# RobustH

Authors: Dawei Zhao; Fuguang Ding; Lili Zhou; Wenying Zhang; He Xu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

Due to the input time-delay existing in most thrust systems of the ships, the robustH
∞ controller is designed for the ship dynamic positioning (DP) system with time-delay. The input delay system is turned to a neutral time-delay system by a state-derivative control law. The less conservative result is derived for the neutral system with state-derivative feedback by the delay-decomposition approach and linear matrix inequality (LMI). Finally, the numerical simulations demonstrate the asymptotic stability and robustness of the controller and verify that the designed DP controller is effective in the varying environment disturbances of wind, waves, and ocean currents.


---

## Body

## 1. Introduction

The dynamic positioning (DP) systems of ships keep floating structures in fixed position or predetermined track for marine operation purpose exclusively by means of active thrusters [1]. Conventional DP control systems are designed based on linearization PID controllers in each degree of freedom combined with notch-filters. Subsequently linear optimal controller combined with Kalman filter theory was proposed [2, 3]. In recent years, the nonlinear controllers and estimators were employed to handle the inherent nonlinear characteristics of the vessel dynamics [4, 5]. In [6–8], Lyapunov methods and back-stepping technique were used to design a passive nonlinear observer to estimate the vessel velocity. Many scholars have studied the DP controllers but they did not consider the time-delay in the ship thrust system.Time-delays as a source of instability and poor performance often appear in many dynamic systems, so it is very important to make a stability analysis of systems with time-delays. In vessel’s DP control, the time-delays generally exist in thruster driver, including main propellers, tunnel thrusters, azimuth thrusters, and rudders. Compared with the rapidly changed instructions from the controller, the response time of thrusters, which generally lasts several seconds, is longer. Therefore, it is crucial to take account of the time-delay during the design of the DP controller. In the past, many useful approaches have been applied to guarantee the stabilization and theH
∞ performance of systems. Recently, more attention has been paid to the delay-dependent system which, in general, can be considered as less conservative than the delay-independent ones. By using the Lyapunov-Razumikhin functional approach or the Lyapunov-Krasovskii functional approach, several stability criteria have been proposed for the delay-dependent systems [9–18]. Based on a model transformation and Park’s inequality, a delay-dependent stability criterion for the uncertain neutral systems with time-varying discrete delay was proposed in [10]. Reference [11] proposed a delay-dependent stability criterion for time-varying delay systems by means of an augmented model. In [12], the triple integral Lyapunov-Krasovskii functional approaches which utilized more information about the states and the delay states were proposed. Scholar Min Wu proposed the free weighting matrix method to avoid the conservatism of the transformation results, but it requires more possible decidable variables which make the computation quite complex [13, 14]. References [15–17] proposed the delay-decomposition approach, that the delay interval is uniformly divided into N equidistant subintervals, and proposed a new delay-dependent stability criterion. The above researches for the stability of time-delays system are to increase the upper bound of the stability condition and reduce the conservatism of the algorithm. The H
∞ controllers are often utilized to stabilize the unstable systems or improve the system’s performance [18]. Reference [19] proposed an H
∞ observer-based controller for the uncertain neutral time-delay systems. Shariati proposed an H
∞ controller for a linear system with time-delay, which used composite state-derivative feedback [20]. Reference [21] proposed a nonfragile controller for the discrete-time descriptor systems with multiple state delays. Reference [22] proposed a Lyapunov function with a piecewise analysis method for the semi-Markovian jump systems with mode-dependent delays. The state-derivative feedback can effectively improve the performance of the controller and it is generally used in practical systems such as the mechanical system and the ship DP system, where the accelerometers and the velocity sensors are used for measuring the system’s motion. Moreover, the acceleration feedback can suppress varying disturbances, and the use of acceleration feedback in DP control problems has been discussed in [8].In this paper, the robustH
∞ control problem for the neutral time-delay system has been considered. The less conservative results were derived for the neutral system with state-derivative feedback by a delay-decomposition approach and LMI method. The robust H
∞ controller is used in the ship DP control systems, and the designed closed-loop control law with speed and acceleration feedback enables DP controller to perform better and resist external environmental disturbances greatly.This paper is organized as follows. In Section2, the problem formulation for the DP system is discussed. In Section 3.1, the H
∞ performance analysis is proposed. In Section 3.2, the H
∞ state-derivative feedback controller is designed for the neutral time-delay system. Simulation results are provided in Section 4, which demonstrate the effectiveness and the advantages of the proposed controller under the ocean disturbances. Conclusions are given in Section 5.


