# Robust Sampled-DataH

Authors: Mingjie Cai; Zhengrong Xiang; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper presents a robust sampled-dataH
∞ control for vibration attenuation of offshore platforms with missing measurements subject to external wave force. It is well known that the phenomena of missing measurements are unavoidable due to various reasons, such as sensor aging and sensor temporal failure, which may degrade the control system performance or even cause instability. For active vibration control of offshore platforms with sampling measurements, to deal with the missing measurements, a robust sampled-data H
∞  control method is proposed in this paper. The robust sampled-data H
∞ state feedback controller is obtained in terms of the solvability of certain linear matrix inequalities (LMIs). Finally, simulations on an offshore platform are exploited to demonstrate the effectiveness of the proposed method. The simulation results show that the designed robust sampled-data H
∞ control scheme is effective to attenuate the external wave force in the presence of missing measurements.


---

## Body

## 1. Introduction

In the modern world, the offshore platforms, especially the oil and gas production platforms, play a more and more important role. The offshore platforms generally undergo various disturbances coming from the hostile environment that they are located in, such as wave, wind, ice, and earthquake, and the self-excited nonlinear hydrodynamic force [1, 2]. These external loads inevitably induce large continuous vibrations which make the offshore platform deformation, fatigue damage, and even unsafety. To prevent fatigue damage and ensure safety and production efficiency, the vibrations of the offshore platforms should be limited.Up to now, a large number of researches have been made to improve the control performance of the system via active control scheme. For example, for offshore steel jacket platforms with an active tuned mass damper (TMD) mechanism, the multiloop feedback design method [1], the nonlinear control scheme, and the robust state feedback control scheme [2] have been developed to reduce the internal oscillation amplitudes of the offshore platforms. By using an active mass damper (AMD), some optimal control based schemes have been applied to improve the performance of the jacket platforms [3–6]. Recently, the dynamic output feedback control scheme [7] and the integral sliding mode control method [8, 9] have been presented to improve the performance of the offshore platforms. More recently, a delay-dependent state feedback controller has been developed to stabilize the offshore platforms subject to self-excited hydrodynamic force and actuator time-delays [10]. In [11], by artificially introducing a proper time-delay into control channel, a delayed H
∞ controller is designed to attenuate the wave-induced vibration of the offshore platform and thereby improve the control performance of the system. It is indicated that the aforementioned active control schemes are effective ways to deal with the vibration problem of offshore platforms subject to nonlinear wave force.It is well known that computers are usually used as digital controllers to control continuous-time systems in modern control systems [12] with the rapid progress of computer and digital technologies. With recent focus on wireless monitoring and control of offshore platforms [13–18] based on networked control technique [19], studying sampled-data control problem for offshore platforms is becoming significant, and the vibration control methods adopted in [1–11] are no longer applicable to offshore platforms with sampling measurements. On the other hand, because of various reasons, such as sensor aging and sensor temporal failure, the phenomena of missing measurements are unavoidable, which may degrade the control system performance or even cause instability [20–26]. Thus, it is necessary to consider the missing measurements encountered in practical issues for vibration control. However, to the best of our knowledge, no results have been made on sampled-data H
∞ control for vibration attenuation of offshore platforms in presence of missing measurements, which motivates our work in this paper.This paper is concerned with the robust sampled-dataH
∞ controller design for vibration attenuation of offshore platforms subject to missing measurements and external wave force. The issue of vibration attenuation is transformed into an H
∞ disturbance attenuation problem of the system. A Lyapunov functional approach is used to solve the H
∞ disturbance attenuation problem, and the controller design is formulated in terms of linear matrix inequalities (LMIs). To validate the effectiveness of the proposed approach, the designed controllers are applied to reduce the vibration of an offshore platform. Simulation results show good vibration attenuation performance in spite of involving missing measurements.The rest of this paper is organized as follows. In Section2, the description of an offshore platform with an AMD mechanism is given first. Then, the formulation of robust sampled-data H
∞ control problem with missing measurements is presented. In Section 3, the robust sampled-data H
∞ controller design problem involving missing measurements is solved. An illustrative example is given in Section 4, and we conclude the paper in Section 5.Notation. The notation used in the paper is fairly standard. The superscript “T” stands for matrix transposition; R denotes the space of real numbers; R
n denotes the n-dimensional Euclidean space; the notation P
>
0 (≥0) means that the matrix P is real symmetric and positive definite (semidefinite); I and 0 represent the identity matrix and zero matrix of appropriate dimensions, respectively; ℒ is the infinitesimal operator; and diag
⁡
{
·
} stands for a block-diagonal matrix. In addition, E
{
x
} means expectation of the stochastic variable x. The space of square-integrable vector functions over [
0
,
∞
) is denoted by L
2
[
0
,
∞
) and for f
(
t
)
∈
L
2
[
0
,
∞
), its norm is given by ∥
f
(
t
)
∥
2
=
∫
t
=
0
∞
f
T
(
t
)
f
(
t
)
d
t. For simplicity, the symmetric term in a symmetric matrix is denoted by “*.”


