iss-based robust adaptive fuzzy algorithm for maintaining a ship’s track
TRANSCRIPT
Journal of Marine Science and Application, Vol.6, No.4, December 2007, pp.1-7
�������������� �����������
ISS-based robust adaptive fuzzy algorithm for maintaining a
ship’s track
LI Tie-shan1,2
, YAN Shu-jia2
, and QIAO Wen-ming2,3
1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China
2. Navigation College, Dalian Maritime University, Dalian 116026, China
3. Navigation School, Shandong Jiaotong College, Weihai 250023, China
Abstract: This paper focuses on the problem of linear track keeping for marine surface vessels. The
influence exerted by sea currents on the kinematic equation of ships is considered first. The input-to-state
stability (ISS) theory used to verify the system is input-to-state stable. Combining the Nussbaum gain
with backstepping techniques, a robust adaptive fuzzy algorithm is presented by employing fuzzy
systems as an approximator for unknown nonlinearities in the system. It is proved that the proposed
algorithm that guarantees all signals in the closed-loop system are ultimately bounded. Consequently, a
ship’s linear track-keeping control can be implemented. Simulation results using Dalian Maritime
University’s ocean-going training ship 'YULONG' are presented to validate the effectiveness of the
proposed algorithm.
Keywords: ISS theory; adaptive control; fuzzy system; ship track-keeping
CLC number: U664.82 Document code: A Article ID: 1671-9433(2007)04-0001-07
1 Introduction1
In practice, long distance navigation tasks are
frequently carried out while traveling via way-points
at constant cruise speed. To save traveling time,
distance and fuel, ship straight-path tracking is an
important practice and has received considerable
attention[1]
. When the linear course is to be tracked,
only the yaw moment, supplied by the ship rudder
control system, is served as control input to drive the
sway displacement, sway velocity and heading angle
to zero while the sway axis is not actuated, and the
surge velocity is maintained by the main thruster
control system. This configuration is by far the most
common among marine surface vessels. So the goal of
waypoint tracking is to control both the heading angle
and the sway displacement by using the yaw torque
only. The main difficulty of the problem is due to the
underactuated nature of ships, especially in the
presence of external disturbances.
In literature, many approaches have been introduced
to treat this control problem in the last few years. As a
proof, a large number of papers published in
Received date: 2007-04-23.
Foundation item: Supported by the National Natural Science
Foundation of China under Grant No. 10572094.
literatures have been dedicated to the field of
nonlinear ship control, e.g. Refs.[2-7] and references
therein. Refer to Ref.[8] for further details. Recently, a
nonlinear model-based and guidance based controller
was presented for path following by Breivik and
Fossen [9]
. It can be easily extended from fully
actuated to under-actuated vessels exposed to a
constant environmental force. In Ref.[8], combined
Nussbaum gain technique by backstepping approach,
an adaptive robust approach was designed on the
straight-path tracking problem for under-actuated
ships with parametric uncertainties and bounded
exogenous disturbances.
However, a common problem exists in all the works
mentioned above, that is, the influence exerted by sea
current on ship’s kinematics equations, especially the
sway direction, is not considered. In literatures, what
concerned about is only the influence induced by
current on ship’s dynamics equations. Whereas, as we
know, the sea current plays an important role in the
sway displacement (or the cross-track error) of ships
during the path following.
In this paper, a robust adaptive fuzzy algorithm is
explored for ship straight-path tracking problem. The
interested ship motion equations include both of the
� � � � � � Journal of Marine Science and Application, Vol.6, No.4, December 2007 2
unknown nonlinear uncertainties and the bounded
external disturbances induced by current, wave and
wind, especially with the influence exerted by sea
current on ship kinematic equation. Based upon ISS
theory, employing Mamdani fuzzy systems to
approximate unstructured uncertain function in the
dynamics equation of ships, the algorithm is
developed by combing the backstepping approach and
Nussbaum-type gain technique. In such a way, the
straight-path tracking problem considered for ships is
tackled elegantly. Simulation results validate the
effectiveness of the proposed scheme.