## 2. Problem Formulation and Preliminaries

Here, the model of ship DP system with time-delay is established. The equations of motion of surface vessels can be described by the following model [1]:(1)
M
v
˙
+
D
v
=
u
+
w
,
n
˙
=
J
ψ
ν
,
M
=
m
-
X
u
˙
0
0
0
m
-
Y
v
˙
m
x
G
-
Y
r
˙
0
m
x
G
-
Y
r
˙
I
z
-
N
r
˙
,
D
=
-
X
u
0
0
0
-
Y
v
-
Y
r
0
-
Y
r
-
N
r
,where n
=
X
Y
ψ is the position and orientation of the vessel with respect to an inertial reference coordinate system and ν
=
u
υ
r is a vector of velocities given in the body-fixed coordinate system; J
(
ψ
) is the transformation matrix between the inertial and body-fixed coordinate frames; u is a control vector of forces and moments provided by the propulsion system. Generally, time-delay exists in the propulsion system, which means the input delay exists in the ship control system; w is the disturbance input of the system. The inertia matrix M includes hydrodynamic added inertia; D is a strictly positive damping matrix due to linear wave drift damping and laminar skin friction; m is the mass of the ship; I
z is the moment of inertia. X
u
˙, Y
v
˙, N
r
˙, and Y
r
˙ are the accessional mass caused by liquid forces, respectively.The time-delays mainly existing in the ship’s propulsion system are considered in this paper, and the formula (1) is transformed into the state-space model of the vessel with time-delay as follows:(2)
x
˙
t
=
A
x
t
+
B
u
t
-
τ
t
+
E
w
t
,
z
t
=
C
x
t
+
D
1
u
t
-
τ
t
,
A
=
0
I
-
M
-
1
G
-
M
-
1
D
,
B
=
0
M
-
1
,
E
=
0
M
-
1
,where x
(
t
)
=
[
n
T
,
ν
T
]
T is the system state including the position, heading, and velocities of vessel; u
(
t
-
τ
(
t
)
) is the system input and also is the control vector of forces and moments with time-delay provided by the propulsion system; τ
(
t
) is the time-delay in the propulsion system; B is a constant matrix describing the actuator configuration; w
(
t
) is the disturbance input of the system caused by the environment disturbances from wind, currents, and waves; z
(
t
) is the controlled system output. The matrices C, D
1 are assumed to be known.In ship DP system, a variety of sensors are installed to measure the state of the ship motion, such as global position system (GPS) for measuring the ship’s position, attitude sensor for measuring the pitch, roll and heave, and so forth, and gyrocompass for measuring the ship’s heading. Using the measured information for feedback control is a practical approach of the ship motion control. Velocity feedback in the DP controller can improve the performance of the control system. High-precision commercial inertial measurement units, which are used to measure the acceleration in the positioning control of vessels to suppress varying disturbances, can be easily interfaced with the existing control systems. Lindegaard used the acceleration feedback to increase the performance of DP systems [8]. Acceleration feedback (AFB) will provide a virtual mass in addition to the physical mass of the vessel. Therefore, the vessel becomes less sensitive to the environmental excitations.Using the state-derivative feedback control law in the position control as well as the velocity and acceleration feedback can suppress varying disturbances. The aim of this paper is to design the composite control law as follows:(3)
u
=
K
1
x
+
K
2
x
˙
,where K
1 and K
2 are the controller gain to be designed; the design goal is to make the system (2) asymptotically stable and meet certain performance index, where x
˙
(
t
)
=
[
n
˙
T
,
ν
˙
T
]
T, n
˙
T is the velocity vector of vessel, and ν
˙ is the acceleration velocity of vessel. The velocity and acceleration feedback loop is established by measuring the velocity and acceleration in the DP control of vessels to suppress varying disturbances in the sea environment. By applying the feedback control law (3), the input time-delay system (2) is turned into a time-delay closed-loop system of the neutral type which is described as follows:(4)
x
˙
t
=
A
x
t
+
B
K
1
x
t
-
τ
t
+
B
K
2
x
˙
t
-
τ
t
+
E
w
t
,
z
t
=
C
x
t
+
D
1
K
1
x
t
-
τ
t
+
D
1
K
2
x
˙
t
-
τ
t
.Neutral time-delay systems contain delays both in its state and its derivatives of the state. As it is shown in (4), both the coefficients of x
(
t
-
τ
) and x
˙
(
t
-
τ
) are dependent on the controller’s parameters. Thus, our objective is to design a state feedback controller u
=
K
1
x
+
K
2
x
˙, in which for a given τ
(
t
)
>
0, the following requirements are satisfied:(1)
the closed-loop system (4) with w
(
t
)
=
0 is asymptotically stable;
(2)
under zero initial condition, the closed-loop system satisfiesz
2
<
γ
w
2 for any nonzero w
∈
L
2
[
0
,
∞
], where γ
>
0 is a prescribed scalar.In order to obtain the main results, we state the following well-known lemmas.Lemma 1 (see [17]).
For any constant matrix0
<
R
=
R
T
∈
R
n
×
n, scalar γ
>
0, vector function ω
:
[
0
,
γ
]
→
R
n such that the integrations concerned are well defined, then(5)
∫
0
γ
ω
s
d
s
T
R
∫
0
γ
ω
T
s
d
s
≤
γ
-
1
∫
0
γ
ω
s
R
ω
T
s
d
s
.Lemma 2 (see [18]).
For given matricesA
11, A
12, A
21, A
22 with the appropriate dimensions, A
11
A
21
A
12
A
22
<
0 holds if and only if A
22
<
0, A
11
-
A
12
A
22
-
1
A
12
T
<
0.


## 3. Main Results

In this section, the delay-dependent stability criterion for the neutral system (4) will be presented by means of the Lyapunov-Krasovskii functional approach. In order to derive less conservative stability criterion for the system (4), we employ the delay-decomposition approach for the neutral time-delay system and divide τ
(
t
) into several components by an integer N with the length of each component denoted by η
(
t
); that is,(6)
η
t
=
τ
t
N
.Therefore, η
(
t
) represents a partition of the time-varying delay τ
(
t
), which satisfies(7)
0
≤
η
t
≤
η
,
0
≤
η
˙
t
≤
ν
.


