# Neural Prescribed Performance Control for Uncertain Marine Surface Vessels without Accurate Initial Errors

Authors: Wenjie Si; Xunde Dong
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper deals with the problems concerned with the trajectory tracking control with prescribed performance for marine surface vessels without velocity measurements in uncertain dynamical environments, in the presence of parametric uncertainties, unknown disturbances, and unknown dead-zone. First, only the ship position and heading measurements are available and a high-gain observer is used to estimate the unmeasurable velocities. Second, by utilizing the prescribed performance control, the prescribed tracking control performance can be ensured, while the requirement for the initial error is removed via the preprocessing. At last, based on neural network approximation in combination with backstepping and Lyapunov synthesis, a robust adaptive neural control scheme is developed to handle the uncertainties and input dead-zone characteristics. Under the designed adaptive controller for marine surface vessels, all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB), and the prescribed transient and steady tracking control performance is guaranteed. Simulation studies are performed to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

As the demand for offshore exploration and operation, ocean surface vessels have been widely used in the marine industry. With the development of marine industrial technology, control design of marine surface vessels has become a hot topic of research [1, 2]. Control of ocean surface ships is a difficult question; the working environment is often complex (ocean currents and sea breeze); external unpredictable disturbances may degrade control system performance and even damage the stability. Therefore, the tracking control of the marine vessels has attracted much attention.Various tracking control approaches have been presented for marine surface vessels; the sliding mode tracking control schemes were developed in [3, 4] based on explicit models. For the systems with unknown system dynamics, both fuzzy logic systems (FLSs) and neural networks (NNs) have been proved to be useful in the control design, where their universal approximation properties are employed to model unknown nonlinear functions [5–12]. Recently, many approximation-based adaptive control schemes have also been proposed to handle the control problem for uncertain ocean surface ships [13–15]. The output feedback control and the full-state feedback control were designed based on approximation-based adaptive backstepping of the ship dynamics in [16]. A neural learning control method was presented in [17] to solve the problem of tracking control of unknown ship systems. In [18], the unknown ship dynamics was learned by using deterministic learning theory and then learning problem from neural output feedback control of uncertain ship dynamics was studied. The prescribed performance control was introduced in [19] to deal with the neural learning control of ocean surface vessels via deterministic learning. However, in the working environments of marine surface vessels, the initial errors cannot be obtained in advance, and the control method given by [19] is invalid in most working environments. To solve the aforementioned problem, we propose a new performance function for uncertain marine surface vessels with unknown initial errors.Dead-zone is one of important input nonlinearities which appears in a wide range of practical engineering. The existence of dead-zone nonlinearities degrades the performance of control system and even may lead to system instability. In [20, 21], the robust adaptive control methods were used for nonlinear systems with parametric uncertainties subject to the input deal-zone, and the systems must satisfy linear parameterized condition. Recently, in order to deal with unknown nonlinear systems with input dead-zone when the knowledge of system functions is unavailable, many adaptive controllers have been proposed by utilizing universal approximation capability of neural networks or some fuzzy logic systems [21, 22]. A robust adaptive NN control design method was proposed in [23] for a kind of strict-feedback nonlinear systems with uncertainties and input dead-zone. An adaptive fuzzy output feedback control was studied in [24] for switched nonlinear systems with uncertainties. In [25], the problem of the adaptive fuzzy backstepping output feedback tracking control was investigated for multi-input and multioutput (MIMO) stochastic nonlinear systems. The problem of adaptive decentralized NN control was investigated in [26] for large-scale stochastic nonlinear time-delay systems with input dead-zone.In the control design, the tracking error is only required to converge to a small residual set, while the transient and steady-state tracking performance is not considered. The practical engineering often requires certain prespecified performance. More recently, the prescribed performance control (PPC) has been proposed in [27]. Furthermore, in [28, 29], the PPC was used for the position tracking control of robot. Combining PPC with dynamic surface control (DSC), a fuzzy control scheme was studied in [30] to ensure the performance of tracking control. When states were unmeasurable, an output feedback control was presented in [31] for large-scale nonlinear time-delay systems. The prescribed performance control technology was extended to MIMO systems [32, 33]. However, to our best knowledge, by using prescribed performance control, no tracking control methods exit for uncertain ocean surface vessels without the need for the initial error conditions.Motivated by the aforementioned discussion, to guarantee the predefined performance for ocean surface vessels without velocity measurements in the presence of unknown input dead-zone, we will design an adaptive neural output feedback control scheme. RBF NNs are used to approximate the unknown nonlinearities. The prescribed performance function is designed to ensure the performance of the prescribed tracking control without any consideration for accurate initial errors. Then, based on the backstepping and Lyapunov theory, we propose an adaptive neural tracking control method to ensure the boundedness of the closed-loop system.Compared with previous works, our paper has the following advantages.(1) The unmeasurable velocities of the ocean surface ship are estimated by employing a high-gain observer. To prevent peaking of the high-gain observer at the initial phase, a new method is used different from the saturation functions employed in [16, 34]. (2) A performance function is given to ensure the tracking control performance, and the requirement for the exact initial error is removed. (3) With the proposed controller, the tracking control for uncertain surface vessels is achieved with only position sensors and the control performance of the system is guaranteed by prescribed performance control.The rest of the paper is organized as follows. Section2 presents the preliminaries and problem formulation. An adaptive neural tracking control scheme for uncertain ocean surface vessels with prescribed performance control is given in Section 3. The simulation studies are presented in Section 4 to demonstrate the effectiveness of the proposed method. Section 5 concludes this paper.


