Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 528695, 9 pages
http://dx.doi.org/10.1155/2013/528695
Research Article
𝐻∞ Control of Discrete-Time Singularly Perturbed Systems via
Static Output Feedback
Dan Liu, Lei Liu, and Ying Yang
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering,
Peking University, Beijing 100871, China
Correspondence should be addressed to Ying Yang; yy@mech.pku.edu.cn
Received 19 December 2012; Accepted 22 February 2013
Academic Editor: Guanglu Zhou
Copyright © 2013 Dan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper concentrates on 𝐻∞ control problems of discrete-time singularly perturbed systems via static output feedback. Two
methods of designing an 𝐻∞ controller, which ensures that the resulting closed-loop system is asymptotically stable and meets a
prescribed 𝐻∞ norm bound, are presented in terms of LMIs. Though based on the same matrix transformation, the two approaches
are turned into different optimal problems. The first result is given by an 𝜖-independent LMI, while the second result is related to 𝜖.
Furthermore, a stability upper bound of the singular perturbation parameter is obtained. The validity of the proposed two results
is demonstrated by a numerical example.
1. Introduction
Singularly perturbed systems widely exist in industrial processes, such as aircraft and racket systems, power systems,
and nuclear reactor systems. These kind of systems usually
embrace complicated dynamic phenomena which are characterized by slow and fast modes with multiple time-scales. This
property causes high dimensionality and ill-conditioning
problems. In control theory, a parameter-related state-space
model is frequently used to describe a singularly perturbed
system. With important practical meaning, the stability
bounds of the singular perturbation parameter have been
extensively studied by many researchers. In early times, a
traditional method of decomposing the original system into
fast and slow subsystems was frequently used, see [1, 2]. In
[3], a method to testify the stability of singularly perturbed
systems without the fast-slow decomposition was established.
The stability bound was proved to have close relationship with
the system matrix, which contributes to analyse some robust
control problems. Furthermore, two algorithms to compute
and improve the stability bound were developed in [4]. More
stability problems are discussed in [5–10] and the references
therein.
In recent years, computer science is increasingly applied
to industrial processes. Therefore, discrete-time singularly
perturbed systems have attracted much attention. An 𝐻∞
control problem for uncertain discrete-time singularly perturbed systems via state feedback was studied in [11], where
two methods of designing 𝐻∞ controllers were given in
terms of LMIs. Fast sampling discrete-time singularly perturbed systems were taken into consideration in [12], and
the obtained results were generalized to robust controller
design. In [13], a new sufficient condition which guaranteed
the existence of state feedback controllers and made the
closed-loop system asymptotically stable while satisfying
a prescribed 𝐻∞ norm requirement was proposed. This
condition was also given in the form of LMIs but was proved
to be less conservative than that in [11]. It will be more perfect
if a theoretical proof was given. Interested readers can refer
to [14–16] for more information of discrete-time singularly
perturbed systems.
Though state feedback can achieve desired properties, it
requires the availability of all state variables, which cannot be
satisfied in most of the practical systems. On the other hand,
dynamic output feedback usually increases the dimension of
the original system. Therefore, static output feedback plays
an important role in control theory considering that it is
the simplest control technique in a closed-loop sense and
can be easily realized with a cost not as high as that in
state feedback case [17]. The primal point involved in static
2
Abstract and Applied Analysis
output feedback is the decoupling problem. A variety of
approaches are developed to solve such issues. In [18], special
inequality and some tuning scalars were used to transfer
nonlinear matrix inequality to a linear one. A stabilizing state
feedback controller gain and some matrix transformations
were introduced to deal with the nonlinear inequalities in
[19]. This method is effective and easy to implement. Thus,
the same decoupling technique is adopted in this paper.
Based on those reasons and motivated by the above
studies, we aim to design an 𝐻∞ controller via static output
feedback to stabilize a discrete-time singularly perturbed
system and guarantee that the transfer function of the
resulting closed-loop system satisfies a prescribed 𝐻∞ norm
bound.
The rest of this paper is organized as follows. Section 2
states the system description and some useful lemmas. In
Section 3, two LMI-based methods are proposed to design
a static output feedback controller for the system presented
in Section 2. A numerical example is given to demonstrate
the effectiveness of the proposed results in Section 4. Finally,
conclusions are given in Section 5.
The following notation will be adopted throughout this
paper. 𝐼 denotes an identity matrix with appropriate dimension. 𝐴𝑇 denotes the transpose of matrix 𝐴. Sym{𝐴} denotes
𝐴 + 𝐴𝑇 . For a symmetric block matrix, (∗ ) stands for the
blocks induced by symmetry.
then the resulting closed-loop system can be obtained as
follows:
̃𝜖 𝑥𝑘 + 𝐵1𝜖 𝑤𝑘 ,
𝑥𝑘+1 = 𝐴
̃1 𝑥𝑘 + 𝐷11 𝑤𝑘 ,
𝑧𝑘 = 𝐶
𝑦𝑘 = 𝐶2 𝑥𝑘 ,
where
̃
̃
̃𝜖 = [𝐼𝑛1 + 𝜖𝐴11 𝜖𝐴12 ] ,
𝐴
̃21
̃22
𝐴
𝐴
̃ 𝐴
̃
𝐴
𝐴 𝐴
𝐵
[̃11 ̃12 ] = [ 11 12 ] + [ 21 ] 𝐹 [𝐶21 𝐶22 ] ,
𝐴 21 𝐴 22
𝐵22
𝐴21 𝐴22
̃11 𝐶
̃12 ] = [𝐶11 𝐶12 ] + 𝐷12 𝐹 [𝐶21 𝐶22 ] .
