# Course Control of Underactuated Ship Based on Nonlinear Robust Neural Network Backstepping Method

Authors: Junjia Yuan; Hao Meng; Qidan Zhu; Jiajia Zhou
Journal: Computational Intelligence and Neuroscience
 (2016)

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## Abstract

The problem of course control for underactuated surface ship is addressed in this paper. Firstly, neural networks are adopted to determine the parameters of the unknown part of ideal virtual backstepping control, even the weight values of neural network are updated by adaptive technique. Then uniform stability for the convergence of course tracking errors has been proven through Lyapunov stability theory. Finally, simulation experiments are carried out to illustrate the effectiveness of proposed control method.


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## Body

## 1. Introduction

Tracking control performance for surface vessel along the predefined route has been an essential control problem for marine autopilot system design, and it has received considerable attractions from control community. In 1922, proportional-integral-derivative (PID) autopilot for ship steering was presented by Nicholas Minosky [1]. PID controller greatly improved the performance of autopilots. Until the 1980s almost all makes of autopilots were based on these controllers. One challenge for tracking control of surface vessel based on above method is that the systems are often underactuated by the sway motion due to weight, complexity, and efficiency considerations and exhibit nonholonomic constraints, which meets Brocket’s theorem that there is no continuous or even smooth time-invariant state feedback law that can stabilize the system to the origin [2]. Another challenge is that the vessel model itself exhibits severe nonlinear characteristic and model uncertainties induced by the ocean environment [3, 4].For the ship with nonlinear maneuvering characteristics and without uncertainties, a state feedback linearization control law was designed [5], while feedback linearization with saturation and slew rate limiting actuators was discussed [6]. Later, combined with a genetic algorithm, the backstepping method was employed to develop a nonlinear ship course controller by Witkowska and Smierzchalski [7], where the ship course parameters were automatically tuned to the optimal values with the aid of a genetic algorithm. Even considering the ship steering model with both constant parametric uncertainties and input disturbance with unknown bound, a robust adaptive nonlinear control law was presented based on projection approach and Lyapunov stability theory [8]. Recently many papers have tackled these problems based on Lyapunov theory [9–12]. In [13–15] a global tracking controller for underactuated ship is addressed with nonzero off-diagonal terms, the reference trajectory is generated by using a virtual target guidance algorithm, and the controller designed is facilitated by an introduction of changing the ship outputs, several coordinate transformations, and backstepping method. And the controller design is heavily depending on accurate dynamic model; the robustness against disturbance has not been addressed. A method using backstepping adaptive dynamical sliding mode control is presented for path following control of USV in [16], the control system takes account of the modeling errors and disturbances, and simplified tracking error dynamics are obtained by assuming that the sway velocity is small which can be neglected in the controller design and only for straight line path tracking can be achieved. The LOS based guidance law is also used in the controller design which causes the complexity of computing high-order derivative of virtual control. In [17], a transformation of vessel kinematics to the Serret-Frenet frame is introduced by exploring an extra degree of freedom by controlling explicitly the progression rate of the virtual target along the path and overcomes the major singular problem; approach angle is introduced for controller design via backstepping method. Neural networks are introduced to enhance system stability and transient performance, which can handle the known dynamics and uncertainties of systems well [18–20]. Particularly in [12] a single hidden layer neural network (SHLNN) is adopted to obtain the adaptive signal online, but the choice of the single hidden layer neural network is limited by the number of hidden layer node selections that will affect the online learning speed and accuracy and cannot produce a better estimation effect on the fast changing disturbances.Therefore, a solution to the course control of underactuated surface vessel is addressed in this paper. In view of the characteristics of the underactuated performance, the backstepping control method is used to deal with above problem. The direct adaptive neural network is adopted to design control law by using the RBF neural network to overcome the problem that the ideal virtual control cannot be used directly in practice. The weights of the neural network are updated by adaptive technique to guarantee the stability of the closed-loop system through Lyapunov stability theory. Simulation results are illustrated to verify the performance of the proposed adaptive neural network controller with good precision.


## 2. Adaptive Robust Neural Network Controller Design

### 2.1. Problem Description

Consider the following nonlinear systems:(1)x˙i=fix-i+gix-ixi+1+di,1≤i≤n-1,x˙n=fnx-n+gnx˙nu+dn,n≥2,y=x1,where x-i=x1,x2,…,xi is system state, u is control input, and y is system output. The control objective is to design an adaptive neural network controller and make y track yd. yd meets the smooth bounded reference model as follows:(2)x˙di=fdixd,1≤i≤myd=xd1,m≥n,where xd=xd1,xd2,…,xdmT∈Rm is state constant, yd∈R represents system output, and fdi·,  i=1,2,…,m, denote nonlinear function, assuming that the reference model for each state is bounded as xd∈Ωd,  ∀t≥0.Assumption 1.
There is an unknown constantpi∗ to meet, ∀(x-n,t)∈Rn×R+,  di(x-n,t)≤pi∗ρi(x-i), and ρi(x-i) is a known positive smooth function.