## 2. Problem Formulation and Preliminaries

### 2.1. Ship Dynamics

Consider multiple-input-multiple-output (MIMO) systems for a three degrees of freedom surface vessel subject to unknown model uncertainties and input dead-zone. The dynamics of the surface vessel is described by [1, 35](1)η˙=Jηντ=Mν˙+Cνν+Dνν+gη+Δη,ν,where η=[x,y,ψ]T∈R3, (x,y) is the surface vessel position, and ψ is the surface vessel heading; J(η) is the rotation matrix; ν=[u,v,r]T∈R3, u,v,r are velocities of the surge, the sway, and the yaw, respectively; τ∈R3 is the vector of control input, M denotes the inertia matrix of the ship, C(ν) is the total Coriolis and centripetal acceleration matrix, D(ν) is the damping matrix, g(η)=[g1(η),g2(η),g3(η)]T is the vector of buoyancy/gravitational forces, and Δ(η,ν)=[Δ1(η,ν),Δ2(η,ν),Δ3(η,ν]T is used to model the uncertainties with Δi(η,ν),(i=1,2,3) being unknown.Precisely,M=MT>0 is given by [35]; J(η), C(ν), and D(ν) are given as follows:(2)Jη=cos⁡ψ-sin⁡ψ0sin⁡ψcos⁡ψ0001,Cν=00c13ν00c23ν-c13ν-c23ν1,Dν=d11ν000d22νd23ν0d32νd33ν,where c13(ν), c23(ν), d11(ν), d22(ν), d23(ν), d32(ν), and d33(ν) are unknown.We have the description of the ship dynamics (1) subject to the unknown dead-zone nonlinearity as follows:(3)η˙=Jηνν˙=M-1-Cνν-Dνν-gη+τ-Δη,ντ=Duτ,where η∈R3 is the system output, τ∈R3 is the system input and the dead-zone output, and the actuator dead-zone characteristic is described as D(uτ) with uτ being the input of the unknown dead-zone.The main goal of this control scheme is to present an adaptive NN tracking controller for the system described by (3) to ensure that the system output η∈R3 can track the reference signal yd(t)∈R3 and all signals in the closed-loop system remain bounded.Assumption 1.
The system outputη(t)∈R3 and its first-order derivative η˙(t)∈R3 are continuous and bounded, such that η<Y0 and η˙<Y1, where Y0 and Y1 are positive constants.Assumption 2.
The reference signalyd(t)∈R3 and its nth order ydn(t)∈R3 are continuous and bounded.


### 2.2. Dead-Zone Characteristic

The actuator dead-zone nonlinearity can be described as follows [36]:(4)τ=Duτ=hruτ-bruτ≥br0bl<uτ<brhluτ-bluτ≤bl,where hr(·) and hl(·) are unknown smooth functions.Assumption 3.
The parameters of the dead-zonebr and bl are unknown constants and satisfy br>0 and bl<0.Assumption 4.
For unknown functionshr(·) and hl(·), there exist unknown constants H0 and H1, such that(5)0<Hr0<h˙ruτ-br<Hr1,r∈br,+∞0<Hl0<h˙luτ-bl<Hl1,r∈-∞,bl.LetDmin=min⁡H0 and Dmax=max⁡H1. The dead-zone nonlinearity can be rewritten as(6)τ=Duτ=Ξuτuτ+duτ,where(7)Ξ=Ξr′uτ≥brΞl′uτ<br,duτ=-Ξr′bruτ≥br-Ξr′uτbr<uτ<bl-Ξl′bluτ≤bl,where Ξr′=h˙r(·) and Ξl′=h˙l(·).It can be obtained thatDmin≤Ξ≤Dmax, d(u)≤d∗≔max⁡-Dmaxdl,Dmaxdr.


