# Neural Network-Based Fault-Tolerant Control of Underactuated Surface Vessels

Authors: Bong Seok Park
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper addresses the problem of trajectory tracking of underactuated surface vessels (USVs) in the presence of thruster failure. Multilayer neural networks (MNNs) are employed to estimate the unknown model parameters and external disturbances. To design a fault-tolerant controller without a fault detection scheme, we use the Nussbaum gain technique. We introduce an additional control to resolve the difficulty arising from having fewer inputs than degrees-of-freedom. Further, an approach angle is proposed to track both a straight and curved path. Stability analysis and simulations are performed to demonstrate the effectiveness of the proposed scheme.


---

## Body

## 1. Introduction

As land resources are depleted, interest in ocean exploration has increased. There are many marine vehicles designed to explore the ocean. Among these, unmanned marine vehicles (UMVs) such as autonomous underwater vehicles (AUVs) and unmanned surface vessels are useful because they can reduce the cost and the risk of personal injury [1]. Therefore, significant research has been undertaken to perform complex tasks using UMVs.An important problem in the control of UMVs is failure occurrence on the actuators [2, 3]. If there is no immediate solution for actuator failure, the UMV must abort the mission and return to the base for repair. This results in a waste of time and money. Therefore, to enhance the reliability while performing its mission in the presence of thruster failure, there are more thrusters than the degrees-of-freedom of the UMV. For this reason, the thruster-force allocation problem is important to overcome thruster failure [4]. The allocation of thruster forces of an AUV that can accommodate both thruster faults and saturation was presented in [5, 6]. A fault diagnosis and accommodation system was proposed using weighted pseudoinverse in [7]. Weighted pseudoinverse and quantum particle swarm optimization were introduced for hybrid fault-tolerant control in [8]. The combination of backstepping technique and fuzzy control was presented for tracking control of fully actuated surface vessels in [9]. However, all these works focused on fully or overactuated systems. This implies that the cost and weight of the system are increased because there must be more devices. Moreover, if a fully actuated system is damaged, we then have an underactuated controller [10]. Therefore, it is necessary to develop a fault-tolerant controller for underactuated systems.Motivated by these observations, a neural network-based fault-tolerant control is presented for underactuated surface vessels (USVs). The proposed algorithm can be applied to AUVs because the equations of motion for surface vessels are the same as those of AUVs in the horizontal plane. For a practical perspective, we assume that the bow and stern are not symmetric and the model parameters for the hydrodynamic terms are unknown. To estimate the unknown model parameters and external disturbances, we employ multilayer neural networks (MNNs) [11]. Unlike other fault-tolerant control methods [12–14], we do not require a fault detection scheme; we design the fault-tolerant controller using the Nussbaum gain technique [15]. Further, an additional control is introduced to resolve the difficulty of controller design of underactuated systems, and an approach angle is proposed to track any trajectory including both a straight- and curved-line path. It is proved that all the signals of the closed-loop system are ultimately, uniformly bounded. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.The main contributions of our paper are as follows. (1) The fault-tolerant controller is designed without the fault detection scheme. (2) For underactuated surface vessels, we develop the additional control input and analyze the stability including the sway dynamics. (3) The approach angle which is composed only of position information of USVs is introduced. With the help of a novel approach angle, we can track any trajectory including both a straight- and curved-line path. (4) For a practical application, the model parameters for the hydrodynamic terms and external disturbances are assumed to be unknown. These uncertainties are estimated by MNNs.Throughout this paper, the following notations are used: (1)∥
·
∥ denotes any suitable vector norm; (2) t
r
(
·
) is the trace; (3) ∥
·
∥
F is the Frobenius norm defined by ∥
A
∥
F
=
t
r
(
A
T
A
); (4) W
i and W
^
i denote bounded ideal neural weights and the estimates of neural weights, respectively; (5) W
~
i
=
W
i
-
W
^
i denotes the error between W
i and W
^
i; (6) λ
m
i
n
(
A
) and λ
m
a
x
(
A
) denote the minimum and maximum eigenvalues of matrix A, respectively; (7) d
i
a
g
(
·
) is a diagonal matrix.