[𝐶
The 𝐻∞ problem studied in this paper can be described
as follows: given a scalar 𝛾 > 0 and a discrete-time singularly
perturbed system (1), design a static output feedback controller in the form of (2) such that the closed-loop system (3)
is asymptotically stable and its transfer function from 𝜔 to 𝑧,
−1
2. Problem Formulation
(1)
𝑦𝑘 = 𝐶2 𝑥𝑘 ,
11
where 𝑥𝑘 = [ 𝑥𝑥1𝑘
], 𝐴 𝜖 = [ 𝐼𝑛1𝐴+𝜖𝐴
2𝑘
21
𝐶𝑇
𝑇
𝐶𝑇
𝑇
𝜖𝐴 12
𝐴 22
11
], 𝐵1𝜖 = [ 𝜖𝐵
𝐵12 ], 𝐵2𝜖 =
22
in which 𝑥1𝑘 ∈ 𝑅𝑛1 , 𝑥2𝑘 ∈ 𝑅𝑛2 , and 𝑛1 + 𝑛2 = 𝑛, 𝑢𝑘 ∈ 𝑅𝑚1 is
the control input, 𝑤𝑘 ∈ 𝑅𝑚2 is the disturbance input which
belongs to 𝐿 2 [0, ∞), 𝑦𝑘 ∈ 𝑅𝑞𝑙 is the measurement output,
𝑧𝑘 ∈ 𝑅𝑞2 is the controlled output. The scalar 𝜖 > 0 denotes
the singular perturbation parameter.
In the rest of this paper, we will assume that system (1) is
completely controllable and observable. We will also assume
that not all of its state variables are available. Therefore, the
following static output feedback control law is taken into
consideration:
𝑢𝑘 = 𝐹𝑦𝑘 ,
satisfies ‖𝐺‖∞ < 𝛾.
The following lemmas will be used in establishing our
main results.
(i) ‖𝐺(𝑧)‖∞ < 1 and 𝐴 are stable in the discrete-time sense
(|𝜆 𝑖 (𝐴)| < 1);
(ii) there exists 𝑋 = 𝑋𝑇 > 0 such that
𝑛
11
21
21
[ 𝜖𝐵
𝐵22 ], 𝐶1 = [ 𝐶𝑇 ] , 𝐶2 = [ 𝐶𝑇 ] , 𝑥𝑘 ∈ 𝑅 is the state vector,
12
(5)
Lemma 1 (see [19]). Consider a discrete-time transfer function
𝐺(𝑧) = 𝐶(𝑧𝐼 − 𝐴)−1 𝐵 + 𝐷. The following statements are
equivalent:
𝑥𝑘+1 = 𝐴 𝜖 𝑥𝑘 + 𝐵1𝜖 𝑤𝑘 + 𝐵2𝜖 𝑢𝑘 ,
𝑧𝑘 = 𝐶1 𝑥𝑘 + 𝐷11 𝑤𝑘 + 𝐷12 𝑢𝑘 ,
(4)
̃ 1 = [𝐶
̃12 ] ,
̃11 𝐶
𝐶
̃𝜖 ) 𝐵1𝜖 + 𝐷11 ,
̃1 (𝑧𝐼 − 𝐴
𝐺 (𝑧) = 𝐶
In this paper, we consider a class of linear fast sampling
discrete-time singularly perturbed systems of the following
form:
(3)
(2)
𝐴𝑇 𝑋𝐴 − 𝑋 𝐴𝑇 𝑋𝐵 𝐶𝑇
[
]
[ 𝐵𝑇 𝑋𝐴
𝐵𝑇 𝑋𝐵 − 𝐼 𝐷𝑇 ]
[
]<0
𝐶
𝐷
−𝐼 ]
[
(6)
holds.
Lemma 2 (see [20]). The following two statements are equivalent.
(i) Let 𝐴, 𝐵, and 𝐶 be given such that the LMI
𝐴 𝐵
𝑋
[ T ] + Sym{[ ] [𝐶T −𝐼]} < 0
𝑌
𝐵 0
is feasible in 𝑋 and 𝑌.
(7)
Abstract and Applied Analysis
3
(ii) Let 𝐴, 𝐵, and 𝐶 be given such that the following LMI
𝐴 + 𝐵𝐶T + 𝐶𝐵T < 0
and matrices 𝑃12 , 𝐿, 𝑋𝑖𝑗 , 𝑌𝑘𝑗 , and 𝑖 = 1, . . . , 5, 𝑘, 𝑗 = 1, 2 with
appropriate dimensions, such that the following LMI holds:
(8)
𝑇
Ξ−𝑋𝐹0𝑇 𝑇−(𝑋𝐹0𝑇 𝑇)
∗
∗
[
[
[
holds.
Lemma 3 (see [20]). The following two statements are equivalent.
