# Disturbance Observer-Based Adaptive Neural Network Control of Marine Vessel Systems with Time-Varying Output Constraints

Authors: Wei Zhao; Li Tang; Yan-Jun Liu
Journal: Complexity
 (2020)

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## Abstract

This article investigates an adaptive neural network (NN) control algorithm for marine surface vessels with time-varying output constraints and unknown external disturbances. The nonlinear state-dependent transformation (NSDT) is introduced to eliminate the feasibility conditions of virtual controller. Moreover, the barrier Lyapunov function (BLF) is used to achieve time-varying output constraints. As an important approximation tool, the NN is employed to approximate uncertain and continuous functions. Subsequently, the disturbance observer is structured to observe time-varying constraints and unknown external disturbances. The novel strategy can guarantee that all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, the simulation results verify the benefit of the proposed method.


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## Body

## 1. Introduction

In recent years, the adaptive control [1, 2] of nonlinear systems [3–7] is attracted more and more attention. As two main approximation tools, neural networks (NNs) and fuzzy logic systems (FLSs) [8, 9] are used to dispose some uncertain variables and unknown functions. For example, in [10], by using NNs, the neural network adaptive fault-tolerant control method under multiobjective constraints is designed. An effective finite-frequency H−/H∞ fault detection method is proposed for the descriptor system in [11]. It is noted that the aforementioned studies neglect the influence of constraints on nonlinear systems.In general, because of physical limitations and safety factors of the system, multifarious constraints exist in most practical systems, such as flexible string system [12], flexible-joint robot manipulators [13], nonuniform gantry crane systems [14], and flexible aerial refueling hose [15]. The barrier Lyapunov function (BLF) is presented in [16] for strict feedback systems. Since then, as a main tool, the BLF plays a crucial role in dealing with constraints. Subsequently, full state constraints adaptive controller is designed based on BLF and backstepping technique in [17]. In order to handle the control problems of state-constrained nonlinear systems better, in [18, 19], the authors present an integral BLF. It is worth noting that none of the above articles involve time-varying constraints. In [20–22], the adaptive controller of time-varying full-state constraint is constructed. Thus, the constant constraints expand the time-varying constraints. However, there is a limitation in the adaptive control design based on BLF, i.e., the feasibility of the virtual controller, which will increase the cost of control design. Hence, how to eliminate the feasibility of the controller is an urgent problem. It is noted that few of them involve the feasibility conditions of intermediate controllers to marine vessel systems, which is also a challenge of this paper.More recently, artificial intelligence has made impressive progress. More and more adaptive tracking control of marine vessel is widely studied. Subsequently, a number of significant results are proposed around the adaptive control design of the ship. The adaptive NN controller for ships is designed in [23]. Afterward, much effort has been made in addressing the situation where the ship system parameters are unknown. For the sliding mode method based on the model, an adaptive control approach is proposed in [24]. The precise recognition and learning control of marine ships under an unknown dynamic environment is further developed in [25]. Based on the above description, we propose an adaptive NN control method of marine vessel with time-varying output constraints.In this article, we try our best to propose an adaptive NN control method for the marine vessel system with disturbance observer and time-varying output constraints. Furthermore, NNs are developed to deal with uncertain parameters and unknown functions. As is known to all, in the existing literature, this kind of marine vessel system with time-varying state constraints is rarely dealt with by removing the feasibility of virtual controller. A novel transformation function is proposed to ensure that the states do not violate the bounds. The main work and contributions of this article are as follows:(1)
As is known to all, most of the existing constraint control methods involve feasibility conditions. In this article, a new coordinate transformation is designed to resolve the time-varying position constraint of marine surface vessel without involving feasibility conditions. The coordinate transformation is used to get a control scheme that can avoid the feasibility condition, which is conducive to the realization of the control scheme.(2)
An exponential convergence disturbance observer is designed to deal with time-varying constraints and uncertain external disturbances, which are created by wind, waves, currents, and so on.The specific organization of this paper is as follows. First, the dynamic model of the surface vessels is introduced. Then, the detailed procedure of BLF derivation for dealing with time-varying output constraints is given, in which the disturbance observer, the adaptive controller, and the adaptive laws are designed. Finally, simulation results verify the effectiveness of the proposed method.