### 2.2. Direct Adaptive Neural Network Controller Design

In view of the problems and solutions described in the last section, the direct adaptive neural network controller for nonlinear systems with RBF neural network is chosen. Detailed design steps will be described in the following.Step 1.
Letz1=x1-xd1, z2=x2-α1, and then(3)z˙1=f1x1+g1x1x2+d1-x˙d1.
Consider the following Lyapunov function:(4)V1=12g1x1z12+12W~1TΓ1-1W~1,where W~1=W^1-W1∗, W1∗ represents the ideal weight vector of neural network, W^1 represents the estimated value of the neural network weight vector, W~1 represents the estimation error of weight vector, Γ1=Γ1T>0 is the adaptive gain matrix, and the derivation of V1 can be computed as(5)V˙1=z1z˙1g1x1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1g1x1f1x1+g1x1x2+d1-x˙d1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1z2+α1+f1x1-x˙d1g1x1+z1d1g1x1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1.
According to Assumption1, we can get(6)V˙1≤z1z2+α1+f1x1-x˙d1g1x1+z12ρ122g12x1+P1∗22+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1z2+α1+f1x1-x˙d1g1x1+z1ρ122g12x1+P1∗22+g˙1x1z122g12x1+W~1TΓ1-1W^˙1.
There is an ideal virtual feedback control law:(7)α1∗=-c1z1-f1x1-x˙d1g1x1+z1ρ122g12x1,where c1>0 is designed controller parameter.
Because of the unknown smooth functionsf1(x1) and g1(x1), we cannot actually get the ideal feedback control law α1∗; from (7) we can see that the unknown part f1x1-x˙d1/g1x1 is smooth function of x1 and x˙d1, so that(8)h1Z1≜f1x1-x˙d1g1x1+z1ρ122g12x1,Z1≜x1,x˙d1T⊂R2.
RBF neural networkW1TS1(Z1) is used to approximate the unknown function h1(Z1), and α1∗ can be expressed as(9)α1∗=-c1z1-W1∗TS1Z1-e1,where e1≤e1∗ is estimated error and meets e1∗>0.
BecauseW1∗ is unknown, the virtual control law is selected as follows:(10)α1=-c1z1-W^1TS1Z1and then(11)V˙1≤z1z2-c1z12+g˙1x1z122g12x1+z1e1+P1∗22-W~1TS1z1+W~1TΓ1-1W^˙1.
Adaptive law can be chosen as follows:(12)W^˙1=W~˙1=Γ1S1Z1z1-σ1W^1,where σ1>0 and then(13)V˙1≤z1z2-c1z12+g˙1x1z122g12x1+z1e1+P1∗22-σ1W~1TW^1.
Letc1=c10+c11, where c10>0 and c11>0, and then the upper equation becomes(14)V˙1≤z1z2-c10+g˙12g12z12-c11z12+z1e1+P1∗22-σ1W~1TW^1.
According to the complete square formula,(15)-σ1W~1TW^1=-σ1W~1TW~1+W1∗≤-σ1W~12+σ1W~1W1∗≤-σ1W~122+σ1W1∗22,-c11z12+z1e1≤-c11z12+z1e1≤e124c11≤e1∗24c11.
Because-(c10+(g˙1/2g12))z12≤-(c10-(g1d/2g1m2))z12, we can make (c10∗≜c10-(g1d/2g1m2))>0 by choosing the appropriate c10 and obtain the following inequality:(16)V˙1≤z1z2-c10∗z12-σ1W~122+σ1W1∗22+e1∗24c11+P1∗22.The cross couplingz1z2 in (16) will be eliminated in the next step.Step 2.
Letz2=x2-α1; then(17)z˙2=f2x-2+g2x-2x3+d2-α˙1.
From (10) we can see that α1 is a function of x1, xd, and W^1, and  α˙1 can be written as(18)α˙1=∂α1∂x1x˙1+∂α1∂xdx˙d+∂α1∂W^1W^˙1=∂α1∂x1g1x1x2+f1x1+φ1,where ϕ1=(∂α1/∂xd)x˙d+(∂α1/∂W^1)Γ1(S1(Z1)z1-σ1W^1) can be calculated.
Consider the following Lyapunov function:(19)V2=V1+12g2x-2z22+12W~2TΓ2-1W~2,where Γ2=Γ2T>0 is an adaptive gain matrix.
Then the derivation ofV2 can be calculated as(20)V˙2=V˙1+z2z˙2g2x-2+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2g2x-2f2x-2+g2x-2x3+d2-α˙1+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z2d2g2x-2+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2.
According to Assumption1 we can get(21)V˙2≤V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z22ρ222g22x-2+P2∗22+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z2ρ222g22x-2+P2∗22+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2.
There is an ideal feedback control law:(22)α2∗=-z1-c2z2-f2x-2-α˙1g2x-2+z2ρ222g22x-2,where c2>0 is a designed controller parameter.
Because of the unknown smooth functionsf2(x-2) and g2(x-2), we cannot actually get the ideal feedback control law α2∗; from (22) we can see that the unknown part is a smooth function of x-2 and α˙1; let(23)h2Z2≜f2x-2-α˙1g2x-2+z2ρ222g22x-2,where Z2≜x-2T,(∂α1/∂x1),ϕ1T⊂R4. RBF neural network W2TS2(Z2) is used to approximate the unknown function h2(Z2), and α2∗ can be expressed as(24)α2∗=-z1-c2z2-W2∗TS2Z2-e2,where W2∗ is expressed as the ideal constant weight vector and e2≤e2∗ is the estimated error and meets e2∗>0.
BecauseW2∗ is unknown, select the following virtual control law:(25)α2=-z1-c2z2-W^2TS2Z2,where W^2 is the estimated value of W2∗; then(26)V˙2≤V˙1-z1z2+z2z3-c2z22+g˙2x-2z222g22x-2+z2e2+P2∗22-W~2TS2z2+W~2TΓ2-1W^˙2,where W~2=W^2-W2∗.
Adaptive law can be chosen as(27)W^˙2=W~˙2=Γ2S2Z2z2-σ2W^2,where σ2>0; then(28)V˙2≤V˙1-z1z2+z2z3-c2z22+g˙2x-2z222g22x-2+z2e2+P2∗22-σ2W~2TW^2.
Letc2=c20+c21,  c20,c21>0; then the upper equation becomes(29)V˙2≤V˙1-z1z2+z2z3-c20+g˙2x-22g22x-2z22-c21z22+z2e2+P2∗22-σ2W~2TW^2.
According to the complete square formula,(30)-σ2W~2TW^2=-σ2W~2TW~2+W2∗≤-σ2W~22+σ2W~2W2∗≤-σ2W~222+σ2W2∗22,-c21z22+z2e2≤-c21z22+z2e2≤e224c21≤e2∗24c21.
Because-(c20+(g˙2/2g22))z22≤-(c20-(g2d/2g2m2))z22, then we can make (c20∗≜c20-(g2d/2g2m2))>0 by selecting the proper c20; then(31)V˙2≤V˙1-z1z2+z2z3-c20∗z22-σ2W~222+σ2W2∗22+e2∗24c21+P2∗22≤z2z3-∑k=12ck0∗zk2-∑k=12σkW~k22+∑k=12σkWk∗22+∑k=12ek∗24ck1.The cross couplingz2z3 in (31) will be eliminated in the next step.Step
i
(
3
≤
i
≤
n
-
1