2 Preliminaries
2.1 Nussbaum gain technique
In this paper, Nussbaum gain proposed originally in
Ref.[10] is employed to cope with the unknown
control gain with an unknown sign.
Lemma 1[11]
: Let ( )V ⋅ and ( )κ ⋅ be smooth functions
defined on [0, t f ) with V(t)≥0, [0, )f
t t∀ ∈ , ( )N ⋅ be an
even smooth Nussbaum-type function. If the following
inequality holds:
( ) ( )( ) ( )1 1 1 1
0
0 0
e e d e e d ,
t t
c t c t c t c t
V t c g x Nτ κ κ τ κ τ
− −
≤ + +∫ ∫� �
where c0 represents some suitable constant, c
1>0,
g(x(t)) is a time-varying function, then V(t), ( )tκ
and ( )( ) ( )0
d
t
g x Nτ κ κ τ∫� must be bounded in [0,tf ).
Remark 1: According to proposition 2 in Ryan [12]
, if the
solutions to the closed-loop system exist, then tf =∞.
2.2 Mamdani fuzzy system
Considering a Mamdani fuzzy system to uniformly
approximate a continuous multi-dimensional function
f (x) with a complicated formulation, x is input vector
with n independent x = (x1 … xn)
T
. Generally, the fuzzy
system can be constructed by the following K(K > 1)
fuzzy rules:
Ri : IF x1 is A
i
h1 AND x
2 is A
i
h2 AND…AND xn is A
i
hn,
THEN yi is Bi
h1,h2…hn , i = 1,2, … ,K, where, Ai
hj, j =
1,…,n are unknown constants. Bi
h1,h2…hn ( i = 1,2,…,K)
represent output fuzzy sets. The product fuzzy
inference is employed to evaluate the ANDs in the
fuzzy rules. After being defuzzyfied by a typical
center average defuzzifier, the output of the Mamdani
fuzzy system is
( ) ( ) ( )T
1
K
i i
i
F x y x xξ
=
= = =∑y θ ξ , (1)
where
( ) ( ) ( ) ( )
T
1 2
... .
K
x x x xξ ξ ξ= ⎡ ⎤⎣ ⎦
ξ
[ ]T
1
,...
K
θ θ=θ is a vector of weights, and
( ) ( ) ( )1 1
1j j
Kn i n i
i j h j j h ji
x x xξ μ μ= =
=
⎡ ⎤= Π Π⎣ ⎦
∑
is called a fuzzy-based function.
Lemma 2[13]
: Suppose that the input universal of
discourse U is a compact set in Rr
. Then, for any
given real continuous function f (x) on U and 0ε∀ > ,
referred to as approximation error, there exists a fuzzy
system in the form of expression (1) such that:
( ) ( )sup
x U
f x F x ε
∈
− ≤ . (2)
2.3 Input to state stability (ISS)
The concepts of ISS and ISS-Lyapunov function
proposed in Ref.[14] have recently been used in
various control problems such as nonlinear
stabilization, robust control and observer designs. In
order to ease the discussion of control design, the
definition of input-to-state stability is reviewed in the
following.
Definition 1: For the system ( , )x f x u=� , it is said to
be input-to state stable (ISS) if there exist a function g
of class K, called the nonlinear L∞ gain, and a function
b of class KL such that, for any initial condition x(0),
each measurable essentially bounded control u(t)
defined for all t≥0, the associated solutions x(t) are
defined on [0,∞) and satisfy:
( ) ( )( ) ( )
0
0 0
, sup
t t
x t x t t t u
τ
β γ τ
≤ ≤
⎛ ⎞≤ − +
⎜ ⎟
⎝ ⎠
.