Φ−𝑋−𝑇𝑇𝐹0 𝑌T 𝑋𝐶2𝑇 +𝑇𝑇𝐿
]
𝑌𝐶2𝑇 ]
−𝑌−𝑌𝑇
∗
−𝐺−𝐺T ] (13)
< 0,
(i) Let 𝐴, 𝐵, and 𝐶 be given such that the LMI
where
𝐴
𝐵 + 𝐶𝐺T
[ T
]
T
𝐵 + 𝐺𝐶 −𝐺 − 𝐺T
=[
𝐴 𝐵
0
] + Sym{[ ] 𝐺 [𝐶T −𝐼]} < 0
𝐼
𝐵T 0
(9)
𝑋 = [𝑋𝑖𝑗 ] ,
𝑌 = [𝑌𝑘𝑗 ] ,
𝑖 = 1, . . . , 5, 𝑘, 𝑗 = 1, 2,
𝑇
Δ 110 = 𝑃11 𝐴𝑇110 + 𝑃12 𝐴𝑇120 + (𝑃11 𝐴𝑇110 + 𝑃12 𝐴𝑇120 ) ,
Δ 120 = 𝑃11 𝐴𝑇210 + 𝑃12 𝐴𝑇220 − 𝑃12 ,
is feasible in 𝐺.
(ii) Let 𝐴, 𝐵, and 𝐶 be given such that the LMIs
𝐴<0
(10)
𝐴 + 𝐵𝐶T + 𝐶𝐵T < 0
0 𝑃22
[𝑃11 𝑃12 ]
[
]
]
Φ=[
[ 0 0 ],
[0 0]
[0 0]
𝑇
0
[ 𝐵21 ]
[ ]
]
𝑇=[
[ 𝐵22 ] ,
[𝐷12 ]
[ 0 ]
hold.
3. Main Results
In this section, two LMI-based methods of designing a
static output feedback controller are proposed to ensure
asymptotical stability of a closed-loop discrete-time singularly perturbed system (3). The first result is given in the form
of an 𝜖-independent LMI, while the second result is presented
by two LMIs which are related to the singular perturbation
parameter 𝜖. Furthermore, we can obtain the stability upper
bound of 𝜖.
Before giving the results, let 𝐹0 be a stabilizing state
feedback controller gain of the system (1). In other words, the
matrix 𝐹0 is chosen to make matrix 𝐴 𝜖0 = 𝐴 𝜖 + 𝐵2𝜖 𝐹0 stable.
Then, we introduce the following transformations which will
be used in the rest of this paper:
𝐴 0 = 𝐴 + 𝐵2𝜖 𝐹0 ,
𝐶20 = 𝐶2 + 𝐷12 𝐹0 ,
𝐹𝐶2 = 𝑆 + 𝐹0 ,
(11)
where
𝐴0 = [
𝐶20 =
𝐴 110 𝐴 120
],
𝐴 210 𝐴 220
𝑇
𝑇
𝐶210
[ 𝑇 ] ,
𝐶220
[
]
𝑆=
𝐴 𝐴
𝐴 = [ 11 12 ] ,
𝐴 21 𝐴 22
𝑇
𝑆1𝑇
[ 𝑇] ,
𝑆2
[ ]
𝐹0 =
𝑇 𝑇
𝐹01
[ 𝑇] .
𝐹02
[
(12)
]
Theorem 4. For a discrete-time singularly perturbed system in
the form of (1), given a stabilizing state feedback controller with
gain 𝐹0 , if there exist matrices 𝑃11 > 0, 𝑃22 > 0, and 𝐺 > 0
T
𝑃22 𝐶120
−𝑃22 𝑃22 𝐴T120 𝑃22 𝐴T220
𝑇
T
[ ∗
Δ 120 𝑃11 𝐶110 + 𝑃12 𝐶120
Δ 110
[
[
Ξ=[ ∗
∗
−𝑃22
0
[ ∗
∗
∗
−𝛾𝐼
∗
∗
∗
[ ∗
0
𝐵11 ]
]
𝐵12 ]
],
𝐷11 ]
−𝛾𝐼]
(14)
then there exists a scalar 𝜖∗ > 0 such that for every 𝜖 ∈ (0, 𝜖∗ ],
the system (3) is asymptotically stable, and its transfer function
satisfies ‖𝐺‖∞ < 𝛾 with the static output feedback controller in
the form of (2) whose control gain is given by 𝐹 = 𝐿𝐺−𝑇 .
Proof. Based on Lemma 1, the closed-loop system (3) is
asymptotically stable, and its transfer function satisfies
‖𝐺‖∞ < 𝛾 if there exists a positive definite matrix 𝑃 such that
the following LMI holds:
̃𝑇 0
̃𝑇 𝑃𝐶
−𝑃 𝑃𝐴
[ ∗ −𝑃𝜖 0 1 𝐵 ]
[
1𝜖 ] < 0,
[∗
∗ −𝛾𝐼 𝐷11 ]
∗
∗ −𝛾𝐼]
[∗
(15)
then for all 𝜖 ∈ (0, 𝜖∗ ], let
𝑃 (𝜖) = [
𝜖𝑃11 𝜖𝑃12
]>0
∗ 𝑃22
(16)
4
Abstract and Applied Analysis
satisfy (15), and we have
where
𝑇
𝑇
𝑇
̃ + 𝑃12 𝐴
̃ ,
Δ 11 = 𝑃11 𝐴
11
12
𝜖Γ1
−𝜖𝑃11 −𝜖𝑃12 𝜖𝑃11 + 𝜖2 Δ 11 𝜖Δ 12
[ ∗
−𝑃22
Δ 21 (𝜖)
Δ 22 (𝜖) Γ2 (𝜖)
[
[ ∗
−𝜖𝑃12
0
∗
−𝜖𝑃11
[
[
∗
∗
−𝑃22
0
[ ∗
[
[ ∗
∗
∗
∗
−𝛾𝐼
∗
∗
∗
∗
[ ∗
0
0 ]
]
𝜖𝐵11 ]
]
] < 0,
𝐵12 ]
]
𝐷11 ]
−𝛾𝐼 ]
(17)
𝑇
𝑇
𝑇
̃ ) + 𝜖2 𝑃𝑇 𝐴
̃
Δ 21 (𝜖) = 𝜖 (𝑃12
+ 𝑃22 𝐴
12
12 11 ,
𝑇
𝑇
𝑇
̃ + 𝑃12 𝐴
̃ ,
Δ 12 = 𝑃11 𝐴
21
22
𝑇
𝑇̃
̃ ,
𝐶11 + 𝑃22 𝐶
Γ2 (𝜖) = 𝜖𝑃12
12
𝑇
𝑇
𝑇̃
̃ .
