# H

Authors: Bu Xuhui; Wang Hongqi; Zheng Zheng; Qian Wei
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The problem ofH
∞ control for network-based 2D systems with missing measurements is considered. A stochastic variable satisfying the Bernoulli random binary distribution is utilized to characterize the missing measurements. Our attention is focused on the design of a state feedback controller such that the closed-loop 2D stochastic system is mean-square asymptotic stability and has an  H
∞ disturbance attenuation performance. A sufficient condition is established by means of linear matrix inequalities (LMIs) technique, and formulas can be given for the control law design. The result is also extended to more general cases where the system matrices contain uncertain parameters. Numerical examples are also given to illustrate the effectiveness of proposed approach.


---

## Body

## 1. Introduction

Two-dimensional (2D) systems have attracted considerable research interest over the past few decades due to their wide applications in the areas such as multidimensional digital filtering, linear image processing, signal processing, process control, and iterative learning control [1–5]. Thus the stability and stabilization of 2D systems have attracted a lot of interests; see, for example, [6–13] and the references therein. H
∞ optimization is a powerful tool that can be used to design a robust controller or filter, which has been proved to be one of the most important strategies. Recently, such problems on 2D systems have stirred a great deal of research attention. For example, several versions of 2D bounded real lemma have been established in [2, 6, 14]. The problem of H
∞ control for 2D systems with state delays has been considered in [15]. The problem of H
∞ filtering for 2D systems has been studied extensively in [16]. An H
∞ controller is designed for a class of 2D nonlinear discrete systems with sector nonlinearity in [17, 18].Notice that all the above-mentioned results are based on an implicit assumption that the communication between the physical plant and controller is perfect; that is, the signals transmitted from the plant will arrive at the controller simultaneously and perfectly. However, in many practical situations, there may be a nonzero probability that all the signals can be measured during their transmission. In other words, the systems may have missing measurements. Moreover, networked systems are becoming more and more popular for the reason that they have several advantages over traditional systems, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability [19–22]. If network is introduced to controller design, the data packet dropout phenomenon, which appears in a typical network environment, will naturally induce missing measurements from the plant to the controller. In 1D system, the problems of stability, stabilization, filtering, or state estimation for networked control systems have been widely researched [23–30]. However, in the network-based 2D system case, only few results have been available. For instance, the problem of robust H
∞ filtering for 2D systems described by the Fornasini-Marchesini (FM) second model with missing measurements is considered in [31]. To the best of the authors’ knowledge, the H
∞ control for 2D systems represented by the Roesser model which is structurally quite distinct from FM second model has not been addressed in the literature so far.Motivated by the aforementioned observations, this paper considers the problem ofH
∞ control for network-based 2D systems presented by the Roesser model with missing measurements. We also notice that some dynamical processes such as gas absorption, water stream heating, and air drying can be described by a 2D Roesser model. In practical, these systems are often implemented by distribute control systems (DCSs) or field-bus control systems (FCSs), where control loops that are closed over a communication network. Hence, the considered topic in this paper is of practical significance. Compared with existing results, this paper proposes a state feedback controller design method for 2D systems in the framework of networked control systems. Meanwhile, the introduction of the random missing measurements renders the 2D system to be stochastic instead of a deterministic one. To analyze the stability, we introduce the stochastic mean-square asymptotically stable and stochastic H
∞ disturbance attenuation level. The controller is also designed under the framework of 2D stochastic systems.The remainder of this paper is organized as follows. In Section2, the mathematical description and design objectives of this paper are presented. In Section 3, a sufficient condition of mean-square asymptotic stability with H
∞ performance for such 2D stochastic systems is derived by means of LMI technique, and then formulas can be given for the control law design. In Section 4, the design result is extended to the 2D systems with uncertain parameters. Numerical examples are given in Section 5 and conclusions are drawn in Section 6.Notation 1.
The superscript “T” denotes the matrix transposition, R
ndenotes then-dimensional Euclidean space, I denotes the identity matrix, 0 denotes the zero vector or matrix with the required dimensions, and diag
⁡
{
·
}denotes the standard (block) diagonal matrix whose off-diagonal elements are zero. In symmetric block matrices, an asterisk*is used to denote the term that is induced by symmetry. The notation ∥
·
∥ refers to the Euclidean vector norm and λ
min
⁡
(
·
), λ
max
⁡
(
·
)denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively.E
{
x
}, E
{
x
∣
y
}mean the expectation of x and the expectation of xconditional ony. Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation

