Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 903126, 28 pages
doi:10.1155/2011/903126
Research Article
On Output Feedback Multiobjective Control for
Singularly Perturbed Systems
Mehdi Ghasem Moghadam and
Mohammad Taghi Hamidi Beheshti
Control and Communication Networks Laboratory, Electrical Engineering Department,
Tarbiat Modares University, Tehran, Iran
Correspondence should be addressed to
Mohammad Taghi Hamidi Beheshti, mbehesht@modares.ac.ir
Received 6 September 2010; Revised 30 November 2010; Accepted 6 January 2011
Academic Editor: Alexei Mailybaev
Copyright q 2011 M. Ghasem Moghadam and Ṁ. T. H. Beheshti. 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.
A new design procedure for a robust H2 and H∞ control of continuous-time singularly perturbed
systems via dynamic output feedback is presented. By formulating all objectives in terms of a
common Lyapunov function, the controller will be designed through solving a set of inequalities.
Therefore, a dynamic output feedback controller is developed such that H∞ and H2 performance
of the resulting closed-loop system is less than or equal to some prescribed value. Also, H∞ and H2
performance for a given upperbound of singular perturbation parameter ε ∈ 0, ε∗ are guaranteed.
It is shown that the ε-dependent controller is well defined for any ε ∈ 0, ε∗ and can be reduced to
an ε-independent one so long as ε is sufficiently small. Finally, numerical simulations are provided
to validate the proposed controller. Numerical simulations coincide with the theoretical analysis.
1. Introduction
It is well known that the multiple time-scale systems, otherwise known as singularly
perturbed systems, often raise serious numerical problems in the control engineering field. In
the past three decades, singularly perturbed systems have been intensively studied by many
researchers 1–8.
In practice, many systems involve dynamics operating on two or more time-scales
3. In this case, standard control techniques lead to ill-conditioning problems. Singular
perturbation methods can be used to avoid such numerical problems 1. By utilizing the time
scale properties, the system is decomposed into several reduced order subsystems. Reduced
order controllers are designed for each subsystems.
In the framework of singularly perturbed systems, H∞ control has been investigated
by many researchers, and various approaches have been proposed in this field 9–14.
2
Mathematical Problems in Engineering
However, to the best of our knowledge, the problem of multiobjective control for linear
singular perturbed systems is still an open problem. By multiobjective control, we refer to
synthesis problems with a mix of time- and frequency-domain specifications ranging from
H2 and H∞ performances to regional pole placement, asymptotic tracking or regulation, and
settling time or saturation constraints.
In 12, the H∞ control problem is concerned via state feedback for fast sampling
discrete-time singularly perturbed systems. A new H∞ controller design method is given
in terms of solutions to linear matrix inequalities LMIs, which eliminate the regularity
restrictions attached to the Riccati-based solution. In 13, the H∞ control problem via state
feedback for fast sampling discrete-time singularly perturbed systems is investigated. In
fact, a new sufficient condition which ensures the existence of state feedback controllers is
presented such that the resulting closed-loop system is asymptotically stable. In addition, the
results were extended to robust controller design for fast sampling discrete-time singularly
perturbed systems with polytopic uncertainties. Presented condition, in terms of a linear
matrix inequality LMI, is independent of the singular perturbation parameter.
Undoubtedly, output feedback stabilization is one of the most important problems in
control theory and applications. In many real systems, the state vector is not always accessible
and only partial information is available via measured output. Furthermore, the reliability of
systems and the simplicity of implementation are other reasons of interest in output control
which is adopted to stabilize a system.
LMI’s have emerged as a powerful formulation and design technique for a variety
of linear control problems. Since solving LMI’s is a convex optimization problem, such
formulations offer a numerically tractable means of attacking problems that lack an analytical
solution. Consequently, reducing a control design problem to an LMI can be considered as a
practical solution to this. Since the nonconvex formulation of the output feedback control, its
conditions are restrictive or not numerically tractable 15. It has been an open question how
to make these conditions tractable by means of the existing software. Many research results
on such a question have been reported in 15–20.
In 1, 2, the dynamic output feedback control of singular perturbation systems
has been investigated. However, to the best of our knowledge, the design of dynamic
output feedback for robust controller via LMI optimization is still an open problem. The
main contribution of this paper is to solve the problem of multiobjective control for linear
singularly perturbed systems. Considered problem consists of H∞ control, H2 performance,
and singular perturbation bound design. For given an H∞ performance bound γ or H2
performance bound υ, and an upperbound ε∗ for the singular perturbation, an ε-dependent
dynamic output feedback controller will be constructed, such that for all ε ∈ 0, ε∗ . Due to
this, the closed-loop system is admissible and the H∞ norm H2 norm of the closed-loop
system is less than a prescribed γ prescribed υ. Sufficient conditions for such a controller
are obtained in form of strict LMIs. A mixed control H2 /H∞ problem for singular systems
is also considered in this paper. It is shown that the designed controller is well-defined for
for all ε ∈ 0, ε∗ . It is shown that if ε is sufficiently small, the controller can be reduced to an
ε-independent one. Numerical examples are given to illustrate the main results.