𝐴21 + 𝑃22 𝐴
Δ 22 (𝜖) = 𝜖𝑃12
22
(18)
Using the Schur complement to (17), we conclude that
̃𝑇 + 𝜖2 Θ12
̃𝑇 + 𝜖Θ13
̃𝑇 + 𝜖Θ14
𝜖𝑃22 𝐴
𝑃22 𝐴
𝑃22 𝐶
−𝑃22 + 𝜖Θ11
12
22
12
2
𝑇
2
2
∗
𝜖 (Δ 11 + Δ 11 ) + 𝜖 Θ22 𝜖 (Δ 12 − 𝑃12 ) + 𝜖 Θ23 𝜖Γ1 + 𝜖2 Θ24
∗
∗
−𝑃22 + 𝜖Θ33
𝜖Θ34
∗
∗
∗
−𝛾𝐼 + 𝜖Θ44
∗
∗
∗
∗
[
[
[
[
[
[
[
where
𝑇
̃ + 𝑃12 𝐶
̃ ,
Γ1 = 𝑃11 𝐶
11
12
0
]
𝜖𝐵11 ]
]
< 0,
𝐵12 ]
]
𝐷11 ]
−𝛾𝐼 ]
(19)
Applying Lemma 2 to (24), we get
Θ11 =
Θ13 =
𝑇 −1
𝑃12
𝑃11 𝑃12 ,
Θ12 =
𝑇 −1
̃𝑇 ,
−𝑃12
𝑃11 𝑃12 𝐴
22
𝑇 −1
̃𝑇 ,
−𝑃12
𝑃11 𝑃12 𝐴
12
Θ14 =
−1
Δ 11 ,
Θ22 = Δ𝑇11 𝑃11
−1
Θ23 = Δ𝑇11 𝑃11
Δ 12 ,
−1
Θ24 = Δ𝑇11 𝑃11
Γ1 ,
[
𝑇 −1
̃𝑇 ,
−𝑃12
𝑃11 𝑃12 𝐶
12
(20)
𝑇
{ 𝑋𝑆
}
Ξ Φ−𝑋
] [𝑇 0] < 0.
[
] + Sym{
[
}
∗ −𝑌 − 𝑌𝑇
𝑌𝑆𝑇
]
{[
}
−1
Θ44 = Γ1𝑇 𝑃11
Γ1 .
Pre- and post-multiplying (19) by diag(𝐼, (1/𝜖)𝐼, 𝐼, 𝐼, 𝐼), we
have
Π + 𝜖Θ < 0,
Ξ − 𝑋𝐹0𝑇 𝑇 − (𝑋𝐹0𝑇 𝑇)
[
∗
where
Θ11 Θ12 Θ13
[ ∗ Θ22 Θ23
[
Θ=[
[ ∗ ∗ Θ33
[∗ ∗ ∗
[∗ ∗ ∗
Θ14
Θ24
Θ34
Θ44
∗
0
]
𝐵11 ]
]
],
𝐵12 ]
]
𝐷11 ]
−𝛾𝐼]
where Ξ, Φ, 𝑆, and 𝑇 are given as before.
𝑇
Ξ − 𝑋𝐹0𝑇 𝑇 − (𝑋𝐹0𝑇 𝑇)
∗
∗
[
[
[
(23)
To turn (23) into an LMI, we will use the same idea
proposed in [17]. By substituting the transformation defined
before, we get another form of (23):
𝑇
Φ − 𝑋 − 𝑇𝑇 𝐹0 𝑌𝑇
]
−𝑌 − 𝑌𝑇
(27)
Using Lemma 3 to (27), we have
(22)
because 𝜖 is small enough, we can simplify (21) as in the
following form:
Ξ + Φ(𝑇𝑇 𝑆) + (𝑇𝑇 𝑆) Φ𝑇 < 0,
𝑇
𝑇
}
{ 𝑋𝐶2
] [𝐹𝑇 𝑇 0] < 0.
[
+ Sym{
}
𝑌𝐶2𝑇
]
}
{[
0
0]
]
0]
].
0]
0]
Π < 0.
(26)
Taking the expression of 𝑆 into account, we have
(21)
̃𝑇
̃𝑇 𝑃22 𝐶
̃𝑇
𝑃22 𝐴
𝑃22 𝐴
−𝑃
12
22
12
[ 22
[ ∗ Δ + Δ𝑇 Δ − 𝑃
Γ
[
11
12
12
1
11
Π=[
[ ∗
∗
−𝑃
0
22
[
[ ∗
∗
∗
−𝛾𝐼
∗
∗
∗
∗
[
(25)
It is obvious that (25) can be rewritten as follows:
−1
Θ33 = Δ𝑇12 𝑃11
Δ 12 ,
−1
Θ34 = Δ𝑇12 𝑃11
Γ1 ,
Ξ Φ
𝑋
] + Sym{[ ] [𝑆𝑇 𝑇 −𝐼]} < 0.