## 2. Underactuated Surface Vessel Model

The kinematics and dynamics of USVs are described as follows [16]: (1)
η
˙
=
J
ψ
ν
,
M
ν
˙
=
-
C
ν
ν
-
D
ν
ν
+
τ
d
+
τ
,where η
=
[
x
,
y
,
ψ
]
T denotes position (
x
,
y
) and yaw angle ψ of the USV in the earth-fixed frame; ν
=
[
u
,
v
,
r
]
T denotes surge, sway, and yaw velocities of the USV in the body-fixed frame, respectively, τ
d
=
[
τ
d
,
u
,
τ
d
,
v
,
τ
d
,
r
]
T denotes the external forces such as wind and ocean currents, and τ
=
[
τ
u
,
0
,
τ
r
]
T is the control vector of the surge force τ
u and the yaw moment τ
r. In the above equation, matrices J
(
ψ
), D
(
ν
), C
(
ν
), and M are given as follows:(2)
J
ψ
=
c
o
s
ψ
-
s
i
n
ψ
0
s
i
n
ψ
c
o
s
ψ
0
0
0
1
,
D
ν
=
d
11
u
0
0
0
d
22
v
,
r
d
23
v
,
r
0
d
32
v
,
r
d
33
v
,
r
,
C
ν
=
0
0
-
m
22
v
-
m
t
r
0
0
m
11
u
m
22
v
+
m
t
r
-
m
11
u
0
,
M
=
m
11
0
0
0
m
22
m
23
0
m
32
m
33
,where (3)
m
11
=
m
-
X
u
˙
,
m
22
=
m
-
Y
v
˙
,
m
23
=
m
x
g
-
Y
r
˙
,
m
32
=
m
x
g
-
N
v
˙
,
m
33
=
I
z
-
N
r
˙
,
m
t
=
m
23
+
m
32
2
,
d
11
u
=
-
X
u
+
X
u
u
u
,
d
22
v
,
r
=
-
Y
v
+
Y
v
v
v
+
Y
r
v
r
,
d
23
v
,
r
=
-
Y
r
+
Y
v
r
v
+
Y
r
r
r
,
d
32
v
,
r
=
-
N
v
+
N
v
v
v
+
N
r
v
r
,
d
33
v
,
r
=
-
N
r
+
N
v
r
v
+
N
r
r
r
.
X
u, X
u
|
u
|, Y
v, Y
|
v
|
v, Y
|
r
|
v, Y
r, Y
|
v
|
r, Y
|
r
|
r, N
v, N
|
v
|
v, N
|
r
|
v, N
r, N
|
v
|
r, and N
|
r
|
r are the linear and quadratic drag coefficients, m is the mass of the USV, X
u
˙, Y
v
˙, Y
r
˙, N
v
˙, and N
r
˙ are the added masses, x
g is the X
b-coordinate of the USV center of gravity in the body-fixed frame, I
z is the inertia with respect to the vertical axis, and M is invertible.Remark 1.
Unlike other works, we assume that the parametersm
23 and m
32 are not zero. This means that the sway dynamics is influenced by the yaw moment τ
r and we must control sway and yaw velocities simultaneously using only the yaw moment τ
r. When the parameters m
23 and m
32 are not zero, controller design difficulties are increased.To address the problem stated in Remark1, we utilize the following state transformations [17]:(4)
x
-
=
x
+
ϵ
c
o
s
ψ
,
y
-
=
y
+
ϵ
s
i
n
ψ
,
v
-
=
v
+
ϵ
r
,where ϵ
=
m
23
/
m
22. The transformed equation (4) implies that the virtual center of mass is positioned by ϵ in the longitudinal direction. Using (4), the USV dynamics (1) can be expressed by defining state variables z
1
=
[
z
11
,
z
12
,
z
13
]
T
=
[
x
-
,
y
-
,
ψ
]
T and z
2
=
[
z
21
,
z
22
,
z
23
]
T
=
[
u
,
v
-
,
r
]
T: (5)
z
˙
1
=
J
ψ
z
2
,
z
˙
2
=
φ
z
2
+
τ
-
,
y
=
z
1
,where (6)
φ
z
2
=
φ
1
,
φ
2
,
φ
3
T
,
φ
1
=
m
22
v
r
+
m
t
r
2
-
d
11
u
u
+
τ
d
,
u
m
11
,
φ
2
=
-
m
11
m
22
u
r
-
d
22
v
,
r
m
22
v
-
d
23
v
,
r
m
22
r
+
1
m
22
τ
d
,
v
,
φ
3
=
1
Δ
m
11
m
22
-
m
22
2
u
v
+
m
11
m
32
-
m
t
m
22
u
r
-
d
33
v
,
r
r
+
d
32
v
,
r
v
m
22
+
d
23
v
,
r
r
+
d
22
v
,
r
v
m
32
-
m
32
τ
d
,
v
+
m
22
τ
d
,
r
m
11
m
22
-
m
22
2
,
Δ
=
m
22
m
33
-
m
23
m
32
,
τ
-
=
τ
-
1
,
0
,
τ
-
2
T
=
τ
u
m
11
,
0
,
m
22
τ
r
Δ
T
.In these expressions, we assume that φ
u, φ
v, and φ
r are unknown time-varying functions because they cannot be obtained precisely by measurement or calculation.When faults occur in the control vectorτ
-, we can express them as follows: (7)
τ
-
=
F
p
τ
n
+
f
b
,where τ
n
=
[
τ
n
,
1
,
0
,
τ
n
,
2
]
T is a nominal control vector, F
p
=
d
i
a
g
(
f
1
,
0
,
f
2
)
∈
R
3
×
3 denotes the partial loss of control vector, and f
b
=
[
f
b
,
1
,
0
,
f
b
,
2
]
T is an unknown bias vector. f
1 and f
2 are defined as(8)
f
i
=
0
<
f
i
<
1
;
i
f
t
h
e
t
h
r
u
s
t
e
r
s
a
r
e
i
n
p
a
r
t
i
a
l
f
a
i
l
u
r
e
f
i
=
1
;
i
f
t
h
e
t
h
r
u
s
t
e
r
s
a
r
e
n
o
t
i
n
f
a
i
l
u
r
e
.The control objective is to design a fault-tolerant controller such that it tracks the reference trajectoryη
d
=
[
x
d
,
y
d
,
ψ
d
] even though faults exist in the control vector of the USV. The reference trajectory is given as (9)
x
˙
d
=
u
d
c
o
s
ψ
d
-
v
d
s
i
n
ψ
d
,
y
˙
d
=
u
d
s
i
n
ψ
d
+
v
d
c
o
s
ψ
d
,
ψ
˙
d
=
r
d
,where x
d and y
d denote the desired position; ψ
d denotes the desired yaw angle, and u
d, v
d, and r
d are the desired velocities of the USV. To generate the reference trajectory, we can use path planning techniques presented in [18, 19].Assumption 1.
The desired velocitiesu
d, v
d, and r
d are bounded. Further, the first derivatives u
˙
d, v
˙
d, and r
˙
d are available and bounded.