### 3.1. Robust Stabilization

In this section, we focus on the analysis of theH
∞ performance.Theorem 3.
Given scalarτ
≥
0, the closed-loop system (4) with w
(
t
)
=
0 is asymptotically stable for a continuous time-varying state delay τ
(
t
) satisfying 0
≤
τ
(
t
)
≤
τ, if there exist some positive definite symmetric matrices P
1
>
0, S
2
>
0, Q
1
>
0, Q
2
>
0, Z
1
>
0, Z
2
>
0, Z
3
>
0  M
1
>
0, M
2
>
0, such that the following symmetric linear matrix inequalities hold:(8)
Ω
11
B
K
1
B
K
2
P
1
-
1
Z
1
0
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
∗
Ω
22
0
Z
2
Z
3
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
∗
∗
-
S
2
0
0
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
∗
∗
∗
Ω
44
0
0
0
0
0
∗
∗
∗
∗
Ω
55
0
0
0
0
∗
∗
∗
∗
∗
-
S
2
0
0
0
∗
∗
∗
∗
∗
∗
-
η
2
Z
1
0
0
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
2
0
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
3
<
0
,where(9)
Ω
11
=
2
A
P
1
-
1
+
M
1
-
M
2
,
Ω
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Ω
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Ω
55
=
-
1
-
3
v
Q
3
-
Z
3
.The asterisk represents the symmetric form of the matrix (8).Proof.
Choose a suitable Lyapunov-Krasovskii function candidate for the system (4) as the following form:(10)
V
t
=
V
1
t
+
V
2
t
+
V
3
t
+
V
4
t
.Here,(11)
V
1
t
=
x
T
P
1
x
t
,
V
2
t
=
∫
t
-
τ
t
x
˙
T
s
S
2
x
˙
s
d
s
,
V
3
t
=
∑
i
=
1
N
∫
t
-
i
η
t
t
-
i
-
1
η
t
x
T
s
Q
i
x
s
d
s
,
V
4
t
=
∑
i
=
1
N
∫
-
i
η
-
i
-
1
η
∫
t
+
θ
t
η
x
˙
T
s
Z
i
x
˙
s
d
s
d
θ
,where(12)
P
1
=
P
1
T
>
0
,
S
2
=
S
2
T
>
0
,
Q
1
=
Q
1
T
>
0
,
Q
2
=
Q
2
T
>
0
,
Z
1
=
Z
1
T
>
0
,
Z
2
=
Z
2
T
>
0
,
Z
3
=
Z
3
T
>
0
.
Now calculate the derivative ofV
(
t
) along the state trajectory of the system (4). Employing Lemma 1, the following upper bound for V
˙
(
t
) can be obtained as follows:(13)
V
˙
1
t
=
2
x
T
t
P
1
x
˙
t
=
2
x
T
t
P
1
A
x
t
+
B
K
1
x
t
-
τ
t
hhhhhhhhhhh
+
B
K
2
x
˙
t
-
τ
t
+
E
ω
t
,
V
˙
2
t
=
x
˙
T
t
S
2
x
˙
t
-
x
˙
T
t
-
τ
S
2
x
˙
t
-
τ
,
V
˙
3
t
=
∑
i
=
1
N
1
-
i
-
1
η
˙
t
×
x
T
t
-
i
-
1
η
t
hhhhhih
·
Q
i
x
t
-
i
-
1
η
t
hhhhhhih
-
1
-
i
η
˙
t
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
≤
∑
i
=
1
N
1
+
i
-
1
ν
×
x
T
t
-
i
-
1
η
t
hhhhhhh
·
Q
i
x
t
-
i
-
1
η
t
hhhhhhhh
-
1
-
i
ν
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
,
V
˙
4
t
=
η
2
x
˙
T
t
∑
i
=
1
N
Z
i
x
˙
t
-
∑
i
=
1
N
∫
t
-
i
η
t
-
i
-
1
η
η
x
˙
T
θ
Z
i
x
˙
θ
d
θ
≤
η
2
x
˙
T
t
∑
i
=
1
N
Z
i
x
˙
t
hhh
-
∑
i
=
1
N
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
T
hhhhhhhh
×
Z
i
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
hhhhhhhh
-
1
-
i
η
˙
t
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
T
.By simple derivations, get N
=
3; the upper bound for V
˙
(
t
) is obtained as follows:(14)
V
˙
t
≤
x
˙
T
t
S
2
x
˙
t
-
x
˙
T
t
-
τ
S
2
x
˙
t
-
τ
+
2
x
T
t
P
1
A
x
t
+
B
K
1
x
t
-
τ
t
hhhhhhhhhhhh
+
B
K
2
x
˙
t
-
τ
t
+
E
ω
t
+
x
T
t
Q
1
x
t
-
1
-
v
x
T
t
-
η
t
Q
1
x
t
-
η
t
+
1
+
v
x
T
t
-
η
t
Q
2
x
t
-
η
t
-
1
-
2
v
x
T
t
-
2
η
t
Q
2
x
t
-
2
η
t
+
1
+
2
v
x
T
x
-
2
η
t
Q
3
x
t
-
2
η
t
-
1
-
3
v
x
T
t
-
3
η
t
Q
3
x
t
-
3
η
t
+
η
2
x
˙
T
t
Z
1
+
Z
2
+
Z
3
x
˙
t
-
x
T
t
Z
1
x
t
+
2
x
T
t
Z
1
x
t
-
η
t
-
x
T
t
-
η
t
Z
1
x
t
-
η
t
-
x
T
t
-
η
t
Z
2
x
t
-
η
t
+
2
x
T
t
-
2
η
t
Z
2
x
t
-
η
t
-
x
T
t
-
2
η
t
Z
2
x
t
-
2
η
t
-
x
T
t
-
2
η
t
Z
3
x
t
-
2
η
t
+
2
x
T
t
-
3
η
Z
3
x
t
-
2
η
t
-
x
T
t
-
3
η
t
Z
3
x
t
-
3
η
t
≤
ς
T
t
ψ
ς
t
,where(15)
ς
T
t
=
x
T
t
,
x
T
t
-
τ
,
x
˙
T
t
-
τ
,
x
T
t
-
η
,
x
T
t
-
3
η
,
ψ
=
Σ
11
Σ
12
Σ
13
Z
1
0
∗
Σ
22
Σ
23
Z
2
Z
3
∗
∗
Σ
33
0
0
∗
∗
∗
Σ
44
0
∗
∗
∗
∗
Σ
55
,where(16)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
+
A
T
θ
A
,
Σ
12
=
P
1
B
K
1
+
A
T
θ
B
K
1
,
Σ
13
=
P
1
B
K
2
+
A
T
θ
B
K
2
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
+
K
1
T
B
T
θ
B
K
1
,
Σ
23
=
K
1
T
B
T
θ
B
K
2
,
Σ
33
=
-
S
2
+
K
2
T
B
T
θ
B
K
2
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,where θ
=
S
2
+
η
2
(
Z
1
+
Z
2
+
Z
3
).
Remark  1. The decomposition number N is a positive integer and N is not less than 2. The conservatism will be reduced gradually as the N is increasing, and the time-delay tends to the certain upper boundary. When N equals 2 or 3, the conservatism is small enough.
By using Schur complement, the following matrix inequality is the equivalence of the conditionψ
<
0:(17)
Ξ
=
Ξ
11
Ξ
12
Ξ
12
T
Ξ
22
<
0
,where(18)
Ξ
11
=
Σ
11
P
1
B
K
1
P
1
B
K
2
Z
1
0
∗
Σ
22
0
Z
2
Z
3
∗
∗
-
S
2
0
0
∗
∗
∗
Σ
44
0
∗
∗
∗
∗
Σ
55
,
Ξ
12
=
Δ
Δ
Δ
Δ
,where(19)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Δ
=
A
B
K
1
B
K
2
0
0
T
,
Ξ
22
=
diag
-
S
2
-
η
2
Z
1
-
η
2
Z
2
-
η
2
Z
3
.Premultiplying and postmultiplying the matrix Ξ in (17) by Φ
Tand Φ, where Φ
=
diag
(
P
1
-
1
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
) and applying some change of variables, M
1
=
P
1
-
1
Q
1
P
1
-
1, M
2
=
P
1
-
1
Z
1
P
1
-
1, we obtain result (8); hence, we have V
˙
(
t
)
<
0, then system (4) with w
(
t
)
=
0 is asymptotically stable. This completes the proof.