∗ 0
𝑌
(24)
Φ − 𝑋 − 𝑇𝑇 𝐹0 𝑌𝑇 𝑋𝐶2𝑇
]
−𝑌 − 𝑌𝑇
𝑌𝐶2𝑇 ]
∗
0 ]
(28)
{ 0
}
+ Sym{[0] 𝐺 [𝐹𝑇 𝑇 0 −𝐼]} < 0.
{[𝐼]
}
By letting 𝐿 = 𝐹𝐺𝑇 , we can finally get (13). This completes the
proof.
Both the static output feedback gain matrix 𝐹 and the
minimum 𝐻∞ norm 𝛾 can be obtained by solving the
following optimization problem:
min
𝛾,
𝑃𝑚𝑗 ,𝑋𝑖𝑗 ,𝑌𝑘𝑗 ,𝐿,𝐺,
(OP1)
Abstract and Applied Analysis
5
where 𝑖 = 1, . . . , 5, 𝑚 ≤ 𝑗, 𝑚, 𝑘, 𝑗 = 1, 2, subject to 𝛾 > 0,
𝑃11 > 0, 𝑃22 > 0, and 𝐺 > 0 and the LMI in (13).
Note that the result in Theorem 4 is finally obtained by
solving a 𝛾-related optimal problem. While in the following
part, a different approach of designing a static output feedback controller is given by solving an optimal problem in
which 𝜖 is involved.
Theorem 5. Given a scalar 𝛾 > 0 and a stabilizing state
feedback controller with gain 𝐹0 , if there exist matrices 𝑃11 > 0,
𝑃22 > 0, 𝐺 > 0, and 𝑄𝑚𝑚 > 0, 𝑚 = 1, . . . , 4 and matrices
̃1, 𝑋
̃ 2 , 𝑌,
̃ 𝑋𝑖 , 𝑄𝑚𝑛 , 𝑛 < 𝑚, 𝑖, 𝑚, 𝑛 = 1, . . . , 4, with
𝑃12 , 𝐿, 𝑌, 𝑋
appropriate dimensions, such that the following set of LMIs
hold:
[
[
[
[
̃1
𝑄
̃2
𝑄
̃1
−𝑋
̃ 1 𝐵21 + 𝐶2𝑇 𝐿𝑇 − 𝐹0𝑇 𝐺T
𝑋
̃3
𝑄
̃4
𝑄
T
̃2
𝑝12
−𝑋
̃ 2 𝐵21
𝑋
̃ T1
−𝑋
𝑝12 −
T ̃
𝑋1 + 𝐿𝐶2 − 𝐺𝐹0
[𝐵21
T
̃ 2T
𝑋
̃ −𝑌
̃
−𝑌
T
T ̃
𝑌
𝐵21
T
T ̃
𝑋2
𝐵21
̃ 21
𝑌𝐵
−𝐺 − 𝐺T
]
Π43
𝑄42 ]
T
0 0
𝑄41
]
[ T
[𝑄 0 0]
42
],
=[
]
[ T
[𝑄 0 0]
43
[𝑄44 0 0]
T
+𝑃22 𝐴 210
𝑃22 𝐴 220
𝑃12
]
[ ∗
T
]
=[
[𝜖 𝑃11 +𝑃11 𝐴 110 +𝑃12 𝐴 210 𝑃11 𝐴 120 +𝑃12 𝐴 220 ] ,
𝐶110
[
𝐶120
]
𝑃22 0 0
[𝑃 𝑃 0]
Φ = [ 12 11 ] ,
0
𝐼]
−𝜖∗ 𝑄11
Π44
∗
∗
[ ∗
∗
∗
[−𝜖 𝑄21 −𝜖 𝑄22
=[
[−𝜖∗ 𝑄 −𝜖∗ 𝑄 −𝜖∗ 𝑄
[
31
32
33
∗
∗
∗
∗
∗
∗
]
]
],
]
]
∗
[−𝜖 𝑄41 −𝜖 𝑄42 −𝜖 𝑄43 −𝜖 𝑄44 ]
< 0,
[
[
[
[
[
[
[
[
[
Π31
T
𝑄21
𝑄22 ]
]
],
𝑄32 ]
[0
]
]
]
]
T
Π41
𝑄11
[𝑄
[
= [ 21
[𝑄31
[𝑄41
Π11
∗
∗
∗
∗
∗
0
−𝛾𝐼
∗
∗
∗
∗
Π31
Π32
Π33
∗
∗
∗
Π41
Π42
Π43
Π44
∗
∗
−𝑋1T
−𝑋2T
Φ − 𝑋3T
−𝑋4T
−𝑌 − 𝑌T
∗
ΨT 𝑋2T
ΨT 𝑋3T
ΨT 𝑋4T
ΨT 𝑌T
[ΨT 𝑋1T − 𝐿𝐶2 + 𝐺𝐹0
]
]
]
]
]
]
]
]
]
(29)
where
T
T
−𝑄41
−𝑄31
[ T
]
̃ 2 = [−𝑄 −𝑄 ] ,
𝑄
42 ]
[ 32
[−𝑄33 −𝑄43 ]
[
]
̃ 1 = [−𝑄21 −𝑄22 ] ,
𝑄
[
]
−𝑄
−𝑄
32 ]
[ 31
T
T
̃3 = [
𝑄
[
T
(−𝑄41 + 𝑃12
𝐴 110 )
T
−𝑄44
[
T
̃4 = [
𝑄
[
T
−𝑄44
[
Π11 = [
−𝜖∗ 𝑃11
∗
T
−𝑃12
−𝑃22
],
]
] ,
]
T
(−𝑄43 + 𝑃12
𝐵11 )
T
]
] ,
]
𝑃22 𝐵12
Π32 = [𝑃11 𝐵11 + 𝑃12 𝐵12 ] ,
𝐷11
[
]
−𝑃22
∗
∗
Π33 = [−𝑃12 −𝜖∗ 𝑃11 ∗ ] ,
0
−𝛾𝐼]
[ 0
𝐵22
Ψ = [ 𝐵21 ] ,
[𝐷12 ]
(30)
−𝐺 − 𝐺𝑇 ]
< 0,
T
−𝑄11 −𝑄21
Π42
𝑇
𝑄31
[ 𝑇]
[𝑄 ]
32 ]
=[
[ ],
[𝑄33 ]
[𝑄43 ]
then for any singular perturbation parameter 𝜖 ∈ (0, 1/𝜖∗ ], by
introducing a static output feedback controller in the form of
(2), the closed-loop system (3) is asymptotically stable, and its
transfer function satisfies ‖𝐺‖∞ < 𝛾 with the controller gain
𝐹 = 𝐺−1 𝐿.