3 Problem formulation
Now consider the following nonlinear ship straight
track-keeping control equation[7]
:
( )
sin sin ,
,
,
c c
y U V
r
r f r bu
ψ ψ
ψ
Δ
⎧ = +
⎪
=⎨
⎪′= + +
⎩
�
�
�
(3)
where y, ψ, r and U denote the sway displacement
(crosstrack error), heading angle, yaw rate and cruise
speed respectively, b denotes the control rudder angle;
Δ′ denotes uncertain external perturbations induced by
LI Tie-shan, et al: ISS-based robust adaptive fuzzy algorithm for maintaining a ship’s track 3
wave and wind, f (r) is an unknown nonlinear function
for r, Vc, ψ
c represent constant current speed and
direction respectively, b is the control gain.
Remark 2: The current influence on ship’s kinematic
equation in sway direction was not considered in
literatures mentioned above.
Assumptions are imposed for system (3) as follows:
Assumption 1: The high-frequency gain b and its sign
are completely unknown.
Assumption 2: The disturbance Δ′ is slowly time-
varying, bounded, and its exact bound is unknown.
But it satisfies
Δ′ ≤ λф(x), (4)
where λ is an unknown positive constant, and ф is a
known nonnegative smooth function.
The design goal in this paper is to explore a robust
adaptive fuzzy controller for system (3), such that the
cross track error y, yaw angle y and yaw rate r can be
stabilized respectively.
4 Design procedure and stability analysis
4.1 A useful proposition
At first, a useful proposition is proposed as follows:
Proposition 1: The system (3) is a minimum-phase
system with a stable zero dynamics.
Proof: define coordinate transformation [7]
( )
1
2
arcsin ,
1
ky
x
ky
ψ
⎛ ⎞
⎜ ⎟= +
⎜ ⎟+
⎝ ⎠
(5)
where k is a positive constant, then Eq.(3) can be
changed into
( )
( )
( )
1 1 2 1
2 2 2 2 2
1
2
1
,
,
sin arcsin sin ,
1
,
c c
x f x
x f x g u
ky
y U x V
ky
x
Δ
Δ
ψ
ζ
⎧ = ⋅ + +
⎪
= + +⎪
⎪⎪ ⎛ ⎞⎛ ⎞⎨
⎜ ⎟⎜ ⎟= − +⎪
⎜ ⎟⎜ ⎟⎜ ⎟⎪ +
⎝ ⎠⎝ ⎠⎪
=⎪⎩
�
�
�
(6)
where ( )
( )
1 2
sin
1
k
f U
ky
ψ⋅ =
+
, ( ) ( )2
f f r⋅ = ,
( )
1 2
sin
1
c c
k
V
ky
Δ ψ=
+
, Δ2=Δ′ , g
2=b.
Remark 3: Now notice that the stabilization of (3)
becomes that of (6) via the global transformation (5).
Obviously, the relative order of system (6) is 2, and its
zero dynamics is
( )
( )
( )
2
2
sin
1
, .
1
c c
c c
kU
y y V
ky
kU
y V
ky
ψ
γ ψ
= − + =
+
− +
+
�
(7)
From Eq.(7) it can be seen that the zero dynamics is
exponential stable when without outer disturbances,
i.e. γ(Vc, y
c) = 0. In addition, (V
c,y
c)�L
∞(the current
considered is constant), then γ�L∞, which implies
that Eq.(7) is ISS from γ to y. Consequently, the
system (6) as well as (3) is a minimum phase system
with a stable zero dynamics.
Now, only if the sub-system (x1, x
2)
( )
T
1 1 1 2 1
2 2 2 2 2
,x f x
x f x g u
Δ
Δ
⎧ ′= + +⎪
⎨
= + +⎪⎩
θ�
�
(8)
is stabilized, then system (6) as well as (3) is also
stabilized.