## 2. Preliminaries

We consider the dynamics of 3 degree-of-freedom (3DOF) surface ships with model uncertainty and external disturbance. The dynamic model between the vessel’s position and velocity can be expressed as follows:(1)η˙=Rην,where the vector η=ηx,ηy,ηψT∈R3 stands for the output of earth-frame positions and heading. ηx,ηy∈R2 is the position of the vessel, and ηψ∈R denotes the heading, respectively. The vector ν=νx,νy,νψT∈R3 represents the velocity vector of the vessel. The rotation matrix Rη can be described as follows:(2)Rη=cosηψ−sinηψ0sinηψcosηψ0001,R−1η=RTη.In this paper, the output of position and heading of marine vessel system is constrained by(3)−Fk1t≤η≤Fk2t,k=1,2,3,where(4)Fk1t=F11t,F21t,F31tT,Fk2t=F12t,F22t,F32tT.The relationship between the vessel’s velocity and the force acting on the vessel can be described by the dynamic model of vessel [26] as follows:(5)Mν˙+Aνν+Bνν+fη=τ+g,where gt=g1t,g2t,g3tT is unknown and time-varying external disturbance caused by wind and waves, and fη is unknown restorative force caused by the ship’s gravity and buoyancy. The control input is τ=τ1,τ2,τ3T. Besides, M=MT∈R3×3 denotes the positive definite inertia matrix, Aν∈R3×3 represents the torques of Coriolis and centripetal, and Bν∈R3×3 stands for damping matrix. They are, respectively,(6)M=a11000a22a230a32a33,Aν=00−a22νy−a23νψ00a11νxa22νy+a23νψ−a11νx0,Bν=b11000b22b230b32b33,where(7)b11=−X1−X11vx−X111vx2,b22=−Y2−Y22vy−Y32vψ,b23=−Y3−Y23vy−Y33vψ,b32=−N2−N22vy−N32vψ,b33=−N3−N23vy−N33vψ,with a11, a22, a23, a32, a33, X1, X11, X111, Y2, Y22, Y32, Y3, Y23, Y33, N2, N22, N32, N3, N23, and N33 are design parameters.Definex1=η and x2=ν, and then (1) and (5) can be transformed into the following equation:(8)x˙1=Rx1x2,x˙2=M−1τ+g−fx1−Ax2x2−Bx2x2.The main task of this article is to design controller and adaptive laws so that the system output of vesselx1=x11,x12,x13T tracks the expected trajectory ϖd=ϖd1,ϖd2,ϖdT as effectively as possible. Meanwhile, all signals are bounded in the closed-loop systems and their constraints are not violated forever.As important approximation methods in nonlinear adaptive control, the radial basis function neural networks (RBFNNs) are usually utilized to deal with unknown and continuous functions. In this article, the RBFNNs are applied to approximate the unknown and continuous functionFZ. The following relation holds:(9)FZ=θTφZ+ςZ,where ςZ represents approximation error and satisfies ςZ≤ς¯, which will be used in the following derivation. Z∈ΩZ⊂Rq is the input vector, the optimal weight matrix is θT∈Rl,andl≥1 denotes the number of neurons. The known continuous basis function vector is φZ=φ1Z,φ2Z,…,φlZT and φiZi=1,2,…, is the Gaussian function with(10)φiZ=exp−Z−μi2ηi2,i=1,2,…,l,where μi=μ1,…,μlT,l>1 stands for the center of the receptive field, and the width of the Gaussian basis function is ηi.Remark 1.