Consider the following 2D discrete system given by(1)
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
=
A
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
+
B
u
(
i
,
j
)
+
B
1
w
(
i
,
j
)
,
wherex
h
(
i
,
j
)
∈
R
n
1, x
v
(
i
,
j
)
∈
R
n
2, and u
(
i
,
j
)
∈
R
mrepresent the horizontal state, vertical state, and control input, respectively.w
(
i
,
j
)
∈
R
pdenotes the noise input, which belongs toℓ
2.A, B, B
1are real matrices with appropriate dimension.We make the following assumption on the boundary condition.Assumption 1.
The boundary condition is assumed to satisfy(2)
lim
⁡
N
→
∞
E
{
∑
k
=
0
N
(
|
x
k
,
0
|
2
+
|
x
0
,
k
|
2
)
}
<
∞
.Now, we consider the following state feedback control law:(3)
u
(
i
,
j
)
=
G
x
~
(
i
,
j
)
,
wherex
~
(
i
,
j
)is the measurement of state signals,Gis appropriately dimensioned controller matrix to be determined. When the feedback control is implemented via a networked control system, the datax
h
(
i
,
j
)
,
x
v
(
i
,
j
)are transferred as two separate packets from the remote plant to the controller. In this process, the data may be missed due to the network transmission failure, resulting in what we call missing measurement. It is assumed that the data packet dropout can be described by a stochastic variable; that is,
(4)
x
~
(
i
,
j
)
=
α
i
,
j
x
(
i
,
j
)
,
wherex
(
i
,
j
)
=
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
], the stochastic parameters{
α
i
,
j
}is a Bernoulli distributed white sequence taking the values of 0 and 1 with
(5)
Prob
{
α
i
,
j
=
1
}
=
E
{
α
i
,
j
}
=
α
,
Prob
{
α
i
,
j
=
0
}
=
1
-
E
{
α
i
,
j
}
=
1
-
α
,
and0
≤
α
≤
1is a known constant.Notice that the introduction of the stochastic variable{
α
i
,
j
}renders the 2D system to be stochastic instead of a deterministic one. Before proceeding further, we need to introduce the following definition of stochastic stability for the 2D system, which will be essential for our derivation.Definition 2 (see [31]).
The 2D system (1) is said to be mean-square asymptotically stable if under the zero input and for every bounded initial conditionx
h
(
i
,
0
), x
v
(
0
,
j
), the following is satisfied
(6)
lim
⁡
i
+
j
→
∞
E
{
∥
x
(
i
,
j
)
∥
2
}
=
0
.Definition 3.
Given a scalarγ
>
0, the 2D system (1) is said to have an H
∞ disturbance attenuation lev γ, if it is mean-square asymptotically stable and under zero initial conditions,∥
x
∥
E
<
γ
∥
w
∥
2is satisfied for any external disturbance w
(
i
,
j
)
∈
ℓ
2, where
(7)
∥
x
∥
E
=
E
{
∑
i
=
0
∞
∑
j
=
0
∞
∥
x
(
i
,
j
)
∥
2
}
,
∥
w
∥
2
=
E
{
∑
i
=
0
∞
∑
j
=
0
∞
∥
w
(
i
,
j
)
∥
2
}
.To this end, the design objective of this paper can be described as follows.Problem Statement. For any initial condition satisfying Assumption (1) and missing measurements described as (5), design a state feedback law (3) such that the closed-loop 2D system is mean-square asymptotically stable and has an H
∞ disturbance attenuation levelγ.


## 3. Main Results

In this section, we assume that the system matricesA
,
B
,
B
1and controller gain matrixGare known, and then we study the condition under which the closed-loop 2D system is mean-square asymptotically stable with a guaranteed H
∞ performance. Then, a feasible controller gain matrix can be given based on the condition.From system (1), (3), and (4), we can obtain the following closed-loop system:
(8)
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
=
(
A
+
α
i
,
j
B
G
)
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
+
B
1
w
(
i
,
j
)
.
Defineα
~
i
,
j
=
α
i
,
j
-
α; it is obvious that
(9)
E
{
α
~
i
,
j
}
=
0
,
E
{
α
~
i
,
j
α
~
i
,
j
}
=
α
(
1
-
α
)
.
Then, the 2D closed-loop system can be rewritten as
(10)
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]

=

(
A
+
α
B
G

+

α
~
i
,
j
B
G
)
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]