The paper is organized as follows. Section 2 gives the problem statement and
motivations. Section 3 presents the main results. The theorems for H2 , H∞ and multiobjective
H 2 /H∞ design via output feedback control are presented in this section. Due to proposed
theorems, robust H2 , H∞ , and multiobjective performance of continuous-time singularly
perturbed systems via dynamic output feedback for a linear systems will be accessible.
Mathematical Problems in Engineering
3
Section 4 illustrates numerical simulations for the proposed theorems. Finally, conclusions
in Section 5 close the paper.
Notation. A star ∗ in a matrix indicates a transpose
quantity.
For
example: ∗ A > 0 stands
∗ stands for A BT .
for AT A > 0, or in a symmetric matrix A
B C
B C
2. Problem Statement
Consider the following singularly perturbed system with slow and fast dynamics described
in the standard “singularly perturbed” form:
z1
ẋ1
A1 x1 A2 x2 B1 u Bw1 w,
εẋ2
A3 x1 A4 x2 B2 u Bw2 w,
Cz11 x1 Cz12 x2 Dzu1 u Dzw1 w,
z2
Cz21 x1 Cz22 x2 Dzu2 u,
y
Cy1 x1 Cy2 x2 Dyw w,
2.1
where x1 t ∈ Rn1 and x2 t ∈ Rn2 form the state vector, ut ∈ Rp is the control input vector,
yt ∈ Rm1 is the output, wt is a vector of exogenous inputs such as reference signals,
disturbance signals, sensor noise and z1 , z2 are regulated outputs.
By introducing the following notations:
x
A1 A2
A
B1
,
B2
B
Cz1
x1
,
x2
A3 A4
Cy1 Cy2 ,
Cy
Cz11 Cz12 ,
Cz2
Bw
,
Bw1
Bw2
2.2
Cz21 Cz22 ,
.
The system 2.1 can be rewritten into the following compact form:
Eε ẋ
z1
Ax Bu Bw w,
Cz1 x Dzu1 u Dzw1 w,
z2
Cz2 x Dzu2 u,
y
Cy x Dyw w.
2.3
4
Mathematical Problems in Engineering
By applying dynamic output feedback controller in the following form:
ẋc1
Ac1 xc1 Ac2 xc2 Bc1 y,
εẋc2
Ac3 xc1 Ac4 xc2 Bc2 y,
u
2.4
⎡ ⎤
xc1
Cc2 ⎣ ⎦ Dc y.
xc2
Cc1
The controller 2.4 can be rewritten into the following compact form:
Ac xc Bc y,
Eε ẋc
u
2.5
Cc xc Dc y,
where
⎤
⎡
Ac1 Ac2
⎦,
⎣
Ac3 Ac4
Ac
Bc
Bc1
,
Bc2
Cc
Cc1 Cc2 .
2.6
The closed loop system is
Eε 0 ẋ
0 I
zi
where Dcl 2
A BDc Cy
ẋc
Czi Dzui Dc Cy
BCc
x
Bw BDc Dyw
Eε−1 Bc Cy Eε−1 Ac xc
Eε−1 Bc Dyw
x
Dzwi Dzui Dc Dyw wt,
Dzui Cc
xc
wt,
2.7
for i
1, 2,
0 and
Ee
diagEε , I,
Eε diagI, εI,
xcl
A BDc Cy Bw Cc
,
Acl
Eε−1 Bc Cy Eε−1 Ac
Bw BDc Dyw
,
Bcl
Eε−1 Bc Dyw
Ccl i
Czi Dzui Dc Cy Dzui Cc ,
Dcl 1
T T T
x xc ,
Dzw1 Dzu1 Dc Dyw .
Note, following definitions and lemmas are useful in next sections.
2.8
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5
Definition 2.1. For a linear time-invariant operator G : ω ∈ L2 R → z ∈ L2 R , G is L2
stable if ω ∈ L2 R implies z ∈ L2 R. Here, G is said to have L2 gain less than or equal to
γ > 0 if and only if
T
t
zt2 dt ≤ γ 2
0
wt2 dt
2.9
0
for all T ∈ R .
Lemma 2.2 see 21. For system G : A, B, C, D, the L2 gain will be less than γ > 0 if there exist
a positive definite matrix X X T > 0 such that
⎡
⎤
XA AT X XB CT
⎢
⎥
⎢ BT X
−γI DT ⎥
⎣
⎦ < 0.
C
D −γI
2.10
Definition 2.3. For a linear time-invariant operator G : w → z, H2 norm G is defined by
G22
∞
1
trace
2π
−∞
∗
G jω G jω dω.
2.11
Lemma 2.4 see 21. For stable system G : A, B, C the H2 performance will be less than v if
there exist a matrix Z and a positive definite matrix X X T > 0 such that the following LMIs are
feasible:
AT X XA XB
BT X
X CT
C Z
−I
< 0,
> 0,
2.12
traceZ < ν.
Lemma 2.5 see 22. For a positive scalar ε∗ and symmetric matrices S1 , S2 , and S3 with
appropriate dimensions, inequality
S1 εS2 ε2 S3 > 0
2.13
holds for all ε ∈ 0, ε∗ , if
S1 ≥ 0,
S1 ε∗ S2 > 0,
S1 ε∗ S2 ε∗ 2 S3 > 0.