## 2. Problem Formulation

In this section, a dynamic model of an idealized two-degree-of-freedom system with an AMD mechanism will be presented and the active vibration attenuation problem for the system with sampling measurements will be formulated by applying a state feedback control scheme.Consider an offshore platform shown in Figure1 [11]; the dynamic equations of the offshore platform can be described as
(1)
m
1
z
¨
1
(
t
)
=
-
(
m
1
ω
1
2
+
m
2
ω
2
2
)
z
1
(
t
)
+
m
2
ω
2
2
z
2
(
t
)
+
f
(
t
)
-
u
(
t
)
-
2
(
m
1
ξ
1
ω
1
+
m
2
ξ
2
ω
2
)
z
˙
1
(
t
)
+
2
m
2
ξ
2
ω
2
z
˙
2
(
t
)
,
m
2
z
¨
2
(
t
)
=
m
2
ω
2
2
z
1
(
t
)
+
2
m
2
ξ
2
ω
2
z
˙
1
(
t
)
-
m
2
ω
2
2
z
2
(
t
)
+
u
(
t
)
-
2
m
2
ξ
2
ω
2
z
˙
2
(
t
)
,
where z
1
(
t
) and z
2
(
t
) are displacements of the deck motion of the offshore platform and the AMD, respectively; m
1, ω
1, and ξ
1 are the modal mass, natural frequency, and damping ratio of the offshore platform, respectively; m
2, ω
2, and ξ
2 are the mass, natural frequency, and damping ratio of the AMD, respectively; u
(
t
) is the active control of the system; f
(
t
) is the external wave force acting on the offshore platform and can be numerically calculated by referring to [6].Figure 1
Schematic diagram of offshore platform with AMD.Define the following state variables:(2)
x
1
(
t
)
=
z
1
(
t
)
,
x
2
(
t
)
=
z
2
(
t
)
,
x
3
(
t
)
=
z
˙
1
(
t
)
,
x
4
(
t
)
=
z
˙
2
(
t
)
,
and let
(3)
x
(
t
)
=
[
x
1
(
t
)
x
2
(
t
)
x
3
(
t
)
x
4
(
t
)
]
T
.Then, the dynamic model of the offshore platform (1) can be written as
(4)
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
+
D
f
(
t
)
,
x
(
0
)
=
x
0
,
where
(5)
A
=
[
0
0
1
0
0
0
0
1
-
(
ω
1
2
+
ω
2
2
m
2
m
1
)
ω
2
2
m
2
m
1
-
2
(
ξ
1
ω
1
+
ξ
2
ω
2
m
2
m
1
)
2
ξ
2
ω
2
m
2
m
1
ω
2
2
-
ω
2
2
2
ξ
2
ω
2
-
2
ξ
2
ω
2
]
,
B
=
[
0
0
-
1
m
1
1
m
2
]
T
,
D
=
[
0
0
1
m
1
0
]
T
.To guarantee the performance index from the wave forcef
(
t
) to the control output ξ
(
t
) to be realized with the specified requirement, the displacement response x
1
(
t
) and the velocity response x
3
(
t
) of the offshore platform are chosen as the control output. In this situation, the control output equation is given as
(6)
ξ
(
t
)
=
C
1
x
(
t
)
,
where
(7)
C
1
=
[
1
0
0
0
0
0
1
0
]
.In practice, for offshore platform system, only sampled measurements of state variables are available at discrete instants of time. Without loss of generality, it is assumed that the state variables of the active vibration system are measured at time instants0
=
t
0
<
t
1
<
⋯
<
t
k
<
t
k
+
1
…; that is, only x
(
t
k
) is available for interval [
t
k
,
t
k
+
1
), where t
k
+
1
-
t
k
≤
h and h is the maximum sampling interval. We are interested in designing a state feedback controller, which may contain missing measurements described by
(8)
u
(
t
)
=
u
(
t
k
)
=
r
(
t
k
)
K
x
(
t
k
)
,
t
k
≤
t
<
t
k
+
1
,
where K is the controller gain matrix to be designed and r
(
t
k
)
∈
R is the stochastic variable having the probabilistic density function q
(
r
(
t
k
)
) on the interval [
0,1
] with mathematical expectation b and variance σ
2; that is,
(9)
E
{
r
(
t
k
)
}
=
b
,
E
{
r
2
(
t
k
)
}
=
[
E
{
r
(
t
k
)
}
]
2
+
σ
2
=
b
2
+
σ
2
,
where b
∈
R and σ
∈
R are known positive scalars.Remark 1.
In real systems, due to various reasons, such as sensor aging and sensor temporal failure, the measurements missing at one moment might be partial and therefore the missing probability cannot be simply described by 0 or 1 [24]. It is easy to see that the widely used Bernoulli distribution is included here as a special case.
According to (4) and (8), the closed-loop system is given by
(10)
x
˙
(
t
)
=
A
x
(
t
)
+
r
(
t
k
)
B
K
x
(
t
k
)
+
D
f
(
t
)
,
ξ
(
t
)
=
C
1
x
(
t
)
,
t
k
≤
t
<
t
k
+
1
.
It should be noticed that the closed-loop system (10) is actually a stochastic system, since it contains the stochastic quantity r
(
t
k
). Therefore, in the sequel, we will use the notion of stochastic stability in the mean-square sense.
Here, the issue of vibration attenuation for the offshore platform can be transformed into anH
∞ disturbance attenuation problem of system (10). The objective is to determine the controller (8) such that(1)
the closed-loop system (10) with f
(
t
)
=
0 is asymptotically mean-square stable;
(2)
under zero initial condition, theH
∞ performance
(11)
E
{
∥
ξ
(
t
)
∥
2
}
≤
γ
E
{
∥
f
(
t
)
∥
2
}
,
of the closed-loop system (10) is guaranteed for nonzero f
(
t
)
∈
L
2
[
0
,
∞
) and a prescribed γ
>
0.
To obtain the main results, the following lemma is needed.Lemma 2 (Schur complement [24]).
Given constant matricesS
1
,
S
2, and S
3, where S
1
=
S
1
T and 0
<
S
2
=
S
2
T, then S
1
+
S
3
T
S
2
-
1
S
3
<
0 if and only if
(12)
[
S
1
S
3
T
S
3
-
S
2
]
<
0
o
r
[
-
S
2
S
3
S
3
T
S
1
]
<
0
.


## 3. Main Results

In this section, we will solve the problem of robust sampled-dataH
∞ controller design for systems (4) and (6) with missing measurements by the recently developed input delay approach [27–29]. The key idea behind this approach is that we represent the sampling instant t
k as
(13)
t
k
=
t
-
(
t
-
t
k
)
=
t
-
d
(
t
)
,
where d
(
t
)
=
t
-
t
k. Then, from (8) and (13), we obtain
(14)
u
(
t
)
=
u
(
t
k
)
=
u
(
t
-
d
(
t
)
)
=
r
(
t
-
d
(
t
)
)
K
x
(
t
-
d
(
t
)
)
,
t
k
≤
t
<
t
k
+
1
,
where u
(
t
k
) is a discrete-time control signal and the time-varying delay d
(
t
) is piece-wise linear and satisfies d
(
t
)
=
t
-
t
k
≤
h and d
˙
(
t
)
=
1 for t
≠
t
k. By using the input delay approach, the closed-loop system (10) can be transformed into a time-delay system as follows:
(15)
x
˙
(
t
)
=
A
x
(
t
)
+
r
(
t
-
d
(
t
)
)
B
K
x
(
t
-
d
(
t
)
)
+
D
f
(
t
)
,
ξ
(
t
)
=
C
1
x
(
t
)
.In the following, we will determine the controller gain matrixK such that system (15) is asymptotically mean-square stable and satisfies the H
∞ disturbance attenuation in (11).Remark 3.
Recently, the vibration control problem for offshore platforms with constant input delay has been addressed in [10, 11]. It is worth pointing out that the transformed system in our problem contains nondifferentiable time-varying delay in the states, which hinders the results in [10, 11] to be directly applied to the problem considered here.