+

B
1
w
(
i
,
j
)
.Theorem 4.
The 2D closed-loop system (10) is mean-square asymptotically stable with a givenH
∞disturbance attenuation levelγ, if there exists a positive define symmetric matrixP
=
diag
⁡
{
P
h
,
P
v
}
>
0, satisfying
(11)
Ξ
=
[
Ξ
11
Ξ
12
*
Ξ
22
]
<
0
,
where
(12)
Ξ
11
=
(
A
+
α
B
G
)
T
P
(
A
+
α
B
G
)
+
θ
2
(
B
G
)
T
P
B
G
+
I
-
P
,
Ξ
12
=
(
A
+
α
B
G
)
T
P
B
1
,
Ξ
22
=
B
1
T
P
B
1
-
γ
2
I
,
θ
2
=
α
(
1
-
α
)
.Proof.
We first prove the mean-square asymptotically stability of 2D system (10) with zero disturbancew
(
i
,
j
)
=
0
.In this case, the system becomes
(13)
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
=
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
.
Define
(14)
W
1
=
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
∣
x
~
}
,
W
2
=
x
~
T
P
x
~
,
wherex
~
=
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
].
Consider the following index(15)
J
≜
W
1
-
W
2
.
Substituting (13) into the above index, we can obtain
(16)
J
=
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
∣
x
~
}
-
x
~
T
P
x
~
=
E
{
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
]
T
P
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
]
∣
x
~
}
-
x
~
T
P
x
~
=
x
~
T
[
(
A
+
α
B
G
)
T
P
(
A
+
α
B
G
)
+
θ
2
(
B
G
)
T
P
B
G
-
P
]
x
~
=
x
~
T
Ψ
x
~
,
where
(17)
Ψ
=
(
A
+
α
B
G
)
T
P
(
A
+
α
B
G
)
+
θ
2
(
B
G
)
T
P
B
G
-
P
.
From (11), it is easy to see thatΨ
<
0. Hence, for allx
~
≠
0, we have
(18)
W
1
-
W
2
W
2
=
-
x
~
T
(
-
Ψ
)
x
~
x
~
T
P
x
~
≤
-
λ
min
⁡
(
-
Ψ
)
λ
max
⁡
(
P
)
=
δ
-
1
,
whereδ
=
1
-
(
λ
min
⁡
(
-
Ψ
)
/
λ
max
⁡
(
P
)
).
Notice that(
λ
min
⁡
(
-
Ψ
)
/
λ
max
⁡
(
P
)
)
>
0; we haveδ
<
1. From (18), it is also easy to see that
(19)
δ
≥
W
1
W
2
>
0
.
Hence,δ
∈
(
0,1
)and it is independent of x
~. Thus, we obtainW
1
≤
δ
W
2, and taking expectation of both sides yields
(20)
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
}
≤
δ
E
{
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
T
P
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
}
.
That is,
(21)
E
{
x
v
(
k
+
1,0
)
T
P
v
x
v
(
k
+
1,0
)
}
=
E
{
x
v
(
k
+
1,0
)
T
P
v
x
v
(
k
+
1,0
)
}
E
{
x
h
(
k
+
1,0
)
T
P
h
x
h
(
k
+
1,0
)
+
x
v
(
k
,
1
)
T
P
v
x
v
(
k
,
1
)
}
≤
δ
E
{
x
h
(
k
,
0
)
T
P
h
x
h
(
k
,
0
)
+
x
v
(
k
,
0
)
T
P
v
x
v
(
k
,
0
)
}
E
{
x
h
(
k
,
1
)
T
P
h
x
h
(
k
,
1
)
+
x
v
(
k
-
1,2
)
T
P
v
x
v
(
k
-
1,2
)
}
≤
δ
E
{
x
h
(
k
-
1,1
)
T
P
h
x
h
(
k
-
1,1
)
h
h
h
h
h
h
i
+
x
v
(
k
-
1,1
)
T
P
v
x
v
(
k
-
1,1
)
}
E
{
x
h
(
k
-
1,2
)
T
P
h
x
h
(
k
-
1,2
)
+
x
v
(
k
-
2,3
)
T
P
v
x
v
(
k
-
2,3
)
}
≤
δ
E
{
x
h
(
k
-
2,2
)
T
P
h
x
h
(
k
-
2,2
)
j
j
j
j
j
j
j
j
j
j
+
x
v
(
k
-
2,2
)
T
P
v
x
v
(
k
-
2,2
)
}
⋮
E
{
x
h
(
1
,
k
)
T
P
h
x
h
(
1
,
k
)
+
x
v
(
0
,
k
+
1
)
T
P
v
x
v
(
0
,
k
+
1
)
}
≤
δ
E
{
x
h
(
0
,
k
)
T
P
h
x
h
(
0
,
k
)
+
x
v
(
0
,
k
)
T
P
v
x
v
(
0
,
k
)
}
E
{
x
h
(
0
,
k
+
1
)
T
P
h
x
h
(
0
,
k
+
1
)
}
=
{
x
h
(
0
,
k
+
1
)
T
P
h
x
h
(
0
,
k
+
1
)
}
.
Adding both sides of the inequality system (21) yields
(22)
E
{
∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
h
x
h
(
k
+
1
-
j
,
j
)
0
+
x
v
(
k
+
1
-
j
,
j
)
T
P
v
x
v
(
k
+
1
-
j
,
j
)
]



∑
j
=
0
k
+
1
}
≤
δ
E
{
∑
j
=
0
k
[
x
h
(
k
-
j
,
j
)
T
P
h
x
h
(
k
-
j
,
j
)
+
x
v
(
k
-
j
,
j
)
T
P
v
x
v
(
k
-
j
,
j
)
]



∑
j
=
0
k
+
1
}
+
E
{
x
v
(
k
+
1,0
)
T
P
v
x
v
(
k
+
1,0
)
}
+
E
{
x
h
(
k
+
1,0
)
T
P
h
x
h
(
k
+
1,0
)
}
.
Using this relationship iteratively, we can obtain
(23)
E
{
∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
h
x
h
(
k
+
1
-
j
,
j
)
0
+
x
v
(
k
+
1
-
j
,
j
)
T
P
v
x
v
(
k
+
1
-
j
,
j
)
]



∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
h
x
h
(
k
+
1
-
j
,
j
)
}
≤
δ
k
+
1
E
{
x
h
(
0,0
)
T
P
h
x
h
(
0,0
)
+
x
v
(
0,0
)
T
P
v
x
v
(
0,0
)
}
+
E
{
∑
j
=
0
k
δ
j
[
x
v
(
k
+
1
-
j
,
0
)
T
P
v
x
v
(
k
+
1
-
j
,
0
)
+
x
h
(
0
,
k
+
1
-
j
)
T
P
h
x
h
(
0
,
k
+
1
-
j
)
]



∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
h
x
h
(
k
+
1
-
j
,
j
)
}
=
E
{
∑
j
=
0
k
+
1
δ
j
[
x
v
(
k
+
1
-
j
,
0
)
T
P
v
x
v
(
k
+
1
-
j
,
0
)
+
x
h
(
0
,
k
+
1
-
j
)
T
P
h
x
h
(
0
,
k
+
1
-
j
)
]



∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
h
x
h
(
k
+
1
-
j
,
j
)
}
,
which implies
(24)
E
{
∑
j
=
0
k
+
1
∥
x
(
k
+
1
-
j
,
j
)
∥
2
}
≤
κ
∑
j
=
0
k
+
1
δ
j
E
{
∥
x
h
(
0
,
k
+
1
-
j
)
∥
2
∥
x
v
(
k
+
1
-
j
,
0
)
∥
2
0
+
∥
x
h
(
0
,
k
+
1
-
j
)
∥
2
}
,
where
(25)
κ
:
=
λ
max
⁡
(
P
)
λ
min
⁡
(
P
)
.
Now, denoteχ
k
:
=
∑
j
=
0
k
∥
x
(
k
-
j
,
j
)
∥
2; then, upon the inequality (24) we have
(26)
E
{
χ
0
}
≤
κ
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
E
{
χ
1
}
≤
κ
[
δ
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
0
+
E
{
∥
x
v
(
1,0
)
∥
2
+
∥
x
h
(
0,1
)
∥
2
}
]
E
{
χ
2
}
≤
κ
[
δ
2
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
0
+
δ
E
{
∥
x
v
(
1,0
)
∥
2
+
∥
x
h
(
0,1
)
∥
2
}
0
+
E
{
∥
x
v
(
2,0
)
∥
2
+
∥
x
h
(
0,2
)
∥
2
}
]
⋮
E
{
χ
N
}
≤
κ
[
δ
N
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
0
+
δ
N
-
1
E
{
∥
x
v
(
1,0
)
∥
2
+
∥
x
h
(
0,1
)
∥
2
}
0
+
⋯
+
E
{
∥
x
v
(
N
,
0
)
∥
2
+
∥
x
h
(
0
,
N
)
∥
2
}
]
.
Adding both sides of the inequalities yields
(27)
∑
k
=
0
N
E
{
χ
k
}
≤
κ
(
1
+
δ
+
⋯
+
δ
N
)
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
+
κ
(
1
+
δ
+
⋯
+
δ
N
-
1
)
E
{
∥
x
v
(
1,0
)
∥
2
+
∥
x
h
(
0,1
)
∥
2
}
+
⋯
+
κ
E
{
∥
x
v
(
N
,
0
)
∥
2
+
∥
x
h
(
0
,
N
)
∥
2
}
≤
κ
(
1
+
δ
+
⋯
+
δ
N
)
E
{
∥
x
v
(
0,0
)
∥
2
+
∥
x
h
(
0,0
)
∥
2
}
+
κ
(
1
+
δ
+
⋯
+
δ
N
)
E
{
∥
x
v
(
1,0
)
∥
2
+
∥
x
h
(
0,1
)
∥
2
}
+
⋯
+
κ
(
1
+
δ
+
⋯
+
δ
N
)
E
{
∥
x
h
(
0
,
N
)
∥
2
∥
x
v
(
N
,
0
)
∥
2
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
+
∥
x
h
(
0
,
N
)
∥
2
}
=
κ
×
1
-
δ
N
1
-
δ
E
{
∑
i
=
0
N
[
∥
x
v
(
k
,
0
)
∥
2
+
∥
x
h
(
0
,
k
)
∥
2
]
}
.
Then, under Assumption 1, the right side of this inequality is bounded, which means that lim
⁡
k
→
∞
E
{
χ
k
}
=
0; that is, lim
⁡
i
+
j
→
∞
E
{
∥
x
(
i
,
j
)
∥
2
}
=
0. Then the 2D stochastic system (10) is mean-square asymptotically stable.
Now, theH
∞performance for the 2D stochastic system (10) will be established. To this end, assume zero initial boundary conditions; that is, x
h
(
0
,
i
)
=
0
,
x
v
(
i
,
0
)
=
0for alli. In this case, the index Jbecomes
(28)
J
=
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
∣
x
~
}
-
x
~
T
P
x
~
=
E
{
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
+
B
1
w
(
i
,
j
)
]
T
P
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
+
B
1
w
(
i
,
j
)
]
∣
x
~
}
-
x
~
T
P
x
~
.
Another index is introduced as
(29)
Π
≜
J
+
x
~
T
x
~
-
γ
2
w
T
w
=
E
{
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
+
B
1
w
(
i
,
j
)
]
T
P
[
(
A
+
α
B
G
+
α
~
i