### 3.2. RobustH
∞ Control

Theorem 4.
Given scalarτ
≥
0, the closed-loop system (4) is asymptotically stable and T
z
w
∞
<
γ for a continuous time-varying state delay τ
(
t
) satisfying 0
≤
τ
(
t
)
≤
τ, if there exist positive definite symmetric matrices P
1
>
0, S
2
>
0, Q
1
>
0, Q
2
>
0, Z
1
>
0, Z
2
>
0, Z
3
>
0  M
1
>
0, M
2
>
0. Moreover, the H
∞ composite state-derivative feedback control law is given by u
=
K
1
x
+
K
2
x
˙, such that the following symmetric linear matrix inequalities hold:(20)
Ω
11
B
K
1
B
K
2
P
1
-
1
Z
1
0
E
0
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
C
T
∗
Ω
22
0
Z
2
Z
3
0
0
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
D
1
T
∗
∗
-
S
2
0
0
0
0
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
D
1
T
∗
∗
∗
Ω
44
0
0
0
0
0
0
0
0
∗
∗
∗
∗
Ω
55
0
0
E
T
E
T
E
T
E
T
0
∗
∗
∗
∗
∗
-
γ
2
I
0
0
0
0
0
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
0
0
0
0
0
∗
∗
∗
∗
∗
∗
∗
-
S
2
0
0
0
0
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
1
0
0
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
2
0
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
3
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
I
<
0
,where the asterisk represents the symmetric form of the matrix (20):(21)
Ω
11
=
2
A
P
1
-
1
+
M
1
-
M
2
,
Ω
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Ω
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Ω
55
=
-
1
-
3
v
Q
3
-
Z
3
.Proof.
Choose the same Lyapunov-Krasovskii function given in (10) for the system (4) as the following form. Assuming the zero initial condition, that is, ϕ
(
t
)
=
0, ∀
t
∈
[
-
τ
,
0
], we can get V
(
q
(
t
)
)
|
t
=
0
=
0. For a prescribed γ
>
0, take account of the performance index as follows:(22)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
d
T
d
d
τ
,where(23)
d
t
=
w
t
w
˙
t
.Therefore, the performance index (22) can be rewritten as(24)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
d
τ
.Then for the random nonzero external disturbances w
∈
L
2
[
0
,
∞
], since V
(
t
)
|
t
=
0
=
0 and V
(
t
)
|
t
→
∞
≥
0, we can obtain(25)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
+
V
˙
t
d
τ
+
V
t
t
=
0
-
V
t
t
→
∞
≤
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
+
V
˙
t
d
τ
.Introducing (4) and (14) into the above, we can obtain the following inequality:(26)
J
z
d
w
≤
∫
0
∞
ξ
T
t
Σ
ξ
t
d
t
.With defined(27)
ξ
T
t
=
x
T
t
,
x
T
t
-
τ
,
x
˙
T
t
-
τ
,
x
T
t
-
η
,
h
h
h
h
h
h
h
h
h
h
x
T
t
-
3
η
,
w
t
,
w
˙
t
,
Σ
=
Σ
11
Σ
12
Σ
13
Z
1
0
Σ
16
0
∗
Σ
22
Σ
23
Z
2
Z
3
Σ
26
0
∗
∗
Σ
33
0
0
Σ
36
0
∗
∗
∗
Σ
44
0
0
0
∗
∗
∗
∗
Σ
55
0
0
∗
∗
∗
∗
∗
Σ
66
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
<
0
,where(28)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
+
C
T
C
+
A
T
θ
A
,
Σ
12
=
P
1
B
K
1
+
C
T
D
1
K
1
+
A
T
θ
B
K
1
,
Σ
13
=
P
1
B
K
2
+
C
T
D
1
K
2
+
A
T
θ
B
K
2
,
Σ
16
=
P
1
E
+
A
T
θ
E
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
+
K
1
T
D
1
T
D
1
K
1
+
K
1
T
B
T
θ
B
K
1
Σ
23
=
K
1
T
D
1
T
D
1
K
2
+
K
1
T
B
T
θ
B
K
2
,
Σ
26
=
K
1
T
B
T
θ
E
,
Σ
33
=
-
S
2
+
K
2
T
D
1
T
D
1
K
2
+
K
2
T
B
T
θ
B
K
2
,
Σ
36
=
K
2
T
B
T
θ
E
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Σ
66
=
-
γ
2
+
E
T
θ
E
,where θ
=
S
2
+
η
2
(
Z
1
+
Z
2
+
Z
3
)
By using Schur complement, the following matrix inequality (29) is the equivalent of the condition of Σ
<
0:(29)
Ξ
=
Ξ
11
Ξ
12
Ξ
12
T
Ξ
22
<
0
,where(30)
Ξ
11
=
Σ
11
P
1
B
K
1
P
1
B
K
2
Z
1
0
P
1
E
0
∗
Σ
22
-
1
2
S
2
Z
2
Z
3
0
0
∗
∗
0
0
0
0
0
∗
∗
∗
Σ
44
0
0
0
∗
∗
∗
∗
Σ
55
0
0
∗
∗
∗
∗
∗
-
γ
2
I
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
,
Ξ
12
=
Δ
1
Δ
1
Δ
1
Δ
1
Δ
2
,where(31)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Δ
1
=
A
B
K
1
B
K
2
0
0
E
0
T
,
Δ
2
=
C
D
1
K
1
D
1
K
2
0
0
0
0
T
,
Ξ
22
=
diag
-
S
2
-
η
2
Z
1
-
η
2
Z
2
-
η
2
Z
3
-
I
.Premultiplying and postmultiplying the matrix Ξ in (29) by diag
(
P
1
-
1
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
), and applying some change of variables, M
1
=
P
1
-
1
Q
1
P
1
-
1, M
2
=
P
1
-
1
Z
1
P
1
-
1, it is easy to know that the LMI (20) is satisfied, then(32)
J
z
d
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
d
τ
≤
0
.Therefore(33)
∫
0
∞
z
T
z
d
τ
≤
γ
2
∫
0
∞
d
T
d
d
τ
.The proof is completed.