). The derivative of zi=xi-αi-1 can be calculated as(32)z˙i=fix-i+gix-ixi+1-α˙i-1, where(33)α˙i-1=∑k=1i-1∂αi-1∂xkgkx-kxk+1+fkx-k+φi-1,ϕi-1=∑k=1i-1∂αi-1∂xdx˙d+∑k=1i-1∂αi-1∂W^kΓkSkZkzk-σkW^k.Consider the following Lyapunov function:(34)Vi=Vi-1+12gix-izi2+12W~iTΓi-1W~i,where Γi=ΓiT>0 is an adaptive gain matrix.Then the derivation ofVi can be calculated as(35)V˙i=V˙i-1+ziz˙igix-i+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zigix-ifix-i+gix-ixi+1+di-α˙i-1+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zidigix-i+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i.According to Assumption1 we can get(36)V˙i≤V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zi2ρi22gi2x-i+Pi∗22+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zi2ρi22gi2x-i+Pi∗22+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i.There is an ideal feedback control law as(37)αi∗=-zi-1-cizi-fix-i-α˙i-1gix-i+ziρi22gi2x-i,where ci>0 is designed controller parameter.Because of the unknown smooth functionsfi(x-i) and gi(x-i), we cannot actually get the ideal feedback control law αi∗; from (37) we can see that the unknown part is a smooth function of x-i and α˙i-1, and let(38)hiZi≜fix-i-α˙i-1gix-i+ziρi22gi2x-i,where(39)Zi≜x-iT,∂αi-1∂x1,…,∂αi-1∂xi-1,φi-1T⊂R2i.By introducing the direct variable(∂αi-1/∂x1),…,(∂αi-1/∂xi-1), φi-1, we can make the number of neural networks minimized. RBF neural network WiTSi(Zi) is used to approximate the unknown function hi(Zi), and αi∗ can be expressed as(40)αi∗=-zi-1-cizi-Wi∗TSiZi-ei,where ei≤ei∗ is estimated error and meets ei∗>0.BecauseWi∗ is unknown, select the following virtual control law:(41)αi=-zi-1-cizi-W^iTSiZi,where Wi∗ is the estimated value of W^i; then(42)V˙i≤V˙i-1-zi-1zi+zizi+1-cizi2+g˙ix-izi22gi2x-i+ziei+Pi∗22-W~iTSizi+W~iTΓi-1W^˙i,where W~i=W^i-Wi∗.The following adaptive law can be selected as(43)W^˙i=W~˙i=ΓiSiZizi-σiW^i,where σi>0; then(44)V˙i≤V˙i-1-zi-1zi+zizi+1-cizi2+g˙ix-izi22gi2x-i+ziei+Pi∗22-σiW~iTW^i.Letci=ci0+ci1,  ci0,ci1>0; then (44) can be rewritten as(45)V˙i≤V˙i-1-zi-1zi+zizi+1-ci0+g˙ix-i2gi2x-izi2-ci1zi2+ziei+Pi∗22-σiW~iTW^i.According to the complete square formula,(46)-σiW~iTW^i=-σiW~iTW~i+Wi∗≤-σiW~i2+σiW~iWi∗≤-σiW~i22+σiWi∗22,-ci1zi2+ziei≤-ci1zi2+ziei≤ei24ci1≤ei∗24ci1.Because-(ci0+(g˙i/2gi2))zi2≤-(ci0-(gid/2gim2))zi2, then we can make (ci0∗≜ci0-(gid/2gim2))>0 by selecting the proper ci0; then(47)V˙i≤V˙i-1-zi-1zi+zizi+1-ci0∗zi2-σiW~i22+σiWi∗22+ei∗24ci1+Pi∗22≤zizi+1-∑k=1ick0∗zk2-∑k=1iσkW~k22+∑k=1iσkWk∗22+∑k=1iek∗24ck1+∑k=1iPk∗22.The cross couplingzizi+1 in (47) will be eliminated in the next step.Step
n. The derivative of zn=xn-αn-1 can be calculated as(48)z˙n=fnx-n+gnx-n-1u-α˙n-1,where(49)α˙n-1=∑k=1n-1∂αn-1∂xkgkx-kxk+1+fkx-k+ϕn-1,where (50)ϕn-1=∑k=1n-1∂αn-1∂xdx˙d+∑k=1n-1∂αn-1∂W^kΓkSkZkzk-σkW^k.Consider the following Lyapunov function:(51)Vn=Vn-1+12gnx-nzn2+12W~nTΓn-1W~n,where Γn=ΓnT>0 is an adaptive gain matrix. Then the derivation of Vn can be calculated as(52)V˙n=V˙n-1+znz˙ngix-i+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+zngnx-nfnx-n+gnx-nu+dn-α˙n-1+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+znzn+1+u+fnx-n-α˙n-1gnx-n+zndngnx-n+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n.According to Assumption1 we can get(53)V˙n≤V˙n-1+znzn+1+u+fix-i-α˙n-1gix-i+zn2ρn22gn2x-n+Pn∗22+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+znzn+1+u+fnx-n-α˙n-1gnx-n+zn2ρn22gn2x-n+Pn∗22+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n.There is an ideal feedback control law as(54)u∗=-zi-1-cizi-fix-i-α˙i-1gix-i+ziρi22gi2x-i,where cn>0 is designed controller parameter.Because of the unknown smooth functionsfn(x-n) and gi(x-i), we cannot actually get the ideal feedback control law u∗; from (54) we can see the unknown part is a smooth function of x-n and α˙n-1, and let(55)hnZi≜fnx-n-α˙n-1gnx-n+znρn22gn2x-n,where Zn≜x-nT,∂αn-1/∂x1,…,∂αn-1/∂xn-1,ϕn-1T⊂R2n.RBF neural networkWnTSn(Zn) is used to approximate the unknown function hn(Zn), and u∗ can be expressed as(56)u∗=-zn-1-cnzn-Wn∗TSnZn-en,where en≤en∗ is estimated error and meets en∗>0.BecauseWn∗ is unknown, select the following virtual control law:(57)u=-zn-1-cnzn-W^nTSnZn,where W^i is the estimated value of Wi∗; then(58)V˙n≤V˙n-1-zn-1zn+znzn+1-cnzn2+g˙nx-nzn22gn2x-n+znen+Pn∗22-W~nTSnzn+W~nTΓn-1W^˙n,where W~n=W^n-Wn∗.The following adaptive law can be selected as(59)W^˙n=W~˙n=ΓnSnZnzn-σnW^n,where σn>0; then(60)V˙n≤V˙n-1-zn-1zn+znzn+1-cnzn2+g˙nx-nzn22gn2x-n+znen+Pn∗22-σnW~nTW^n.Letcn=cn0+cn1,  cn0,cn1>0; (60) can be rewritten as(61)V˙n≤V˙n-1-zn-1zn+znzn+1-cn0+g˙nx-n2gn2x-nzn2-cn1zn2+znen+Pn∗22-σnW~nTW^n.According to the complete square formula,(62)-σnW~nTW^n=-σnW~nTW~n+Wn∗≤-σnW~n2+σnW~nWn∗≤-σnW~n22+σnWn∗22,-cn1zn2+znen≤-cn1zn2+znen≤en24cn1≤en∗24cn1.Because-(cn0+(g˙n/2gn2))zn2≤-(cn0-(gnd/2gnm2))zn2, then we can make (cn0∗≜cn0-(gnd/2gnm2))>0 by selecting the proper cn0; then(63)V˙n≤-∑k=1nck0∗zk2-∑k=1nσkW~k22+∑k=1nσkWk∗22+∑k=1nek∗24ck1+∑k=1nPk∗22.Letδ≜∑k=1n(σkWk∗2/2)+∑k=1n(ek∗2/4ck1)+∑k=1n(pk∗2/2),ck0∗≥(γ/2gkm),ck0>(γ/2gkm)+(gkd/2gkm2),k=1,2,…,n,  where γ>0,σk≥γλmax{Γk-1},k=1,2,…,n; then(64)V˙n≤-∑k=1nck0∗zk2-∑k=1nσkW~k22+δ≤-∑k=1nγ2gkmzk2-∑k=1nγW~kTΓk-1W~k2gkm+δ≤-γ∑k=1n12gkzk2+∑k=1nW~kTΓk-1W~k2+δ≤-γVn+δ.The stability and control performance of the closed-loop adaptive system are demonstrated by the following theorem.Theorem 2.
In the initial conditions, by formula (1), reference model (2), control law (57), and neural network weight update rate in (12), (27), (43), and (59), supposing that there is a large enough set of closed sets Ωi∈R2i,  i=1,2,…,n, for any given moment t≥0, making Zi∈Ωi, the following conclusions can be obtained as follows:(1)
The signal of the whole closed-loop system is bounded, and the state variablex-n and the neural network estimation errors W^1T,…,W^nT will eventually converge to the closed set as follows:(65)Ωs1≜x-n,W^1,…,W^n∣V<δγ,xd∈Ωd.(2)
By choosing the proper control parameters, the output tracking errory(t)-yd1(t) is close to a small neighborhood of zero [21].