,
j
B
G
)
x
~
+
B
1
w
(
i
,
j
)
]
T
∣
x
~
}
-
x
~
T
P
x
~
+
x
~
T
x
~
-
γ
2
w
T
w
=
ζ
T
Ξ
ζ
,
whereζ
=
[
x
~
T
w
~
T
]
T.
From condition (11), we haveΠ
<
0for anyζ
≠
0; that is,
(30)
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
∣
x
~
}
<
x
~
T
P
x
~
-
x
~
T
x
~
+
γ
2
w
~
T
w
~
.
Taking the expectation of both sides yields(31)
E
{
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
T
P
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
∣
x
~
}
<
E
{
x
~
T
P
x
~
-
x
~
T
x
~
}
+
γ
2
w
~
T
w
~
.
Due to the relationship (31), it can be established that
(32)
E
{
x
v
(
k
+
1,0
)
T
P
2
x
v
(
k
+
1,0
)
}
=
E
{
x
v
(
k
+
1,0
)
T
P
2
x
v
(
k
+
1,0
)
}
E
{
x
h
(
k
+
1,0
)
T
P
1
x
h
(
k
+
1,0
)
+
x
v
(
k
,
1
)
T
P
2
x
v
(
k
,
1
)
}
≤
E
{
x
h
(
k
,
0
)
T
P
1
x
h
(
k
,
0
)
+
x
v
(
k
,
0
)
T
P
2
x
v
(
k
,
0
)
}
-
E
{
x
~
(
k
,
0
)
T
x
~
(
k
,
0
)
}
+
γ
2
w
(
k
,
0
)
T
w
(
k
,
0
)
E
{
x
h
(
k
,
1
)
T
P
1
x
h
(
k
,
1
)
+
x
v
(
k
-
1,2
)
T
P
2
x
v
(
k
-
1,2
)
}
≤
E
{
x
h
(
k
-
1,1
)
T
P
1
x
h
(
k
-
1,1
)
+
x
v
(
k
-
1,1
)
T
P
2
x
v
(
k
-
1,1
)
}
-
E
{
x
~
(
k
-
1,1
)
T
x
~
(
k
-
1,1
)
}
+
γ
2
w
(
k
-
1,1
)
T
w
(
k
-
1,1
)
E
{
x
h
(
k
-
1,2
)
T
P
1
x
h
(
k
-
1,2
)
+
x
v
(
k
-
2,3
)
T
P
2
x
v
(
k
-
2,3
)
}
≤
E
{
x
h
(
k
-
2,2
)
T
P
1
x
h
(
k
-
2,2
)
+
x
v
(
k
-
2,2
)
T
P
2
x
v
(
k
-
2,2
)
}
-
E
{
x
~
(
k
-
2,2
)
T
x
~
(
k
-
2,2
)
}
+
γ
2
w
(
k
-
2,2
)
T
w
(
k
-
2,2
)
⋮
E
{
x
h
(
1
,
k
)
T
P
1
x
h
(
1
,
k
)
+
x
v
(
0
,
k
+
1
)
T
P
2
x
v
(
0
,
k
+
1
)
}
≤
E
{
x
h
(
0
,
k
)
T
P
1
x
h
(
0
,
k
)
+
x
v
(
0
,
k
)
T
P
2
x
v
(
0
,
k
)
}
-
E
{
x
~
(
0
,
k
)
T
x
~
(
0
,
k
)
}
+
γ
2
w
(
0
,
k
)
T
w
(
0
,
k
)
E
{
x
h
(
0
,
k
+
1
)
T
P
1
x
h
(
0
,
k
+
1
)
}
=
{
x
h
(
0
,
k
+
1
)
T
P
1
x
h
(
0
,
k
+
1
)
}
.
Adding both sides of the inequality system, we have
(33)
E
{
∑
j
=
0
k
+
1
[
x
h
(
k
+
1
-
j
,
j
)
T
P
1
x
h
(
k
+
1
-
j
,
j
)
+
x
v
(
k
+
1
-
j
,
j
)
T
P
2
x
v
(
k
+
1
-
j
,
j
)
]
∑
j
=
0
k
+
1
}
<
{
∑
j
=
0
k
[
x
h
(
k
-
j
,
j
)
T
P
1
x
h
(
k
-
j
,
j
)
jjjjjjjjjjjjjjjjjjjj
+
x
v
(
k
-
j
,
j
)
T
P
2
x
v
(
k
-
j
,
j
)
]
∑
j
=
1
k
+
1
}
+
E
{
x
v
(
k
+
1,0
)
T
P
2
x
v
(
k
+
1,0
)
}
+
E
{
x
h
(
0
,
k
+
1
)
T
P
1
x
h
(
0
,
k
+
1
)
T
}
-
E
{
∑
j
=
0
k
[
x
~
(
k
-
j
,
j
)
T
x
~
(
k
-
j
,
j
)
}
+
γ
2
∑
j
=
0
i
w
(
k
-
j
,
j
)
T
w
(
k
-
j
,
j
)
.
Summing up both sides of this inequality fromk
=
0tok
=
N, we have
(34)
E
{
∑
k
=
0
N
∑
j
=
0
k
x
~
(
k
-
j
,
j
)
T
x
~
(
k
-
j
,
j
)
}
<
γ
2
∑
k
=
0
N
∑
j
=
0
k
w
(
k
-
j
,
j
)
T
w
(
k
-
j
,
j
)
+
∑
k
=
0
N
(
E
{
x
v
(
k
+
1,0
)
T
P
2
x
v
(
k
+
1,0
)
}
j
j
j
j
j
j
j
j
j
j
+
E
{
x
h
(
0
,
k
+
1
)
T
P
1
x
h
(
0
,
k
+
1
)
T
}
)
-
E
{
∑
j
=
0
N
[
x
h
(
N
+
1
-
j
,
j
)
T
P
1
x
h
(
N
+
1
-
j
,
j
)
jjjjjjjjjjjjjjjj
+
x
v
(
N
+
1
-
j
,
j
)
T
P
2
x
v
(
N
+
1
-
j
,
j
)