Remark 4: Since the structure of the first equation in
(8) is known, which is derived from ship kinematic
equation via coordinate transformation, so the function
f1 in Eq. (8) can be decomposed into
T
1 1 1
f f ′= θ , where
( )
1 2
sin
1
k
f
ky
ψ′=
+
, θ1 represents the parameter
perturbation of U. And 1
Δ can be decomposed into
1 1 1
Δ λφ= , where
( )
1 2
1
k
ky
φ =
+
, the magnitude of outer
disturbance λ1=V
csinψ
c is an unknown constant.
4.2 Controller design
Now begin the controller design on subsystem (8)
with the backstepping technique in the following. The
structure of system (8) suggests to design the
controller in two stages by applying the backstepping
approach.
Step 1: Let z1 = x
1, z
2 = x
2-a
1. And define Lyapunov
function V1 as follows:
2 T 1 1 *2
1 1 1 1 1 1 1
1 1 1
.
2 2 2
V z λ
− −
= + Γ +θ θ γ� � �
(9)
The derivative of V1 is
� � � � � � Journal of Marine Science and Application, Vol.6, No.4, December 2007 4
( )
( )
( ) ( )
T 1 1 * *
1 1 1 1 1 1 1 1 1
T T 1
1 1 1 2 1 1 1 1 1 1
1 * * T *
1 1 1 1 1 1 2 1 1 1
T 1 * 1 *
1 1 1 1 1 1 1 1 1 1
ˆ ˆ
ˆ
ˆ ˆ ˆ
ˆ ˆ,
z z
z f z
z f z
f z z
λ λ
α λ φ
λ λ α λ φ
λ λ φ
− −
−
−
− −
= − − =
+ + + − −
= + + + −
− − −
V θ Γ θ γ
θ θ Γ θ
γ θ
θ Γ θ γ
� �
� ��
�
�
�
�
�
� �
� �
(10)
where1 1
φ φ= ,*
1 1
λ λ= ,* * *
1 1 1
ˆ
λ λ λ= +
�
,1 1 1
ˆ
= +θ θ θ�
.
Now take the intermediate stabilizing function
T *
1 1 1 1 1 1 1
ˆ ˆ.c z fα λ φ= − − −θ (11)
By substituting it into (10), we can get
( )
( )
2 T 1
1 1 2 1 1 1 1 1 1 1
* 1 *
1 1 1 1 1
ˆ
ˆ .
V z z c z f z
z
θ
λ λ φ
−
−
= − − − −
−
θ Γ
γ
�
��
�
�
(12)
Step 2: Define Lyapunov function V2as follows
2 T 1 1 *2
2 1 2 2 2 2 2 2
1 1 1
.
2 2 2
V V z λ
− −
= + + +θ Γ θ γ� � �
(13)
Due to
( )
( )
( )
1 1 1
1 1 2
1
T1 1
1 2 1 1 1 2 2
1
1
T1 1 1
1 2 1 1 1 2
1
1 1
1 1 2
1
sin sin
sin
,
c c
x y t
x y
g x f x
x
U V
y
g x f x
x y
x y
α α α
α ψ β
ψ
α α
β
ψ
α
ψ ψ
α α α
ψ
ψ
α α
φ λ β
∂ ∂ ∂
= + + − =
∂ ∂ ∂
∂ ∂′⎡ ⎤ + + Δ − + +