2.14
6
Mathematical Problems in Engineering
3. Main Result
Here, we address stability, H∞ stability, H2 performance, and multiobjective H2 /H∞
performance for a singularly perturbed system via dynamic output feedback control.
3.1. Stability Problem
Consider closed loop system 2.7 without disturbance wt:
Ee ẋcl
A BDc Cy
BCc
Eε−1 Bc Cy
Eε−1 Ac
3.1
xcl ,
where A, B, Cy were defined in 2.2. In the following theorem, we propose design
procedure for obtaining controllers parameters such that the closed loop system 3.1
becomes asymptotically stable.
Theorem 3.1. Given an upperbound ε∗ for the singular perturbation ε, if there exist matrices
Ak , Bk , Ck , Dk , Y11 ,Y12 , Y22 , X11 , X12 , and X22 satisfying the following LMIs
⎡
Υ11 0 ≤ 0,
3.2
Υ11 ε∗ < 0,
3.3
ε∗ Y12
Y11
⎢ ∗ T ∗
⎢ ε Y ε Y22
⎢
12
⎢
I 0
⎣
0 ε∗ I
I
0
⎤
⎥
⎥
0 ε∗ I
⎥ > 0,
X11 ε∗ X12 ⎥
⎦
3.4
T
ε∗ X12
ε∗ X22
Y11
I
I
X11
3.5
> 0,
where
⎡
∗
Υ11 ε Y11 ε∗ Y12
⎢A T
⎢
Y12
⎢
⎢
⎣
Y22
∗
⎤
BCk ∗
A BDk Cy T
X11 X12
A Bk Cy ∗
ε∗ X12 X22
ATk
⎥
⎥
⎥.
⎥
⎦
3.6
Then, for any ε ∈ 0, ε∗ , the closed-loop singularly perturbed system 3.1 is asymptotically stable
Mathematical Problems in Engineering
7
via dynamic output controller 2.4, also controller parameters are obtained from following equations:
Dc
Ck − Dc CQ11 ,
Cc
T
ϑε−1 Bk − P11
BDc ,
Bc
Ac
Dk ,
3.7
T
T
ϑε−1 Ak − P11
BCc ,
A BDc Cy Q11 ϑε Bc Cy Q11 P11
where
P11
X11 εX12
T
X12
X22
ϑε
ϑ1
,
Y11 εY12
Q11
T
Y12
Y22
,
I − ϑ1 − εϑ2
−ϑ3
−εϑ4
I − εϑ2T − ϑ5
,
3.8
T
X12 Y12
,
ϑ3
X11 Y12 X12 Y22 ,
T
T
X12
Y11 X22 Y12
,
ϑ5
X22 Y22 .
X11 Y11 ,
ϑ4
ϑ2
Proof. Choose the Lyapunov function as
3.9
xclT tEe Pε xcl t,
V t
where
3.10
Ee Pε PεT Ee > 0,
P11 P12
.
Pε
T
P12
Eε P22
3.11
From 3.10, it is concluded that
Ee Pε
PεT Ee
⇒
Eε P11 Eε P12
T
P12
Eε
T
P11 Eε Eε P12
T
P12
Eε
P22
T
P22
3.12
and from 3.12, P11 has following structure:
Eε P11
T
P11
Eε
⇒ P11
X11 εX12
T
X12
X22
.
3.13
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Mathematical Problems in Engineering
Now, define invert of Ee Pε as follow
Ee Pε −1 Pε−1 Ee−1 ,
Q11 Eε−1 Q12
−1
Pε
Qε
T
Q12
Q22
3.14
and Q11 has following structure:
−1
Qε Ee T
Ee−1 Q11
⇒
Q11 Eε−1 Eε−1 Q12
T −1
Q12
Eε
T
Eε−1 Q12
Eε−1 Q11
T −1
Q12
Eε
Q22
Q22
,
Q11
Y11 εY12
T
Y12
Y22 . 3.15
Using the following equality:
Eε P11 Eε P12 Q11 Eε−1 Eε−1 Q12
T
P12
Eε
P22
T −1
Q12
Eε
I 0
0 I
Q22
3.16
,
we also have the constraint that
T
Eε P11 Q11 P12 Q12
Eε−1
T
I ⇒ P11 Q11 P12 Q12
3.17
I.
Here, we define new matrices Π1 and Π2 as follows:
Π1
Q11 I
,
T
Q12
0
Pε Π1
3.18
Π2
3.19
and Π2 is obtained from 3.17 as follows:
P11
P12
T
P12
Eε P22
Q11 I
T
P11 Q11 P12 Q12
P11
T
T
T
P12
Eε Q11 P22 Q12
P12
Eε
T
Q12
0
,
Π2
I
P11
T
0 P12
Eε .
3.20
The derivative of the Lyapunov function 3.9 is
ẋclT Ee Pε xcl xclT Ee Pε ẋcl ,
V̇ < 0 ⇒ xclT ATcl Pε PεT Acl xcl < 0,
V̇
where Acl is defined in 2.8.
3.21
Mathematical Problems in Engineering
9
Sufficient condition to satisfy 3.21 is
ATcl Pε PεT Acl < 0.