Proof. According to Lemma 1, we know that the closed-loop
system (3) is asymptotically stable, and its transfer function is
less than 𝛾 if there exists a positive definite matrix 𝑃 such that
the following LMI holds:
−𝑃 ∗
∗ ∗
]
[
[ 0 −𝛾𝐼 ∗ ∗ ]
]
[
] < 0.
[
̃𝜖 𝑃𝐵1𝜖 −𝑃 ∗ ]
[𝑃𝐴
]
[
̃1 𝐷11 0 −𝛾𝐼
𝐶
]
[
(31)
For every singular perturbation parameter 𝜖 ∈ (0, 1/𝜖∗ ], let
1
[ 𝜖 𝑃11 ∗ ]
𝑃 (𝜖) = [
]>0
𝑇
𝑃
𝑃
22 ]
[ 12
satisfy (31), and we get
(32)
6
Abstract and Applied Analysis
1
∗
∗
∗
𝑃
[
𝜖 11
[
𝑇
−𝑃22
∗
∗
−𝑃12
[
[
0
0
−𝛾𝐼
∗
[
[1
1
[ 𝑃 +𝑃 𝐴
̃
̃
̃
̃
𝑃11 𝐵11 + 𝑃12 𝐵12 − 𝑃11
[ 𝜖 11
11 11 + 𝑃12 𝐴21 𝑃11 𝐴12 + 𝑃12 𝐴22
𝜖𝑇
[ 𝑇
𝑇
𝑇
[ 𝑃 + 𝜖𝑃 𝐴
̃
̃
̃
̃
𝑃12
12
12 11 + 𝑃22 𝐴21 𝜖𝑃12 𝐴12 + 𝑃22 𝐴22 𝜖𝑃12 𝐵11 + 𝑃22 𝐵12
̃11
̃12
𝐶
𝐶
𝐷11
0
[
0
𝐼
0
0
0
0
0
0
𝐼
0
0
0
0
0
0
0
𝐼
0
∗
∗
∗
∗
∗
]
]
]
]
]
] < 0.
∗ ∗ ]
]
]
𝑃22 ∗ ]
0 −𝛾𝐼]
(33)
and for all 𝜖 ∈ (0, 1/𝜖∗ ],
Pre- and post-multiplying (33) by
𝐼
[0
[
[0
[
[0
[
[0
[0
∗
0
0
0
𝐼
0
0
0
0]
]
0]
]
0]
]
0]
𝐼]
Θ11 (𝜖∗ )
(34)
and its transpose, respectively, then for every 𝜖 ∈ (0, 1/𝜖∗ ], we
have
1
∗
[Θ11 ( 𝜖 )
]
[
] + [𝜖Ω 0] < 0,
[
0 0
1
1 ]
Θ21 ( ) Θ22 ( )
[
𝜖
𝜖 ]
(35)
where
0
∗
∗
0
0
∗
0
0
0
𝑇̃
𝑇̃
𝑇
𝐴
𝐴
𝑃
𝑃
𝑃
[ 12 11 12 12 12 𝐵11
[
Ω=[
[
∗
∗]
]
∗],
0]
then we can easily conclude that
1
∗
∗
Θ ( )
[ 11 𝜖
]
[
]
[
]
1
1
[Θ ( ) Θ ( ) ∗ ] < 0.
22
[ 21 𝜖
]
𝜖
[
]
[
]
1
𝑄
0
− 𝑄
[
𝜖 ]
1
∗
]
[Θ11 ( 𝜖 )
] + [𝜖𝑄 0] < 0.
[
[
0 0
1
1 ]
Θ ( ) Θ22 ( )
[ 21 𝜖
𝜖 ]
1
∗
∗
∗
− 𝑃11
]
𝜖 𝑇
−𝑃22
∗
∗ ]
−𝑃12
],
0
0
−𝛾𝐼
∗ ]
𝑇
̃21 𝑃22 𝐴
̃22 𝑃22 𝐵12 −𝑃22 ]
[𝑃12 + 𝑃22 𝐴
(36)
1
1
∗
− 𝑃
],
Θ22 ( ) = [ 𝜖 11
𝜖
0
−𝛾𝐼
(39)
(40)
Taking (37) into consideration, we have (35).