## 3.1. Robust Stabilization

In this section, we focus on the analysis of theH
∞ performance.Theorem 3.
Given scalarτ
≥
0, the closed-loop system (4) with w
(
t
)
=
0 is asymptotically stable for a continuous time-varying state delay τ
(
t
) satisfying 0
≤
τ
(
t
)
≤
τ, if there exist some positive definite symmetric matrices P
1
>
0, S
2
>
0, Q
1
>
0, Q
2
>
0, Z
1
>
0, Z
2
>
0, Z
3
>
0  M
1
>
0, M
2
>
0, such that the following symmetric linear matrix inequalities hold:(8)
Ω
11
B
K
1
B
K
2
P
1
-
1
Z
1
0
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
∗
Ω
22
0
Z
2
Z
3
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
∗
∗
-
S
2
0
0
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
∗
∗
∗
Ω
44
0
0
0
0
0
∗
∗
∗
∗
Ω
55
0
0
0
0
∗
∗
∗
∗
∗
-
S
2
0
0
0
∗
∗
∗
∗
∗
∗
-
η
2
Z
1
0
0
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
2
0
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
3
<
0
,where(9)
Ω
11
=
2
A
P
1
-
1
+
M
1
-
M
2
,
Ω
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Ω
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Ω
55
=
-
1
-
3
v
Q
3
-
Z
3
.The asterisk represents the symmetric form of the matrix (8).Proof.
Choose a suitable Lyapunov-Krasovskii function candidate for the system (4) as the following form:(10)
V
t
=
V
1
t
+
V
2
t
+
V
3
t
+
V
4
t
.Here,(11)
V
1
t
=
x
T
P
1
x
t
,
V
2
t
=
∫
t
-
τ
t
x
˙
T
s
S
2
x
˙
s
d
s
,
V
3
t
=
∑
i
=
1
N
∫
t
-
i
η
t
t
-
i
-
1
η
t
x
T
s
Q
i
x
s
d
s
,
V
4
t
=
∑
i
=
1
N
∫
-
i
η
-
i
-
1
η
∫
t
+
θ
t
η
x
˙
T
s
Z
i
x
˙
s
d
s
d
θ
,where(12)
P
1
=
P
1
T
>
0
,
S
2
=
S
2
T
>
0
,
Q
1
=
Q
1
T
>
0
,
Q
2
=
Q
2
T
>
0
,
Z
1
=
Z
1
T
>
0
,
Z
2
=
Z
2
T
>
0
,
Z
3
=
Z
3
T
>
0
.
Now calculate the derivative ofV
(
t
) along the state trajectory of the system (4). Employing Lemma 1, the following upper bound for V
˙
(
t
) can be obtained as follows:(13)
V
˙
1
t
=
2
x
T
t
P
1
x
˙
t
=
2
x
T
t
P
1
A
x
t
+
B
K
1
x
t
-
τ
t
hhhhhhhhhhh
+
B
K
2
x
˙
t
-
τ
t
+
E
ω
t
,
V
˙
2
t
=
x
˙
T
t
S
2
x
˙
t
-
x
˙
T
t
-
τ
S
2
x
˙
t
-
τ
,
V
˙
3
t
=
∑
i
=
1
N
1
-
i
-
1
η
˙
t
×
x
T
t
-
i
-
1
η
t
hhhhhih
·
Q
i
x
t
-
i
-
1
η
t
hhhhhhih
-
1
-
i
η
˙
t
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
≤
∑
i
=
1
N
1
+
i
-
1
ν
×
x
T
t
-
i
-
1
η
t
hhhhhhh
·
Q
i
x
t
-
i
-
1
η
t
hhhhhhhh
-
1
-
i
ν
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
,
V
˙
4
t
=
η
2
x
˙
T
t
∑
i
=
1
N
Z
i
x
˙
t
-
∑
i
=
1
N
∫
t
-
i
η
t
-
i
-
1
η
η
x
˙
T
θ
Z
i
x
˙
θ
d
θ
≤
η
2
x
˙
T
t
∑
i
=
1
N
Z
i
x
˙
t
hhh
-
∑
i
=
1
N
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