⎣ ⎦∂ ∂
∂
+ =
∂
⎡ ⎤∂ ∂ ∂′ + + + +
⎢ ⎥∂ ∂ ∂
⎣ ⎦
⎛ ⎞∂ ∂
+ −⎜ ⎟∂ ∂
⎝ ⎠
θ
θ θ
� �� �
(14)
where1 1
2 1 1
1
ˆ ˆ
ˆ ˆ
α α
β λ
λ
⎛ ⎞∂ ∂
= − +⎜ ⎟⎜ ⎟∂ ∂
⎝ ⎠1
θ
θ
� �
, then
( ) ( )( )
( )
( )
2 2 1 2 2 2
1 1
1
1
T1 1 1
1 2 1 1 1 2
1
2 2 2 2
sin
sin
,
c c
z x g x u f x
V
x y
g x f x
x y
g x u f
α Δ
α α
Δ ψ
α α α
ψ
ψ
β Δ
= − = + + −
⎛ ⎞∂ ∂
+ −⎜ ⎟∂ ∂
⎝ ⎠
⎡ ⎤∂ ∂ ∂′+ + + +⎢ ⎥
∂ ∂ ∂⎣ ⎦
′′ ′= + +
θ θ
���
(15)
where
( )
1 1
2 2 2
1
1 1
2 1 1
1
sin
1
,
c c
k
V
x yky
x y
α α
Δ Δ ψ
α α
Δ φ λ
⎛ ⎞⎛ ⎞∂ ∂
⎜ ⎟′ ⎜ ⎟= − + =
⎜ ⎟⎜ ⎟∂ ∂+⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞∂ ∂
− +⎜ ⎟⎜ ⎟⎜ ⎟
∂ ∂⎝ ⎠⎝ ⎠
( ) ( )
( )
( )
1 1
2 2 2 2 2 1 1 1
1
1 1
2 2 2 2 2
T1 1
2 1 1 1
1
sin
sin .
T
f f x x f
x y
x f x x
x f
x y
α α
β ψ
α α
β
ψ ψ
α α
θ ψ
⎡∂ ∂′′ ′= + − + + +
⎢∂ ∂
⎣
∂ ∂⎤ ⎛ ⎞
= − + −⎜ ⎟⎥
∂ ∂⎦ ⎝ ⎠
⎡ ⎤∂ ∂′ + +
⎢ ⎥∂ ∂
⎣ ⎦
θ θ
θ
Define ( )2 2 2
F f ′′=Z ,
T
1 1
2 2 2
1
, , ,x
x y
α α
β
⎡ ⎤∂ ∂
=⎢ ⎥
∂ ∂⎣ ⎦
Z by
using Mamdani fuzzy system to approximate F2(Z
2)
according to Lemma 2, then
( ) ( )2
T
2 2 2 2 2 Z
F ξ ε= +Z θ Z , (16)
where ( ) ( ) ( ) ( )
T
2 2 2
2 2 1 2 2 2 2
, ,...,
K
ξ ξ ξ⎡ ⎤=⎣ ⎦
ξ Z Z Z Z ,
T
T 2 2
2 1
...
K
θ θ⎡ ⎤=⎣ ⎦
θ ,2
Z
ε is the approximating error, which
has an upper bound, 2 2
*
Z Z
ε ε≤ , 2
*
Z
ε is its known
upper bound.
Then
( ) ( )2
T
2 2 2 2 2 2
.
Z
z g x u ξ ε Δ′= + + +θ Z�
������
(17)
The derivative of V2 is
( )( )
( )( )
( )( ) ( )
2
T 1 1 * *
2 1 2 2 2 2 2 2 2 2
T 2
1 2 2 2 2 2 2
T 1 1 * * 2
2 2 2 2 2 2 1 1
T 2 *
2 1 2 2 2 2 2
T 1 1 * 1 *
1 1 1 1 1 1 1 1 1 1
2
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
Z
T
V V z z
V z g u
c z
z z g u
z z
λ λ
ξ ε λ φ
λ λ
ξ λ φ
ξ λ λ φ
− −
− −
− −
= + − − =
+ + + + −
− = − +
+ + + −
− − − −
θ Γ θ γ
θ Z
θ Γ θ γ
θ Z
θ Γ θ Z γ
θ
� �
� �� �
�
�
� �
� �
� �
� �
�
( )( ) ( )1 2 * 1 *
2 2 2 2 2 2 2 2 2
ˆ ˆ,z zθ ξ λ λ φ
− −
− − −Γ Z γ
� �
�
(18)
where ( )2
* *
2 1 2
max , ,Z
λ λ λ ε= ,�
� ( )1 1
2 2 1
1
1
x y
α α
φ φ φ
⎛ ⎞∂ ∂
⋅ = + + +⎜ ⎟⎜ ⎟∂ ∂
⎝ ⎠
.