## 3. Fault-Tolerant Control

### 3.1. Neural Network Estimation

In this paper, MNN is used to estimate the unknown functionφ
(
z
2
). From the universal approximation property for neural networks (NNs) [20], the nonlinear function φ
(
z
2
) can be written as(10)
φ
z
2
=
W
T
σ
V
T
z
2
+
ε
,where ε is the NN reconstruction error and σ
(
·
) is the sigmoid function defined as σ
(
z
)
=
1
/
1
+
e
-
z.Assumption 2.
W and V are bounded such that ∥
W
∥
F
≤
W
M and ∥
V
∥
F
≤
V
M for positive constants W
M and V
M, respectively, and ε is bounded such that ∥
ε
∥
≤
ε
M for a positive constant ε
M.The Taylor series expansion ofσ
(
·
) in (10) for a given z
2 can be written as (11)
σ
V
T
z
2
=
σ
V
^
T
z
2
+
σ
-
V
^
T
z
2
V
~
T
z
2
+
H
V
~
T
z
2
2
,where σ
-
=
(
d
σ
(
V
T
z
2
)
/
d
(
V
T
z
2
)
)
V
T
z
2
=
V
^
T
z
2, H
(
V
~
T
z
2
)
2 denotes higher order terms, and V
~
=
V
-
V
^. From (10) and (11), we can obtain the following result: (12)
φ
z
2
-
φ
^
z
2
=
W
~
T
σ
V
^
T
z
2
+
W
^
T
σ
-
V
^
T
z
2
V
~
T
z
2
+
P
,where P
=
W
~
T
σ
-
(
V
^
T
z
2
)
V
~
T
z
2
+
W
T
H
(
V
~
T
z
2
)
2
+
ε.Property 1.
The unknown termP in (12) is bounded such that ∥
P
∥
≤
δ
T
Θ, where δ
=
[
c
0
,
c
1
,
c
2
,
c
3
,
c
4
]
T, Θ
=
[
1
,
∥
z
2
∥
,
∥
W
^
∥
F
∥
z
2
∥
,
∥
V
^
∥
F
∥
z
2
∥
,
∥
W
^
∥
F
∥
V
^
∥
F
∥
z
2
∥
]
T, and c
j
(
j
=
0
,
…
,
4
) is a positive bounding constant.Proof.
See  the appendix.