T
hhhhhhhh
×
Z
i
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
hhhhhhhh
-
1
-
i
η
˙
t
x
T
t
-
i
η
t
Q
i
x
t
-
i
η
t
x
t
-
i
-
1
η
t
-
x
t
-
i
η
t
T
.By simple derivations, get N
=
3; the upper bound for V
˙
(
t
) is obtained as follows:(14)
V
˙
t
≤
x
˙
T
t
S
2
x
˙
t
-
x
˙
T
t
-
τ
S
2
x
˙
t
-
τ
+
2
x
T
t
P
1
A
x
t
+
B
K
1
x
t
-
τ
t
hhhhhhhhhhhh
+
B
K
2
x
˙
t
-
τ
t
+
E
ω
t
+
x
T
t
Q
1
x
t
-
1
-
v
x
T
t
-
η
t
Q
1
x
t
-
η
t
+
1
+
v
x
T
t
-
η
t
Q
2
x
t
-
η
t
-
1
-
2
v
x
T
t
-
2
η
t
Q
2
x
t
-
2
η
t
+
1
+
2
v
x
T
x
-
2
η
t
Q
3
x
t
-
2
η
t
-
1
-
3
v
x
T
t
-
3
η
t
Q
3
x
t
-
3
η
t
+
η
2
x
˙
T
t
Z
1
+
Z
2
+
Z
3
x
˙
t
-
x
T
t
Z
1
x
t
+
2
x
T
t
Z
1
x
t
-
η
t
-
x
T
t
-
η
t
Z
1
x
t
-
η
t
-
x
T
t
-
η
t
Z
2
x
t
-
η
t
+
2
x
T
t
-
2
η
t
Z
2
x
t
-
η
t
-
x
T
t
-
2
η
t
Z
2
x
t
-
2
η
t
-
x
T
t
-
2
η
t
Z
3
x
t
-
2
η
t
+
2
x
T
t
-
3
η
Z
3
x
t
-
2
η
t
-
x
T
t
-
3
η
t
Z
3
x
t
-
3
η
t
≤
ς
T
t
ψ
ς
t
,where(15)
ς
T
t
=
x
T
t
,
x
T
t
-
τ
,
x
˙
T
t
-
τ
,
x
T
t
-
η
,
x
T
t
-
3
η
,
ψ
=
Σ
11
Σ
12
Σ
13
Z
1
0
∗
Σ
22
Σ
23
Z
2
Z
3
∗
∗
Σ
33
0
0
∗
∗
∗
Σ
44
0
∗
∗
∗
∗
Σ
55
,where(16)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
+
A
T
θ
A
,
Σ
12
=
P
1
B
K
1
+
A
T
θ
B
K
1
,
Σ
13
=
P
1
B
K
2
+
A
T
θ
B
K
2
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
+
K
1
T
B
T
θ
B
K
1
,
Σ
23
=
K
1
T
B
T
θ
B
K
2
,
Σ
33
=
-
S
2
+
K
2
T
B
T
θ
B
K
2
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,where θ
=
S
2
+
η
2
(
Z
1
+
Z
2
+
Z
3
).
Remark  1. The decomposition number N is a positive integer and N is not less than 2. The conservatism will be reduced gradually as the N is increasing, and the time-delay tends to the certain upper boundary. When N equals 2 or 3, the conservatism is small enough.
By using Schur complement, the following matrix inequality is the equivalence of the conditionψ
<
0:(17)
Ξ
=
Ξ
11
Ξ
12
Ξ
12
T
Ξ
22
<
0
,where(18)
Ξ
11
=
Σ
11
P
1
B
K
1
P
1
B
K
2
Z
1
0
∗
Σ
22
0
Z
2
Z
3
∗
∗
-
S
2
0
0
∗
∗
∗
Σ
44
0
∗
∗
∗
∗
Σ
55
,
Ξ
12
=
Δ
Δ
Δ
Δ
,where(19)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Δ
=
A
B
K
1
B
K
2
0
0
T
,
Ξ
22
=
diag
-
S
2
-
η
2
Z
1
-
η
2
Z
2
-
η
2
Z
3
.Premultiplying and postmultiplying the matrix Ξ in (17) by Φ
Tand Φ, where Φ
=
diag
(
P
1
-
1
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
) and applying some change of variables, M
1
=
P
1
-
1
Q
1
P
1
-
1, M
2
=
P
1
-
1
Z
1
P
1
-
1, we obtain result (8); hence, we have V
˙
(
t
)
<
0, then system (4) with w
(
t
)
=
0 is asymptotically stable. This completes the proof.