## 2.1. Problem Description

Consider the following nonlinear systems:(1)x˙i=fix-i+gix-ixi+1+di,1≤i≤n-1,x˙n=fnx-n+gnx˙nu+dn,n≥2,y=x1,where x-i=x1,x2,…,xi is system state, u is control input, and y is system output. The control objective is to design an adaptive neural network controller and make y track yd. yd meets the smooth bounded reference model as follows:(2)x˙di=fdixd,1≤i≤myd=xd1,m≥n,where xd=xd1,xd2,…,xdmT∈Rm is state constant, yd∈R represents system output, and fdi·,  i=1,2,…,m, denote nonlinear function, assuming that the reference model for each state is bounded as xd∈Ωd,  ∀t≥0.Assumption 1.
There is an unknown constantpi∗ to meet, ∀(x-n,t)∈Rn×R+,  di(x-n,t)≤pi∗ρi(x-i), and ρi(x-i) is a known positive smooth function.


## 2.2. Direct Adaptive Neural Network Controller Design

In view of the problems and solutions described in the last section, the direct adaptive neural network controller for nonlinear systems with RBF neural network is chosen. Detailed design steps will be described in the following.Step 1.
Letz1=x1-xd1, z2=x2-α1, and then(3)z˙1=f1x1+g1x1x2+d1-x˙d1.
Consider the following Lyapunov function:(4)V1=12g1x1z12+12W~1TΓ1-1W~1,where W~1=W^1-W1∗, W1∗ represents the ideal weight vector of neural network, W^1 represents the estimated value of the neural network weight vector, W~1 represents the estimation error of weight vector, Γ1=Γ1T>0 is the adaptive gain matrix, and the derivation of V1 can be computed as(5)V˙1=z1z˙1g1x1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1g1x1f1x1+g1x1x2+d1-x˙d1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1z2+α1+f1x1-x˙d1g1x1+z1d1g1x1+g˙1x1z122g12x1+W~1TΓ1-1W^˙1.
According to Assumption1, we can get(6)V˙1≤z1z2+α1+f1x1-x˙d1g1x1+z12ρ122g12x1+P1∗22+g˙1x1z122g12x1+W~1TΓ1-1W^˙1=z1z2+α1+f1x1-x˙d1g1x1+z1ρ122g12x1+P1∗22+g˙1x1z122g12x1+W~1TΓ1-1W^˙1.
There is an ideal virtual feedback control law:(7)α1∗=-c1z1-f1x1-x˙d1g1x1+z1ρ122g12x1,where c1>0 is designed controller parameter.
Because of the unknown smooth functionsf1(x1) and g1(x1), we cannot actually get the ideal feedback control law α1∗; from (7) we can see that the unknown part f1x1-x˙d1/g1x1 is smooth function of x1 and x˙d1, so that(8)h1Z1≜f1x1-x˙d1g1x1+z1ρ122g12x1,Z1≜x1,x˙d1T⊂R2.
RBF neural networkW1TS1(Z1) is used to approximate the unknown function h1(Z1), and α1∗ can be expressed as(9)α1∗=-c1z1-W1∗TS1Z1-e1,where e1≤e1∗ is estimated error and meets e1∗>0.
BecauseW1∗ is unknown, the virtual control law is selected as follows:(10)α1=-c1z1-W^1TS1Z1and then(11)V˙1≤z1z2-c1z12+g˙1x1z122g12x1+z1e1+P1∗22-W~1TS1z1+W~1TΓ1-1W^˙1.
Adaptive law can be chosen as follows:(12)W^˙1=W~˙1=Γ1S1Z1z1-σ1W^1,where σ1>0 and then(13)V˙1≤z1z2-c1z12+g˙1x1z122g12x1+z1e1+P1∗22-σ1W~1TW^1.
Letc1=c10+c11, where c10>0 and c11>0, and then the upper equation becomes(14)V˙1≤z1z2-c10+g˙12g12z12-c11z12+z1e1+P1∗22-σ1W~1TW^1.
According to the complete square formula,(15)-σ1W~1TW^1=-σ1W~1TW~1+W1∗≤-σ1W~12+σ1W~1W1∗≤-σ1W~122+σ1W1∗22,-c11z12+z1e1≤-c11z12+z1e1≤e124c11≤e1∗24c11.
Because-(c10+(g˙1/2g12))z12≤-(c10-(g1d/2g1m2))z12, we can make (c10∗≜c10-(g1d/2g1m2))>0 by choosing the appropriate c10 and obtain the following inequality:(16)V˙1≤z1z2-c10∗z12-σ1W~122+σ1W1∗22+e1∗24c11+P1∗22.The cross couplingz1z2 in (16) will be eliminated in the next step.Step 2.
Letz2=x2-α1; then(17)z˙2=f2x-2+g2x-2x3+d2-α˙1.
From (10) we can see that α1 is a function of x1, xd, and W^1, and  α˙1 can be written as(18)α˙1=∂α1∂x1x˙1+∂α1∂xdx˙d+∂α1∂W^1W^˙1=∂α1∂x1g1x1x2+f1x1+φ1,where ϕ1=(∂α1/∂xd)x˙d+(∂α1/∂W^1)Γ1(S1(Z1)z1-σ1W^1) can be calculated.
Consider the following Lyapunov function:(19)V2=V1+12g2x-2z22+12W~2TΓ2-1W~2,where Γ2=Γ2T>0 is an adaptive gain matrix.
Then the derivation ofV2 can be calculated as(20)V˙2=V˙1+z2z˙2g2x-2+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2g2x-2f2x-2+g2x-2x3+d2-α˙1+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z2d2g2x-2+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2.