### 3.2. Neural Network-Based Fault-Tolerant Control

In this section, we design the neural network-based fault-tolerant control system. To avoid the difficulty of detecting the faults, the controller is proposed using the Nussbaum gain technique. The following lemma is used to design the neural network-based fault-tolerant control law.Lemma 2 (see [21]).
LetV
(
·
) and ζ
(
·
) be smooth functions defined on [
0
,
t
f
) with V
(
t
)
≥
0, ∀
t
∈
[
0
,
t
f
). For t
∈
[
0
,
t
f
), if the following inequality holds, (13)
V
t
≤
α
2
+
e
-
α
3
t
∫
0
t
‍
g
λ
N
ζ
ζ
˙
e
α
3
λ
d
λ
+
e
-
α
3
t
∫
0
t
‍
ζ
˙
e
α
3
λ
d
λ
,where α
2 and α
3 are positive constants, N
(
ζ
)
=
e
ζ
2
c
o
s
(
(
π
/
2
)
ζ
) is a Nussbaum-type function, and g
(
t
) is a time-varying parameter which takes values in the closed intervals Ω
:
=
[
ω
-
,
ω
+
], then V
(
t
), ζ
(
t
), and ∫
0
t
‍
g
(
λ
)
N
(
ζ
)
ζ
˙
d
λ must be bounded on [
0
,
t
f
).We now design the neural network-based fault-tolerant control law using the dynamic surface design approach.Step 1.
Define the following errors:(14)
e
1
=
z
11
-
x
-
d
,
e
2
=
z
12
-
y
-
d
,
e
3
=
z
13
-
ψ
a
,where x
-
d
=
x
d
+
ϵ
c
o
s
ψ
d and y
-
d
=
y
d
+
ϵ
s
i
n
ψ
d. An approach angle ψ
a is defined as (15)
ψ
a
=
a
r
c
t
a
n
e
2
e
1
t
a
n
h
D
2
γ
1
+
ψ
d
1
-
t
a
n
h
D
2
γ
1
,where D
=
e
1
2
+
e
2
2 and γ
1 is a positive constant.Remark 3.
In this paper, we use an approach angleψ
a as an alternative for the desired yaw angle ψ
d. Using the approach angle ψ
a, we can track any trajectory including both straight-line and curved-line paths. There are other works using the approach angle; however, our approach angle is different from the other studies because it requires only position information.Using (5) and (9), the time derivative of (14) is given as (16)
e
˙
1
=
z
21
c
o
s
z
13
-
z
22
s
i
n
z
13
-
u
d
c
o
s
ψ
d
+
v
-
d
s
i
n
ψ
d
,
e
˙
2
=
z
21
s
i
n
z
13
+
z
22
c
o
s
z
13
-
u
d
s
i
n
ψ
d
-
v
-
d
c
o
s
ψ
d
,
e
˙
3
=
z
23
-
ψ
˙
a
.To stabilize the error dynamics of (16), we choose the virtual controls ϕ
1, ϕ
2, and ϕ
3 as(17)
ϕ
1
=
-
k
1
e
1
c
o
s
z
13
-
k
1
e
2
s
i
n
z
13
+
u
d
c
o
s
z
13
-
ψ
d
+
v
-
d
s
i
n
z
13
-
ψ
d
,
ϕ
2
=
k
1
e
1
s
i
n
z
13
-
k
1
e
2
c
o
s
z
13
-
u
d
s
i
n
z
13
-
ψ
d
+
v
-
d
c
o
s
z
13
-
ψ
d
,
ϕ
3
=
-
k
2
e
3
+
ψ
˙
a
,where k
1 and k
2 are design parameters.Step 2.
In this step, we use the NNs to estimate uncertain termsφ
1, φ
2, and φ
3. From (12), we can derive the following results: (18)
φ
i
-
φ
^
i
=
W
~
i
T
σ
V
^
i
z
2
+
W
^
i
T
σ
-
V
^
i
T
z
2
V
~
i
T
z
2
+
P
i
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
=
1,2
,
3
,where φ
^
i is the estimate of φ
i and P
i
=
W
~
i
T
σ
-
(
V
^
i
T
z
2
)
V
~
i
T
z
2
+
W
i
T
H
(
V
~
i
T
z
2
)
2
+
ε
i. By Property 1, P
i is bounded such that ∥
P
i
∥
≤
δ
i
T
Θ
i, where δ
i is the unknown constant vector and Θ
i
=
[
1
,
∥
z
2
∥
,
∥
W
^
i
∥
∥
z
2
∥
,
∥
V
^
i
∥
∥
z
2
∥
,
∥
W
^
i
∥
∥
V
^
i
∥
∥
z
2
∥
]
T.Assumption 3.
W
i and V
i are bounded such that ∥
W
i
∥
≤
W
i
,
M and ∥
V
i
∥
≤
V
i
,
M for positive constants W
i
,
M and V
i
,
M, respectively, and ε
i is bounded such that ∥
ε
i
∥
≤
ε
i
,
M for a positive constant ε
i
,
M.Define the following errors:(19)
e
4
=
z
21
-
ϕ
1
,
f
-
γ
2
t
a
n
h
β
1
,
e
5
=
z
22
-
ϕ
2
,
f
-
γ
2
t
a
n
h
β
2
,
e
6
=
z
23
-
ϕ
3
,
f
-
γ
2
t
a
n
h
β
3
,where γ
2 is a positive constant. ϕ
1
,
f, ϕ
2
,
f, and ϕ
3
,
f are signals obtained by the following first-order filters: (20)
κ
i
ϕ
˙
i
,
f
+
ϕ
i
,
f
=
ϕ
i
,
ϕ
i
,
f
0
=
ϕ
i
0
,
i
=
1,2
,
3
,where κ
i is a positive constant. Here, β
1, β
2, and β
3 are used to solve the problem that there is no actual control on the sway dynamics as seen from (5) and obtained by the following differential equations: (21)
β
˙
1
=
-
c
o
s
h
2
β
1
γ
2
T
u
t
a
n
h
β
2
,
β
˙
2
=
c
o
s
h
2
β
2
γ
2
δ
^
2
T
Θ
2
s
g
n
e
5
k
3
e
5
-
e
1
s
i
n
z
13
+
e
2
c
o
s
z
13
+
φ
^
2
+
δ
^
2
T
Θ
2
s
g
n
e
5
-
ϕ
˙
2
,
f
,
β
˙
3
=
-
c
o
s
h
2
β
3
γ
2
T
r
t
a
n
h
β
2
,where T
u, T
r, k
3
>
0 are design parameters and δ
^
2 is the estimate of δ
2. To design the unified control algorithm from the error dynamics of (19), we introduce a function h
(
ρ
(
t
)
) as (22)
h
ρ
t
=
1
,
∥
y
~
t
∥
>
ρ
t
0
,
∥
y
~
t
∥
≤
ρ
t
,where ρ
(
t
) is the threshold which is the criterion for judgment.Remark 4.
The thresholdρ
(
t
) can be derived from the fault detection algorithm. However, the proposed algorithm does not require the threshold ρ
(
t
) because we design the neural network-based fault-tolerant controller under the assumption that the function h
(
ρ
(
t
)
) in (22) is unknown.Using (5), (7), and (22), the time derivative of (19) is represented as (23)
e
˙
4
=
φ
-
1
+
g
1
t
τ
n
,
1
+
T
u
t
a
n
h
β
2
-
ϕ
˙
1
,
f
,
e
˙
5
=
φ
2
-
k
4
e
5
+
e
1
s
i
n
z
13
-
e
2
c
o
s
z
13
-
φ
^
2
-
δ
^
2
T
Θ
2
s
g
n
e
5
,
e
˙
6
=
φ
-
3
+
g
2
t
τ
n
,
2
+
T
r
t
a
n
h
β
2
-
ϕ
˙
3
,
f
,where(24)
φ
-
1
=
φ
1
+
h
ρ
t
f
b
,
1
,
φ
-
3
=
φ
3
+
h
ρ
t
f
b
,
2
,
g
1
t
=
1
-
h
ρ
t
+
h
ρ
t
f
1
,
g
2
t
=
1
-
h
ρ
t
+
h
ρ
t
f
2
.Property 2.
g
1
(
t
) and g
2
(
t
) are bounded such that 0
<
g
1
(
t
)
≤
1 and 0
<
g
2
(
t
)
≤
1.Proof.
See   the appendix.From (23), we choose the actual controls τ
n
,
1 and τ
n
,
2 as follows: (25)
τ
n
,
1
=
N
ζ
1
θ
1
,
θ
1
=
φ
-
^
1
+
k
3
e
4
-
ϕ
˙
1
,
f
+
T
u
t
a
n
h
β
2
+
e
1
c
o
s
z
13
+
e
2
s
i
n
z
13
+
δ
^
1
T
Θ
1
s
g
n
e
4
,
ζ
˙
1
=
e
4
θ
1
,
τ
n
,
2
=
N
ζ
2
θ
2
,
θ
2
=
φ
-
^
3
+
k
3
e
6
-
ϕ
˙
3
,
f
+
T
r
t
a
n
h
β
2
+
δ
^
3
T
Θ
3
s
g
n
e
6
+
e
3
,
ζ
˙
2
=
e
6
θ
2
,where N
(
ζ
1
) and N
(
ζ
2
) are Nussbaum-type functions. For the stability analysis, we define the boundary layer error as (26)
ϖ
i
=
ϕ
i
,
f
-
ϕ
i
,
i
=
1,2
,
3
.Let us consider the following Lyapunov functions:(27)
V
1
=
1
2
e
1
2
+
e
2
2
+
e
4
2
+
e
5
2
+
∑
i
=
1
2
‍
ϖ
i
2
+
∑
i
=
1
2
‍
W
~
i
T
Λ
i
-
1
W
~
i
+
t
r
V
~
i
T
Λ
i
-
1
V
~
i
+
δ
~
i
T
Λ
i
-
1
δ
~
i
,
V
2
=
1
2
e
3
2
+
e
6
2
+
ϖ
3
2
+
W
~
3
T
Λ
3
-
1
W
~
3
+
t
r
V
~
3
T
Λ
3
-
1
V
~
3
+
δ
~
3
T
Λ
3
-
1
δ
~
3
,where Λ
i is positive definite matrix.Theorem 5.
Consider the USV model (5) in the presence of the faults. For any initial conditions V
1
(
0
)
≤
μ
1 and V
2
(
0
)
≤
μ
2, where μ
1 and μ
2 are any positive constants, if Assumptions 1–3 are satisfied and one applies the controllers (17) and (25) with the adaptation laws defined as (28)
W
^
i
=
Λ
i
e
i
+
3
σ
V
^
i
T
z
2
-
ι
i
Λ
i
W
^
i
,
V
^
i
=
Λ
i
z
2
σ
-
V
^
i
T
z
2
W
^
i
e
i
+
3
T
-
ι
i
Λ
i
V
^
i
,
δ
^
˙
i
=
Γ
i
Θ
i
e
i
+
3
-
ι
i
Γ
i
δ
^
i
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
=
1,2
,
3
,where ι
i is a positive constant, then one can ensure that (29)
l
i
m
t
→
∞
∥
η
-
η
d
∥
=
ϑ
,where ϑ is a positive constant.Proof.
See  the  appendix.