3.22
Equation 3.22 will hold if and only if
ΠT1 ATcl Pε Π1 ΠT1 PεT Acl Π1 < 0,
3.23
where Π1 is defined in 3.18.
From 3.19, equation 3.23 can be rewritten as
ΠT1 ATcl Π2 ΠT2 Acl Π1 < 0.
3.24
With substituting Π1 and Acl from 3.18 and 2.8, respectively, we have
I
A BDc C
0
Eε−1 Bc C
T
P11
Eε P12
Q11 I
BCc
Eε−1 Ac
T
Q12
0
∗ < 0.
3.25
Now, we define the new variables:
Ck
Bk
Ak
T
Dc CQ11 Cc Q12
,
T
P11
BDc Eε P12 Eε−1 Bc ,
3.26
T
T
T
T
P11
BCc Q12
Eε P12 Eε−1 Ac Q12
.
A BDc Cy Q11 Eε P12 Eε−1 Bc Cy Q11 P11
Then, from 3.13, 3.15, and 3.26, equation 3.25 can be rewritten as
⎡
Υ11 ε
Y11 εY12
⎤
BCk ∗
A BDk Cy ⎢A T
⎢
Y12 Y22
⎢
T
⎢
X11 X12
⎣
A Bk Cy ∗
∗
εX12 X22
ATk
⎥
⎥
⎥ < 0.
⎥
⎦
3.27
Also, the condition 3.10 holds if and only if
⎡
Y11 εY12
⎢
⎢ εY T εY22
12
ΠT1 Ee Pε Π1 > 0 ⇒ ⎢
⎢ I 0 ⎣
0 εI
I 0
⎤
⎥
⎥
0 εI
⎥
> 0.
X11 εX12 ⎥
⎦
3.28
T
εX12
εX22
According to Lemma 2.5, the conditions 3.27 and 3.28 are valid for all ε ∈ 0, ε∗ , If
3.2, 3.3, 3.4, and 3.5 are satisfied.
10
Mathematical Problems in Engineering
For computing controller parameters we obtain P11 and Q11 from solving LMIs in 3.2,
3.3, 3.5, and 3.6. Then from constraint 3.17 with assumption Q12 I we have
P12
I − P11 Q11
I−
X11 εX12
Y11 εY12
T
X12
X22
T
Y12
Y22
Eε P12 Eε−1
I−
ϑ1 εϑ2
εϑ3
ϑ4
εϑ2T ϑ5
,
I − ϑ1 − εϑ2
−ϑ3
−εϑ4
I − εϑ2T − ϑ5
3.29
,
where
X11 Y11 ,
ϑ2
T
X12 Y12
,
X11 Y12 X12 Y22 ,
ϑ4
T
T
X12
Y11 X22 Y12
,
ϑ1
ϑ3
ϑ5
3.30
X22 Y22 .
Also From 3.29 and 3.26 we can obtain controller parameters from 3.7. Also
from
c Bc
3.7 controllers parameters are always well defined for all ε ∈ 0, ε∗ and limε → 0 A
Cc Dc
become
Dc0
Ck − Dc CQ110 ,
Cc0
Bc0
Ac0
Dk ,
−1
T
Bk − P110
ϑε0
BDc ,
3.31
−1
T
T
Ak − P110
ϑε0
BCc0 ,
A BDc Cy Q11 ϑε0 Bc Cy Q110 P110
where ϑ1 , ϑ3 , ϑ5 are defined in 3.30 and
P110
X11
0
Y11
0
,
Q110
,
T
T
X12
Y12
X22
Y22
I − ϑ1 −ϑ3
.
ϑε0
0
I − ϑ5
3.32
This completes the proof.
Remark 3.2. Throughout the paper, it is assumed that the singular perturbation parameter
ε is available for feedback. Indeed, in many singular perturbation systems, the singular
perturbation parameter ε can be measured. In these cases, ε is available for feedback,
which has attracted much attention. For example, ε-dependent controllers were designed
Mathematical Problems in Engineering
11
for singular perturbation systems in 22–24. Since ε is usually very small, an ε-dependent
controller may be ill-conditioning as ε tends to zero. Thus, it is a key task to ensure the
obtained controller to be well defined. This problem will be discussed later.
3.2. H∞ Performance
Consider the closed loop system 2.7 with regulated output z1 . In following theorem,
we proposed a procedure for obtaining controller parameters such that the closed loop
singular perturbed system 2.7 with regulated output z1 becomes asymptotically stable and
guarantees the H∞ performance with attenuation parameter γ.
Theorem 3.3. Given an H∞ performance bound γ and an upperbound ε∗ for the singular perturbation
ε, if there exist matrices Ak , Bk , Ck , Dk , Y11 , Y12 , Y22 , X11 , X12 , and X22 satisfying the following LMIs
⎡
ψ11 0
⎢
⎢
⎣
⎡
ψ11 ε∗ ⎢
⎢
⎣
Υ11 0
∗
Υ11 ε∗ ∗
⎤
Υ12 0
⎥
−γI Dzw1 Dzu1 Dk Dyw ⎥
⎦ ≤ 0,
−γI
∗
3.33
⎤
Υ12 ε∗ ⎥
−γI Dzw1 Dzu1 Dk Dyw ⎥
⎦ < 0,
−γI
∗
where Υ11 ε∗ defined in 3.6 Υ12 ε∗ is
⎡
∗
Υ12 ε Bw BDk Dyw
⎢
⎢
⎢
⎢ X11 X T ⎣
12
Bw Bk Dyw
∗
ε X12 X22
Y11
T
Y12
ε∗ Y12 Y22
T
Cz1
⎤
CzT
T
CkT Dzu1
⎥
⎥
T
CyT DkT Dzu1
⎥.