According to Assumption1 we can get(21)V˙2≤V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z22ρ222g22x-2+P2∗22+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2=V˙1+z2z3+α2+f2x-2-α˙1g2x-2+z2ρ222g22x-2+P2∗22+g˙2x-2z222g22x-2+W~2TΓ2-1W^˙2.
There is an ideal feedback control law:(22)α2∗=-z1-c2z2-f2x-2-α˙1g2x-2+z2ρ222g22x-2,where c2>0 is a designed controller parameter.
Because of the unknown smooth functionsf2(x-2) and g2(x-2), we cannot actually get the ideal feedback control law α2∗; from (22) we can see that the unknown part is a smooth function of x-2 and α˙1; let(23)h2Z2≜f2x-2-α˙1g2x-2+z2ρ222g22x-2,where Z2≜x-2T,(∂α1/∂x1),ϕ1T⊂R4. RBF neural network W2TS2(Z2) is used to approximate the unknown function h2(Z2), and α2∗ can be expressed as(24)α2∗=-z1-c2z2-W2∗TS2Z2-e2,where W2∗ is expressed as the ideal constant weight vector and e2≤e2∗ is the estimated error and meets e2∗>0.
BecauseW2∗ is unknown, select the following virtual control law:(25)α2=-z1-c2z2-W^2TS2Z2,where W^2 is the estimated value of W2∗; then(26)V˙2≤V˙1-z1z2+z2z3-c2z22+g˙2x-2z222g22x-2+z2e2+P2∗22-W~2TS2z2+W~2TΓ2-1W^˙2,where W~2=W^2-W2∗.
Adaptive law can be chosen as(27)W^˙2=W~˙2=Γ2S2Z2z2-σ2W^2,where σ2>0; then(28)V˙2≤V˙1-z1z2+z2z3-c2z22+g˙2x-2z222g22x-2+z2e2+P2∗22-σ2W~2TW^2.
Letc2=c20+c21,  c20,c21>0; then the upper equation becomes(29)V˙2≤V˙1-z1z2+z2z3-c20+g˙2x-22g22x-2z22-c21z22+z2e2+P2∗22-σ2W~2TW^2.
According to the complete square formula,(30)-σ2W~2TW^2=-σ2W~2TW~2+W2∗≤-σ2W~22+σ2W~2W2∗≤-σ2W~222+σ2W2∗22,-c21z22+z2e2≤-c21z22+z2e2≤e224c21≤e2∗24c21.
Because-(c20+(g˙2/2g22))z22≤-(c20-(g2d/2g2m2))z22, then we can make (c20∗≜c20-(g2d/2g2m2))>0 by selecting the proper c20; then(31)V˙2≤V˙1-z1z2+z2z3-c20∗z22-σ2W~222+σ2W2∗22+e2∗24c21+P2∗22≤z2z3-∑k=12ck0∗zk2-∑k=12σkW~k22+∑k=12σkWk∗22+∑k=12ek∗24ck1.The cross couplingz2z3 in (31) will be eliminated in the next step.Step
i
(
3
≤
i
≤
n
-
1
). The derivative of zi=xi-αi-1 can be calculated as(32)z˙i=fix-i+gix-ixi+1-α˙i-1, where(33)α˙i-1=∑k=1i-1∂αi-1∂xkgkx-kxk+1+fkx-k+φi-1,ϕi-1=∑k=1i-1∂αi-1∂xdx˙d+∑k=1i-1∂αi-1∂W^kΓkSkZkzk-σkW^k.Consider the following Lyapunov function:(34)Vi=Vi-1+12gix-izi2+12W~iTΓi-1W~i,where Γi=ΓiT>0 is an adaptive gain matrix.Then the derivation ofVi can be calculated as(35)V˙i=V˙i-1+ziz˙igix-i+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zigix-ifix-i+gix-ixi+1+di-α˙i-1+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zidigix-i+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i.According to Assumption1 we can get(36)V˙i≤V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zi2ρi22gi2x-i+Pi∗22+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i=V˙i-1+zizi+1+αi+fix-i-α˙i-1gix-i+zi2ρi22gi2x-i+Pi∗22+g˙ix-izi22gi2x-i+W~iTΓi-1W^˙i.There is an ideal feedback control law as(37)αi∗=-zi-1-cizi-fix-i-α˙i-1gix-i+ziρi22gi2x-i,where ci>0 is designed controller parameter.Because of the unknown smooth functionsfi(x-i) and gi(x-i), we cannot actually get the ideal feedback control law αi∗; from (37) we can see that the unknown part is a smooth function of x-i and α˙i-1, and let(38)hiZi≜fix-i-α˙i-1gix-i+ziρi22gi2x-i,where(39)Zi≜x-iT,∂αi-1∂x1,…,∂αi-1∂xi-1,φi-1T⊂R2i.By introducing the direct variable(∂αi-1/∂x1),…,(∂αi-1/∂xi-1), φi-1, we can make the number of neural networks minimized. RBF neural network WiTSi(Zi) is used to approximate the unknown function hi(Zi), and αi∗ can be expressed as(40)αi∗=-zi-1-cizi-Wi∗TSiZi-ei,where ei≤ei∗ is estimated error and meets ei∗>0.BecauseWi∗ is unknown, select the following virtual control law:(41)αi=-zi-1-cizi-W^iTSiZi,where Wi∗ is the estimated value of W^i; then(42)V˙i≤V˙i-1-zi-1zi+zizi+1-cizi2+g˙ix-izi22gi2x-i+ziei+Pi∗22-W~iTSizi+W~iTΓi-1W^˙i,where W~i=W^i-Wi∗.The following adaptive law can be selected as(43)W^˙i=W~˙i=ΓiSiZizi-σiW^i,where σi>0; then(44)V˙i≤V˙i-1-zi-1zi+zizi+1-cizi2+g˙ix-izi22gi2x-i+ziei+Pi∗22-σiW~iTW^i.Letci=ci0+ci1,  ci0,ci1>0; then (44) can be rewritten as(45)V˙i≤V˙i-1-zi-1zi+zizi+1-ci0+g˙ix-i2gi2x-izi2-ci1zi2+ziei+Pi∗22-σiW~iTW^i.According to the complete square formula,(46)-σiW~iTW^i=-σiW~iTW~i+Wi∗≤-σiW~i2+σiW~iWi∗≤-σiW~i22+σiWi∗22,-ci1zi2+ziei≤-ci1zi2+ziei≤ei24ci1≤ei∗24ci1.Because-(ci0+(g˙i/2gi2))zi2≤-(ci0-(gid/2gim2))zi2, then we can make (ci0∗≜ci0-(gid/2gim2))>0 by selecting the proper ci0; then(47)V˙i≤V˙i-1-zi-1zi+zizi+1-ci0∗zi2-σiW~i22+σiWi∗22+ei∗24ci1+Pi∗22≤zizi+1-∑k=1ick0∗zk2-∑k=1iσkW~k22+∑k=1iσkWk∗22+∑k=1iek∗24ck1+∑k=1iPk∗22.The cross couplingzizi+1 in (47) will be eliminated in the next step.Step