As usual, the neural network chosen in this article has some special properties. Firstly, with the increase of the number of neurons, approximation errorςZ will gradually decrease. Secondly, the approximate area of the NNs is uniformly covered by the receptive field μi. Finally, based on the equation ηi=2dmax/l, we can get the width of the Gaussian function and dmax denotes the maximal Euclid distance of the approximation region.
It is necessary to research the marine vessel position constraints because marine environmental disturbances are time-varying and marine conditions are complex. In [27], for the nonlinear system control algorithm based on BLF, the state constraint is converted into the constraint of tracking error, which increases the restriction on the initial conditions of the system. It should be noted that most of the previous research studies on constraints were carried out in simple cases. To deal with time-varying output constraints gracefully, we consider the following nonlinear state-dependent transformation (NSDT):(11)μ1k=x1ktFk1t+x1ktFk2t−x1kt,where the initial states x1k0∈D1k, in which k=1,2,3. From (11), it should be pointed that μ1k tends to infinity when x1k is close to D1k, for any initial condition x1k0∈D1k. That is(12)μ1k⟶±∞if and only ifx1k⟶−Fk1orx1k⟶Fk2.
Therefore, it is noted that for anyx1k0∈D1k, if μ1k∈L∞,t≥0, then x1k∈D1k can be guaranteed for any t≥0, that is only need μ1k, k=1,2,3, are bounded, and time-varying output constraints will not be violated. In other words, the issue of output constraints attributed to making sure that μ1k for t≥0 is bounded.Assumption 1.
Time-varying functionFk1,k=1,2,3, and Fk2,k=1,2,3, are bounded and continuous, as well as its first derivative is also bounded and continuous.
The time derivative ofμ1k is given as(13)μ˙1k=η1kx˙1k+γ1k,where(14)η1k=Fk1Fk2+x1k2Fk1+x1k2Fk2−x1k2,γ1k=−F˙k1Fk2+Fk1F˙k2x1k+F˙k1−F˙k2x1k2Fk1+x1k2Fk2−x1k2,for k=1,2,3.
Then, (13) can also be rewritten as follows:(15)μ˙k=ηkx˙k+γk,where(16)μk=μ11,μ12,μ13T∈R,k=1,2,3,ηk=diagη1k∈R3×3,k=1,2,3,γk=γk1,γk2,γk3T∈R,k=1,2,3.
In this paper, feasibility condition in the control design is removed. Accordingly, it is much better to implement control methods. Most of the existing vessel constraint methods dispose the constant state constraints. For comprehensive consideration, the adaptive NN control of time-varying output constraints is studied in this article.
The different coordinate transformation is introduced as follows:(17)z1=μ1−βd,(18)z2=x2−α1,with(19)βd=βd1,βd2,βd3T∈R,βdk=ϖdktFk1t+ϖdktFk2t−ϖdkt,k=1,2,3,where α1 is the intermediate controller and it will be defined later. It should be clear that βdk is bounded in the set of D1k.Assumption 2.
The expected tracking signalϖdk,k=1,2,3, is bounded, and the derivative of ϖdk with respect to time is also bounded.