### 2.3. Prescribed Performance

This section introduces the concept of the prescribed performance. Then, a performance function is given and it will be chosen to be used in the control design.Definition 5 (see [27]).
A smooth functionρ(t):R+→R+ is called a performance function, if(i)
ρ(t) is a strictly positive decreasing function;(ii)
limt→+∞⁡ρ(t)=ρ∞>0, where ρ∞ is a positive constant.According to Definition5, we choose the performance function as follows:(8)μt=μ0-μ∞exp⁡-κt+μ∞,where μ0, μ∞ and κ are design parameters. It is obvious that μ(t) satisfies that limt→∞⁡μ(t)=μ∞. According to the description of the performance function μ(t), the tracking error e1=y-yd has the following performance bound (as shown in Figure 1):(9)-Mμt<e1t<μt,ife1t≥0-μt<e1t<Mμt,ife1t<0,where 0≤M≤1, t≥0.Figure 1
Prescribed performance of the errore1(t).To represent (9) by an unconstrained form, the following state transformation is employed [27]:(10)e1t=μtRζ1,where(11)Rζ1=Meζ1-e-ζ1Meζ1+e-ζ1,ife1t≥0Rζ1=Meζ1-e-ζ1eζ1+Me-ζ1,ife1t<0and ζ1 is called the transformed error. According to (11), the derivation of ζ1 is as follows:(12)
ζ
˙
1
=
∂
R
-
1
∂
ω
ω
˙
=
1
2
1
+
M
ω
+
M
1
-
ω
ω
˙
,
if
e
1
t
≥
0
1
2
1
+
M
ω
+
1
M
-
ω
ω
˙
,
if
e
1
t
<
0
.Based one1(t)=x1(t)-yd, one has(13)ω˙=1μte˙1-μ˙tμte1t,ζ˙1=1μ∂R-1∂ωe˙1-μ˙tμte1t=p1e˙1-μ˙tμte1twith(14)p1=1μ∂R-1∂ω.


### 2.4. RBF Neural Networks

It has been shown that neural networks are good at modeling unknown nonlinear functions in control design [37]. In the study, RBF NN will be employed to model unknown continuous function f(Z):Rq→R over a compact set ΩZ⊂Rq for a given arbitrary accuracy ϵ∗>0 as follows:(15)fZ=W∗TSZ+ϵZ,where Z∈ΩZ is the input vector with q being the input dimension of the neural networks. W=[w1,w2,…,wl]T∈Rl is the neural weight vector with l>1 being the node number of the neural network, and W∗ denotes the ideal constant weight vector. ϵ(Z) is the approximation error, ϵ(Z)≤ϵ∗. S(Z)=[s1(Z),s2(Z),…,sl(Z)]T is the vector of basis function; si(Z) commonly is Gaussian function which has the following form:(16)siZ=exp⁡-Z-ξiTZ-ξiη2,i=1,…,l,where ξi=[ξi1,ξi2,…,ξiq]T is the center of the receptive field and η is the bandwidth of Gaussian function [37].The ideal constant weight vector is defined as(17)W∗≔arg⁡minW^∈Rl⁡supZ∈ΩZ⁡fZ-W^TSZ,where W^ is the estimation of W∗.Lemma 6 (see [38]).
Consider the Gaussian RBF neural networks (15). Let ϱ≔1/2mini≠j⁡ξi-ξj. q is the dimension of the neural network input Z and η is the width of the Gaussian function (16); then the following inequality holds:(18)SZ≤∑k=0∞3qk+2q-1e-2ϱ2k2/η2≔s∗,where s∗ is an upper bound of S(Z) and is limited and independent of neural input Z and the neural weights dimension l.