n. The derivative of zn=xn-αn-1 can be calculated as(48)z˙n=fnx-n+gnx-n-1u-α˙n-1,where(49)α˙n-1=∑k=1n-1∂αn-1∂xkgkx-kxk+1+fkx-k+ϕn-1,where (50)ϕn-1=∑k=1n-1∂αn-1∂xdx˙d+∑k=1n-1∂αn-1∂W^kΓkSkZkzk-σkW^k.Consider the following Lyapunov function:(51)Vn=Vn-1+12gnx-nzn2+12W~nTΓn-1W~n,where Γn=ΓnT>0 is an adaptive gain matrix. Then the derivation of Vn can be calculated as(52)V˙n=V˙n-1+znz˙ngix-i+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+zngnx-nfnx-n+gnx-nu+dn-α˙n-1+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+znzn+1+u+fnx-n-α˙n-1gnx-n+zndngnx-n+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n.According to Assumption1 we can get(53)V˙n≤V˙n-1+znzn+1+u+fix-i-α˙n-1gix-i+zn2ρn22gn2x-n+Pn∗22+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n=V˙n-1+znzn+1+u+fnx-n-α˙n-1gnx-n+zn2ρn22gn2x-n+Pn∗22+g˙nx-nzn22gn2x-n+W~nTΓn-1W^˙n.There is an ideal feedback control law as(54)u∗=-zi-1-cizi-fix-i-α˙i-1gix-i+ziρi22gi2x-i,where cn>0 is designed controller parameter.Because of the unknown smooth functionsfn(x-n) and gi(x-i), we cannot actually get the ideal feedback control law u∗; from (54) we can see the unknown part is a smooth function of x-n and α˙n-1, and let(55)hnZi≜fnx-n-α˙n-1gnx-n+znρn22gn2x-n,where Zn≜x-nT,∂αn-1/∂x1,…,∂αn-1/∂xn-1,ϕn-1T⊂R2n.RBF neural networkWnTSn(Zn) is used to approximate the unknown function hn(Zn), and u∗ can be expressed as(56)u∗=-zn-1-cnzn-Wn∗TSnZn-en,where en≤en∗ is estimated error and meets en∗>0.BecauseWn∗ is unknown, select the following virtual control law:(57)u=-zn-1-cnzn-W^nTSnZn,where W^i is the estimated value of Wi∗; then(58)V˙n≤V˙n-1-zn-1zn+znzn+1-cnzn2+g˙nx-nzn22gn2x-n+znen+Pn∗22-W~nTSnzn+W~nTΓn-1W^˙n,where W~n=W^n-Wn∗.The following adaptive law can be selected as(59)W^˙n=W~˙n=ΓnSnZnzn-σnW^n,where σn>0; then(60)V˙n≤V˙n-1-zn-1zn+znzn+1-cnzn2+g˙nx-nzn22gn2x-n+znen+Pn∗22-σnW~nTW^n.Letcn=cn0+cn1,  cn0,cn1>0; (60) can be rewritten as(61)V˙n≤V˙n-1-zn-1zn+znzn+1-cn0+g˙nx-n2gn2x-nzn2-cn1zn2+znen+Pn∗22-σnW~nTW^n.According to the complete square formula,(62)-σnW~nTW^n=-σnW~nTW~n+Wn∗≤-σnW~n2+σnW~nWn∗≤-σnW~n22+σnWn∗22,-cn1zn2+znen≤-cn1zn2+znen≤en24cn1≤en∗24cn1.Because-(cn0+(g˙n/2gn2))zn2≤-(cn0-(gnd/2gnm2))zn2, then we can make (cn0∗≜cn0-(gnd/2gnm2))>0 by selecting the proper cn0; then(63)V˙n≤-∑k=1nck0∗zk2-∑k=1nσkW~k22+∑k=1nσkWk∗22+∑k=1nek∗24ck1+∑k=1nPk∗22.Letδ≜∑k=1n(σkWk∗2/2)+∑k=1n(ek∗2/4ck1)+∑k=1n(pk∗2/2),ck0∗≥(γ/2gkm),ck0>(γ/2gkm)+(gkd/2gkm2),k=1,2,…,n,  where γ>0,σk≥γλmax{Γk-1},k=1,2,…,n; then(64)V˙n≤-∑k=1nck0∗zk2-∑k=1nσkW~k22+δ≤-∑k=1nγ2gkmzk2-∑k=1nγW~kTΓk-1W~k2gkm+δ≤-γ∑k=1n12gkzk2+∑k=1nW~kTΓk-1W~k2+δ≤-γVn+δ.The stability and control performance of the closed-loop adaptive system are demonstrated by the following theorem.Theorem 2.
In the initial conditions, by formula (1), reference model (2), control law (57), and neural network weight update rate in (12), (27), (43), and (59), supposing that there is a large enough set of closed sets Ωi∈R2i,  i=1,2,…,n, for any given moment t≥0, making Zi∈Ωi, the following conclusions can be obtained as follows:(1)
The signal of the whole closed-loop system is bounded, and the state variablex-n and the neural network estimation errors W^1T,…,W^nT will eventually converge to the closed set as follows:(65)Ωs1≜x-n,W^1,…,W^n∣V<δγ,xd∈Ωd.(2)
By choosing the proper control parameters, the output tracking errory(t)-yd1(t) is close to a small neighborhood of zero [21].