To deal with the unknown virtual control gain by
employing Nussbaum-type gain technique, the
following controller and adaptive laws are defined
( )2 2
,u N κ ς= (19)
2 2 2
,zκ ς=�
�
���������������(20)
( )T *ˆ ˆ
tanh ,
z
c z z
φ
ς ξ λ φ
ε
⎛ ⎞
= + + + ⎜ ⎟
⎝ ⎠
Z
2 2 2
2 2 2 1 2 2 2 2
2
θ
�
(21)�
( )0
1 1 1 1 1 1 1
ˆ ˆ,f z σ
⎡ ⎤= − −⎣ ⎦
θ Γ θ θ
�
��������(22)
( ) ( )2 0
2 2 2 2 2 2 2
ˆ ˆ,zξ σ
⎡ ⎤= − −⎣ ⎦
θ Γ Z θ θ
�
�����(23)
LI Tie-shan, et al: ISS-based robust adaptive fuzzy algorithm for maintaining a ship’s track 5
( )* * 0ˆ ˆ
tanh , 1,2i i
i i i i i i i i
i
z
z iλ
φ
λ γ φ γ σ λ λ
ε
⎛ ⎞
= − − =⎜ ⎟
⎝ ⎠
�
��(24)
To substitute them into (18), we can get
( )( )
( ) ( )( )
( ) ( )( )
( )
2 T 2 *
2 1 1 2 1 2 2 2 2
T 1 1
2 2 2 2 2 2 1 1 1 1 1
* 1 * T 1 2
1 1 1 1 1 2 2 2 2 2
22 2
* 1 * 2
2 2 2 2 21 1
22
2 21
ˆ ˆ
ˆ
ˆ ˆ
1ˆ
2
1
2
i i i ii i
i ii
V c z z z
g N z z
z z
z c z
g Nλ
ξ λ φ
κ ς κ κ ξ
λ λ φ ξ
λ λ φ σ θ
σ λ δ κ
−
− −
−
= =
=
= − + + + +
+ − − − −
− − − −
− ≤ − − −
+ +
∑ ∑
∑
θ Z
θ Γ θ Z
γ θ Γ θ Z
γ
�
�
�
� �
� �
� �
�
� �
�
( )( )
( )( )
2 2 2 2
2 2 2 2
1
1 ,
d V
g N
κ
κ κ δ
+ ≤ − +
+ +
�
�
�
(25)�
where
( )2 1 2
1
max
min , , , , ( 1,2)
i
i i
i
d c c iλ
σ
σ γ
λ−
⎧ ⎫⎪ ⎪
= =⎨ ⎬
Γ⎪ ⎪⎩ ⎭
,
22 2* 0
21 1
22* 0
1
1
0.2785
2
1
.
2
i i i i ii i
i i ii
λ
δ λ ε σ θ θ
σ λ λ
= =
=
= + − +
−
∑ ∑
∑
�
By setting2 2 2
dρ δ= , and multiplying (25) by 2
e
d t
,
then integrating its result over [0, t], we have
( ) ( )( )2 2
2 2 2 2 2 2
0
0 e 1 e d .
t
d t d
V V g N
τ
ρ κ κ τ
−
≤ + + +∫� (26)
Now the main result is proposed:
Theorem 1: With respect to system (8), the proposed
robust fuzzy adaptive controller (19) and the
intermediate virtual control law (11) with the
parameter adaptive laws (20)~(24) can render all
signals and states in the closed-loop bounded.
5 Simulation results
Now, the proposed algorithm derived from (8) is
applied to the ship linear track-keeping control system
(3) to demonstrate its effectiveness.