### 2.5. High-Gain Observer

The system outputη=[x,y,ψ]T∈R3 is available for measurement, and ν=[u,v,r]T∈R3 is unavailable. To solve the unmeasured states problem, we need a high-gain observer used in [39].Lemma 7 (see [16]).
Consider the following linear system:(19)ελπ˙1=π2,ελπ˙2=-λπ2-π1+η,where ελ is a small constant, π1 and π2 are system states, and λ is chosen to ensure that the linear system is stable. According to Assumption (1), the following properties hold:(i)
(20)π2ελ-η˙=-ελϕ¨,where ϕ=π2+λπ1.(ii)
There exist two constantst∗>0 and h0>0, which rely on the parameters Y0, Y1, ελ, and λ, such that ϕ¨≤h0, ∀t>t∗.Considering thatη˙=J(η)ν in system (1) and the property JT(η)J(η)=I, we can use ν^=JT(η)π2/ελ to estimate the state variable ν.Lemma 8 (see [39]).
For the RBF neural networks, ifZ=Z^-εϕ, ε>0, ϕ is a bounded vector; then(21)SZ^=SZ+εSt,where St is the bounded function, St<st∗; st∗ is a positive constant.


## 2.1. Ship Dynamics

Consider multiple-input-multiple-output (MIMO) systems for a three degrees of freedom surface vessel subject to unknown model uncertainties and input dead-zone. The dynamics of the surface vessel is described by [1, 35](1)η˙=Jηντ=Mν˙+Cνν+Dνν+gη+Δη,ν,where η=[x,y,ψ]T∈R3, (x,y) is the surface vessel position, and ψ is the surface vessel heading; J(η) is the rotation matrix; ν=[u,v,r]T∈R3, u,v,r are velocities of the surge, the sway, and the yaw, respectively; τ∈R3 is the vector of control input, M denotes the inertia matrix of the ship, C(ν) is the total Coriolis and centripetal acceleration matrix, D(ν) is the damping matrix, g(η)=[g1(η),g2(η),g3(η)]T is the vector of buoyancy/gravitational forces, and Δ(η,ν)=[Δ1(η,ν),Δ2(η,ν),Δ3(η,ν]T is used to model the uncertainties with Δi(η,ν),(i=1,2,3) being unknown.Precisely,M=MT>0 is given by [35]; J(η), C(ν), and D(ν) are given as follows:(2)Jη=cos⁡ψ-sin⁡ψ0sin⁡ψcos⁡ψ0001,Cν=00c13ν00c23ν-c13ν-c23ν1,Dν=d11ν000d22νd23ν0d32νd33ν,where c13(ν), c23(ν), d11(ν), d22(ν), d23(ν), d32(ν), and d33(ν) are unknown.We have the description of the ship dynamics (1) subject to the unknown dead-zone nonlinearity as follows:(3)η˙=Jηνν˙=M-1-Cνν-Dνν-gη+τ-Δη,ντ=Duτ,where η∈R3 is the system output, τ∈R3 is the system input and the dead-zone output, and the actuator dead-zone characteristic is described as D(uτ) with uτ being the input of the unknown dead-zone.The main goal of this control scheme is to present an adaptive NN tracking controller for the system described by (3) to ensure that the system output η∈R3 can track the reference signal yd(t)∈R3 and all signals in the closed-loop system remain bounded.Assumption 1.
The system outputη(t)∈R3 and its first-order derivative η˙(t)∈R3 are continuous and bounded, such that η<Y0 and η˙<Y1, where Y0 and Y1 are positive constants.Assumption 2.
The reference signalyd(t)∈R3 and its nth order ydn(t)∈R3 are continuous and bounded.


## 2.2. Dead-Zone Characteristic

The actuator dead-zone nonlinearity can be described as follows [36]:(4)τ=Duτ=hruτ-bruτ≥br0bl<uτ<brhluτ-bluτ≤bl,where hr(·) and hl(·) are unknown smooth functions.Assumption 3.
The parameters of the dead-zonebr and bl are unknown constants and satisfy br>0 and bl<0.Assumption 4.
For unknown functionshr(·) and hl(·), there exist unknown constants H0 and H1, such that(5)0<Hr0<h˙ruτ-br<Hr1,r∈br,+∞0<Hl0<h˙luτ-bl<Hl1,r∈-∞,bl.LetDmin=min⁡H0 and Dmax=max⁡H1. The dead-zone nonlinearity can be rewritten as(6)τ=Duτ=Ξuτuτ+duτ,where(7)Ξ=Ξr′uτ≥brΞl′uτ<br,duτ=-Ξr′bruτ≥br-Ξr′uτbr<uτ<bl-Ξl′bluτ≤bl,where Ξr′=h˙r(·) and Ξl′=h˙l(·).It can be obtained thatDmin≤Ξ≤Dmax, d(u)≤d∗≔max⁡-Dmaxdl,Dmaxdr.