x
h
(
N
+
1
-
j
,
j
)
T
P
1
x
h
(
N
+
1
-
j
,
j
)
]
∏
j
h
k
h
j
k
j
k
}
+
E
{
x
v
(
0,0
)
T
P
2
x
v
(
0,0
)
}
+
E
{
x
h
(
0,0
)
T
P
1
x
h
(
0,0
)
T
}
;
that is,
(35)
E
{
∑
k
=
0
∞
∑
j
=
0
k
x
~
(
k
-
j
,
j
)
T
x
~
(
k
-
j
,
j
)
}
<
γ
2
∑
k
=
0
∞
∑
j
=
0
k
w
(
k
-
j
,
j
)
T
w
(
k
-
j
,
j
)
+
∑
k
=
0
∞
(
P
1
x
h
(
0
,
k
)
T
E
{
x
v
(
k
,
0
)
T
P
2
x
v
(
k
,
0
)
}
j
j
j
j
j
j
j
j
+
E
{
x
h
(
0
,
k
)
T
P
1
x
h
(
0
,
k
)
T
}
)
.
Considering the zero initial boundary conditions, (35) means
(36)
∥
x
~
∥
E
<
γ
2
∥
w
∥
2
.
This completes the proof.Remark 5.
Theorem4 provides a sufficient condition of the mean-square asymptotic stability andH
∞disturbance attenuation levelγfor 2D systems with missing measurements. If the communication links existing between the plant and the controller are perfect, that is, there is no packet dropout during their transmission, thenα
=
1andθ
=
0. In this case, the condition in Theorem 4 becomes the condition obtained in [6] for 2D deterministic system. From this point of view, Theorem 4 can be seen as an extension of [6] to 2D systems with missing measurement.
Theorem4 gives a mean-square asymptotic stability condition withH
∞ disturbance attenuation levelγfor system (10) where the controller gain matrixGis known. However, our eventual purpose is to determine a suitableGby system matricesA
,
B
,
B
1and parameterα.The following well-known lemma is needed in the proof of our main result.Lemma 6 (Schur complement).
AssumeW
,
L
,
V are given matrices with appropriate dimensions, whereWandVare positive definite symmetric matrices. Then
(37)
L
T
V
L
-
W
<
0
,
if and only if
(38)
[
-
W
L
T
L
-
V
-
1
]
<
0
,
or
(39)
[
-
V
-
1
L
L
T
-
W
]
<
0
.Based on the above lemma, we can give our main result.Theorem 7.
For the 2D closed-loop system (10), if there exist a positive define symmetric matrixQand a matrixMsatisfying
(40)
Ω
=
[
-
Q
0
Q
A
T
+
α
M
T
B
T
θ
M
T
B
T
Q
-
γ
2
I
B
1
T
0
0
-
Q
0
0
*
-
Q
0
-
I
]
<
0
,
then the 2D closed-loop system (10) is mean-square asymptotically stability and has anH
∞disturbance attenuation levelγ. In this case, a suitable state feedback control law can be given asG
=
M
Q
-
1.Proof.
The condition in Theorem4 can be rewritten as
(41)
Θ
T
[
P
P
I
]
Θ
+
[
-
P
-
γ
2
I
]
<
0
,
where
(42)
Θ
T
=
[
(
A
+
α
B
G
)
T
θ
(
B
G
)
T
I
B
1
T
0
0
]
.
By applying Lemma6, condition (41) is equivalent to the following LMI condition:
(43)
[
-
P
0
(
A
+
α
B
G
)
T
θ
(
B
G
)
T
I
-
γ
2
I
B
1
T
0
0
-
P
-
1
0
0
*
-
P
-
1
0
-
I
]
<
0
.
DefineQ
=
P
-
1, and pro- and postmultiplying diag
⁡
(
Q
,
I
,
I
,
I
,
I
,
I
)for the above condition give
(44)
[
-
Q
0
Q
A
T
+
α
(
G
Q
)
T
B
T
θ
(
G
Q
)
T
B
T
Q
-
γ
2
I
B
1
T
0
0
-
Q
0
0
*
-
Q
0
-
I
]
<
0
.
SetG
Q
=
Mto obtain the LMI of (40) and the proof is complete.Remark 8.
Theorem7 provides an LMI condition for the mean-square asymptotic stability andH
∞disturbance attenuation levelγof 2D stochastic system, which can be solved by LMI Toolbox. Then by (40), we also can give a suitable state feedback control law. It is noted that LMI (40) only contains few elements; hence, the computation is not big. However, when the system matrices get bigger, the solution of LMI (40) may be time-consuming due to complicated computation. In practical system, the system dimension is not very big (often less than 5). Hence, the computation complexity is acceptable.Remark 9.
For a fixedγ, the feasibility of (40) is a suboptimalH
∞controller. Whenγis not fixed, the minimization ofγ that satisfies (40) can be searched. Hence, an optimalH
∞controller can be obtained by solving the following optimization problem:
(45)
min
⁡
Q
,
M
γ
2
s
.
t
(40).