Define five fuzzy sets for each of the variables in (16),
which are characterized by the following membership
functions
( )
( )2
1
1
exp
0.2
x
xμ
⎛ ⎞+
⎜ ⎟= −
⎜ ⎟
⎝ ⎠
, ( )
( )2
2
0.5
exp
0.2
x
xμ
⎛ ⎞+
⎜ ⎟= −
⎜ ⎟
⎝ ⎠
,
( )
( )2
3
exp
0.2
x
xμ
⎛ ⎞
⎜ ⎟= −
⎜ ⎟
⎝ ⎠
, ( )
( )2
4
0.5
exp
0.2
x
xμ
⎛ ⎞−
⎜ ⎟= −
⎜ ⎟
⎝ ⎠
,
( )
( )2
5
1
exp
0.2
x
xμ
⎛ ⎞−
⎜ ⎟= −
⎜ ⎟
⎝ ⎠
.�
In this section, simulation results are based on an
oceangoing training vessel YULONG[7]
. The design
parameters are chosen as k = 0.005, γ1
= 0.05, γ2 =
0.25,1
10λ
σ = , 2
5λ
σ = , 1
0.05Γ = ,
( )2
diag 0.05,0.05,0.05,0.05Γ = , 1
0.5σ = ,
( )2
diag 5,5,5,5σ = , 1
0.1ε = , 2
0.1ε = .
( ) [ ]2
ˆ0 1,1∈ −θ (we take 0.1 here). The initial conditions
are y0 =500 m, ψ
0 =10° . The external disturbance
signals are chosen as 1
Δ = 0.5, 2
Δ = 0.1.
Nussbaum function N (κ ) =κ2
cos(κ ),κ (0) = 0.5π.
Simulation results using Matlab Simulink illustrate the
properties of the proposed algorithm with Figs. 1~5.
Fig.1 Simulation comparison under constant current-ship
trajectory in x-y plane
(a) Track error y
� � � � � � Journal of Marine Science and Application, Vol.6, No.4, December 2007 6
(b) Heading ψ
(c) Yawing rate r
(d) Rudder angle u
Fig.2 Time response under the controller in this paper
(a) Track error y
(b) Heading ψ
(c) Yawing rate r
(d) Rudder angle u
Fig.3 Time response under the controller in Ref.[8]
(a) Ship speed1
ˆ
θ �
(b) Magnitudes of external disturbance: 1-*
1
ˆ
λ ���� *
2
ˆ
λ
Fig.4 Parameters estimates
LI Tie-shan, et al: ISS-based robust adaptive fuzzy algorithm for maintaining a ship’s track 7
(a) Nussbaum gain N(k)
(b) Nussbaum parameter k
Fig.5 Nussbaum gain and its parameters
The above illustrations show that, when the influence
exerted by sea current on ship’s kinematic equation is
considered, the cross-track error y (in sway motion)
can be stabilized with the controller presented in this
paper, whereas it cannot be done with the algorithms
in most literatures.
6 Conclusions�
The problem of ship straight-path tracking control is
studied in this paper. Especially, the interested ship
motion equations include the influence exerted by sea
current on ship kinematic equation. The Mamdani
fuzzy system is employed to approximate those
uncertainties in the considered system. Based upon
ISS theory, a robust adaptive fuzzy algorithm is
developed by combing the backstepping approach and
Nussbaum-type gain technique. As a result, the
interested problem is solved. The results provide a
valuable reference to ship engineering practice.
�
References �
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LI Tie-shan was born in 1968. He is an associate
professor of Dalian Maritime University. His
current research interests include adaptive
control, robust control and fuzzy control for
nonlinear systems and their applications to
marine ship motion control.
YAN Shu-jia was born in 1982. She is a master
student of Dalian Maritime University. Her
current research interests include robust control
etc.
QIAO Wen-ming was born in 1969. He is an
associate professor of Shandong Jiaotong
University. His current research interests include
traffic information engineering and control.