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Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 471680, 21 pages
doi:10.1155/2010/471680
Research Article
Robust Reliable Stabilization of Switched
Nonlinear Systems with Time-Varying Delays and
Delayed Switching
Zhengrong Xiang and Qingwei Chen
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China
Correspondence should be addressed to Zhengrong Xiang, [email protected]
Received 29 September 2009; Revised 18 April 2010; Accepted 11 May 2010
Academic Editor: John Burns
Copyright q 2010 Z. Xiang and Q. Chen. 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 is concerned with the problem of robust reliable stabilization of switched nonlinear
systems with time-varying delays and delayed switching is investigated. The parameter
uncertainties are allowed to be norm-bounded. The switching instants of the controller experience
delays with respect to those of the system. The purpose of this problem is to design a reliable
state feedback controller such that, for all admissible parameter uncertainties and actuator failure,
the system state of the closed-loop system is exponentially stable. We show that the addressed
problem can be solved by means of algebraic matrix inequalities. The explicit expression of the
desired robust controllers is derived in terms of linear matrix inequalities LMIs.
1. Introduction
A switched system is composed of a family of continuous-time or discrete-time subsystems
and a rule specifying the switching among them. Switched systems have received increasing
attentions in the past few years, since many real-word systems such as mechanical systems,
automotive industry, aircraft, and air traffic control systems, chemical processes can be
modelled as switched systems see 1–3. A large number of results have been reported
for such systems see 4–8.
On the other hand, time delay systems have continuously been receiveing considerable
attention over the past decades. The main reason is that many kinds of engineering systems,
for instance, long-distance transportation systems, hydraulic pressure systems, network
control systems, and so on, include time delay phenomena in their dynamics. Many valuable
results have been obtained for switched systems with time delay see 9–15. On the other
hand, the actuators may be subjected to failures in real practice, therefore, it is of practical
2
Mathematical Problems in Engineering
importance to design a control system which can tolerate faults of actuators. Several design
approaches to the reliable controller have been proposed for linear and nonlinear systems
see 16–19, and these results have been extended to switched systems see 20–22.
Recently, the asynchronous switching control problem of switched systems has stirred
renewed research interests, and a varity of switched systems have been investigated by
different approaches 23–26. However, to the best of the authors’ knowledge, the issue
of reliable stabilization of switched nonlinear systems with time-varying delay under
asynchronous switching has not been fully investigated, which motivated the present
study.
In this paper, we are interested in designing the robust reliable controller for uncertain
switched nonlinear system with time-varying delays and delayed switching. The remainder
of the paper is organized as follows. In Section 2, problem formulation is presented and the
failure model of actuator in switched system is introduced briefly. In addition, some necessary
lemmas are given. In Section 3, based on the average dwell-time approach, controller design
for switched nonlinear system with time-varying delays and delayed switching is developed,
and sufficient conditions for the existence of the controller are formulated in terms of a set of
matrix inequalities. Concluding remarks are given in Section 4.
Notation. Throughout this paper, the superscript “T ” denotes the transpose, · denotes the
Euclidean norm. λmax P and λmin P denote the maximum and minimum eigenvalues of
matrix P , respectively, I is an identity matrix with appropriate dimension. diag{ai } denotes
diagonal matrix with the diagonal elements ai , i 1, 2, . . . , n. The asterisk ∗ in a matrix is
used to denote term that is induced by symmetry. The set of positive integers is represented
by Z .
2. Problem Formulation and Preliminaries
Consider the following uncertain nonlinear switched system with actuator fault
σt xt A
dσt x t − dσt t Bσt uf t fσt xt, t,
ẋt A
xt ϕt,
t ∈ t0 − d, t0 ,
2.1
where xt ∈ Rn is the state vector, uf t ∈ Rl is the control input of actuator fault, ϕt
is a continuous vector-valued function. The function σt : t0 , ∞ → N {1, 2, . . . , N} is
the switching signal which is deterministic, piecewise constant and right continuous, that is,
σ : {t0 , σt0 , t1 , σt1 , . . .}, k ∈ Z , where t0 ≥ 0 is the initial time, and tk denotes the
kth switching instant. dσt t denotes the time-varying state delay satisfying 0 < dσt t ≤
d, ḋσt t ≤ τ for constants d and τ. Moreover, σt i means that the ith subsystem is
activated. N denotes the number of subsystems. fi xt, t i ∈ N are nonlinear functions
satisfying
fi xt, t ≤ Ui xt,
where Ui are known real constant matrices.