## 2.3. Prescribed Performance

This section introduces the concept of the prescribed performance. Then, a performance function is given and it will be chosen to be used in the control design.Definition 5 (see [27]).
A smooth functionρ(t):R+→R+ is called a performance function, if(i)
ρ(t) is a strictly positive decreasing function;(ii)
limt→+∞⁡ρ(t)=ρ∞>0, where ρ∞ is a positive constant.According to Definition5, we choose the performance function as follows:(8)μt=μ0-μ∞exp⁡-κt+μ∞,where μ0, μ∞ and κ are design parameters. It is obvious that μ(t) satisfies that limt→∞⁡μ(t)=μ∞. According to the description of the performance function μ(t), the tracking error e1=y-yd has the following performance bound (as shown in Figure 1):(9)-Mμt<e1t<μt,ife1t≥0-μt<e1t<Mμt,ife1t<0,where 0≤M≤1, t≥0.Figure 1
Prescribed performance of the errore1(t).To represent (9) by an unconstrained form, the following state transformation is employed [27]:(10)e1t=μtRζ1,where(11)Rζ1=Meζ1-e-ζ1Meζ1+e-ζ1,ife1t≥0Rζ1=Meζ1-e-ζ1eζ1+Me-ζ1,ife1t<0and ζ1 is called the transformed error. According to (11), the derivation of ζ1 is as follows:(12)
ζ
˙
1
=
∂
R
-
1
∂
ω
ω
˙
=
1
2
1
+
M
ω
+
M
1
-
ω
ω
˙
,
if
e
1
t
≥
0
1
2
1
+
M
ω
+
1
M
-
ω
ω
˙
,
if
e
1
t
<
0
.Based one1(t)=x1(t)-yd, one has(13)ω˙=1μte˙1-μ˙tμte1t,ζ˙1=1μ∂R-1∂ωe˙1-μ˙tμte1t=p1e˙1-μ˙tμte1twith(14)p1=1μ∂R-1∂ω.


## 2.4. RBF Neural Networks

It has been shown that neural networks are good at modeling unknown nonlinear functions in control design [37]. In the study, RBF NN will be employed to model unknown continuous function f(Z):Rq→R over a compact set ΩZ⊂Rq for a given arbitrary accuracy ϵ∗>0 as follows:(15)fZ=W∗TSZ+ϵZ,where Z∈ΩZ is the input vector with q being the input dimension of the neural networks. W=[w1,w2,…,wl]T∈Rl is the neural weight vector with l>1 being the node number of the neural network, and W∗ denotes the ideal constant weight vector. ϵ(Z) is the approximation error, ϵ(Z)≤ϵ∗. S(Z)=[s1(Z),s2(Z),…,sl(Z)]T is the vector of basis function; si(Z) commonly is Gaussian function which has the following form:(16)siZ=exp⁡-Z-ξiTZ-ξiη2,i=1,…,l,where ξi=[ξi1,ξi2,…,ξiq]T is the center of the receptive field and η is the bandwidth of Gaussian function [37].The ideal constant weight vector is defined as(17)W∗≔arg⁡minW^∈Rl⁡supZ∈ΩZ⁡fZ-W^TSZ,where W^ is the estimation of W∗.Lemma 6 (see [38]).
Consider the Gaussian RBF neural networks (15). Let ϱ≔1/2mini≠j⁡ξi-ξj. q is the dimension of the neural network input Z and η is the width of the Gaussian function (16); then the following inequality holds:(18)SZ≤∑k=0∞3qk+2q-1e-2ϱ2k2/η2≔s∗,where s∗ is an upper bound of S(Z) and is limited and independent of neural input Z and the neural weights dimension l.


## 2.5. High-Gain Observer

The system outputη=[x,y,ψ]T∈R3 is available for measurement, and ν=[u,v,r]T∈R3 is unavailable. To solve the unmeasured states problem, we need a high-gain observer used in [39].Lemma 7 (see [16]).
Consider the following linear system:(19)ελπ˙1=π2,ελπ˙2=-λπ2-π1+η,where ελ is a small constant, π1 and π2 are system states, and λ is chosen to ensure that the linear system is stable. According to Assumption (1), the following properties hold:(i)
(20)π2ελ-η˙=-ελϕ¨,where ϕ=π2+λπ1.(ii)
There exist two constantst∗>0 and h0>0, which rely on the parameters Y0, Y1, ελ, and λ, such that ϕ¨≤h0, ∀t>t∗.Considering thatη˙=J(η)ν in system (1) and the property JT(η)J(η)=I, we can use ν^=JT(η)π2/ελ to estimate the state variable ν.Lemma 8 (see [39]).
For the RBF neural networks, ifZ=Z^-εϕ, ε>0, ϕ is a bounded vector; then(21)SZ^=SZ+εSt,where St is the bounded function, St<st∗; st∗ is a positive constant.