## 3.1. Neural Network Estimation

In this paper, MNN is used to estimate the unknown functionφ
(
z
2
). From the universal approximation property for neural networks (NNs) [20], the nonlinear function φ
(
z
2
) can be written as(10)
φ
z
2
=
W
T
σ
V
T
z
2
+
ε
,where ε is the NN reconstruction error and σ
(
·
) is the sigmoid function defined as σ
(
z
)
=
1
/
1
+
e
-
z.Assumption 2.
W and V are bounded such that ∥
W
∥
F
≤
W
M and ∥
V
∥
F
≤
V
M for positive constants W
M and V
M, respectively, and ε is bounded such that ∥
ε
∥
≤
ε
M for a positive constant ε
M.The Taylor series expansion ofσ
(
·
) in (10) for a given z
2 can be written as (11)
σ
V
T
z
2
=
σ
V
^
T
z
2
+
σ
-
V
^
T
z
2
V
~
T
z
2
+
H
V
~
T
z
2
2
,where σ
-
=
(
d
σ
(
V
T
z
2
)
/
d
(
V
T
z
2
)
)
V
T
z
2
=
V
^
T
z
2, H
(
V
~
T
z
2
)
2 denotes higher order terms, and V
~
=
V
-
V
^. From (10) and (11), we can obtain the following result: (12)
φ
z
2
-
φ
^
z
2
=
W
~
T
σ
V
^
T
z
2
+
W
^
T
σ
-
V
^
T
z
2
V
~
T
z
2
+
P
,where P
=
W
~
T
σ
-
(
V
^
T
z
2
)
V
~
T
z
2
+
W
T
H
(
V
~
T
z
2
)
2
+
ε.Property 1.
The unknown termP in (12) is bounded such that ∥
P
∥
≤
δ
T
Θ, where δ
=
[
c
0
,
c
1
,
c
2
,
c
3
,
c
4
]
T, Θ
=
[
1
,
∥
z
2
∥
,
∥
W
^
∥
F
∥
z
2
∥
,
∥
V
^
∥
F
∥
z
2
∥
,
∥
W
^
∥
F
∥
V
^
∥
F
∥
z
2
∥
]
T, and c
j
(
j
=
0
,
…
,
4
) is a positive bounding constant.Proof.
See  the appendix.