## 4. RobustH
∞ Control for Uncertain 2D Systems

In this section, we extend the above design to the case of robustH
∞ control. Consider the following 2D system with uncertain parameter perturbations
(46)
[
x
h
(
i
+
1
,
j
)
x
v
(
i
,
j
+
1
)
]
=
(
A
+
Δ
A
)
[
x
h
(
i
,
j
)
x
v
(
i
,
j
)
]
+
(
B
+
Δ
B
)
u
(
i
,
j
)
+
(
B
1
+
Δ
B
1
)
w
(
i
,
j
)
,
whereΔ
A
,
Δ
B
,
Δ
B
1denote admissible uncertain perturbations of matricesA, B, andB
1, which can be represented as
(47)
Δ
A
=
E
Σ
F
1
,
Δ
B
=
E
Σ
F
2
,
Δ
B
1
=
E
Σ
F
3
,
whereE
,
F
1
,
F
2
,
F
3are known real constant matrices characterizing the structures of uncertain perturbations andΣis an uncertain perturbation of the system that satisfiesΣ
T
Σ
≤
I.Lemma 10.
AssumeX
,
Yare matrices or vectors with appropriate dimensions. For any scalarε
>
0and all matrices Δwith appropriate dimensions satisfyingΔ
Δ
T
≤
I, the following inequality holds:
(48)
X
Δ
Y
+
Y
T
Δ
T
X
T
≤
ε
X
X
T
+
ε
-
1
Y
T
Y
.
The main result of this section is given in the following theorem.Theorem 11.
For the uncertain 2D system (46) used the state feedback control law (3) with missing measurement, if there exist a positive define symmetric matrixQ, a matrixM, and scalarsε
>
0satisfying
(49)
[
-
Q
0
Ω
1
θ
M
T
B
T
Q
Ω
2
θ
M
T
F
2
T
-
γ
2
I
B
1
T
0
0
F
3
T
0
ε
E
E
T
-
Q
0
0
0
0
ε
E
E
T
-
Q
0
0
0
-
I
0
0
*
-
ε
I
0
-
ε
I
]
<
0
,
where
(50)
Ω
1
=
Q
A
T
+
α
M
T
B
T
,
Ω
2
=
Q
F
1
T
+
α
M
T
F
2
T
,
then the uncertain 2D system is mean-square asymptotically stability and has an H
∞ disturbance attenuation levelγ. In this case, a suitable state feedback control law can be given asG
=
M
Q
-
1.Proof.
By applying Theorem7, the uncertain 2D system with state feedback control law (3) is mean-square asymptotically stable and has anH
∞disturbance attenuation levelγ, and there exist a positive define symmetric matrix Q, a matrix M satisfying (51)
Ω
=
[
-
Q
0
Q
(
A
+
Δ
A
)
T
+
α
M
T
(
B
+
Δ
B
)
T
θ
M
T
(
B
+
Δ
B
)
T
Q
-
γ
2
I
(
B
1
+
Δ
B
1
)
T
0
0
-
Q
0
0
*
-
Q
0
-
I
]
<
0
.That is,
(52)
[
-
Q
0
Ω
1
θ
M
T
B
T
Q
-
γ
2
I
B
1
T
0
0
-
Q
0
0
*
-
Q
0
-
I
]
+
[
0
0
E
0
0
]
Σ
[
Ω
2
F
3
T
0
0
0
]
T
+
[
Ω
2
F
3
T
0
0
0
]
Σ
T
[
0
0
E
0
0
]
T
+
[
0
0
0
E
0
]
Σ
[
θ
M
T
F
2
T
0
0
0
0
]
T
+
[
θ
M
T
F
2
T
0
0
0
0
]
Σ
T
[
0
0
0
E
0
]
T
<
0
.
Therefore, using Lemma 10, there exists a scalarε
>
0such that
(53)
[
-
Q
0
Ω
1
θ
M
T
B
T
Q
-
γ
2
I
B
1
T
0
0
-
Q
0
0
*
-
Q
0
-
I
]
+
ε
[
0
0
E
0
0
]
[
0
0
E
0
0
]
T
+
ε
-
1
[
Ω
2
F
3
T
0
0
0
]
T
[
Ω
2
F
3
T
0
0
0
]
+
ε
[
0
0
0
E
0
]
T
[
0
0
0
E
0
]
+
ε
-
1
[
θ
M
T
F
2
T
0
0
0
0
]
T
[
θ
M
T
F
2
T
0
0
0
0
]
<
0
.
Note that the above condition is equivalent to
(54)
[
-
Q
0
Ω
1
θ
M
T
B
T
Q
-
γ
2
I
B
1
T
0
0
ε
E
E
T
-
Q
0
0
*
ε
E
E
T
-
Q
0
-
I
]
+
ε
-
1
[
Ω
2
θ
M
T
F
2
T
F
3
T
0
0
0
0
0
0
0
]
[
(
Ω
2
)
T
F
3
0
0
0
θ
F
2
M
0
0
0
0
]
<
0
.
By the Schur complement, LMI (54) implies LMI in Theorem 11 holds.
This completes the proof.In addition, by solving the following optimization problem(55)
min
⁡
Q
,
M
,
ε
γ
2
s
.
t
(49).
we can obtain a robust optimalH
∞ controller for the uncertain 2D stochastic systems (46).Remark 12.
This paper considers the problem ofH
∞ stabilization for a class of 2D systems with missing measurements. Here, we describe the missing measurements as a Bernoulli random binary distribution, which renders the 2D systems to be stochastic ones. Under this framework, we give the definition of stochastic mean-square stability and H
∞ performance, and then a state feedback control design approach is addressed. The proposed design approach is systematic for 2D stochastic system. The results in this paper can be extended to solve other problems such as network-induced delay and H
∞ filter design.