### 3.1.H
∞ Performance Analysis

The following theorem provides a sufficient condition for system (15) to be asymptotically mean-square stable and satisfy the H
∞ disturbance attenuation in (11).Theorem 4.
Given scalarsh
>
0, b
>
0, σ
>
0 and controller gain matrix K, the closed-loop system (15) with f
(
t
)
=
0 is asymptotically mean-square stable and the H
∞ disturbance attenuation in (11) is satisfied for the wave force f
(
t
)
∈
L
2
[
0
,
∞
) and a prescribed γ
>
0, if there exist matrices P
>
0, Q
>
0, S
11
>
0, S
12, S
13, S
22
>
0, and S
23 with appropriate dimensions such that the following LMIs hold:
(16)
S
=
[
S
11
S
12
S
13
*
S
22
S
23
*
*
Q
]
≥
0
,
(17)
[
Ψ
11
Ψ
12
P
D
h
A
T
Q
0
C
1
T
*
Ψ
22
0
h
b
K
T
B
T
Q
h
σ
K
T
B
T
Q
0
*
*
-
γ
2
I
h
D
T
Q
0
0
*
*
*
-
Q
0
0
*
*
*
*
-
Q
0
*
*
*
*
*
-
I
]
<
0
,
where
(18)
Ψ
11
=
P
A
+
A
T
P
+
h
S
11
+
S
13
+
S
13
T
,
Ψ
12
=
b
P
B
K
+
h
S
12
-
S
13
+
S
23
T
,
Ψ
22
=
h
S
22
-
S
23
-
S
23
T
.Proof.
Considering a Lyapunov-Krasovskii functional as follows:(19)
V
(
t
)
=
V
1
(
t
)
+
V
2
(
t
)
=
x
T
(
t
)
P
x
(
t
)
+
∫
-
h
0
∫
t
+
θ
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
d
θ
,
where P
>
0 and Q
>
0 are matrices to be determined.
The infinitesimal operatorℒ of V
(
t
) is defined as [26]
(20)
ℒ
V
(
t
)
=
lim
⁡
Δ
→
0
+
1
Δ
{
E
{
V
(
t
+
Δ
)
-
V
(
t
)
}
}
.
Combining (19) and (20) yields
(21)
ℒ
V
(
t
)
=
ℒ
V
1
(
t
)
+
ℒ
V
2
(
t
)
,
where
(22)
ℒ
V
1
=
2
E
{
x
T
(
t
)
P
x
˙
(
t
)
}
=
2
E
{
x
T
(
t
)
P
(
A
x
(
t
)
+
r
(
t
)
B
K
x
(
t
-
d
(
t
)
)
+
D
f
(
t
)
)
}
=
x
T
(
t
)
(
P
A
+
A
T
P
)
x
(
t
)
+
2
x
T
(
t
)
b
P
B
K
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
P
D
f
(
t
)
.
(23)
ℒ
V
2
=
h
E
{
x
˙
T
(
t
)
Q
x
˙
(
t
)
}
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
h
(
x
T
(
t
)
A
T
Q
A
x
(
t
)
+
x
T
(
t
)
(
b
A
T
Q
B
K
+
(
b
A
T
Q
B
K
)
T
)
x
(
t
-
d
(
t
)
)
+
x
T
(
t
)
A
T
Q
D
f
(
t
)
+
x
T
(
t
-
d
(
t
)
)
α
K
T
B
T
Q
B
K
x
(
t
-
d
(
t
)
)
+
x
T
(
t
-
d
(
t
)
)
b
K
T
B
T
Q
D
f
(
t
)
+
f
T
(
t
)
D
T
Q
D
f
(
t
)
)
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
.
where α
=
E
{
r
2
(
t
-
d
(
t
)
)
}
=
E
{
r
2
(
t
)
}
=
b
2
+
σ
2.
From (16), we have
(24)
0
≤
E
∫
t
-
d
(
t
)
t
[
x
(
t
)
x
(
t
-
d
(
t
)
)
x
˙
(
s
)
]
T
[
S
11
S
12
S
13
*
S
22
S
23
*
*
Q
]
×
[
x
(
t
)
x
(
t
-
d
(
t
)
)
x
˙
(
s
)
]
d
s
=
d
(
t
)
x
T
(
t
)
S
11
x
(
t
)
+
2
d
(
t
)
x
T
(
t
)
S
12
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
S
13
∫
t
-
d
(
t
)
t
x
˙
(
s
)
d
s
+
d
(
t
)
x
T
(
t
-
d
(
t
)
)
S
22
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
-
d
(
t
)
)
S
23
E
{
∫
t
-
d
(
t
)
t
x
˙
(
s
)
d
s
}
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
≤
h
x
T
(
t
)
S
11
x
(
t
)
+
2
h
x
T
(
t
)
S
12
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
S
13
(
x
(
t
)
-
x
(
t
-
d
(
t
)
)
)
+
h
x
T
(
t
-
d
(
t
)
)
S
22
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
-
d
(
t
)
)
S
23
(
x
(
t
)
-
x
(
t
-
d
(
t
)
)
)
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
x
T
(
t
)
(
h
S
11
+
S
13
+
S
13
T
)
x
(
t
)
+
2
x
T
(
t
)
(
h
S
12
-
S
13
+
S
23
T
)
x
(
t
-
d
(
t
)
)
+
x
T
(
t
-
d
(
t
)
)
(
h
S
22
-
S
23
-
S
23
T
)
x
(
t
-
d
(
t
)
)
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
.
Combining (21)–(24), one yields
(25)
ℒ
V
(
t
)
+
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
≤
[
x
(
t
)
x
(
t
-
d
(
t
)
)
f
(
t
)
]
T
Π
[
x
(
t
)
x
(
t
-
d
(
t
)
)
f
(
t
)
]
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
η
T
(
t
)
Π
η
(
t
)
-
E
{
∫
t
-
h
t
-
d
(
t
)
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
,
where
(26)
η
(
t
)
=
[
x
T
(
t
)
x
T
(
t
-
d
(
t
)
)
f
T
(
t
)
]
T
,
Π
=
[
Φ
11
Φ
12
P
D
+
h
A
T
Q
D
*
Φ
22
h
b
K
T
B
T
Q
D
*
*
h
D
T
R
D
-
γ
2
I
]
,
Φ
11
=
P
A
+
A
T
P
+
h
A
T
Q
A
+
h
S
11
+
S
13
+
S
13
T
+
C
1
T
C
1
,
Φ
12
=
b
P
B
K
+
h
b
A
T
Q
B
K
+
h
S
12
-
S
13
+
S
23
T
,
Φ
22
=
h
α
K
T
B
T
Q
B
K
+
h
S
22
-
S
23
-
S
23
T
.
By Lemma2, it can be obtained from (17) that
(27)
Π
<
0
.
It follows from (25)–(27) that
(28)
ℒ
V
(
t
)
+
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
<
0
.
Whenf
(
t
)
=
0, it is easy to get from (28) that ℒ
V
(
t
)
<
0, which means that system (15) with f
(
t
)
=
0 is asymptotically mean-square stable.
On the other hand, integrating both sides of (28) from 0 to ∞ and noting the fact that V
(
0
)
=
0 under zero initial condition and V
(
∞
)
≥
0, we obtain
(29)
∫
0
∞
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
d
t
<
0
,
which indicates that the H
∞ disturbance attenuation (11) is guaranteed.
This completes the proof.