## 3.2. RobustH
∞ Control

Theorem 4.
Given scalarτ
≥
0, the closed-loop system (4) is asymptotically stable and T
z
w
∞
<
γ for a continuous time-varying state delay τ
(
t
) satisfying 0
≤
τ
(
t
)
≤
τ, if there exist positive definite symmetric matrices P
1
>
0, S
2
>
0, Q
1
>
0, Q
2
>
0, Z
1
>
0, Z
2
>
0, Z
3
>
0  M
1
>
0, M
2
>
0. Moreover, the H
∞ composite state-derivative feedback control law is given by u
=
K
1
x
+
K
2
x
˙, such that the following symmetric linear matrix inequalities hold:(20)
Ω
11
B
K
1
B
K
2
P
1
-
1
Z
1
0
E
0
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
A
T
P
1
-
1
C
T
∗
Ω
22
0
Z
2
Z
3
0
0
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
B
T
K
1
T
D
1
T
∗
∗
-
S
2
0
0
0
0
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
B
T
K
2
T
D
1
T
∗
∗
∗
Ω
44
0
0
0
0
0
0
0
0
∗
∗
∗
∗
Ω
55
0
0
E
T
E
T
E
T
E
T
0
∗
∗
∗
∗
∗
-
γ
2
I
0
0
0
0
0
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
0
0
0
0
0
∗
∗
∗
∗
∗
∗
∗
-
S
2
0
0
0
0
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
1
0
0
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
2
0
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
η
2
Z
3
0
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
-
I
<
0
,where the asterisk represents the symmetric form of the matrix (20):(21)
Ω
11
=
2
A
P
1
-
1
+
M
1
-
M
2
,
Ω
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Ω
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Ω
55
=
-
1
-
3
v
Q
3
-
Z
3
.Proof.
Choose the same Lyapunov-Krasovskii function given in (10) for the system (4) as the following form. Assuming the zero initial condition, that is, ϕ
(
t
)
=
0, ∀
t
∈
[
-
τ
,
0
], we can get V
(
q
(
t
)
)
|
t
=
0
=
0. For a prescribed γ
>
0, take account of the performance index as follows:(22)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
d
T
d
d
τ
,where(23)
d
t
=
w
t
w
˙
t
.Therefore, the performance index (22) can be rewritten as(24)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
d
τ
.Then for the random nonzero external disturbances w
∈
L
2
[
0
,
∞
], since V
(
t
)
|
t
=
0
=
0 and V
(
t
)
|
t
→
∞
≥
0, we can obtain(25)
J
z
d
w
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
+
V
˙
t
d
τ
+
V
t
t
=
0
-
V
t
t
→
∞
≤
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
+
V
˙
t
d
τ
.Introducing (4) and (14) into the above, we can obtain the following inequality:(26)
J
z
d
w
≤
∫
0
∞
ξ
T
t
Σ
ξ
t
d
t
.With defined(27)
ξ
T
t
=
x
T
t
,
x
T
t
-
τ
,
x
˙
T
t
-
τ
,
x
T
t
-
η
,
h
h
h
h
h
h
h
h
h
h
x
T
t
-
3
η
,
w
t
,
w
˙
t
,
Σ
=
Σ
11
Σ
12
Σ
13
Z
1
0
Σ
16
0
∗
Σ
22
Σ
23
Z
2
Z
3
Σ
26
0
∗
∗
Σ
33
0
0
Σ
36
0
∗
∗
∗
Σ
44
0
0
0
∗
∗
∗
∗
Σ
55
0
0
∗
∗
∗
∗
∗
Σ
66
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
<
0
,where(28)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
+
C
T
C
+
A
T
θ
A
,
Σ
12
=
P
1
B
K
1
+
C
T
D
1
K
1
+
A
T
θ
B
K
1
,
Σ
13
=
P
1
B
K
2
+
C
T
D
1
K
2
+
A
T
θ
B
K
2
,
Σ
16
=
P
1
E
+
A
T
θ
E
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
+
K
1
T
D
1
T
D
1
K
1
+
K
1
T
B
T
θ
B
K
1
Σ
23
=
K
1
T
D
1
T
D
1
K
2
+
K
1
T
B
T
θ
B
K
2
,
Σ
26
=
K
1
T
B
T
θ
E
,
Σ
33
=
-
S
2
+
K
2
T
D
1
T
D
1
K
2
+
K
2
T
B
T
θ
B
K
2
,
Σ
36
=
K
2
T
B
T
θ
E
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Σ
66
=
-
γ
2
+
E
T
θ
E
,where θ
=
S
2
+
η
2
(
Z
1
+
Z
2
+
Z
3
)
By using Schur complement, the following matrix inequality (29) is the equivalent of the condition of Σ
<
0:(29)
Ξ
=
Ξ
11
Ξ
12
Ξ
12
T
Ξ
22
<
0
,where(30)
Ξ
11
=
Σ
11
P
1
B
K
1
P
1
B
K
2
Z
1
0
P
1
E
0
∗
Σ
22
-
1
2
S
2
Z
2
Z
3
0
0
∗
∗
0
0
0
0
0
∗
∗
∗
Σ
44
0
0
0
∗
∗
∗
∗
Σ
55
0
0
∗
∗
∗
∗
∗
-
γ
2
I
0
∗
∗
∗
∗
∗
∗
-
γ
2
I
,
Ξ
12
=
Δ
1
Δ
1
Δ
1
Δ
1
Δ
2
,where(31)
Σ
11
=
2
P
1
A
+
Q
1
-
Z
1
,
Σ
22
=
-
1
-
2
ν
Q
2
+
1
+
2
v
Q
3
-
Z
2
-
Z
3
,
Σ
44
=
-
1
-
ν
Q
1
+
1
+
ν
Q
2
-
Z
1
-
Z
2
,
Σ
55
=
-
1
-
3
v
Q
3
-
Z
3
,
Δ
1
=
A
B
K
1
B
K
2
0
0
E
0
T
,
Δ
2
=
C
D
1
K
1
D
1
K
2
0
0
0
0
T
,
Ξ
22
=
diag
-
S
2
-
η
2
Z
1
-
η
2
Z
2
-
η
2
Z
3
-
I
.Premultiplying and postmultiplying the matrix Ξ in (29) by diag
(
P
1
-
1
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
,
I
), and applying some change of variables, M
1
=
P
1
-
1
Q
1
P
1
-
1, M
2
=
P
1
-
1
Z
1
P
1
-
1, it is easy to know that the LMI (20) is satisfied, then(32)
J
z
d
=
∫
0
∞
z
T
z
-
γ
2
w
T
w
-
γ
2
w
˙
T
w
˙
d
τ
≤
0
.Therefore(33)
∫
0
∞
z
T
z
d
τ
≤
γ
2
∫
0
∞
d
T
d
d
τ
.The proof is completed.