⎥
⎦
3.34
Then, for any ε ∈ 0, ε∗ , the closed-loop singularly perturbed system 2.7 is asymptotically stable
and with an H∞ -norm less than or equal to γ, also parameters controller are obtained from 3.7.
Proof. According to Lemma 2.2, the closed-loop singularly perturbed system 2.7 is
asymptotically stable and the L2 gain will be less or equal γ if 3.10 and following inequality
are satisfied:
⎡ T
⎤
Acl Pε PεT Acl PεT Bcl CclT 1
⎢
⎥
⎢
−γI DclT 1 ⎥
∗
⎣
⎦ < 0.
∗
∗ −γI
Also Acl , Bcl , Ccl 1 , and Dcl 1 are defined in 2.8.
3.35
12
Mathematical Problems in Engineering
By pre- and postmultiplying with matrices diagΠ1 , I, I and diagΠT1 , I, I, respectively, it is concluded that:
⎡ T T
⎤
Π1 Acl Π2 ΠT2 Acl Π1 ΠT2 Bcl ΠT1 CclT 1
⎢
⎥
⎢
−γI
DclT 1 ⎥
∗
⎣
⎦ < 0,
−γI
∗
∗
3.36
where Π1 is defined in 3.18. According to proof of Theorem 3.1, ΠT1 ATcl Π2 ΠT2 Acl ΠT1 is
obtained. Also ΠT2 Bcl and ΠT1 CclT 1 are presented as follows:
ΠT2 Bcl
Bw BDc Dyw
0
T
Eε P12
P11
ΠT1 CclT 1
I
T
Q12
Q11
I
Eε−1 Bc Dyw
T
CcT Dzu1
Bw BDk Dyw
T
T
P11
Bw P11
BDk Dyw Eε P12 Eε−1 Bc Dyw
Bw BDk Dyw
,
T
P11
Bw Bk Dyw
T
T
Cz1 CyT DcT Dzu1
0
T
T
T Q11 Cz1
Q11 CyT DcT Dzu1
Q12 CcT Dzu1
3.37
T
T
Cz1
CyT DcT Dzu1
T
T
CkT Dzu1
Q11 Cz1
.
T
T
Cz1
CyT DkT Dzu1
From 3.36, and 3.37, we have
⎡
ψ11 ε
⎢
⎣
Υ11 ε
T
Υ12
ε
⎤
Υ12 ε
⎥
−γI Dzw1 Dzu1 Dk Dyw ⎦ < 0,
−γI
∗
3.38
where Υ11 ε defined in 3.27 and
⎡
Υ12 ε
Bw BDk Dyw
⎢
⎢
⎢
⎢ X11 X T ⎣
12
Bw Bk Dyw
εX12 X22
T
Y11 Y12
εY12 Y22
T
Cz1
T
Cz1
⎤
T
CkT Dzu1
⎥
⎥
T
CyT DkT Dzu1
⎥.
⎥
⎦
3.39
According to Lemma 2.5, the condition 3.38 is satisfied for all ε ∈ 0, ε∗ , if 3.33 is
satisfied. 3.4 and 3.5 compute from procedure similar to proof of Theorem 3.1 and this
completes the proof.
3.3. H2 Performance
Consider the closed loop system 2.7 with regulated output z2 , assume Acl is stable, Dzw2 0
0. In following theorem, we proposed a procedure for obtaining controller
and Dyw
Mathematical Problems in Engineering
13
parameters such that the closed loop singularly perturbed system 2.7 with regulated output
z2 guarantees the H2 performance with attenuation parameter v. First we propose the
following lemma that is effective in proof of Theorem 3.5.
Lemma 3.4. The closed-loop singularly perturbed G : Acl , Bcl , Ccl has the H2 performance with
attenuation parameter v if following LMIs hold
ATcl Pε PεT Acl PεT Bcl
T
Pε Eε PεT Ccl
∗
Z
3.40
< 0,
−I
T
∗
3.41
> 0,
3.42
traceZ < v.
Proof. Consider the following closed-loop singular perturbed system:
Ee−1 Acl xt Ee−1 Bcl vt,
ẋt
3.43
Ccl xt,
z
where Ee is defined in 2.8. From Definition 2.3 we have the following equality:
∞
G2H2
0
T −1
−1
trace Ccl eEe Acl t Ee−1 Bcl BclT Ee−1 eAcl Ee t CclT dt.
3.44
Now we define symmetric matrix Wε as follows:
Wε
∞
−1
eEe
Acl t
0
−1
Ee−1 Bcl BclT Ee−1 eAcl Ee
T
t
dt.
3.45
G2H2 is obtained from the following equality:
G2H2
trace Ccl Wε CclT .