### 3.2. Sampled-DataH
∞ Controller Design

In this section, the controller design problem is solved in the following and the controller gain matrix is given in terms of a solution to LMIs.Theorem 5.
Given scalarsh
>
0, b
>
0 and σ
>
0, the closed-loop system (15) with f
(
t
)
=
0 is asymptotically mean-square stable and the H
∞ disturbance attenuation in (11) is satisfied for the wave force f
(
t
)
∈
L
2
[
0
,
∞
) and a prescribed γ
>
0, if there exist matrices X
>
0, Q
^
>
0, W
>
0, S
-
11
>
0, S
-
12, S
-
13, S
-
22
>
0, and S
-
23 with appropriate dimensions such that the following LMIs hold:
(30)
[
S
-
11
S
-
12
S
-
13
*
S
-
22
S
-
23
*
*
2
X
-
Q
^
]
≥
0
,
[
Ψ
-
11
Ψ
-
12
D
h
X
A
T
0
X
C
1
T
*
Ψ
-
22
0
h
b
W
T
B
T
h
σ
W
T
B
T
0
*
*
-
γ
2
I
h
D
T
0
0
*
*
*
-
Q
^
0
0
*
*
*
*
-
Q
^
0
*
*
*
*
*
-
I
]
<
0
,
where
(31)
Ψ
-
11
=
A
X
+
X
A
T
+
h
S
-
11
+
S
-
13
+
S
-
13
T
,
Ψ
-
12
=
b
B
W
+
h
S
-
12
-
S
-
13
+
S
-
23
T
,
Ψ
-
22
=
h
S
-
22
-
S
-
23
-
S
-
23
T
.
Moreover, if inequalities (30) have a feasible solution, then the controller gain matrix in (8) is given by
(32)
K
=
W
X
-
1
.Proof.
By pre- and postmultiplying (16) by diag
⁡
{
P
-
1
,
P
-
1
,
P
-
1
} and pre- and postmultiplying (17) by diag
⁡
{
P
-
1
,
P
-
1
,
I
,
Q
-
1
,
Q
-
1
,
I
}, (16) and (17) can be converted into the following equivalent inequalities:
(33)
[
S
-
11
S
-
12
S
-
13
*
S
-
22
S
-
23
*
*
Q
-
]
≥
0
,
[
Ψ
-
11
Ψ
-
12
D
h
P
-
1
A
T
0
P
-
1
C
1
T
*
Ψ
-
22
0
h
b
P
-
1
K
T
B
T
h
σ
P
-
1
K
T
B
T
0
*
*
-
γ
2
I
h
D
T
0
0
*
*
*
-
Q
-
1
0
0
*
*
*
*
-
Q
-
1
0
*
*
*
*
*
-
I
]
<
0
,
where
(34)
S
-
11
=
P
-
1
S
11
P
-
1
,
S
-
12
=
P
-
1
S
12
P
-
1
,
S
-
13
=
P
-
1
S
13
P
-
1
,
S
-
22
=
P
-
1
S
22
P
-
1
,
S
-
23
=
P
-
1
S
23
P
-
1
,
Q
-
=
P
-
1
Q
P
-
1
,
Ψ
-
11
=
A
P
-
1
+
P
-
1
A
T
+
h
S
-
11
+
S
-
13
+
S
-
13
T
,
Ψ
-
12
=
b
B
K
P
-
1
+
h
S
-
12
-
S
-
13
+
S
-
23
T
,
Ψ
-
22
=
h
S
-
22
-
S
-
23
-
S
-
23
T
.
Noting that(
Q
-
1
-
P
-
1
)
T
Q
(
Q
-
1
-
P
-
1
)
≥
0, one has
(35)
Q
-
=
P
-
1
Q
P
-
1
≥
2
P
-
1
-
Q
-
1
.
By letting X
=
P
-
1, W
=
K
P
-
1, and Q
^
=
Q
-
1, we obtain that (33) hold if (30) is satisfied. Then, by Theorem 4, we can conclude that the theorem is true. The proof is completed.


## 3.1.H
∞ Performance Analysis

The following theorem provides a sufficient condition for system (15) to be asymptotically mean-square stable and satisfy the H
∞ disturbance attenuation in (11).Theorem 4.
Given scalarsh
>
0, b
>
0, σ
>
0 and controller gain matrix K, the closed-loop system (15) with f
(
t
)
=
0 is asymptotically mean-square stable and the H
∞ disturbance attenuation in (11) is satisfied for the wave force f
(
t
)
∈
L
2
[
0
,
∞
) and a prescribed γ
>
0, if there exist matrices P
>
0, Q
>
0, S
11
>
0, S
12, S
13, S
22
>
0, and S
23 with appropriate dimensions such that the following LMIs hold:
(16)
S
=
[
S
11
S
12
S
13
*
S
22
S
23
*
*
Q
]
≥
0
,
(17)
[
Ψ
11
Ψ
12
P
D
h
A
T
Q
0
C
1
T
*
Ψ
22
0
h
b
K
T
B
T
Q
h
σ
K
T
B
T
Q
0
*
*
-
γ
2
I
h
D
T
Q
0
0
*
*
*
-
Q
0
0
*
*
*
*
-
Q
0
*
*
*
*
*
-
I
]
<
0
,
where
(18)
Ψ
11
=
P
A
+
A
T
P
+
h
S
11
+
S
13
+
S
13
T
,
Ψ
12
=
b
P
B
K
+
h
S
12
-
S
13
+
S
23
T
,
Ψ
22
=
h
S
22
-
S
23
-
S
23
T
.Proof.
Considering a Lyapunov-Krasovskii functional as follows:(19)
V
(
t
)
=
V
1
(
t
)
+
V
2
(
t
)
=
x
T
(
t
)
P
x
(
t
)
+
∫
-
h
0
∫
t
+
θ
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
d
θ
,
where P
>
0 and Q
>
0 are matrices to be determined.
The infinitesimal operatorℒ of V
(
t
) is defined as [26]
(20)
ℒ
V
(
t
)
=
lim
⁡
Δ
→
0
+
1
Δ
{
E
{
V
(
t
+
Δ
)
-
V
(
t
)
}
}
.
Combining (19) and (20) yields
(21)
ℒ
V
(
t
)
=
ℒ
V
1
(
t
)
+
ℒ
V
2
(
t
)
,
where
(22)
ℒ
V
1
=
2
E
{
x
T
(
t
)
P
x
˙
(
t
)
}
=
2
E
{
x
T
(
t
)
P
(
A
x
(
t
)
+
r
(
t
)
B
K
x
(
t
-
d
(
t
)
)
+
D
f
(
t
)
)
}
=
x
T
(
t
)
(
P
A
+
A
T
P
)
x
(
t
)
+
2
x
T
(
t
)
b
P
B
K
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
P
D
f
(
t
)
.
(23)
ℒ
V
2
=
h
E
{
x
˙
T
(
t
)
Q
x
˙
(
t
)
}
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
h
(
x
T
(
t
)
A
T
Q
A
x
(
t
)
+
x
T
(
t
)
(
b
A
T
Q
B
K
+
(
b
A
T
Q
B
K
)
T
)
x
(
t
-
d
(
t
)
)
+
x
T
(
t
)
A
T
Q
D
f
(
t
)
+
x
T
(
t
-
d
(
t
)
)
α
K
T
B
T
Q
B
K
x
(
t
-
d
(
t
)
)
+
x
T
(
t
-
d
(
t
)
)
b
K
T
B
T
Q
D
f
(
t
)
+
f
T
(
t
)
D
T
Q
D
f
(
t
)
)
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
.
where α
=
E
{
r
2
(
t
-
d
(
t
)
)
}
=
E
{
r
2
(
t
)
}
=
b
2
+
σ
2.
From (16), we have
(24)
0
≤
E
∫
t
-
d
(
t
)
t
[
x
(
t
)
x
(
t
-
d
(
t
)
)
x
˙
(
s
)
]
T
[
S
11
S
12
S
13
*
S
22
S
23
*
*
Q
]
×
[
x
(
t
)
x
(
t
-
d
(
t
)
)
x
˙
(
s
)
]
d
s
=
d
(
t
)
x
T
(
t
)
S
11
x
(
t
)
+
2
d
(
t
)
x
T
(
t
)
S
12
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
S
13
∫
t
-
d
(
t
)
t
x
˙
(
s
)
d
s
+
d
(
t
)
x
T
(
t
-
d
(
t
)
)
S
22
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
-
d
(
t
)
)
S
23
E
{
∫
t
-
d
(
t
)
t
x
˙
(
s
)
d
s
}
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
≤
h
x
T
(
t
)
S
11
x
(
t
)
+
2
h
x
T
(
t
)
S
12
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
)
S
13
(
x
(
t
)
-
x
(
t
-
d
(
t
)
)
)
+
h
x
T
(
t
-
d
(
t
)
)
S
22
x
(
t
-
d
(
t
)
)
+
2
x
T
(
t
-
d
(
t
)
)
S
23
(
x
(
t
)
-
x
(
t
-
d
(
t
)
)
)
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
x
T
(
t
)
(
h
S
11
+
S
13
+
S
13
T
)
x
(
t
)
+
2
x
T
(
t
)
(
h
S
12
-
S
13
+
S
23
T
)
x
(
t
-
d
(
t
)
)
+
x
T
(
t
-
d
(
t
)
)
(
h
S
22
-
S
23
-
S
23
T
)
x
(
t
-
d
(
t
)
)
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
.
Combining (21)–(24), one yields
(25)
ℒ
V
(
t
)
+
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
≤
[
x
(
t
)
x
(
t
-
d
(
t
)
)
f
(
t
)
]
T
Π
[
x
(
t
)
x
(
t
-
d
(
t
)
)
f
(
t
)
]
-
E
{
∫
t
-
h
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
+
E
{
∫
t
-
d
(
t
)
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
=
η
T
(
t
)
Π
η
(
t
)
-
E
{
∫
t
-
h
t
-
d
(
t
)
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
}
,
where
(26)
η
(
t
)
=
[
x
T
(
t
)
x
T
(
t
-
d
(
t
)
)
f
T
(
t
)
]
T
,
Π
=
[
Φ
11
Φ
12
P
D
+
h
A
T
Q
D
*
Φ
22
h
b
K
T
B
T
Q
D
*
*
h
D
T
R
D
-
γ
2
I
]
,
Φ
11
=
P
A
+
A
T
P
+
h
A
T
Q
A
+
h
S
11
+
S
13
+
S
13
T
+
C
1
T
C
1
,
Φ
12
=
b
P
B
K
+
h
b
A
T
Q
B
K
+
h
S
12
-
S
13
+
S
23
T
,
Φ
22
=
h
α
K
T
B
T
Q
B
K
+
h
S
22
-
S
23
-
S
23
T
.
By Lemma2, it can be obtained from (17) that
(27)
Π
<
0
.
It follows from (25)–(27) that
(28)
ℒ
V
(
t
)
+
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
<
0
.
Whenf
(
t
)
=
0, it is easy to get from (28) that ℒ
V
(
t
)
<
0, which means that system (15) with f
(
t
)
=
0 is asymptotically mean-square stable.
On the other hand, integrating both sides of (28) from 0 to ∞ and noting the fact that V
(
0
)
=
0 under zero initial condition and V
(
∞
)
≥
0, we obtain
(29)
∫
0
∞
E
{
ξ
T
(
t
)
ξ
(
t
)
-
γ
2
f
T
(
t
)
f
(
t
)
}
d
t
<
0
,
which indicates that the H
∞ disturbance attenuation (11) is guaranteed.
This completes the proof.