## 4. Simulations

In order to validate the performance of the designedH
∞ controller, the certain DP ship simulations and the marine simulation environments are accomplished in MATLAB7.0. The main parameters of the ship are referenced to the Marine Systems Simulator (MSS) developed by the Norwegian university of Science and Technology. The ship is named as S-175. The ship’s main parameters are set as follows. Length
=
175
m, beam
=
25.4
m, tonnage
=
2.4009
e
+
4
m
3, and draft
=
9.5
m. Then, the matrices M and D in (1) can be obtained as follows:(34)
M
=
2.6415
e
+
07
0
0
0
33455476
19417351
0
14917351
6.5209
e
+
10
,
D
=
2.2204
e
+
04
0
0
0
2.2204
e
+
05
-
1.7746
e
+
06
0
-
1.7746
e
+
06
7.1506
e
+
08
.The matricesA, B, and E can be calculated according to formula (2). The marine simulation environment includes wind, waves, and currents. The simulation environments are set as the following: the wave model used in the paper is PM spectrum. The wave’s direction is 120 degree; the significant wave height is 5 m; and the peak frequency is 0.8 rad/s. The current direction is 90 degree, and the current speed is 0.2 m/s. The wind speed is 10 m/s; wind direction angle is 225 degree. The initial position of the vessel is n
p
=
0
0
0
°
T. The final desired position is n
d
=
10
10
10
°
T. When the γ is 0.5 and the maximum propeller delay τ is 3.07 sec, the gain matrix of the state feedback controller is calculated through LMI toolbox as follows:(35)
K
1
=
-
1.094
0
0
-
72.071
0
0
0
-
41.886
196.778
0
75.3
1446
0
0.323
-
1.027
0
-
0.055
0.447
,
K
2
=
-
0.001
0
0
0.006
0
0
0
-
0.011
0.011
0
-
0.0075
0.1992
0
-
35.505
98.995
0
4.767
-
8.552
.We compare our results for neutral time-delay system in Theorem4 with the results using the same state-derivative feedback in [20]. Table 1 shows the results of the upper bound of time-delay for different values of γ. It can be seen that the proposed H
∞ state-feedback controller in this paper provides larger delay bounds than the previous results as listed in Table 1.Table 1
The upper bounds ofτ for various values of γ.
γ
0.5
1.0
10.0
τ
1.56
3.61
5.65
τ (Theorem 4), N
=
2
2.86
6.16
10.16
τ (Theorem 4), N
=
3
3.07
6.27
10.28The simulation results of theH
∞ state-feedback controller under the ocean environment are shown in Figures 1–6. The response curve of the ship’s position is shown in Figure 1. The simulation results indicate that the ship can achieve the given position quickly with small overshoot. Under the action of wind, waves, and currents, the ship’s position maintains given position of 0.5 m error range. Figure 2 shows the curves of the thrust without time-delay and with 3 sec of time-delay. Figures 3 and 4 show the position curves of the x direction and the y direction, respectively. Figures 5 and 6 show the ship’s speed of the x and y directions, respectively. Figures 7 and 8 show the ship’s acceleration of two directions. From the results, we can know that when delays exist in the propeller, the designed controller can make the ship stable and achieve the desired position even under the external disturbances (wind, waves, and currents).Figure 1
Position and heading of the vessel usingH
∞ controller.Figure 2
Comparisons of the thrust with and without time-delay.Figure 3
Position of thex direction of the vessel using H
∞ controller.Figure 4
Position of they direction of the vessel using H
∞ controller.Figure 5
Velocity of thex direction of the vessel using H
∞ controller.Figure 6
Velocity of they direction of the vessel using H
∞ controller.Figure 7
Acceleration of thex direction of the vessel using H
∞ controller.Figure 8
Acceleration of they direction of the vessel using H
∞ controller.In order to demonstrate that the proposedH
∞ controller has good delay stability, the simulation results are compared with the results of the PD controller. The simulations with disturbances of wind, waves, and currents are shown in Figures 9–12. Figure 9 shows the position response of the vessel using the H
∞ controller without time-delay. Figure 10 shows that the position response of the vessel using the H
∞ controller with 3 sec of time-delay. It can be seen that the controller has the delay stability under the same parameters sets of the controller with Figure 9. Figure 11 shows the position response of the vessel using the PD controller without time-delay. And Figure 12 shows the position response of the vessel using the PD controller with time-delay under the same parameters sets of the controller with Figure 11. It can be seen when the time-delay is existing the PD controller is vibrating. We can see that the H
∞ controller has better delay stability than the PD controller. The parameters of the PD controller are listed as follows:(36)
K
P
=
10
0
0
0
40
10
10
10
10
,
K
d
=
0.1
0
0
0
0.1
0.1
0.1
0.1
0.1
.Figure 9
Positions of the vessel usingH
∞ controller without time-delay.Figure 10
Positions of the vessel usingH
∞ controller with time-delay.Figure 11
Positions of the vessel using the PD controller without time-delay.Figure 12
Positions of the vessel using the PD controller with time-delay.The results mentioned above can be seen as that the designed controller has not only the delay stability, but also the rejection capability for the external disturbance, that is, the control robustness.


## 5. Conclusions

In this paper, the robustH
∞ controller with time-delay was designed. The state-space model with time-delay for the ship motion was established. The designed closed-loop control law with speed and acceleration feedback enables DP controller to perform better and resist external environmental disturbances. The less conservative results were derived for the neutral system by adopting a delay-decomposition approach and LMI method. Some results for the ship DP systems were conducted to indicate that the delays exist in the thruster system. The simulation results show that the designed controller can make the ship stable and achieve the desired position under the ocean environment (wind, waves, and currents), and the controller thereby makes the performance of the DP system better. We will study the observer-based H
∞ problem for uncertain neutral time-delay systems in our future work.



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*Source: 976925-2015-03-19.xml*