## 3. Adaptive Neural Control Design

The backstepping design is based on the following coordinate changes:(22)z1=ζ1,z2=ν-α1,where α1 is the virtual control signal, which is defined later.Step 1.
Consideringe1(t)=η(t)-yd and e˙1=J(η)ν-y˙d, one has(23)z˙1t=p1Jην-y˙d-μ˙μe1.Noting the following the error variable as z2=ν-α1, we have(24)z˙1t=p1Jηz2+α1-y˙d-μ˙μe1.From the property JT(η)J(η)=I, α1 is defined as(25)α1=JTη-K1p1-1z1+y˙d+μ˙μe1,where K1T=K1≥0 is a design parameter.
We choose a Lyapunov function candidateV1=1/2z1Tz1, and its derivative along (24) yields(26)V˙1=z1Tp1Jηz2-K1p1-1z1=-z1TK1z1+z1Tp1Jηz2,where the last term z1Tp1J(η)z2 will be canceled in the next step.Step 2.
Fromz2=ν-α, we have(27)z˙2=M-1-Cνν-Dνν-gη+τ-Δη,ν-α˙.Noting MT=M≥0 is the design constant obtained by [35], we consider the following Lyapunov function candidate:(28)V2=12z2TMz2.The derivative of V2 along (27) is obtained by(29)V˙2=z2T-Cνν-Dνν-gη+τ-Δη,ν-Mα˙=z2TΞuτuτ+duτ-Cνν-Dνν-gη-Δη,ν-Mα˙.Denote(30)FZ=-Cνν+Dνν+gη+Δη,ν+Mα˙,where F(Z)=[f1(Z),f2(Z),f3(Z)]T∈R3, Z=[η,ν,α,α˙]∈R8. A RBF NN W∗TS is employed to approximate the unknown function F(Z) as follows:(31)FZ=W∗TSZ+ϵZ,ϵZ≤ϵ∗,where W∗TS(Z1)=[W1∗TS1(Z),W2∗TS2(Z),W3∗TS3(Z)]T∈R3 and ϵ(Z)=[ϵ1(Z),ϵ2(Z),ϵ3(Z)]T∈R3 is the approximation error.
We choose the following control as(32)uτ=1C1-K2z^2-W^TSZ^-Jηp1z1,where Z^=[η,ν^,α,α˙]∈R8. W^TS(Z^)=[W^1TS1(Z^),W^2TS2(Z^),W^3TS3(Z^)]T∈R3; C1=Dmax is a design parameter.
From Lemma7 and the property JT(η)J(η)=I, one has(33)z~2=z^2-z2=JTηπ2ελ-α-ν-α=JTηπ2ελ-ν=JTηπ2ελ-η˙=-JTηελϕ¨=β,where β≤ε¯λh0. ε¯λ=JT(η)ελ>0 is constant.
According to Lemmas7 and 8, we have(34)SiZ^=SiZ+εSitW^iTSiZ^=Wi∗TSiZ+Wi∗TεSit+W~iTSiZ^.
Consider the following Lyapunov function candidate(35)V=V1+V2+12∑i=13W~iTΓi-1W~i=12z1Tz1+12z2TMz2+12∑i=13W~iTΓi-1W~i.
The time derivative ofV is given by(36)V˙=-z1TK1z1+z1Tp1Jηz2+z2TΞuτuτ+duτ-Cνν-Dνν-gη-Δη,ν-Mα˙+∑i=13W~iTΓi-1W~˙i.
Define the adaptive law as follows:(37)W^˙=W~˙=ΓSZ^z^2-σW^,where Γ=ΓT>0 and σ are design parameters.
Substituting (32) and (37) into (36) yields(38)V˙≤-z1TK1z1+z1Tp1Jηz2+z2T-K2z^2-W^TSZ^-p1Jηz1+W∗TSZ+ϵ+z2Td∗+W~TSZ^z^2-σW^≤-z1TK1z1-z2TK2z2-z2TK2z~2-∑i=13z2,iWi∗TεSit-∑i=13z2,iW~iTSiZ^+z2Tϵ+z2Td∗+W~TSZ^z^2-σW^≤-z1TK1z1-z2TK2z2-z2TK2z~2-∑i=13z2,iWi∗TεSit+∑i=13z~2,iW~iTSiZ^+z2Tϵ+z2Td∗-σW~TW^.