## 5. Illustrative Examples

In this section, two numerical examples are used to illustrate the effectiveness of the proposed results.Example 1.
Let us consider 2D system (1) with the following parameters:
(56)
A
=
[
0.5
0.3
0.1
0.7
]
,
B
=
[
0.1
0.05
0.2
0
]
,
B
1
=
[
0.1
0
0
0.3
]
.
It is assumed that measurements transmitted between the plant and the controller are imperfect; that is, the state signal may be lost during their transmission. Supposeα
=
0.8; that is, in the communication link, the probability of the data packet missing is 20%. By applying Theorem 7 and solving the optimization problem (45), the minimumH
∞disturbance attenuation level isγ
opt
=
0.32. Meanwhile, we can obtain
(57)
Q
=
[
0.8663
-
0.0405
-
0.0405
0.8554
]
,
M
=
[
-
0.4429
-
3.7218
-
3.7218
2.1220
]
.
Hence, a feasible state feedback control law can be selected as
(58)
G
=
M
Q
-
1
=
[
-
0.7162
-
4.3849
-
4.1895
-
2.2824
]
.
Assume the disturbance is
(59)
w
(
i
,
j
)
=
[
1
10
i
j
1
10
i
j
]
.
Simulation results are shown in Figures1 and 2, where the state response ofx
h
(
i
,
j
)is plotted in Figure 1 andx
v
(
i
,
j
)is plotted in Figure 2. It can be seen from Figures 1 and 2 that the closed-loop 2D system is asymptotically stable. Hence, even though the 2D system is affected by external disturbances and has significant data dropout in the output measurements, the proposed design approach can guarantee the stability and disturbance attenuation ability for the 2D system.Figure 1
State response ofx
h
(
i
,
j
)for Example 1.Figure 2
State response ofx
v
(
i
,
j
)for Example 1.Example 2.
Let us consider uncertain 2D system (46) with the following parameters:
(60)
A
=
[
0.8
0.021
0.01
0.9
]
,
B
=
[
0.03
0
0.1
0.1
]
,
B
1
=
[
0.2
0
0
0.1
]
,
E
=
[
0.01
0.01
0
0.02
]
,
F
1
=
[
0.01
0
0
0.02
]
,
F
2
=
[
0.01
0.01
0.03
0
]
,
F
3
=
[
0.01
0.01
0.01
0.02
]
.
It is also assumed that the probability of the data packet missing is 0.2; that is,α
=
0.8. In this case, by applying Theorem 11 and solving the optimization problem (55), the minimumH
∞disturbance attenuation level isγ
opt
=
0.35. Meanwhile, we can obtain
(61)
Q
=
[
0.5125
-
0.2522
-
0.2522
0.6073
]
,
M
=
[
-
10.4172
10.0783
10.0783
-
16.9824
]
,
ε
=
14.4592
.
Hence, a feasible state feedback control law can be selected as
(62)
G
=
M
Q
-
1
=
[
-
15.2830
10.2485
7.4205
-
24.8822
]
.
The disturbancew
(
i
,
j
)is also given as Example 1. Simulation results are shown in Figures 3 and 4, where the state response ofx
h
(
i
,
j
)is plotted in Figure 3 andx
v
(
i
,
j
)is plotted in Figure 4. It is observed that the closed-loop uncertain 2D system is also asymptotically stable. Hence, the proposed design approach is also effective for uncertain 2D system.Figure 3
State response ofx
h
(
i
,
j
)for Example 2.Figure 4
State response ofx
v
(
i
,
j
)for Example 2.


## 6. Conclusions

In this paper, we have investigated the problem ofH
∞ stabilization for a class of 2D systems described with missing measurements. A sufficient condition has been developed in terms of LMIs, which guarantees mean-square asymptotic stability and H
∞ disturbance attenuation level for closed-loop 2D system. The result is also extended to more general cases where the system matrices contain uncertain parameters. Numerical examples have been provided to illustrate the effectiveness of proposed approach. In our future work, the control and filtering problems for networked-based 2D system with packet dropouts and network-induced delay will be discussed.



---

*Source: 786128-2014-05-11.xml*