## 3.2. Sampled-DataH
∞ Controller Design

In this section, the controller design problem is solved in the following and the controller gain matrix is given in terms of a solution to LMIs.Theorem 5.
Given scalarsh
>
0, b
>
0 and σ
>
0, the closed-loop system (15) with f
(
t
)
=
0 is asymptotically mean-square stable and the H
∞ disturbance attenuation in (11) is satisfied for the wave force f
(
t
)
∈
L
2
[
0
,
∞
) and a prescribed γ
>
0, if there exist matrices X
>
0, Q
^
>
0, W
>
0, S
-
11
>
0, S
-
12, S
-
13, S
-
22
>
0, and S
-
23 with appropriate dimensions such that the following LMIs hold:
(30)
[
S
-
11
S
-
12
S
-
13
*
S
-
22
S
-
23
*
*
2
X
-
Q
^
]
≥
0
,
[
Ψ
-
11
Ψ
-
12
D
h
X
A
T
0
X
C
1
T
*
Ψ
-
22
0
h
b
W
T
B
T
h
σ
W
T
B
T
0
*
*
-
γ
2
I
h
D
T
0
0
*
*
*
-
Q
^
0
0
*
*
*
*
-
Q
^
0
*
*
*
*
*
-
I
]
<
0
,
where
(31)
Ψ
-
11
=
A
X
+
X
A
T
+
h
S
-
11
+
S
-
13
+
S
-
13
T
,
Ψ
-
12
=
b
B
W
+
h
S
-
12
-
S
-
13
+
S
-
23
T
,
Ψ
-
22
=
h
S
-
22
-
S
-
23
-
S
-
23
T
.
Moreover, if inequalities (30) have a feasible solution, then the controller gain matrix in (8) is given by
(32)
K
=
W
X
-
1
.Proof.
By pre- and postmultiplying (16) by diag
⁡
{
P
-
1
,
P
-
1
,
P
-
1
} and pre- and postmultiplying (17) by diag
⁡
{
P
-
1
,
P
-
1
,
I
,
Q
-
1
,
Q
-
1
,
I
}, (16) and (17) can be converted into the following equivalent inequalities:
(33)
[
S
-
11
S
-
12
S
-
13
*
S
-
22
S
-
23
*
*
Q
-
]
≥
0
,
[
Ψ
-
11
Ψ
-
12
D
h
P
-
1
A
T
0
P
-
1
C
1
T
*
Ψ
-
22
0
h
b
P
-
1
K
T
B
T
h
σ
P
-
1
K
T
B
T
0
*
*
-
γ
2
I
h
D
T
0
0
*
*
*
-
Q
-
1
0
0
*
*
*
*
-
Q
-
1
0
*
*
*
*
*
-
I
]
<
0
,
where
(34)
S
-
11
=
P
-
1
S
11
P
-
1
,
S
-
12
=
P
-
1
S
12
P
-
1
,
S
-
13
=
P
-
1
S
13
P
-
1
,
S
-
22
=
P
-
1
S
22
P
-
1
,
S
-
23
=
P
-
1
S
23
P
-
1
,
Q
-
=
P
-
1
Q
P
-
1
,
Ψ
-
11
=
A
P
-
1
+
P
-
1
A
T
+
h
S
-
11
+
S
-
13
+
S
-
13
T
,
Ψ
-
12
=
b
B
K
P
-
1
+
h
S
-
12
-
S
-
13
+
S
-
23
T
,
Ψ
-
22
=
h
S
-
22
-
S
-
23
-
S
-
23
T
.
Noting that(
Q
-
1
-
P
-
1
)
T
Q
(
Q
-
1
-
P
-
1
)
≥
0, one has
(35)
Q
-
=
P
-
1
Q
P
-
1
≥
2
P
-
1
-
Q
-
1
.
By letting X
=
P
-
1, W
=
K
P
-
1, and Q
^
=
Q
-
1, we obtain that (33) hold if (30) is satisfied. Then, by Theorem 4, we can conclude that the theorem is true. The proof is completed.


