee this.Figure 1
Comparison of tracking performance.
(a)
η x ( t ) (b)
η y ( t ) (c)
η Ψ ( t )Figure 2
Tracking errorsε(t) with respect to performance envelopes.
(a)
ε 1 ( t ) (b)
ε 2 ( t ) (c)
ε 3 ( t )Figure 3
(a)ε(t) and (b) τ.
(a) (b)


### 4.2. Scenario  II

In this scenario, the desired trajectoryηr is given by(21)ηr=3cos⁡0.06t+0.632sin⁡0.09tπ+πsin⁡0.13πt-0.52, where η0=2,3,2T is the payload initial position and tr=200 s represents the stopping time.The parametersκ,i=1,2,3 appearing in (4) are 0.7. The parameters in exponential performance functions (5) are ϱ0=2.2,1.2,0.63T, ϱi∞=0.01, and li=0.45,i=1,2,3. Furthermore, the parameters in error transformations function (19) are chosen as ai=0.01,i=1,2,3.The inputs of NNs areZ=ηT,vT,ξT,ξ˙T. The number of nodes is l=512. The centers of the nodes are distributed in -2,2,-2,2,-π,π,-0.1,0.1,-3,3,[-3,3]. The variance of center of Gaussian radial basis function is 5. The initial weights are zeros. The control gain matrices Υ and μ in (17) are chosen as Υ=diag[0.001,0.001,0.0001] and μ=diag[0.001,0.001,0.0001]. The Lyapunov gain matrices are tuned as K1=K2=diag[50,50,50].For comparative studies, we compare our controller to the one without considering the performance bounds. Figure4 shows the positions and heading evolution; Figure 5 shows the tracking errors and Figure 6(a) shows the norm of the error vector. It can be seen that the proposed controller guarantees the tracking errors which are within the predefined bounds and produces better transient and steady-state performance as compared to the one without considering the performance bounds. From Figure 6(b), we can see that, during transient stage, the control input of the adaptive controller without considering the performance bounds oscillates seriously and requires a larger control effort compared with our proposed method, which results in larger overshoots and oscillations of the tracking performances as seen in Figure 5. Therefore, the proposed controller performs better than that without considering the performance bounds in both scenarios.Figure 4
Comparison of tracking performance.
(a)
η x ( t ) (b)
η y ( t ) (c)
η Ψ ( t )Figure 5
Tracking errorsε(t) with respect to performance envelopes.
(a)
ε 1 ( t ) (b)
ε 2 ( t ) (c)
ε 3 ( t )Figure 6
(a)ε(t) and (b) τ.
(a) (b)


## 4.1. Scenario  I

In this scenario, the reference trajectory is given as follows:(20)ηrt,tr=η0+-2.0t3tr3+3.0t2tr2ηf-η0,where η0=4.2,0.8,1.2T is the initial of the vessel, ηf=0,0,0T is the final position, and tr=50 s represents the stopping time.The parametersκ,i=1,2,3 appearing in (4) are set to be 0.8. The parameters in exponential performance functions (5) are ϱi0=0.1, ϱi∞=0.01, and li=5,i=1,2,3. Furthermore, the parameters in error transformations function (19) are chosen as ai=0.008,i=1,2,3.The inputs of NNs areZ=ηT,vT,ξT,ξ˙T. The number of nodes is l=512. The centers of the nodes are evenly distributed in -2,2,-2,2,-π,π,-0.1,0.1,-3,3,[-3,3]. The variance of center of Gaussian radial basis function is 5. The initial weights are zeros. The control gain matrices in (17) are Υ=diag[0.001,0.001,0.0001] and μ=diag[0.001,0.001,0.0001]. The Lyapunov gain matrices are tuned as K1=K2=diag[20,20,20].For comparative studies, we compare our method with the adaptive NN control without prescribed performance bound. Figure1 shows the positions and heading evolution. The tracking errors are illustrated in Figure 2. It is observed that the proposed controller has much better transient and steady-state performance while the tracking errors are within the predefined bounds as compared to the one without considering the predefined tracking error bounds. The norm of the tracking error is shown in Figure 3(a). From Figure 3(b), we can see that, during transient stage, the control input of the controller without predefined error bounds oscillates seriously and requires a larger control effort, which results in larger overshoots and oscillations of the tracking performances as seen in Figure 2, whereas the control input of our proposed method is smooth. In addition, the proposed controller with predefined error bounds converges faster and the tracking errors stay strictly within the predefined bounds, while the one without considering the performance bounds cannot guarantee this.Figure 1
Comparison of tracking performance.
(a)
η x ( t ) (b)
η y ( t ) (c)
η Ψ ( t )Figure 2
Tracking errorsε(t) with respect to performance envelopes.
(a)
ε 1 ( t ) (b)
ε 2 ( t ) (c)
ε 3 ( t )Figure 3
(a)ε(t) and (b) τ.
(a) (b)


## 4.2. Scenario  II

In this scenario, the desired trajectoryηr is given by(21)ηr=3cos⁡0.06t+0.632sin⁡0.09tπ+πsin⁡0.13πt-0.52, where η0=2,3,2T is the payload initial position and tr=200 s represents the stopping time.The parametersκ,i=1,2,3 appearing in (4) are 0.7. The parameters in exponential performance functions (5) are ϱ0=2.2,1.2,0.63T, ϱi∞=0.01, and li=0.45,i=1,2,3. Furthermore, the parameters in error transformations function (19) are chosen as ai=0.01,i=1,2,3.The inputs of NNs areZ=ηT,vT,ξT,ξ˙T. The number of nodes is l=512. The centers of the nodes are distributed in -2,2,-2,2,-π,π,-0.1,0.1,-3,3,[-3,3]. The variance of center of Gaussian radial basis function is 5. The initial weights are zeros. The control gain matrices Υ and μ in (17) are chosen as Υ=diag[0.001,0.001,0.0001] and μ=diag[0.001,0.001,0.0001]. The Lyapunov gain matrices are tuned as K1=K2=diag[50,50,50].For comparative studies, we compare our controller to the one without considering the performance bounds. Figure4 shows the positions and heading evolution; Figure 5 shows the tracking errors and Figure 6(a) shows the norm of the error vector. It can be seen that the proposed controller guarantees the tracking errors which are within the predefined bounds and produces better transient and steady-state performance as compared to the one without considering the performance bounds. From Figure 6(b), we can see that, during transient stage, the control input of the adaptive controller without considering the performance bounds oscillates seriously and requires a larger control effort compared with our proposed method, which results in larger overshoots and oscillations of the tracking performances as seen in Figure 5. Therefore, the proposed controller performs better than that without considering the performance bounds in both scenarios.Figure 4
Comparison of tracking performance.
(a)
η x ( t ) (b)
η y ( t ) (c)
η Ψ ( t )Figure 5
Tracking errorsε(t) with respect to performance envelopes.
(a)
ε 1 ( t ) (b)
ε 2 ( t ) (c)
ε 3 ( t )Figure 6
(a)ε(t) and (b) τ.
(a) (b)


## 5. Conclusions

In this paper, the tracking control of the marine surface vessel is studied and a novel tracking controller with online modeling and predefined performance bounds was developed. The proposed control scheme can work with completely unknown vessel dynamic and unknown disturbances. Moreover, the tracking error was proven to be uniformly bounded. Simulation shows that the resulting tracking error stays strictly in the predefined bounds and the proposed controller performs better than that one without considering the predefined performance bounds.



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*Source: 2595721-2018-03-06.xml*