2.2
Mathematical Problems in Engineering
3
di for i ∈ N are uncertain real-valued matrices with appropriate dimensions, and
i, A
A
have the following form:
i A
di Ai Adi Hi Fi t E1i E2i ,
A
2.3
where Ai , Adi , Bi , H1i , E1i , E2i are known real constant matrices with proper dimensions, and
H1i , E1i , E2i denote the structure of the uncertainties, Fi t are unknown time-varying matrices
which satisfy
FiT tFi t ≤ I.
2.4
The control input of actuator fault uf t can be described as
2.5
uf t Mσt ut,
where ut Kσt xt is the switching controller which will be designed, Mi i ∈ N are the
actuator fault matrices with the following form:
Mi diag{mi1 , mi2 , . . . , mil },
0 ≤ mik ≤ mik ≤ mik , mik ≥ 1, k 1, 2, . . . , l.
2.6
For simplicity, we introduce the following notation
Mi0 diag{m
i1 , m
i2 , . . . , m
il },
Ji diag ji1 , ji2 , . . . , jil ,
Li diag{li1 , li2 , . . . , lil },
2.7
ik /m
ik .
where m
ik 1/2mik mik , jik mik − mik /mik mik , lik mik − m
From 2.6-2.7, we have
Mi Mi0 I Li ,
|Li | ≤ Ji ≤ I,
2.8
where |Li | diag{|li1 |, |li2 |, . . . , |lil |}.
Remark 2.1. mik 1 means normal operation of the kth actuator signal of the ith subsystem.
When mik 0, it covers the case of the complete failure of the kth actuator signal of the ith
1, it corresponds to the case of partial failure of the kth
subsystem. When mik > 0 and mik /
actuator signal of the ith subsystem.
The delayed switching of the controller can be shown in Figure 1.
We can see from Figure 1 that the controller Ki operates the ith subsystem in tk−1 Δk−1 , tk , and operates the jth subsystem in tk , tk Δk .
4
Mathematical Problems in Engineering
Subsystem i
Subsystem j
The switching of the system
Controller Ki
The switching of the controller
tk−1 Δk−1
Controller Ki
tk
Controller Kj
tk Δ k
tk1
Figure 1: Diagram of the delayed switching.
Let σ t denote the switching signal of the controller, the switching instants of the
controller can be described as
t1 Δ1 , t2 Δ2 , . . . , tk Δk , . . . , k ∈ Z ,
2.9
where Δk < infk∈Z tk1 −tk , Δk represents the delayed period, and it is said to be mismatched
period.
Remark 2.2. Mismatched period Δk < infk∈Z tk1 − tk guarantees that there always exists a
period that the controller and the system operate synchronously, and this period is said to be
matched period in the later section.
Due to the delayed switching, the real input of actuator fault can be written as
uf t Mσ t Kσ t xt.
2.10
Under switching controller 2.10, the resulting closed-loop system is given by
ẋt Aσt Bσt Mσ t Kσ t xt Adσt x t − dσt t fσt xt, t,
xt ϕt,
t ∈ t0 − d, t0 .
2.11
System 2.1 without uncertainties and actuator fault can be written as
ẋt Aσt xt Adσt x t − dσt t Bσt ut fσt xt, t,
xt ϕt,
t ∈ t0 − d, t0 .
2.12
Definition 2.3 see 13. If there exists switching signal σt, such that the trajectory of system
2.1 satisfies xt ≤ αxt0 h e−βt−t0 , then system 2.1 is said to be exponentially stable
with convergent rate β, where α ≥ 1, β > 0, t ≥ t0 , xth sup−d≤θ≤0 {xt θ, ẋt θ}.