ConsideringW~=W^-W∗, we have(39)-σW~TW^=-σW~TW∗+W~≤-σW~2+σW~W∗≤12σW∗2-W~2.By completing the squares, we have(40)z2Tϵ≤12z2Tz2+12ϵ∗2z2Td∗≤12z2Tz2+12d∗2-∑i=13z2,iWi∗TεSit≤12z2Tz2+∑i=13Wi∗ε2Sit22∑i=13z~2,iW~iTSiZ^≤∑i=13σiW~i24+∑i=132SiZ^2σi12z~2Tz~2-z2TK2z~2≤12z2Tz2+12K2z~2TK2z~2.Substituting (39)-(40) into (38) yields(41)V˙≤-z1TK1z1-z2TK2-2Iz2-14σW~2+12ϵ∗2+12d∗2+12∑i=13εi2Sit2+σiWi∗2+λmaxK2TK2+diag⁡2s∗2σi12βTβ,where λmax(A) and λmin(A) denote the minimum and maximum eigenvalues of matrix A.
Considering1/2βTβ≤1/2ε¯λ2h02, we have(42)V˙≤-z1TK1z1-z2TK2-2Iz2-14σW~2+12ϵ∗2+12d∗2+12∑i=13εi2Sit2+σiWi∗2+λmaxK2TK2+diag⁡2s∗2σi12ε¯λ2h02,where diag⁡[∗] denotes a block-diagonal matrix.
Then, we have the following inequality:(43)V˙≤-ρ1V+δ1,where(44)ρ1=min⁡2λminK1,2λminK2-2IλmaxM,mini=1,2,…,n⁡σi2λmaxΓi-1δ1=12ϵ∗2+12d∗2+12∑i=13εi2Sit2+σiWi∗2+λmaxK2TK2+diag⁡2s∗2σi12ε¯λ2h02.To ensure ρ1>0, the parameters K1 and K2 are chosen to satisfy that K1>0 and K2-2I>0.Theorem 9.
Consider the ship system composed of system (1), observer (19), the neural controller (32), and the adaptive law (37). Under Assumptions 1 and 2, if the initial conditions are bounded, then all the signals in the closed-loop system are semiglobally uniformly ultimately bounded. Moreover, for all t≥0 the tracking error e1=η-yd is confined within the prescribed performance bounds.Proof.
Multiplying (43) by eρ1t yields(45)ddtVeρ1t≤δ1eρ1t.Integrating (45), one has(46)V≤V0-δ1ρ1e-ρ1t+δ1ρ1≤V0+δ1ρ1.Given Q=2(V(0)+δ1/ρ1), the closed-loop error signals z1, z2, and W~ will converge asymptotically to the compact sets Ωz1, Ωz2, and ΩW, defined by(47)Ωz1≔z1∈R3∣z1≤Q.Ωz2≔z2∈R3∣z2≤QλminM.ΩW≔W~∈Rl×3∣W~≤QλminΓ-1,where l denotes the neural network node number.
Therefore,z1, z2, and W~1,W~2,W~3 are uniformly ultimately bounded. Then W^i=W~i+Wi∗(i=1,2,3) is also bounded. z1 is the transformed error, so e1=η-yd is bounded. From Assumption 2, yd is bounded and α is bounded, and thus η and ν are bounded. The control input τ is bounded since S(Z) is bounded. Thus, all signals in the closed-loop system remain bounded, and the tracking error e1=η-yd not only converges to a small neighborhood of zero, but also achieves the prescribed performances.Remark 10.
To achieve the control design with no need of the initial errore1(0), preprocessing is needed before the operation of the controller and identifying which controller is applicable. In the paper, we have assumed that system state variable ν is unmeasurable. In practice, for most merchant vessels, only the heading and the position are measured. To estimate unmeasured ν, a high-gain observer (19) is introduced. We use the Exponential function to overcome the peaking phenomenon of the high-gain observer.