## 4. Simulation Results

In this section, the parameters of an offshore platform and the wave are given first. Then, the proposed robust sampled-dataH
∞ control (RSDHC) scheme will be applied to control the offshore platform. In order to verify the feasibility and effectiveness of the proposed scheme, simulation results of the numerical example are presented. Besides, the effect of missing signal r
(
t
k
) on the H
∞ control for the offshore platform will be investigated.


### 4.1. The Parameters of the Offshore Platform and the External Wave Force

In Figure1, the masses, natural frequencies, and the damping ratios of the offshore platform and the AMD and other related parameters of the offshore platform are the same as those in [11], which are listed in Table 1, where L denotes the length of the offshore platform and D
~ represents the diameter of the cylinder. Based on the settings in Table 1, the matrices A, B, and D in (3) can be calculated as
(36)
A
=
[
0
0
1.0000
0
0
0
0
1.0000
-
4.2290
0.0403
-
0.0899
0.0080
4.0297
-
4.0297
0.8030
-
0.8030
]
,
B
=
1
0
-
4
×
[
0
0
-
0.0013
0.1278
]
T
,
D
=
1
0
-
6
×
[
0
0
0.1278
0
]
T
.Table 1
Parameters of the offshore platform system with an AMD mechanism.
Description
Symbol
Value
Unit
Length of the offshore platform
L
249
m
Diameter of the cylinder
D
~
1.83
m
Mass of the offshore platform
m
1
7,825,307
kg
Mass of the AMD
m
2
78,253
kg
Natural frequency of the offshore platform
ω
1
2.0466
rad/s
Natural frequency of the AMD
ω
2
2.0074
rad/s
Damping ration of the offshore platform
ξ
1
0.02
—
Damping ration of the AMD
ξ
2
0.2
—To investigate the effectiveness of the proposed control scheme, we need to compute the values of wave force, which can be solved by referring to [6]. The parameters regarding the external wave are taken from [6] and listed in Table 2, where H
s is the significant wave height, d is the water depth, ω
0 is the peak frequency, C
d and C
m are the drag and inertia coefficients, respectively, γ
- is the peakedness coefficient, and ρ is the fluid density. Based on the settings in Table 2, the power spectrum density (PSD) of wave elevation is shown in Figure 2. The shape function ϕ
(
z
) is given as [6]
(37)
ϕ
(
z
)
=
1
-
cos
⁡
(
π
z
2
L
)
,
0
≤
z
≤
L
,
where z is the vertical coordinate with the origin at the sea floor.Table 2
Parameters of the wave acting on the offshore platform.
Description
Symbol
Value
Unit
Significant wave height
H
s
4
m
Water depth
d
13.2
m
Peak frequency of wave
ω
0
0.87
rad/s
Drag coefficient
C
d
1.2
—
Inertia coefficient
C
m
2.0
—
Peakedness coefficient
γ
-
3.3
—
Density of water
ρ
1025.6
kg/m3Figure 2
PSD of wave elevation.Then, one can compute the wave forcef
(
t
) according to [6], which is shown in Figure 3.Figure 3
Wave force acting on the offshore structure.Due to the irregular nature of the wave force adopted in this paper, we investigate both the peak values and the root mean square (RMS) values of displacement and velocity of the offshore platform and the control force [11]. The same terms M
d, M
v, and M
u are used to represent the peak values of displacement, velocity of the offshore platform, and the required control force, respectively. J
d, J
v, and J
u denote the RMS values of displacement, velocity of the offshore platform, and the control force, respectively, where
(38)
M
d
=
max
⁡
{
|
x
1
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
d
=
1
T
s
∫
0
T
x
1
2
(
t
)
d
t
,
M
v
=
max
⁡
{
|
x
3
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
v
=
1
T
s
∫
0
T
x
3
2
(
t
)
d
t
,
M
u
=
max
⁡
{
|
u
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
u
=
1
T
s
∫
0
T
u
2
(
t
)
d
t
,
with T
s as a given measurement period.


### 4.2. Simulation Results of RSDHC for Offshore Platform with Missing Measurements

In this section, simulation results of RSDHC for offshore platform with missing measurements are presented. Leth
=
0.02
s, γ
=
0.3 and let the probabilistic density function of r
(
t
k
) in [
0
,
1
] be described by
(39)
q
(
r
(
t
k
)
)
=
{
0
,
r
(
t
k
)
=
0
,
0.1
,
r
(
t
k
)
=
0.5
,
0.9
,
r
(
t
k
)
=
1
,
from which the expectation and the standard deviation can be easily calculated as b
=
0.95 and σ
=
0.15. Then, by Theorem 5, the controller gain matrix K in (8) can be obtained as
(40)
K
=
1
0
5
×
[
7.9703
0.0595
1.1646
-
0.0747
]
.We give the displacement, velocity, and acceleration curves of the offshore platform system without control and with RSDHC in Figures4, 5, and 6, respectively. The corresponding control curves are presented in Figure 7. When the RSDHC is used to control the offshore platform, the peak values of displacement and velocity of the offshore platform are reduced from 0.1264 m and 0.2419 m/s to 0.0719 m and 0.1149 m/s, respectively. The maximum control force required is about 3.7397
×
1
0
4
N. The RMS values of displacement and the velocity are reduced from 0.0424 m and 0.0826 m/s to 0.0223 m and 0.0365 m/s, respectively, and the RMS value of control force is about 1.4539
×
1
0
4
N.Figure 4
Displacement of offshore platform.Figure 5
Velocity of offshore platform.Figure 6
Acceleration of offshore platform.Figure 7
Control force of offshore platform.The peak and RMS values of displacement, velocity of the offshore platform under no control and RSDHC, and the corresponding control force for different values ofγ are listed in Table 3, where b
=
0.95 and σ
=
0.15. Based on Table 3, one can yield the average reduction of peak and RMS values of the offshore platform under RSDHC with missing measurements, which are presented in Table 4.Table 3
The peak and RMS values of displacement and velocity for different values ofγ.
Peak value
RMS value
M
d (m)
M
v (m/s)
M
u (104 N)
J
d (m)
J
v (m/s)
J
u (104 N)
No control
0.1264
0.2419
0.0424
0.0826
γ
=
0.2
0.0699
0.1114
3.9404
0.0220
0.0355
1.5610
γ
=
0.3
0.0719
0.1149
3.7397
0.0223
0.0365
1.4539
γ
=
0.9
0.0720
0.1150
4.6481
0.0223
0.0365
1.7257Table 4
The average peak and RMS values reduction of displacement and velocity.
Controller
Reduction of peak value
Reduction of RMS value
Displacement (%)
Velocity (%)
Displacement (%)
Velocity (%)
RSDHC
43.62
52.97
47.64
56.21From Figures4–6 and Tables 3-4, it can be found that the proposed RSDHC scheme is effective. The peak and RMS values of the displacement are reduced by about 43 percent and 47 percent of the ones without control, respectively, and the peak and RMS values of the velocity are reduced by 52 percent and 56 percent of the ones without control, respectively. From Figure 6, we can see that the acceleration of offshore platform has been decreased significantly.