3.46
We can be obtained from the following equation:
Ee−1 Acl Wε Wε ATcl Ee−1 Ee−1 Bcl BclT Ee−1
0.
3.47
Suppose that there exist the matrix Xε with following structure:
Xε
X11 Eε−1 X12
X12T
X22
3.48
14
Mathematical Problems in Engineering
that satisfies the following inequality:
Wε < Xε Ee−1
3.49
From 3.49, equation 3.47 can be rewritten as follow:
Ee−1 Acl Xε Ee−1 Ee−1 XεT ATcl Ee−1 Ee−1 Bcl BclT Ee−1 < 0.
3.50
Now, pre- postmultiplying 3.50 with Ee we have
Acl Xε XεT ATcl Bcl BclT < 0.
3.51
Also, from 3.46 and 3.49 we have
G2H2
trace Ccl Wε CclT < trace Ccl Xε Ee−1 CclT < v.
3.52
This is equivalent to the existence of Z such that
Ccl Xε Ee−1 CclT < Z,
3.53
traceZ < v
by using of Schur complement on 3.51 and 3.53 we can conclude
T T
Xε Acl Acl Xε Bcl
−I
∗
Ee Pε CclT
∗
with assumption Xε−1
respectively, we have
Z
< 0,
3.54
3.55
>0
Pε and pre- and postmultiplying 3.54 by diagPεT , I and diagPε , I,
T
Acl Pε PεT Acl Pε Bcl
−I
T
∗
Ee Pε PεT Ccl
∗
Z
> 0,
traceZ < v .
This completes the proof.
< 0,
3.56
Mathematical Problems in Engineering
15
Theorem 3.5. Given an H2 performance bound v and an upperbound ε∗ for the singular perturbation
ε, if there exist matrices Ak , Bk , Ck , Dk , Y11 , Y12 , Y22 , X11 , X12 , and X22 such that traceZ < v
andsatisfying the following LMIs:
12 0
Υ11 0 Υ
ψ11 0
∗
ψ11 ε 0
I
0
0
0
∗
⎡⎡
Y11
ε∗ Y12
⎢⎢
T
⎢⎢
ε∗ Y12
ε∗ Y22
⎢⎢
⎢⎢
I 0
⎢⎣
⎢
⎣
0 ε∗ I
∗
−I
∗
< 0,
⎤
0
⎥
Y11 Y12 T
T
0⎥
Cz2 CkT Dzu2
⎥
⎥
0
Y
22
0⎥
T
T
⎦
Cz2
CyT DkT Dzu2
0
Z
0 X11
0
≤ 0,
12 ε∗ Υ11 ε∗ Υ
∗
⎡⎡
Y11
⎢⎢
⎢⎢
0
⎢⎢
⎢⎢
⎢⎢ I
⎢⎣
⎢
⎣ 0
−I
I
0
⎤
0 ε∗ I
X11 ε∗ X12
T
ε∗ X12
ε∗ X22
⎥
⎥
⎥
⎥
⎦
Y11
ε
∗
Y12
⎤
⎥
⎥
⎥
⎥
⎥ ≥ 0,
⎥
⎥
⎦
3.57
⎤
T
T
Cz2
CkT Dzu2
T
Y12
Y22
T
T
Cz2
CyT DkT Dzu2
⎥
⎥
⎥
⎥ > 0,
⎥
⎥
⎦
Z
12 ε∗ is
where Υ11 ε∗ defined in 3.6 and Υ
⎡
12 ε∗ Υ
Bw
⎤
⎥
⎢
⎢ X11 X T ⎥.
12
⎣
Bw ⎦
ε∗ X12 X22
3.58
Then, for any ε ∈ 0, ε∗ , the closed-loop singularly perturbed system 2.7 is asymptotically stable
and with an H2 -norm less than or equal to v, also parameters controller are obtained from 3.7.
Proof. The proof is similar to proof of Theorem 3.3. From Lemma 3.4, the closed-loop
singularly perturbed system 2.7 has H2 performance less than or equal v if 3.40, 3.41
and 3.42 are satisfied.
From 3.18 and 3.37, multiplying inequalities 3.40 and 3.41, by the matrices
diagΠ1 , I, I and diagΠT1 , I, I, gives
16
Mathematical Problems in Engineering
ψ11 ε
⎡⎡
Y11 εY12
⎢⎢
T
⎢⎢
εY12
εY22
⎢⎢
⎢
⎢
I
0
⎢⎣
⎢
⎣
0 εI
∗
I 0
12 ε
Υ11 ε Υ
−I
∗
⎤
⎥
⎥
0 εI
⎥
X11 εX12 ⎥
⎦
T
εX12
εX22
3.59
< 0,
Y11 Y12
⎤
⎥
T
T
⎥
Cz2
CkT Dzu2
⎥
Y22
⎥ > 0,
⎥
T
T T T
Cz2 Cy Dk Dzu2
⎥
⎦
T
εY12
3.60
Z
where:
⎤
Bw BDk Dyw
⎥
⎢
⎥
⎢ X11 X T 12
⎣
Bw Bk Dyw ⎦
εX12 X22
⎡
12 ε
Υ
3.61
According to Lemma 2.5, the inequalities 3.59 and 3.60 are valid for all ε ∈ 0, ε∗ ,
if 3.57 is satisfied and proof is complete.