## 3. Adaptive Robust Neural Network Control for Ship Course

### 3.1. Problem Formulation

This section introduces a simplified dynamic model of an underactuated surface vehicle with only one control inputδ for heading control. A surface ship usually has three degrees of freedom for path following control in horizontal plane. Assuming that the vessel has three planes of symmetry, for most underactuated vessels have port/starboard symmetry, it can be neglected to simplify the vessel model for controller design. The detailed model which considers the environment disturbances can be set as follows:(66)y˙=Usin⁡ψ,ψ˙=r,r˙=-1Tr-αTr3+KTδ+Δ,y1=y,y2=ψ,where y denotes transverse displacement in the earth inertial coordinates; U=u2+v2 is resultant velocity of ship; ψ is course angle; r is yawing angular velocity; K,T represent performance index for ship steering; α is coefficient of nonlinear term; δ is control rudder angle; y1,y2 represent system output.The control objective is to design the controllerδ to make the control output y, ψ achieve the setting value (yd,ψd). Because the dimension of the system control input is less than the degree of freedom of the system, it is an underactuated system.


### 3.2. Dynamic Controller Design

Selection of coordinate transformation is as follows:(67)we=ψ+arcsin⁡ky1+ky2.The original system can be transformed into a single input single output system:(68)x˙1=ky˙1+ky2+x2,x˙2=-a1x2-a2x23+bu+Δ,where a1=1/T,  a2=α/T,  b=K/T,  x1=we,  x2=r,  u=δ, and the output of whole system is x1.For system model (67) and (68), the controller design is carried out by using backstepping method.Step 1.
Letz1=x1,  xd1=0; then(69)z˙1=ky˙1+ky2+x2.
For the subsystemz1,  α1∗≜x2 is chosen as virtual control input. Select the Lyapunov function Vz1=(1/2)z12, and there is(70)V˙z1=z1z˙1=ky˙1+ky2+x2z1.
Letz2=x2-α1; then x2=z2+α1,(71)V˙z1=ky˙1+ky2+z2+α1z1.
Select the following virtual control law:(72)α1∗=-c1z1-ky˙1+ky2.
V
˙
z
1
=
z
1
z
2
-
c
1
z
1
2, because ky˙/1+(ky)2 is unknown function, h1Z1=ky˙/1+(ky)2, and we will adopt RBF NN to estimate h1Z1 and get h1Z1=W1∗TS1Z1+ε1. But the actual use of the NN for the system is h1Z1=W^1TS1Z1. Actual virtual control input is α1=-c1z1-W^1TS1Z1; then(73)z˙1=ky˙1+ky2+z2+α1=z2-c1z1-W~1S1Z1+ε1,where W~1=W^1-W1∗.
Select Lyapunov function as(74)V1=Vz1+12W~1TΓ-1W~1;then(75)V˙1=V˙z1+W~1Γ-1W^˙1≤z1z2+α1+h1Z1=z1z2-c1z1-W^1S1Z1+W1∗S1Z1+ε1+W~1Γ-1W^˙1=z1z2-c1z1-W~1S1Z1+ε1+W~1Γ-1W^˙1.
The adaptive law of neural network can be designed as(76)W^˙1=W~1=Γ1S1Z1z1-σ1W^1,where σ1>0. Let c1=c10+c11, where c10,c11>0.
Furthermore,(77)V˙1=z1z2-c10z12-c11z12+z1ε1-σ1W~1TW^1;then(78)-σ1W~1TW^1=-σ1W~1TW~1+W1∗≤-σ1W~12+σ1W~1W1∗≤-σ1W~122+σ1W1∗22because(79)-c11z12+z1ε1≤-c11z12+z1ε1≤ε124c11≤ε1∗24c11.
Finally we can get(80)V˙1<z1z2-c10∗z12-σ1W~122+σ1W1∗22+ε1∗24c11.Step 2.
Letz2=x2-α1; derivation of z2 can be calculated as(81)z˙2=f2x-2+g2x-2u+Δ-α˙1=-a1x2-a2x23+bu+Δ-α˙1.
BecauseVz2=1/2bz22, then(82)V˙z2=1bz2z˙2=1bz2-a1x2-a2x23+bu+Δ-α˙1=z2u+1b-a1x2-a2x23-α˙1+Δbz2≤z2u+1b-a1x2-a2x23-α˙1+ρ2z22b+p22,where Δ≤p·ρx, p is unknown parameter, ρx is known nonlinear function, and then(83)u∗=-z1-c2z2-1b-a1x2-a2x23-α˙1+ρ2z22b.
Let(84)h2Z2=1b-a1x2-a2x23-α˙1+ρ2z22b.
Equation (83) can be rewritten as(85)u∗=-z1-c2z2-h2Z2.
In the same way we use RBF NN estimateh2Z2:(86)h2Z2=W2∗TS2Z2+ε2.
The actual use of the NN for the system and controller can be expressed as(87)h2Z2=W^2TS2Z2,u=z1-c2z2-W^2TS2Z2.
Select Lyapunov function as(88)V2=V1+Vz2+12W~2TΓ-1W~2.
The derivation ofV2 can be calculated as(89)V˙2=V˙1+V˙z2+W~1Γ-1W^˙1≤z1z2-c10∗z12-σ1W~122+σ1W1∗22+ε1∗24c11+z2-z1-c2z2-W^2S2Z2+W2∗S2Z2+ε2+p22+W~1Γ-1W^˙1=-∑i=12ci0∗zi2-∑i=12σiW~i22+∑i=12σiWi∗22+∑i=12εi∗24c11+p22.
Therefore, all signals in the close loop of course tracking system are stable, and the tracking errors can be made arbitrarily small by selecting appropriate controller parameters. So the final control law can be designed as(90)u=z1-c2z2-W^2TS2Z2.


## 3.1. Problem Formulation

This section introduces a simplified dynamic model of an underactuated surface vehicle with only one control inputδ for heading control. A surface ship usually has three degrees of freedom for path following control in horizontal plane. Assuming that the vessel has three planes of symmetry, for most underactuated vessels have port/starboard symmetry, it can be neglected to simplify the vessel model for controller design. The detailed model which considers the environment disturbances can be set as follows:(66)y˙=Usin⁡ψ,ψ˙=r,r˙=-1Tr-αTr3+KTδ+Δ,y1=y,y2=ψ,where y denotes transverse displacement in the earth inertial coordinates; U=u2+v2 is resultant velocity of ship; ψ is course angle; r is yawing angular velocity; K,T represent performance index for ship steering; α is coefficient of nonlinear term; δ is control rudder angle; y1,y2 represent system output.The control objective is to design the controllerδ to make the control output y, ψ achieve the setting value (yd,ψd). Because the dimension of the system control input is less than the degree of freedom of the system, it is an underactuated system.