### 4.3. Effects of the Missing Signalr
(
t
k
) on RSDHC for Offshore Platform

Now, we turn to investigate the effects of the artificially introduced missing signalr
(
t
k
) on robust sampled-data H
∞ control for offshore structure. First, we give four missing signals with different probabilistic density functions, and their expectations and standard deviations are computed, which are listed in Table 5. For the cases of the different introduced missing signals, the peak and RMS values of displacement, velocity of the offshore platform, and the control force are computed and listed in Table 6, where γ
=
0.3. Based on Table 6, one can yield the reduction of peak and RMS values of the offshore platform under RSDHC, which are presented in Table 7.Table 5
Probabilistic density functions, expectations, and standard deviations of missing signalr
(
t
k
).
Probabilistic density function
Expectation
Standard deviation
q
1
(
r
1
(
t
k
)
)
0
0.1
0.9
0.95
0.15
r
1
(
t
k
)
0
0.5
1
q
2
(
r
2
(
t
k
)
)
0.2
0.2
0.6
0.7
0.4
r
2
(
t
k
)
0
0.5
1
q
3
(
r
3
(
t
k
)
)
0.3
0.5
0.2
0.45
0.35
r
3
(
t
k
)
0
0.5
1
q
4
(
r
4
(
t
k
)
)
0.6
0.3
0.1
0.25
0.34
r
4
(
t
k
)
0
0.5
1Table 6
The peak and RMS values of displacement and velocity for different missing signalsr(t
k).
Peak value
RMS value
M
d (m)
M
v (m/s)
M
u (104 N)
J
d (m)
J
v (m/s)
J
u (104 N)
No control
0.1264
0.2419
0.0424
0.0826
No missing
0.0690
0.1100
3.7989
0.0218
0.0351
1.5415
b
=
0.95, σ
=
0.15
0.0719
0.1149
3.7397
0.0223
0.0365
1.4539
b
=
0.70, σ
=
0.40
0.0746
0.1191
5.7289
0.0229
0.0377
2.0809
b
=
0.45, σ
=
0.35
0.0821
0.1370
11.570
0.0246
0.0420
3.7421
b
=
0.25, σ
=
0.34
0.0981
0.1760
5.8798
0.0290
0.0530
1.7157Table 7
The peak and RMS values reduction of displacement and velocity.
Reduction of peak value
Reduction of RMS value
Displacement (%)
Velocity (%)
Displacement (%)
Velocity (%)
No missing
45.41
54.53
48.58
57.51
b
=
0.95, σ
=
0.15
43.12
52.50
47.41
55.81
b
=
0.70, σ
=
0.40
40.98
50.76
45.99
54.36
b
=
0.45, σ
=
0.35
35.05
43.37
41.98
49.15
b
=
0.25, σ
=
0.34
22.39
27.24
31.60
35.84From Tables6 and 7, it can be seen that the proposed RSDHC scheme is effective to control the offshore platform, even though the expectation and standard deviation of the missing signal are 0.25 and 0.34, respectively. In addition, we can find that the peak and RMS values reduction of displacement and velocity become smaller with the decrease of b and σ.


## 4.1. The Parameters of the Offshore Platform and the External Wave Force

In Figure1, the masses, natural frequencies, and the damping ratios of the offshore platform and the AMD and other related parameters of the offshore platform are the same as those in [11], which are listed in Table 1, where L denotes the length of the offshore platform and D
~ represents the diameter of the cylinder. Based on the settings in Table 1, the matrices A, B, and D in (3) can be calculated as
(36)
A
=
[
0
0
1.0000
0
0
0
0
1.0000
-
4.2290
0.0403
-
0.0899
0.0080
4.0297
-
4.0297
0.8030
-
0.8030
]
,
B
=
1
0
-
4
×
[
0
0
-
0.0013
0.1278
]
T
,
D
=
1
0
-
6
×
[
0
0
0.1278
0
]
T
.Table 1
Parameters of the offshore platform system with an AMD mechanism.
Description
Symbol
Value
Unit
Length of the offshore platform
L
249
m
Diameter of the cylinder
D
~
1.83
m
Mass of the offshore platform
m
1
7,825,307
kg
Mass of the AMD
m
2
78,253
kg
Natural frequency of the offshore platform
ω
1
2.0466
rad/s
Natural frequency of the AMD
ω
2
2.0074
rad/s
Damping ration of the offshore platform
ξ
1
0.02
—
Damping ration of the AMD
ξ
2
0.2
—To investigate the effectiveness of the proposed control scheme, we need to compute the values of wave force, which can be solved by referring to [6]. The parameters regarding the external wave are taken from [6] and listed in Table 2, where H
s is the significant wave height, d is the water depth, ω
0 is the peak frequency, C
d and C
m are the drag and inertia coefficients, respectively, γ
- is the peakedness coefficient, and ρ is the fluid density. Based on the settings in Table 2, the power spectrum density (PSD) of wave elevation is shown in Figure 2. The shape function ϕ
(
z
) is given as [6]
(37)
ϕ
(
z
)
=
1
-
cos
⁡
(
π
z
2
L
)
,
0
≤
z
≤
L
,
where z is the vertical coordinate with the origin at the sea floor.Table 2
Parameters of the wave acting on the offshore platform.
Description
Symbol
Value
Unit
Significant wave height
H
s
4
m
Water depth
d
13.2
m
Peak frequency of wave
ω
0
0.87
rad/s
Drag coefficient
C
d
1.2
—
Inertia coefficient
C
m
2.0
—
Peakedness coefficient
γ
-
3.3
—
Density of water
ρ
1025.6
kg/m3Figure 2
PSD of wave elevation.Then, one can compute the wave forcef
(
t
) according to [6], which is shown in Figure 3.Figure 3
Wave force acting on the offshore structure.Due to the irregular nature of the wave force adopted in this paper, we investigate both the peak values and the root mean square (RMS) values of displacement and velocity of the offshore platform and the control force [11]. The same terms M
d, M
v, and M
u are used to represent the peak values of displacement, velocity of the offshore platform, and the required control force, respectively. J
d, J
v, and J
u denote the RMS values of displacement, velocity of the offshore platform, and the control force, respectively, where
(38)
M
d
=
max
⁡
{
|
x
1
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
d
=
1
T
s
∫
0
T
x
1
2
(
t
)
d
t
,
M
v
=
max
⁡
{
|
x
3
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
v
=
1
T
s
∫
0
T
x
3
2
(
t
)
d
t
,
M
u
=
max
⁡
{
|
u
(
t
)
|
,
t
∈
[
0
,
T
s
]
}
,
J
u
=
1
T
s
∫
0
T
u
2
(
t
)
d
t
,
with T
s as a given measurement period.