To turn (37) and (40) into LMIs, we still adopt the same
technique used in [17]. By introducing the transformations
defined before, we can rewrite (37) and (40) into the following
form:
̃
̃ 𝑄
0
𝑄
[ ̃ 1 ̃ 2 ] + Sym {[ 𝑇 ] [𝐵21 𝑆 0]} < 0,
𝑃12
𝑄3 𝑄4
̃11 + 𝑃12 𝐴
̃21 ,
Υ11 = 𝑃11 𝐴
Π11 ∗ ∗ ∗
[ 0 −𝛾𝐼 ∗ ∗ ]
]
[
[Π31 Π32 Π33 ∗ ]
[Π41 Π42 Π43 Π44 ]
̃12 + 𝑃12 𝐴
̃22 ,
Υ12 = 𝑃11 𝐴
Υ13 = 𝑃11 𝐵11 + 𝑃12 𝐵12 .
If there exists a positive definite matrix 𝑄 = [𝑄𝑖𝑗 ], 𝑖, 𝑗 =
1, . . . , 4 satisfies
Ω − 𝑄 < 0,
(38)
Applying the Schur complement to (39), we get
[
1
[
Θ11 ( ) = [
[
𝜖
1
1
𝑃 + Υ11 Υ12 Υ13 −𝑃12 ]
[
Θ21 ( ) = 𝜖 11
,
𝜖
̃
̃12 𝐷11 0
𝐶
[ 𝐶11
]
∗
∗
]
[
∗
∗
[Θ21 (𝜖 ) Θ22 (𝜖 ) ∗ ]
] < 0,
[
]
[
∗
𝑄
0
−𝜖 𝑄
]
[
(37)
0
{
}
{
}
{[ 0 ]
}
[
]
0
0
0]
+ Sym {[ ] Ψ [𝑆
< 0.
}
Φ
{
}
{
}
{[ 0 ]
}
(41)
Abstract and Applied Analysis
7
Applying Lemma 2 to (41), we have
Taking Lemma 3 into account, we know that (43) and (44)
are equivalent to
̃2 0
̃1 𝑄
𝑄
̃
{ 𝑋1
}
[̃ ̃
𝑇]
[
̃ 2 ] [𝐵21 𝑆 0 −𝐼] < 0, (42)
[𝑄3 𝑄4 𝑃12 ] + Sym { 𝑋
}
̃]
{[ 𝑌
}
[ 0 𝑃12 0 ]
Π𝑇31
Π𝑇32
Π𝑇41
Π𝑇42
Π11 0
[
[ 0 −𝛾𝐼
[
[
[
[Π31 Π32 Π33 Π𝑇43
[
[
[
[Π41 Π42 Π43 Π44
[
[
0
0 Φ𝑇 0
[
]
0]
]
]
]
Φ]
]
]
]
0]
]
]
0
]
(43)
̃2
̃1
̃1
𝑄
−𝑋
𝑄
]
[
𝑇
[𝑄
̃
̃4
̃2]
𝑄
𝑃12
−𝑋
]
[ 3
]
[
[ 𝑇
𝑇
𝑇]
[−𝑋
̃ 𝑃12 − 𝑋
̃ −𝑌
̃ −𝑌
̃ ]
1
2
]
[
Π𝑇31
Π𝑇41
−𝛾𝐼
Π𝑇32
ΠT34
(44)
Π32
Π33
Π𝑇43
Π42
Π43
Π44
−𝑋2𝑇 Φ𝑇 − 𝑋3𝑇 −𝑋1𝑇
𝑋1 Ψ
}
{
}
{
}
{
[𝑋2 Ψ]
}
{
]
}
{[
]
[
+ Sym {[𝑋3 Ψ] [𝑆 0 0 0 0]} < 0,
}
{
]
}
{
[
}
{
}
{ 𝑋4 Ψ
}
{[ 𝑌Ψ ]
𝑋1 Ψ
]
𝑋2 Ψ]
]
]
]
𝑋3 Ψ]
]
]
]
𝑋4 Ψ]
]
]
𝑌Ψ ]
]
] (47)
]
0
]
respectively.
From the above analysis, we learn that if there exists 𝐺1 =
𝐺2 = 𝐺 which satisfies both (46) and (47), then (29) can
be finally concluded by considering the expression of 𝑆. This
completes the proof.
We can obtain the static output feedback controller gain
matrix 𝐹, as well as the stability upper bound of the singular
perturbation parameter which we record as 1/𝜖∗ by solving
the following optimal problem:
−𝑋1
]
−𝑋2 ]
]
]
]
Φ − 𝑋3 ]
]
]
]
−𝑋4 ]
]
]
𝑇
−𝑌 − 𝑌
]
Π𝑇41
0
}
{
}
{
}
{
[0]
}
{
}
{
[
]
}
{
}
{[0]
[
]
+ Sym {[ ] 𝐺2 [𝑆 0 0 0 0 −𝐼]} < 0,
0
}
{
}
{
[ ]
}
{
}
{
[0]
}
{
}
{
𝐼
}
{[ ]
̃
}
{ 𝑋1 𝐵21
̃ 2 𝐵21 ] [𝑆 0 0] < 0,
+ Sym {[𝑋
}
̃ 21 ]
}
{[ 𝑌𝐵
0
0
Π𝑇31
−𝑋1
[
𝑇
𝑇
[ 0
Π34
−𝑋2
−𝛾𝐼
Π32
[
[
[
[ Π31
Π32
Π33
ΠT43 Φ − 𝑋3
[
[
[
Π42
Π43
Π44
−𝑋4
[ Π41
[
[
[ −𝑋𝑇 −𝑋𝑇 Φ𝑇 − 𝑋𝑇 −𝑋𝑇 −𝑌 − 𝑌𝑇
[
1
2
3
4
[
[
Ψ𝑇 𝑋1𝑇 Ψ𝑇 𝑋2𝑇 Ψ𝑇𝑋3𝑇 Ψ𝑇𝑋4𝑇 𝑌Ψ𝑇
[
Π11
It is obvious that (42) and (43) can be rewritten as
where Ψ is given as before.