## 3.2. Dynamic Controller Design

Selection of coordinate transformation is as follows:(67)we=ψ+arcsin⁡ky1+ky2.The original system can be transformed into a single input single output system:(68)x˙1=ky˙1+ky2+x2,x˙2=-a1x2-a2x23+bu+Δ,where a1=1/T,  a2=α/T,  b=K/T,  x1=we,  x2=r,  u=δ, and the output of whole system is x1.For system model (67) and (68), the controller design is carried out by using backstepping method.Step 1.
Letz1=x1,  xd1=0; then(69)z˙1=ky˙1+ky2+x2.
For the subsystemz1,  α1∗≜x2 is chosen as virtual control input. Select the Lyapunov function Vz1=(1/2)z12, and there is(70)V˙z1=z1z˙1=ky˙1+ky2+x2z1.
Letz2=x2-α1; then x2=z2+α1,(71)V˙z1=ky˙1+ky2+z2+α1z1.
Select the following virtual control law:(72)α1∗=-c1z1-ky˙1+ky2.
V
˙
z
1
=
z
1
z
2
-
c
1
z
1
2, because ky˙/1+(ky)2 is unknown function, h1Z1=ky˙/1+(ky)2, and we will adopt RBF NN to estimate h1Z1 and get h1Z1=W1∗TS1Z1+ε1. But the actual use of the NN for the system is h1Z1=W^1TS1Z1. Actual virtual control input is α1=-c1z1-W^1TS1Z1; then(73)z˙1=ky˙1+ky2+z2+α1=z2-c1z1-W~1S1Z1+ε1,where W~1=W^1-W1∗.
Select Lyapunov function as(74)V1=Vz1+12W~1TΓ-1W~1;then(75)V˙1=V˙z1+W~1Γ-1W^˙1≤z1z2+α1+h1Z1=z1z2-c1z1-W^1S1Z1+W1∗S1Z1+ε1+W~1Γ-1W^˙1=z1z2-c1z1-W~1S1Z1+ε1+W~1Γ-1W^˙1.
The adaptive law of neural network can be designed as(76)W^˙1=W~1=Γ1S1Z1z1-σ1W^1,where σ1>0. Let c1=c10+c11, where c10,c11>0.
Furthermore,(77)V˙1=z1z2-c10z12-c11z12+z1ε1-σ1W~1TW^1;then(78)-σ1W~1TW^1=-σ1W~1TW~1+W1∗≤-σ1W~12+σ1W~1W1∗≤-σ1W~122+σ1W1∗22because(79)-c11z12+z1ε1≤-c11z12+z1ε1≤ε124c11≤ε1∗24c11.
Finally we can get(80)V˙1<z1z2-c10∗z12-σ1W~122+σ1W1∗22+ε1∗24c11.Step 2.
Letz2=x2-α1; derivation of z2 can be calculated as(81)z˙2=f2x-2+g2x-2u+Δ-α˙1=-a1x2-a2x23+bu+Δ-α˙1.
BecauseVz2=1/2bz22, then(82)V˙z2=1bz2z˙2=1bz2-a1x2-a2x23+bu+Δ-α˙1=z2u+1b-a1x2-a2x23-α˙1+Δbz2≤z2u+1b-a1x2-a2x23-α˙1+ρ2z22b+p22,where Δ≤p·ρx, p is unknown parameter, ρx is known nonlinear function, and then(83)u∗=-z1-c2z2-1b-a1x2-a2x23-α˙1+ρ2z22b.
Let(84)h2Z2=1b-a1x2-a2x23-α˙1+ρ2z22b.
Equation (83) can be rewritten as(85)u∗=-z1-c2z2-h2Z2.
In the same way we use RBF NN estimateh2Z2:(86)h2Z2=W2∗TS2Z2+ε2.
The actual use of the NN for the system and controller can be expressed as(87)h2Z2=W^2TS2Z2,u=z1-c2z2-W^2TS2Z2.
Select Lyapunov function as(88)V2=V1+Vz2+12W~2TΓ-1W~2.
The derivation ofV2 can be calculated as(89)V˙2=V˙1+V˙z2+W~1Γ-1W^˙1≤z1z2-c10∗z12-σ1W~122+σ1W1∗22+ε1∗24c11+z2-z1-c2z2-W^2S2Z2+W2∗S2Z2+ε2+p22+W~1Γ-1W^˙1=-∑i=12ci0∗zi2-∑i=12σiW~i22+∑i=12σiWi∗22+∑i=12εi∗24c11+p22.
Therefore, all signals in the close loop of course tracking system are stable, and the tracking errors can be made arbitrarily small by selecting appropriate controller parameters. So the final control law can be designed as(90)u=z1-c2z2-W^2TS2Z2.


## 4. Numerical Simulations and Analysis

The simulation experiment can be operated based on an experimental ship. The nonlinear mathematical model for the ship has been presented in [22], which captures the fundamental characteristics of dynamics and offers good maneuverability in the open-loop test. To illustrate the effectiveness of the theoretical results, the proposed control scheme is implemented and simulated with the above nonlinear model with tracking task.The characteristic parameters of the ship used in the simulation are given asK=0.478, T=216, and α=30. Neural network contains 25 neurons; that is, l1=25; the center vector μl(l=1,2,…,l1) is uniformly distributed in the width -2,2×-2,2×-2,2. Neural network W^2TS2(Z2) contains 135 neurons; that is, l2=125; the center vector μl(l=1,2,…,l2) is uniformly distributed in the width -4,4×-4,4×-4,4×-4,4×-4,4×-4,4×-4,0×-6,6. The controller design parameters are given as follows which satisfy the condition mentioned in design procedure: k=0.1394, c1=4, c2=120, Γ1=diag⁡3, Γ2=diag⁡4, and σ1=4, σ2=2. The initial linear and angular velocity of ship used in the simulation are given as u,v,rT=0.1,0,0T,  x,y,ψT=[10,30,-π/4]T is the initial position and orientation vector of ship, and the desired velocity of ship is given as ud=1 (m/s). We choose the reference trajectory as 10cos⁡ωt.In order to further verify the validity of the proposed control method, the algorithm of this paper is compared with the simulation results in [12]. So the robustness of trajectory tracking controller against the disturbance and model uncertainties can be evaluated. All the simulation results are depicted in Figures 1–4. Figure 1 shows the trajectory tracking of ship with the given path, and the ship can track and converge to the reference path with more accuracy in [12]. Figure 2 plots the position tracking errors; the along-track and cross-track errors asymptotically converge to zero faster. Figure 3 gives the control inputs response. Surge, sway, yaw velocities, and orientation of ship during the trajectory tracking control process are plotted in Figure 4, which gives a clear insight into the model response involved in nonlinear dynamics.Figure 1
Ship tracking performance of proposed control method.Figure 2
Tracking errors of surge and sway.Figure 3
Control force and torque of ship.Figure 4
State changing curves of ship.


## 5. Conclusions

In this paper, we proposed a solution to the course control of underactuated surface vessel. Firstly, the direct adaptive neural network control and its application are introduced. Then the backstepping controller with robust neural network is designed to deal with the uncertain and underactuated characteristics for the ship. Neural networks are adopted to determine the parameters of the unknown part of the ideal virtual control and the ideal control; even the weights of neural network are updated by using adaptive technique. Finally uniform stability for the convergence of tracking errors has been proven through Lyapunov stability theory. The simulation results illustrate the performance of the proposed course tracking controller with good precision.



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*Source: 3013280-2016-05-18.xml*