## 3.2. Neural Network-Based Fault-Tolerant Control

In this section, we design the neural network-based fault-tolerant control system. To avoid the difficulty of detecting the faults, the controller is proposed using the Nussbaum gain technique. The following lemma is used to design the neural network-based fault-tolerant control law.Lemma 2 (see [21]).
LetV
(
·
) and ζ
(
·
) be smooth functions defined on [
0
,
t
f
) with V
(
t
)
≥
0, ∀
t
∈
[
0
,
t
f
). For t
∈
[
0
,
t
f
), if the following inequality holds, (13)
V
t
≤
α
2
+
e
-
α
3
t
∫
0
t
‍
g
λ
N
ζ
ζ
˙
e
α
3
λ
d
λ
+
e
-
α
3
t
∫
0
t
‍
ζ
˙
e
α
3
λ
d
λ
,where α
2 and α
3 are positive constants, N
(
ζ
)
=
e
ζ
2
c
o
s
(
(
π
/
2
)
ζ
) is a Nussbaum-type function, and g
(
t
) is a time-varying parameter which takes values in the closed intervals Ω
:
=
[
ω
-
,
ω
+
], then V
(
t
), ζ
(
t
), and ∫
0
t
‍
g
(
λ
)
N
(
ζ
)
ζ
˙
d
λ must be bounded on [
0
,
t
f
).We now design the neural network-based fault-tolerant control law using the dynamic surface design approach.Step 1.
Define the following errors:(14)
e
1
=
z
11
-
x
-
d
,
e
2
=
z
12
-
y
-
d
,
e
3
=
z
13
-
ψ
a
,where x
-
d
=
x
d
+
ϵ
c
o
s
ψ
d and y
-
d
=
y
d
+
ϵ
s
i
n
ψ
d. An approach angle ψ
a is defined as (15)
ψ
a
=
a
r
c
t
a
n
e
2
e
1
t
a
n
h
D
2
γ
1
+
ψ
d
1
-
t
a
n
h
D
2
γ
1
,where D
=
e
1
2
+
e
2
2 and γ
1 is a positive constant.Remark 3.
In this paper, we use an approach angleψ
a as an alternative for the desired yaw angle ψ
d. Using the approach angle ψ
a, we can track any trajectory including both straight-line and curved-line paths. There are other works using the approach angle; however, our approach angle is different from the other studies because it requires only position information.Using (5) and (9), the time derivative of (14) is given as (16)
e
˙
1
=
z
21
c
o
s
z
13
-
z
22
s
i
n
z
13
-
u
d
c
o
s
ψ
d
+
v
-
d
s
i
n
ψ
d
,
e
˙
2
=
z
21
s
i
n
z
13
+
z
22
c
o
s
z
13
-
u
d
s
i
n
ψ
d
-
v
-
d
c
o
s
ψ
d
,
e
˙
3
=
z
23
-
ψ
˙
a
.To stabilize the error dynamics of (16), we choose the virtual controls ϕ
1, ϕ
2, and ϕ
3 as(17)
ϕ
1
=
-
k
1
e
1
c
o
s
z
13
-
k
1
e
2
s
i
n
z
13
+
u
d
c
o
s
z
13
-
ψ
d
+
v
-
d
s
i
n
z
13
-
ψ
d
,
ϕ
2
=
k
1
e
1
s
i
n
z
13
-
k
1
e
2
c
o
s
z
13
-
u
d
s
i
n
z
13
-
ψ
d
+
v
-
d
c
o
s
z
13
-
ψ
d
,
ϕ
3
=
-
k
2
e
3
+
ψ
˙
a
,where k
1 and k
2 are design parameters.Step 2.
In this step, we use the NNs to estimate uncertain termsφ
1, φ
2, and φ
3. From (12), we can derive the following results: (18)
φ
i
-
φ
^
i
=
W
~
i
T
σ
V
^
i
z
2
+
W
^
i
T
σ
-
V
^
i
T
z
2
V
~
i
T
z
2
+
P
i
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
=
1,2
,
3
,where φ
^
i is the estimate of φ
i and P
i
=
W
~
i
T
σ
-
(
V
^
i
T
z
2
)
V
~
i
T
z
2
+
W
i
T
H
(
V
~
i
T
z
2
)
2
+
ε
i. By Property 1, P
i is bounded such that ∥
P
i
∥
≤
δ
i
T
Θ
i, where δ
i is the unknown constant vector and Θ
i
=
[
1
,
∥
z
2
∥
,
∥
W
^
i
∥
∥
z
2
∥
,
∥
V
^
i
∥
∥
z
2
∥
,
∥
W
^
i
∥
∥
V
^
i
∥
∥
z
2
∥
]
T.Assumption 3.
W
i and V
i are bounded such that ∥
W
i
∥
≤
W
i
,
M and ∥
V
i
∥
≤
V
i
,
M for positive constants W
i
,
M and V
i
,
M, respectively, and ε
i is bounded such that ∥
ε
i
∥
≤
ε
i
,
M for a positive constant ε
i
,
M.Define the following errors:(19)
e
4
=
z
21
-
ϕ
1
,
f
-
γ
2
t
a
n
h
β
1
,
e
5
=
z
22
-
ϕ
2
,
f
-
γ
2
t
a
n
h
β
2
,
e
6
=
z
23
-
ϕ
3
,
f
-
γ
2
t
a
n
h
β
3
,where γ