Then, the time derivative ofβd is given as follows:(20)β˙d=ηdϖ˙d+ρd,where(21)ηd=diagηdk∈R3×3,k=1,2,3,ρdk=ρd1,ρd2,ρd3T∈R,k=1,2,3,ϖdk=ϖd1,ϖd2,ϖd3T∈R,k=1,2,3,ηdk=Fk1Fk2+ϖdk2Fk1+ϖdk2Fk2−ϖdk2,ρdk=−F˙k1Fk2+Fk1F˙k2ϖdk+F˙k1−F˙k2ϖdk2Fk1+ϖdk2Fk2−ϖdk2,for k=1,2,3, and according to the definition of ηdk and ρdk, we know that ηdk and ρdk are known and computable and will be used in the design of the controller in the later.Remark 2.
For uncertain vessel nonlinear system, to make sure that the state constraints are never violated their bounded, the virtual controller must satisfy the feasibility condition:(22)−Fk1≤αk1≤Fk2,k=1,2,3,where αk1 is the virtual controller and −Fk1 and Fk2 are smooth functions. Although the virtual controller αk1 is related to state variables, it also depends on other design parameters. The feasibility conditions of (22) must be satisfied, and only in this way can the control scheme be implemented effectively. However, it is unrealistic and even difficult to find such parameters.Remark 3.
It should be pointed that according to the actual need, the external disturbances are required to be bounded. Note that all of these boundary conditions are not necessary to implement adaptive control, and they can only be used in analysis.Remark 4.
Ifl and m are the nonnegative real numbers, a>1, b is nonzero real number, and 1/a+1/b=1. Then, we can get(23)lm≤laa+mbb.
In this article, the following disturbance observer is constructed to deal with unknown disturbance:(24)g^=γ+K0Mx2,γ˙=−K0γ−K0−Ax2x2−Bx2x2−fx1+τ+K0Mx2,where K0∈R3×3 is an observer gain positive definite matrix, γ is the introduced intermediate variable, and the definition of g^ is given in the following equation:(25)g^˙=γ˙+K0Mx˙2=−K0γ−K0−Ax2x2−Bx2x2−fx1+τ+K0Mx2+K0τ+g−Ax2x2−Bx2x2−fx1=K0g−γ+K0Mx2=K0g−g^.
Then, differentiatingg˜, it yields(26)g˜˙=g˙−g^˙=g˙−K0g−g^=g˙−K0g˜,where g˜=g−g^ represents estimation error vector and g˜=g˜1,g˜2,g˜3T, vector g^=g^1,g^2,g^3T denotes disturbance estimation, and vector g=g1,g2,g3T is external disturbance. Define g˙t≤Cg≤∞, where Cg is a positive constant.
The detailed design steps are as follows:Step1: the following Lyapunov function is chosen:(27)V1=12z1Tz1.Based onz1=μ1−βd in (17), and the definition of system (8), then taking derivation of z1, one obtains(28)z˙1=μ˙1−β˙d=η1Rx1x2+γ1−ηdϖ˙d−ρd=η1Rx1z2+η1Rx1α1+γ1−ηdϖ˙d−ρd.Based on (28), the time derivative of V1 is given by(29)V˙1=z1Tz˙1=z1Tη1Rx1z2+η1Rx1α1+γ1−ηdϖ˙d−ρd.The virtual controller is designed as follows:(30)α1=RTx1η1−1−K1z1−γ1+ηdϖ˙d+ρd.Substituting (30) into (29), we have(31)V˙1=−z1TK1z1+z1Tη1Rx1z2.Step2: the Lyapunov function is designed as follows:(32)V2=12z2TMz2+12g˜Tg˜+12∑i=13θ˜iTΓi−1θ˜i,whereθ˜=θ−θ^ denotes the NN weight error, θ^ is the estimation of weight matrix θ, and Γi=ΓiT is positive definite matrix.
Taking the derivative ofz2 in (18), and based on x˙2=M−1τ+g−fx1−Ax2x2−Bx2x2, it yields(33)z˙2=x˙2−α˙1=M−1τ+g−fx1−Ax2x2−Bx2x2−α˙1.