3.4. Multiobjective H2 /H∞ Performance
Now we have got the LMIs in Theorems 3.3 and 3.5 thus we can solve the multiobjective
H2 /H∞ synthesis problem easily for closed-loop singularly perturbed system 2.7 with
regulated outputs z1 and z2 .
Theorem 3.6. Given an H∞ performance bound γ, H2 performance bound v and an upperbound ε∗
for the singular perturbation ε, if there exist matrices Ak , Bk , Ck , Dk , Y11 , Y12 , Y22 , X11 , X12 , and X22
such that traceZ < v satisfying the LMIs 3.4, 3.5, 3.33, and 3.57. Then, for any ε ∈ 0, ε∗ ,
the closed-loop singularly perturbed system 2.7 is asymptotically stable and with an H∞ -norm less
than or equal to γ, an H2 -norm less than or equal to v, also parameters controller are obtained from
3.7.
Proof. It is from the proof of Theorems 3.3 and 3.5 and omitted for save of brevity.
4. Numerical Example
In this section, we present a numerical results to validate the designed dynamic output
feedback controller for singularly perturbed systems with H∞ or H2 performance.
Mathematical Problems in Engineering
17
Ratio of regulated output energy to
the disturbance energy
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
Time
Figure 1: Simulation for full-order system 4.1 with ε∗
0.1.
0.06
Ratio of regulated output energy to
the disturbance energy
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0
10
20
30
40
50
Time
Figure 2: Simulation for full-order system 4.1 with ε∗
0.001.
18
Mathematical Problems in Engineering
Ratio of regulated output energy to
the disturbance energy
0.03
0.025
0.02
0.015
0.01
0.005
0
10
20
30
40
50
Time
Figure 3: Simulation for reduced order system 4.3.
Regulated output energy to
the disturbance energy
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
Time sec
ε∗
ε∗
0.025
0.1
Figure 4: Simulation for full-order system 4.4 for ε∗
0.1, 0.025.
Mathematical Problems in Engineering
19
15
10
Noise
5
0
−5
−10
−15
0
10
20
30
40
50
Time
Figure 5: Band-limited white noise.
×10−6
1
0.8
Regulated output z2
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8
Time
Figure 6: Regulated output z2 for ε∗
0.001.
10
20
Mathematical Problems in Engineering
×10−4
3
Regulated output z2
2
1
0
−1
−2
−3
−4
0
2
4
6
8
10
Time
Figure 7: Regulated output z2 for ε∗
0.1.
×10−4
8
Regulated output z2
6
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
30
35
40
Time
Figure 8: Regulated output z2 for ε∗
0.001.
45
50
Mathematical Problems in Engineering
21
0.08
Regulated output z2
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
5
0
10
15
20
25
30
35
40
45
50
Time
Figure 9: Regulated output z2 for ε∗
0.1.
20
15
Signal wt
10
5
0
−5
−10
0
2
4
6
8
10
Time
Figure 10: Disturbance signal wt.
4.1. H∞ Performance
Consider the following two-dimensional system and performance index 10
ẋ1 t
εẋ2 t
1
ut wt,
1
3
−1 −2 x2
x1
0.1wt,
z1
2 0.1
x2
x1
wt.
yt
1 0
x2
2
1
x1
2
4.1
22
Mathematical Problems in Engineering
20
Regulated output z2
15
10
5
0
−5
−10
0
2
4
6
8
10
Regulated output energy to the
disturbance energy
Time
Figure 11: Regulated output z2 with ε∗
0.1.
0
8
0.5
0.4
0.3
0.2
0.1
0
2
Figure 12: Simulation for
4
t
0
6
10
Time
zT1 sz1 s/
t
0
wT sws with ε∗
0.1.
Mathematical Problems in Engineering
From Theorem 3.3, for ε∗
are as follows:
23
0.1 minimum of γ is obtained as 0.11942. Controller parameters
Ac
−108.89 −31.42
,
91.35 5.201
Bc
Cc
−.22
,
0.05
4.2
−1793.2 −536.1 ,
Dc
−4.7658.
For upperband ε∗ 0.001, minimum γ is calculated as 0.1. As we expect γε∗
From ε 0, we obtain reduced order dynamic as follows:
ẋ1 t
z1
0.001
≤ γε∗
0.1 .
1.5x1 t 2.5ut 2.5wt,
1.95x1 t 0.05ut 0.15wt,
yt
4.3
x1 t wt.
Here, minimum
value γ is calculated as 0.029. For w
t
t T
z sz1 sds/ 0 wT swsds, respectively.
exhibit
0 1
e−0.1t sin10t, Figures 1, 2, and 3
As a new example for H∞ performance, now consider an F-8 aircraft model 25:
ẋ1
εẋ2
with
A1 A 2
x1
A3 A4
x2
B1
Bw1
u
w,
Bw2
B2
x1
,
z
Cw1 Cw2
x2
1
x1
w
y
Cy1 Cy2
1
x2
4.4
24
Mathematical Problems in Engineering
−0.01357 −0.0644
,
0.06
0
A1
A3
−0.0453775 0
,
0.07125 0
B1
−0.463
,
6.07
Cy1
From Theorem 3.3, for ε∗
are as follows:
Ac
−0.03055
−0.075083 −0.01674
,
B2
Bw2
−4.5525
−11.262499
Cy2
−0.052275
0 0.02
0
0
,
0.019712
,
0.075
4.5
,
,
.