2 is a positive constant. ϕ
1
,
f, ϕ
2
,
f, and ϕ
3
,
f are signals obtained by the following first-order filters: (20)
κ
i
ϕ
˙
i
,
f
+
ϕ
i
,
f
=
ϕ
i
,
ϕ
i
,
f
0
=
ϕ
i
0
,
i
=
1,2
,
3
,where κ
i is a positive constant. Here, β
1, β
2, and β
3 are used to solve the problem that there is no actual control on the sway dynamics as seen from (5) and obtained by the following differential equations: (21)
β
˙
1
=
-
c
o
s
h
2
β
1
γ
2
T
u
t
a
n
h
β
2
,
β
˙
2
=
c
o
s
h
2
β
2
γ
2
δ
^
2
T
Θ
2
s
g
n
e
5
k
3
e
5
-
e
1
s
i
n
z
13
+
e
2
c
o
s
z
13
+
φ
^
2
+
δ
^
2
T
Θ
2
s
g
n
e
5
-
ϕ
˙
2
,
f
,
β
˙
3
=
-
c
o
s
h
2
β
3
γ
2
T
r
t
a
n
h
β
2
,where T
u, T
r, k
3
>
0 are design parameters and δ
^
2 is the estimate of δ
2. To design the unified control algorithm from the error dynamics of (19), we introduce a function h
(
ρ
(
t
)
) as (22)
h
ρ
t
=
1
,
∥
y
~
t
∥
>
ρ
t
0
,
∥
y
~
t
∥
≤
ρ
t
,where ρ
(
t
) is the threshold which is the criterion for judgment.Remark 4.
The thresholdρ
(
t
) can be derived from the fault detection algorithm. However, the proposed algorithm does not require the threshold ρ
(
t
) because we design the neural network-based fault-tolerant controller under the assumption that the function h
(
ρ
(
t
)
) in (22) is unknown.Using (5), (7), and (22), the time derivative of (19) is represented as (23)
e
˙
4
=
φ
-
1
+
g
1
t
τ
n
,
1
+
T
u
t
a
n
h
β
2
-
ϕ
˙
1
,
f
,
e
˙
5
=
φ
2
-
k
4
e
5
+
e
1
s
i
n
z
13
-
e
2
c
o
s
z
13
-
φ
^
2
-
δ
^
2
T
Θ
2
s
g
n
e
5
,
e
˙
6
=
φ
-
3
+
g
2
t
τ
n
,
2
+
T
r
t
a
n
h
β
2
-
ϕ
˙
3
,
f
,where(24)
φ
-
1
=
φ
1
+
h
ρ
t
f
b
,
1
,
φ
-
3
=
φ
3
+
h
ρ
t
f
b
,
2
,
g
1
t
=
1
-
h
ρ
t
+
h
ρ
t
f
1
,
g
2
t
=
1
-
h
ρ
t
+
h
ρ
t
f
2
.Property 2.
g
1
(
t
) and g
2
(
t
) are bounded such that 0
<
g
1
(
t
)
≤
1 and 0
<
g
2
(
t
)
≤
1.Proof.
See   the appendix.From (23), we choose the actual controls τ
n
,
1 and τ
n
,
2 as follows: (25)
τ
n
,
1
=
N
ζ
1
θ
1
,
θ
1
=
φ
-
^
1
+
k
3
e
4
-
ϕ
˙
1
,
f
+
T
u
t
a
n
h
β
2
+
e
1
c
o
s
z
13
+
e
2
s
i
n
z
13
+
δ
^
1
T
Θ
1
s
g
n
e
4
,
ζ
˙
1
=
e
4
θ
1
,
τ
n
,
2
=
N
ζ
2
θ
2
,
θ
2
=
φ
-
^
3
+
k
3
e
6
-
ϕ
˙
3
,
f
+
T
r
t
a
n
h
β
2
+
δ
^
3
T
Θ
3
s
g
n
e
6
+
e
3
,
ζ
˙
2
=
e
6
θ
2
,where N
(
ζ
1
) and N
(
ζ
2
) are Nussbaum-type functions. For the stability analysis, we define the boundary layer error as (26)
ϖ
i
=
ϕ
i
,
f
-
ϕ
i
,
i
=
1,2
,
3
.Let us consider the following Lyapunov functions:(27)
V
1
=
1
2
e
1
2
+
e
2
2
+
e
4
2
+
e
5
2
+
∑
i
=
1
2
‍
ϖ
i
2
+
∑
i
=
1
2
‍
W
~
i
T
Λ
i
-
1
W
~
i
+
t
r
V
~
i
T
Λ
i
-
1
V
~
i
+
δ
~
i
T
Λ
i
-
1
δ
~
i
,
V
2
=
1
2
e
3
2
+
e
6
2
+
ϖ
3
2
+
W
~
3
T
Λ
3
-
1
W
~
3
+
t
r
V
~
3
T
Λ
3
-
1
V
~
3
+
δ
~
3
T
Λ
3
-
1
δ
~
3
,where Λ
i is positive definite matrix.Theorem 5.
Consider the USV model (5) in the presence of the faults. For any initial conditions V
1
(
0
)
≤
μ
1 and V
2
(
0
)
≤
μ
2, where μ
1 and μ
2 are any positive constants, if Assumptions 1–3 are satisfied and one applies the controllers (17) and (25) with the adaptation laws defined as (28)
W
^
i
=
Λ
i
e
i
+
3
σ
V
^
i
T
z
2
-
ι
i
Λ
i
W
^
i
,
V
^
i
=
Λ
i
z
2
σ
-
V
^
i
T
z
2
W
^
i
e
i
+
3
T
-
ι
i
Λ
i
V
^
i
,
δ
^
˙
i
=
Γ
i
Θ
i
e
i
+
3
-
ι
i
Γ
i
δ
^
i
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
=
1,2
,
3
,where ι
i is a positive constant, then one can ensure that (29)
l
i
m
t
→
∞
∥
η
-
η
d
∥
=
ϑ
,where ϑ is a positive constant.Proof.
See  the  appendix.