Substituting (33) into (32), it leads to(34)V˙2=z2TMz˙2+g˜Tg˜˙+∑i=13θ˜iTΓi−1θ˜˙i=z2Tτ+g−fx1−Ax2x2−Bx2x2−Mα˙1+g˜Tg˜˙+∑i=13θ˜iTΓi−1θ˜˙i.
The unknown function is defined by(35)F2Z2=−Ax2x2−Bx2x2−Mα˙1−fx1,where F2Z2 is an unknown and continuous function and Z2=x1T,x2T,α1T,α˙1T are the inputs of the NNs. With the help of the NN approximation capability, the unknown function can be rewritten as follows:(36)F2Z2=θTφZ+ςZ.
The actual controllerτ is constructed as follows:(37)τ=−RTx1η1Tz1−K2z2−g^−θ^TφZ.
Substituting (37) and (36) into (34), one obtains(38)V˙2≤−z2TK2z2−z2TRTx1η1Tz1+z2Tg˜+z2Tθ˜TφZ+z2TςZ+g˜Tg˜˙−∑i=13θ˜iTΓi−1θ^˙i.
Substituting (26) into (38), we have(39)V˙2≤−z2TK2z2−z2TRTx1η1Tz1+z2Tg˜+z2TςZ+z2Tθ˜TφZ+g˜Tg˙−K0g˜−∑i=13θ˜iTΓi−1θ^˙i≤−z2TK2z2−z2TRTx1η1Tz1+z2Tg˜+z2TςZ+z2Tθ˜TφZ+g˜Tg˙−g˜TK0g˜−∑i=13θ˜iTΓi−1θ^˙i.
The adaptive laws are selected as follows:(40)θ^˙i=ΓiφiZiz2,i−δiθ^i,i=1,2,3,where δi>0,i=1,2,3 is the design parameter.
Using Young’s inequality, we have the following inequalities:(41)z2Tg˜≤h1z2Tz2+14h1g˜Tg˜g˜Tg˙≤h2g˜Tg˜+14h2Cgz2TςZ≤12z2Tz2+12ς¯2∑i=13δiθ˜iTθ^i=∑i=13δiθ˜iTθi−∑i=13δiθ˜iTθ˜i≤−12∑i=13δiθ˜i2+12∑i=13δiθi2,where h1 and h2 are the design parameters.
Substituting all the above inequalities into (39), we have(42)V˙2≤−z2TK2z2−z2TRTx1η1Tz1+h1z2Tz2+12z2Tz2+14h1g˜Tg˜+h2g˜Tg˜−g˜TK0g˜−12∑i=13δiθ˜i2+12∑i=13δiθi2+14h2Cg+12ς¯2.
Consider the Lyapunov function as follows:(43)V=V1+V2=12z1Tz1+12z2TMz2+12g˜Tg˜+12∑i=13θ˜iTΓi−1θ˜i.
From (31) and (42), we can obtain(44)V˙≤−z1TK1z1+z1Tη1Rx1z2−z2TK2z2−z2TRTx1η1Tz1+h1z2Tz2+12z2Tz2+14h1g˜Tg˜+h2g˜Tg˜−g˜TK0g˜−12∑i=13δiθ˜i2+12∑i=13δiθi2+14h2Cg+12ς¯2≤−z1TK1z1−z2TK2z2+h1z2Tz2+14h1g˜Tg˜+h2g˜Tg˜+12z2Tz2−g˜TK0g˜−12∑i=13δiθ˜i2+12∑i=13δiθi2+14h2Cg+12ς¯2≤−CV+D,where(45)C=−2minλminK1,λminK2M−1−h1λmaxM−1−12λmaxM−1,λminK0−14h1−h2,12δiλminΓi,D=12∑i=13δiθi2+14h2Cg+12ς¯2,with λmax⋅ and λmin⋅ standing for the largest and minimum eigenvalue of a matrix, respectively.Theorem 1.
Consider the marine vessel nonlinear system with unknown time-varying disturbances (8) under the virtual controller (30), actual controller (37), and adaptive laws (40). Then, we have the following: (1) error signal undulates in a small neighborhood close to zero; (2) all signals in the closed-loop system are SGUUB; and (3) in the meantime, the output signal will not violate the time-varying boundary.Proof.
According to (44), we can obtain(46)0≤Vt≤DC+V0−DCe−Ct,z1≤2DC+2V0−DCe−Ct.
From (44), it is clear that the signals z1, z2, θ˜i,i=1,2,3, and g˜ are SGUUB. Since θ^i,i=1,2,3 is bounded by θ˜i=θi−θ^i with θi being also bounded. As well, b is bounded. Because βdk is bounded in the set D1k, from z1=μ1−βd, we can infer that μ1k∈L∞,k=1,2,3. Based on (12), for any initial condition x1k0∈D1k,k=1,2,3, one deduces that x1kt is bounded. Because ηd, ρd, ϖ˙d, and ϖd are bounded, gradually, it is obtained that the virtual controller α1 is also bounded. According to z2=x2−α1, we can infer that x2 is bounded.