0.025 minimum of γ is obtained as 1.908. Controller parameters
⎡
−17.0078
⎢
⎢ 800.6192
⎢
⎢
⎢ −1964.9
⎣
Bc
Cc
41.1022
−62.0967
−2017.7
3070.05
0.15894
⎤
⎥
−5.964 ⎥
⎥
⎥,
4954.9
−7541
15.033 ⎥
⎦
−109682.3 165444.1 −407.4100
⎡
−0.5966
⎢
⎢ 29.39
⎢
⎢
⎢−72.249
⎣
1586.7
0.5893
⎤
4.6
⎥
−28.98 ⎥
⎥
⎥,
70.9783 ⎥
⎦
−1406.2
−3643.8 7991 −6482 336.66 ,
Dc
For w
0 0
1 0
39387
For upperband ε∗
A4
0.0697
Bw1
A2
−0.0004333
−0.003087 0
,
0.040467 0
−76.567 −4.3010 .
0.1, minimum γ is calculated as 3.96. As we expect γε∗ 0.025 ≤ γε∗
t
t
e−0.1t sin10t, Figure 4 exhibits 0 zT1 sz1 sds/ 0 wT swsds.
0.1 .
Mathematical Problems in Engineering
25
4.2. H2 Performance
Consider singular perturbed system 3.60 as
ẋ1 t
εẋ2 t
x1 t
2
1
ut wt,
−1 −2 x2 t
1
3
x1 t
4 0
2ut,
z2
x2 t
x1 t
.
yt
1 0
x2 t
−2 1
By utilizing Theorem 3.5, minimum value of ν for ε∗
controller parameters are as follows:
Ac
−2558.6 −4.289
2566620 −195.16
Bc
Cc
1.62
−163.03
4.7
0.1 is obtained as 0.074. The
,
4.8
,
−0.034 0.011 ,
Dc
−4.1.
For upperband ε∗ 0.001, minimum v is calculated as 0.0002. As we expect vε∗ 0.001 ≤ vε∗ 0.1 .
Figures 5, 6, and 7 show input noise and regulated output z2 , respectively.
As a new example for H2 performance, now consider singular perturbed system as
ẋ1 t
εẋ2 t
−1 0.5 x1 t
2
1
ut wt,
1 −2 x2 t
1
3
x1 t
2ut,
3 0
z2
x2 t
x1 t
.
yt
1 1
x2 t
By utilizing Theorem 3.5, minimum value of ν for ε∗
controller parameters are as follows:
4.9
0.1 is obtained as 0.003. The
26
Mathematical Problems in Engineering
Ac
5243.9
−14292.8
29566.58 −80500.28
−0.167
,
Bc
−9.4145
−77312.6 6829.2 ,
Cc
,
4.10
−3.8 × 10−9 .
Dc
For upperband ε∗ 0.001, minimum v is calculated as 0.0004. As we expect vε∗ 0.001 ≤ vε∗
Input noise is shown in Figures 5, 8, and 9 show the regulated output z2 , respectively.
0.1 .
4.3. H∞ /H2 Performance
Now, consider singular perturbed system 3.60 as
ẋ1
εẋ2
2
1
u
w,
−1 −2 x2
1
3
x1
0.1w,
z1
2 0.1
x2
x1
2u,
z2
4 0
x2
x1
w.
y
1 0
x2
2
1
x1
4.11
According to Theorem 3.6, due to minimization of ν γ for ε∗ 0.1, the values of ν 0.14 and
γ 0.233 are calculated. By similar calculation, due to minimization of ν γ for ε∗ 0.001,
the values of ν and γ are calculated as 0.088 and 0.2018, respectively.
Here, controller parameters are as the follows when ε∗ 0.1:
Ac
−7.4
−9.17
,
817.41 −0.5
4.16
,
Bc
−52.18
Cc
−0.15 × 10−4 0 ,
Dc
−3.206.
4.12
Mathematical Problems in Engineering
27
Figures 10 and 11 show input signal wt and regulated output z2 . Based on the H∞ /H2 ,
considered controller is designed to minimize effect of signal w on regulated output z2 . Also
the ratio of the regulated output energy to the disturbance energy is shown in Figure 12.
5. Conclusions
In this paper, we addressed robust H2 and H∞ control via dynamic output feedback control
for continuous-time singularly perturbed systems. By formulating all objectives in terms
of a common Lyapunov function, the controller was designed through solving a set of
inequalities. A dynamic output feedback controller was developed such that first, the H∞ and
H2 performances of the resulting closed-loop system is less than or equal to some prescribed
values, and furthermore, these performances are satisfied for all ε ∈ 0, ε∗ . Apart from our
main results, Theorems 3.1 to 3.6 show that the ε-dependent controller is well defined for all
ε ∈ 0, ε∗ and can be reduced to an ε-independent providing ε is sufficiently small. Finally,
numerical simulations were provided to verify the proposed controller.
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