## 4. Simulation Studies

Simulation studies are presented to illustrate the effectiveness of the proposed control method in this section. The following choices of the system (1) are given: c13(ν)=-1.0948r-24.6612v, c23(ν)=25.8u, d11(ν)=1.3274u+5.8664u2+0.7225, d22(ν)=8.05r+36.2823u+0.8612, d23ν=3.450r+0.845v-0.1079, d32=-0.13r-5.0437v-0.1052, d33=0.75r-0.08v+1.9, and g1=g2=g3=0. We assume that these parameters are unknown for the control input τ. The ship inertia matrix M is a constant matrix and, equally with [35],(48)M=25.800000024.66121.094801.09482.7600.Δ(η,ν) is assumed to be the model uncertainties with Δ(η,ν)=[1;0.01u2+0.5;-0.1r3+sin⁡(v)].The dead-zone nonlinearity is given as(49)τ=Du=1.5u-0.4u≥0.40-0.5<u<0.41.3u+0.5u≤-0.5.For the state variablex, the performance function is chosen as(50)μt=0.3-0.02exp⁡-3.4t+0.02,where M=1.For the state variablesy, the performance function is chosen as(51)μt=0.25-0.01exp⁡-2.4t+0.01,where M=1.For the state variableψ, the performance function is chosen as(52)μt=0.25-0.01exp⁡-6.4t+0.01,where M=1.The objective of the control is to put forward an adaptive neural control scheme to ensure that all signals in closed-loop system remain bounded, and the system outputη tracks the ideal reference trajectory yd=[0.8sin⁡t,0.8cos⁡t,-0.8sin⁡(t)]T.During the simulation, the initial conditions areη(0)=[0.1,0.75,-0.0873]T and ν^=[0,0,0]T. W^i(0)=0, i=1,2,3. e1(0)=[x(0)-yd1(0),y(0)-yd2(0),ψ(0)-yd3(0)]T = [0.1,-0.05,-0.0873]T. The design parameters are given as follows: K1=[12,30,24]T, K2=[30,40,30]T, Γ1=diag⁡12, Γ2=diag⁡10, and Γ3=diag⁡17. σ1=0.02, σ2=0.01, and σ3=0.05. The input of the RBF neural networks is Z=[η,ν^,α,α˙]T and W^iTSi(Z) are constructed using 34 nodes with the centers μi evenly spaced on [-1.8,1.8]×[-1.8,1.8]×[-1,1]×[-1,1], the width being ϖi=1.1, i=1,2,3.Figures2–10 show the simulation results for the prescribed performance control of the unknown ship dynamics. Figures 2, 4, and 6 show that the good tracking performances of the closed-loop system are achieved, even though no initial tracking errors are available. In Figures 3, 5, and 7 provide the curves of the tracking errors with PPC. It is obvious that the initial error e1(0) does not need to be known in advance. Figure 8 illustrates the time trajectory of the state variables u, v, and r and their estimations u^,v^,r^ by using the high-gain observer. It can be seen that the peaking effect of the high-gain observer is eliminated. Figure 9 gives the norms of the adaptive laws W^1, W^2, and W^3, which demonstrates the boundedness of the adaptive weights. The control input τ is shown in Figure 10.Figure 2
System outputx and reference signal yd1 with the positive initial error e1(0)>0.Figure 3
Tracking error trajectory ofx with the positive initial error e1(0)>0.Figure 4
System outputy and reference signal yd2 with the negative initial error e2(0)<0.Figure 5
Tracking error trajectory ofy with the negative initial error e2(0)<0.Figure 6
System outputψ and reference signal yd3 with the negative initial error e3(0)<0.Figure 7
Tracking error trajectory ofψ with the negative initial error e3(0)<0.Figure 8
System variablesu,v,r and their estimated value u^,v^,r^.Figure 9
Norms ofW^1, W^2, and W^3.Figure 10
The inputuτi and output τi (i=1,2,3) of the dead-zone.


## 5. Conclusion

In the paper, we investigate the problem of the tracking control with predefined performance for marine surface vessels without velocity measurements in the presence of unknown dead-zone input. RBF neural networks are used to deal with uncertain ship dynamics, a new performance function is designed with no requirement for the exact initial error, and the unmeasurable ship velocity is estimated by using a high-gain observer. It has been shown that the proposed common controller can ensure that all signals in the closed-loop systems are semiglobally uniformly bounded, and tracking error converges to a predefined small neighborhood of zero.



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*Source: 2618323-2017-01-15.xml*