## 4.2. Simulation Results of RSDHC for Offshore Platform with Missing Measurements

In this section, simulation results of RSDHC for offshore platform with missing measurements are presented. Leth
=
0.02
s, γ
=
0.3 and let the probabilistic density function of r
(
t
k
) in [
0
,
1
] be described by
(39)
q
(
r
(
t
k
)
)
=
{
0
,
r
(
t
k
)
=
0
,
0.1
,
r
(
t
k
)
=
0.5
,
0.9
,
r
(
t
k
)
=
1
,
from which the expectation and the standard deviation can be easily calculated as b
=
0.95 and σ
=
0.15. Then, by Theorem 5, the controller gain matrix K in (8) can be obtained as
(40)
K
=
1
0
5
×
[
7.9703
0.0595
1.1646
-
0.0747
]
.We give the displacement, velocity, and acceleration curves of the offshore platform system without control and with RSDHC in Figures4, 5, and 6, respectively. The corresponding control curves are presented in Figure 7. When the RSDHC is used to control the offshore platform, the peak values of displacement and velocity of the offshore platform are reduced from 0.1264 m and 0.2419 m/s to 0.0719 m and 0.1149 m/s, respectively. The maximum control force required is about 3.7397
×
1
0
4
N. The RMS values of displacement and the velocity are reduced from 0.0424 m and 0.0826 m/s to 0.0223 m and 0.0365 m/s, respectively, and the RMS value of control force is about 1.4539
×
1
0
4
N.Figure 4
Displacement of offshore platform.Figure 5
Velocity of offshore platform.Figure 6
Acceleration of offshore platform.Figure 7
Control force of offshore platform.The peak and RMS values of displacement, velocity of the offshore platform under no control and RSDHC, and the corresponding control force for different values ofγ are listed in Table 3, where b
=
0.95 and σ
=
0.15. Based on Table 3, one can yield the average reduction of peak and RMS values of the offshore platform under RSDHC with missing measurements, which are presented in Table 4.Table 3
The peak and RMS values of displacement and velocity for different values ofγ.
Peak value
RMS value
M
d (m)
M
v (m/s)
M
u (104 N)
J
d (m)
J
v (m/s)
J
u (104 N)
No control
0.1264
0.2419
0.0424
0.0826
γ
=
0.2
0.0699
0.1114
3.9404
0.0220
0.0355
1.5610
γ
=
0.3
0.0719
0.1149
3.7397
0.0223
0.0365
1.4539
γ
=
0.9
0.0720
0.1150
4.6481
0.0223
0.0365
1.7257Table 4
The average peak and RMS values reduction of displacement and velocity.
Controller
Reduction of peak value
Reduction of RMS value
Displacement (%)
Velocity (%)
Displacement (%)
Velocity (%)
RSDHC
43.62
52.97
47.64
56.21From Figures4–6 and Tables 3-4, it can be found that the proposed RSDHC scheme is effective. The peak and RMS values of the displacement are reduced by about 43 percent and 47 percent of the ones without control, respectively, and the peak and RMS values of the velocity are reduced by 52 percent and 56 percent of the ones without control, respectively. From Figure 6, we can see that the acceleration of offshore platform has been decreased significantly.


## 4.3. Effects of the Missing Signalr
(
t
k
) on RSDHC for Offshore Platform

Now, we turn to investigate the effects of the artificially introduced missing signalr
(
t
k
) on robust sampled-data H
∞ control for offshore structure. First, we give four missing signals with different probabilistic density functions, and their expectations and standard deviations are computed, which are listed in Table 5. For the cases of the different introduced missing signals, the peak and RMS values of displacement, velocity of the offshore platform, and the control force are computed and listed in Table 6, where γ
=
0.3. Based on Table 6, one can yield the reduction of peak and RMS values of the offshore platform under RSDHC, which are presented in Table 7.Table 5
Probabilistic density functions, expectations, and standard deviations of missing signalr
(
t
k
).
Probabilistic density function
Expectation
Standard deviation
q
1
(
r
1
(
t
k
)
)
0
0.1
0.9
0.95
0.15
r
1
(
t
k
)
0
0.5
1
q
2
(
r
2
(
t
k
)
)
0.2
0.2
0.6
0.7
0.4
r
2
(
t
k
)
0
0.5
1
q
3
(
r
3
(
t
k
)
)
0.3
0.5
0.2
0.45
0.35
r
3
(
t
k
)
0
0.5
1
q
4
(
r
4
(
t
k
)
)
0.6
0.3
0.1
0.25
0.34
r
4
(
t
k
)
0
0.5
1Table 6
The peak and RMS values of displacement and velocity for different missing signalsr(t
k).
Peak value
RMS value
M
d (m)
M
v (m/s)
M
u (104 N)
J
d (m)
J
v (m/s)
J
u (104 N)
No control
0.1264
0.2419
0.0424
0.0826
No missing
0.0690
0.1100
3.7989
0.0218
0.0351
1.5415
b
=
0.95, σ
=
0.15
0.0719
0.1149
3.7397
0.0223
0.0365
1.4539
b
=
0.70, σ
=
0.40
0.0746
0.1191
5.7289
0.0229
0.0377
2.0809
b
=
0.45, σ
=
0.35
0.0821
0.1370
11.570
0.0246
0.0420
3.7421
b
=
0.25, σ
=
0.34
0.0981
0.1760
5.8798
0.0290
0.0530
1.7157Table 7
The peak and RMS values reduction of displacement and velocity.
Reduction of peak value
Reduction of RMS value
Displacement (%)
Velocity (%)
Displacement (%)
Velocity (%)
No missing
45.41
54.53
48.58
57.51
b
=
0.95, σ
=
0.15
43.12
52.50
47.41
55.81
b
=
0.70, σ
=
0.40
40.98
50.76
45.99
54.36
b
=
0.45, σ
=
0.35
35.05
43.37
41.98
49.15
b
=
0.25, σ
=
0.34
22.39
27.24
31.60
35.84From Tables6 and 7, it can be seen that the proposed RSDHC scheme is effective to control the offshore platform, even though the expectation and standard deviation of the missing signal are 0.25 and 0.34, respectively. In addition, we can find that the peak and RMS values reduction of displacement and velocity become smaller with the decrease of b and σ.


## 5. Conclusions

In this paper, we have developed the problem of vibration control for offshore platforms with missing measurements subject to external wave. A robust sampled-dataH
∞ control scheme has been proposed. It is found from the simulation results that the designed scheme is effective to attenuate the external wave force and thereby improve the control performance of the offshore platform. In our further work, we will extend the proposed results to the case of multiple missing measurements of sensors. Also, we can investigate the sampled-data H
2
/
H
∞ control for vibration attenuation of offshore platforms subject to multiple external forces. In addition, we may consider the sampled-data H
∞ control for offshore platforms by using the finite frequency approach [30, 31].



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*Source: 914616-2014-03-04.xml*