## 3. Simulation Example

In this section, to illustrate the effectiveness of the proposed method, we conduct the simulation on the model of Cybership II, which is a 1 : 70 scale supply vessel replica setup in a marine control laboratory in the Norwegian University of Science and Technology [28]. The following expected trajectory is considered:(47)ϖd1=−0.2+0.04sin0.5t,ϖd2=−0.1+0.02cos0.2t,ϖd3=0.002sint,where the matrices are chosen as follows:(48)M=2000023.6−0.80−0.82.76,A=00−23.6νy−0.8νψ0020νx23.6νy+0.8νψ−20νx0,B=B1B2B3T,B1=0.2565+1.5687νx+3.4357νx200T,B2=00.3578+34.5653νy+5.21νψ0.674νy+3.76νψ−0.1079T,B3=0−1.1052−5.0437νy−0.13νψ0.674νy+3.76νψ−0.1079T.The disturbance vector is given as follows:(49)gt=1.4+2sin0.02t+1.5sin0.1t−0.9+2sin0.02t−π6+1.5sin0.3t−sin0.09t+π3−4sin0.01t.The initial state variables are selected asx1=x11,x12,x13T=0.004,0.019,0T and x2=x21,x22,x23T=0.4,0.05,0T. The state variables x11, x12, and x13 satisfy −F11≤x11≤F12, −F21≤x12≤F22, and −F31≤x13≤F32 with F11=1+0.1sint, F12=1+0.1sint, F21=1+0.1sint, F22=1+0.1sint, F13=0.3+0.1sint, and F23=0.3+0.1sint.The initial value of the observer is selected asg^0=0,0,0T. The design parameters are given as δ=0.001,0.001,0.001T. The parameter matrices are given as Γ1=50I, Γ2=100I, Γ3=200I, and K0=diag2,2,2. The control gains are set as K1=diag0.001,0.1,0.1 and K2=diag200,200,0.02.Figures1–11 show the simulation results. Figures 1–3 show the output signal x1, reference signal ϖdk, and constraints interval Fk1 and Fk2 with k=1,2,3, which indicates that the tracking trajectory is perfect, and the state variable does not violate the constraint interval. Figure 4 indicates the trajectory of the state variable x2. Figures 5 and 6 are given to illustrate the trajectories of tracking error z1 and z2, and it can be seen that the error is small enough. The trajectory of actual controller is displayed in Figure 7. From Figure 8, it indicates that the disturbance observer is constructed to deal with the external disturbance better. Meanwhile, Figures 9 and 10 are used to explain the trajectory of adaptive laws. We can clearly see that the adaptive laws are bounded. Figure 11 shows the trajectory of virtual controller. The results show that the adaptive control method is effective.Figure 1
Trajectory ofϖd1, x11, and constraint intervals.Figure 2
Trajectory ofϖd2, x12, and constraint intervals.Figure 3
Trajectory ofϖd3, x13, and constraint intervals.Figure 4
Trajectory of velocity vectorx2.
(a)(b)(c)Figure 5
Trajectories of errorsz1.
(a)(b)(c)Figure 6
Trajectories of errorsz2.
(a)(b)(c)Figure 7
Trajectories of control inputτ.
(a)(b)(c)Figure 8
Trajectories of external disturbancesg1, g2, and g3 and their estimations g^1, g^2, and g^3.
(a)(b)(c)Figure 9
Trajectories ofξ^.
(a)(b)(c)Figure 10
Trajectories ofθ^1, θ^2, and θ^3.
(a)(b)(c)Figure 11
Trajectories of virtual controllerα11, α12, and α13.
(a)(b)(c)


## 4. Conclusion

The adaptive NN control method is studied for vessel nonlinear systems with disturbance observer and time-varying output constraints in this paper. Compared with the existing research on vessel, the method studied in this article is more practical. First, the feasibility condition of virtual controller is eliminated by introducing nonlinear state correlation function and new coordinate transformation. Only in this way, we can deal with control design gracefully. Then, with the help of NNs, the unknown and continuous function is approximated to simplify the design of the controller. Next, the disturbance observer is constructed to dispose the disturbances. Finally, the control performance is tested through simulation examples. In the future work, we will apply the constraint form of this paper to the position and speed constraints of flexible manipulator.



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