## 4. Simulation Results

In this section, we simulate a neural network-based fault-tolerant controller to verify the performance of the proposed control system. The parameters of the underactuated surface vessel are as indicated in [22]. The control gains are chosen as k
1
=
1.2, k
2
=
k
3
=
2, T
u
=
T
r
=
1, γ
1
=
0.0001, γ
2
=
1000, and κ
i
=
0.2, where i
=
1,2
,
3. The reference trajectory is generated by the following desired velocities:(30)
0
≤
t
<
5
:
u
d
,
v
d
,
r
d
=
0.5
s
i
n
π
t
10
,
0,0
,
5
≤
t
<
20
:
u
d
,
v
d
,
r
d
=
0.5,0
,
0
,
20
≤
t
:
u
d
,
v
d
,
r
d
=
0.5,0
,
0.3
.The initial postures for the reference USV and the actual USV are [
x
d
,
y
d
,
ψ
d
]
=
[
0,0
,
0
] and [
x
,
y
,
ψ
]
=
[
-
0.5,0.5,0
], respectively. In this simulation, we use three MNNs to estimate the uncertainties and the MNNs are composed of four hidden layers and three inputs. In addition, the weight tuning parameters of the MNNs are chosen as ι
i
=
0.01, Λ
i
=
d
i
a
g
(
2,2
,
2
), and Γ
i
=
d
i
a
g
(
0.001,0.001,0.001,0.001,0.001
), where i
=
1,2
,
3. To verify the robustness of the proposed neural network-based fault-tolerance controller, we assume that the faults occur after 20 seconds such that F
p
=
d
i
a
g
(
0.6,0
,
0.5
) and f
b
=
[
1.5,0
,
1
].Figure1(a) presents the trajectory tracking result. The tracking errors are depicted in Figures 1(c) and 1(d). As indicated in Figures 1(c) and 1(d), there are tracking errors when faults occur on t
=
20
(
s
e
c
); however these errors are reduced quickly. Further, the tracking errors are bounded for the entire time even though faults exist. From Figure 1(a), we can confirm that the actual USV tracks successfully the reference trajectory including both straight and curved paths. Therefore, we can conclude that the proposed control system possesses robustness for faults, model uncertainties, and external disturbances.Figure 1
Simulation results. (a) Trajectory tracking result. (b) Control inputs (solid:τ
-
1, dash-dot: τ
-
2). (c) Tracking errors (solid: e
1, dotted: e
2, and dash-dot: e
3). (d) Tracking errors (solid: e
4, dotted: e
5, and dash-dot: e
6).
(a)
(b)
(c)
(d)


## 5. Conclusions

This paper presented a neural network-based fault-tolerant controller for an underactuated USV with faults, model uncertainties, and external disturbances. MNNs were employed to estimate the highly nonlinear uncertain terms, and approach angle was implemented to track any reference trajectory including both straight- and curved-line paths. To avoid the difficulty of detecting the faults, the neural network-based fault-tolerant controller was designed using the Nussbaum gain technique. From the simulation results, it was demonstrated that the proposed control system has good tracking performance in the presence of the faults, model uncertainties, and external disturbances. In the future, the actuator saturation problem will be addressed and evaluate the performance through real experiments.



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*Source: 903759-2015-10-04.xml*