(46)
0
}
{
}
{
}
{[0]
[
0
0
−𝐼]
[𝑆
+ Sym {[ ]
< 0,
𝐺
1
}
]
0
}
{
}
{
}
{[ 𝐼 ]
0
𝑋1
}
{
}
{
}
{
[ ]
}
{
𝑋
]
}
{
[
2
}
{
}
{[ ]
]
[
+ Sym {[𝑋3 ] Ψ [𝑆 0 0 0 −𝐼]} < 0.
}
{
}
{
[𝑋 ]
}
{
}
{
[ 4]
}
{
}
{
𝑌
}
{[ ]
Π11
[
[ 0
[
[
[
[ Π31
[
[
[
[ Π41
[
[
−𝑋𝑇
[ 1
̃2
̃ 1 𝐵21
̃1 𝑋
̃1
𝑄
−𝑋
𝑄
𝑇
[ 𝑄
̃3
̃4
̃ 2 𝐵21 ]
̃2 𝑋
𝑄
𝑃
−
𝑋
12
]
[
𝑇
𝑇
[ ̃𝑇
̃ 21 ]
̃ −𝑌
̃ −𝑌
̃ 𝑌𝐵
]
[ −𝑋1 𝑃12 − 𝑋
2
𝑇 ̃𝑇
𝑇 ̃𝑇
𝑇 ̃𝑇
𝐵21 𝑌
0 ]
[𝐵21 𝑋1 𝐵21 𝑋2
min
𝜖∗ ,
̃ 𝑙 ,𝑌,𝐿,𝐺
̃
𝑃𝑖𝑗 ,𝑄𝑚𝑛 ,𝑋𝑘 ,𝑌,𝑋
(OP2)
where 𝑖 ≤ 𝑗, 𝑖, 𝑗, 𝑙 = 1, 2, 𝑚, 𝑛, 𝑘 = 1, . . . , 4, subject to 𝜖∗ > 0,
𝑃𝑖𝑖 > 0, 𝑄𝑛𝑛 > 0, and 𝐺 > 0 and the LMIs in (29).
(45)
Remark 6. The result in Theorem 4 finally turns into (OP1),
which is solved by optimizing 𝛾, the minimum 𝐻∞ norm
of the closed-loop system (3). While results in Theorem 5
are obtained by solving a different optimal problem (OP2),
which optimizes the stability upper bound 1/𝜖∗ with a fixed
performance index 𝛾.
4. A Numerical Example
In this section, a numerical example is presented to illustrate
the effectiveness of the proposed results.
8
Abstract and Applied Analysis
Example 7. Consider a discrete-time singularly perturbed
system described by (1) with
𝐴𝜖 = [
𝐵1𝜖 = [
1 + 0.2129𝜖 1.8140𝜖
],
−0.1814 0.8179
0
],
0.1
1 0
𝐶1 = [1 1] ,
[0 0]
0.31
𝐷12 = [ 0 ] ,
[ 0 ]
𝐵2𝜖 = [
0.1874𝜖
],
0.1812
0.01
𝐷11 = [ 0 ] ,
[ 0 ]
𝐶2 = [
(48)
𝑇
0.4394
] .
0.1372
First of all, by solving the optimal problem (OP1), we can get
the introduced static output feedback controller gain
𝐹 = −0.1078
(49)
and the minimum 𝐻∞ norm of the closed-loop system
𝛾 = 0.8557.
(50)
Then, referring to (OP2), we can solve the corresponding
LMIs with 𝛾 = 1. The obtained static output feedback
controller gain is
𝐹 = −23.3354,
(51)
while the stability upper bound of the singular perturbation
parameter is
1
= 0.2775.
𝜖∗
(52)
Remark 8. Though the performance of the closed-loop system is not as good as the state feedback case, static output
feedback controller plays more important role in implemental
sense with proper performance.
5. Conclusion
In this paper, 𝐻∞ control problems for fast sampling singularly perturbed systems via static output feedback have
been discussed. Rather than adopting the traditional design
method of decomposing the original system into fast and slow
subsystems, two LMI-based sufficient conditions have been
given to guarantee the existence of static output feedback
controllers and the asymptotical stability of the closed-loop
system with a transfer function whose 𝐻∞ norm is less than
𝛾. With LMI toolbox in matlab platform, the obtained LMI
results can be solved easily. The proposed methods simplify
the controller design procedure.
Acknowledgments
This work is supported by the National 973 Program of
China (2012CB821202) and the National Natural Science
Foundation of China (61174052, 90916003).
References
[1] S. Sen and K. B. Datta, “Stability bounds of singularity perturbed
systems,” Transactions on Automatic Control, vol. 38, no. 2, pp.
302–304, 1993.
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