Mathematical Problems in Engineering
5
Definition 2.4 see 27. For any T2 > T1 ≥ 0, let Nσ T1 , T2 denote the switching number of
σt on an interval T1 , T2 . If
Nσ T1 , T2 ≤ N0 T2 − T1
τa
2.13
hold for given N0 ≥ 0, τa > 0, then the constant τa is called the average dwell time and N0 is
the chatter bound.
The following lemmas play an important role in the later development.
Lemma 2.5 see 28. For given vectors a, b and the positive matrix X > 0, there exists the matrix
M with appropriate dimension, such that
T a
X XM
a
.
−2a b ≤ inf
MT X MT X I X −1 XM I b
b
X>0,M
T
2.14
Lemma 2.6 see 29. For matrices X, Y of appropriate dimension and Q > 0, we have
X T Y Y T X ≤ X T QX Y T Q−1 Y.
2.15
Lemma 2.7 see 30. Let U, V, W, and X be real matrices of appropriate dimensions with X
satisfying X X T , then for all V T V ≤ I,
X UV W W T V T UT < 0
2.16
if and only if there exists a scalar ε > 0 such that
X εUUT ε−1 W T W < 0.
2.17
Lemma 2.8 see 31. For matrices R1 , R2 with appropriate dimension, there exists a positive scalar
β > 0, such that
R1 ΣtR2 RT2 ΣT tRT1 ≤ βR1 URT1 β−1 RT2 UR2
2.18
hold, where Σt is time-varying diagonal matrix, U is known real-value matrix satisfying |Σt| ≤ U.
The objective of this paper is to design a reliable controller for system 2.1 with
delayed switching such that the resulting closed-loop system is robust exponentially stable.
6
Mathematical Problems in Engineering
3. Main Results
To obtain the main results of this paper, we first consider the stability of the following
nonlinear delay system
ẋt Axt Ad xt − dt fxt, t,
xt ϕt,
3.1
t ∈ t0 − d, t0 ,
3.2
where xt ∈ Rn is the state vector, ϕt is a continuous vector-valued initial function, A, Ad
are real-valued matrices with appropriate dimensions, f·, · : Rn × R → Rn is unknown
nonlinear functions satisfying
fxt, t ≤ Uxt,
3.3
where U is known real constant matrix.
3.1. Stability Analysis
Lemma 3.1. Consider system 3.1-3.2, for given positive constant α, ρ1 , ρ2 , ε1 , η1 , if there exist
positive definite symmetric matrices P, S, Q, R, and any matrices G, W with appropriate dimensions,
such that
Q ≤ ρ1 I,
P ≤ ρ2 I,
3.4
⎡
Σ
GT
W T Ad
AT Q
0 d WT P
AT Q
0
⎢
⎢
⎢∗ −1−τe−αd R−G− GT
−G
ATd Q
0
0
0
ATd Q
⎢
⎢
⎢∗
∗
−d−1 e−αd Q
0
ATd S
0
0
0
⎢
⎢
−1
⎢∗
∗
∗
−d Q 0
0
0
0
⎢
⎢
⎢∗
∗
∗
∗
−d2 S
0
0
0
⎢
⎢
⎢∗
∗
∗
∗
∗
−S
0
0
⎢
⎢
−1
⎢∗
∗
∗
∗
∗
∗
−d ε1Q
0
⎢
⎢
⎢∗
∗
∗
∗
∗
∗
∗
−d−1 η1 Q
⎣
∗
∗
∗
∗
∗
∗
∗
∗
U
T
⎤
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
< 0,
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
Φ
3.5
holds, then for Lyapunov functional candidate
V xt xT tP xt
t
t−dt
ẋT se−αt−s Rẋsds
0 t
−d
tθ
ẋT re−αt−r Qẋrdr dθ,
3.6
Mathematical Problems in Engineering
7
along the trajectory of system 3.1, there holds the following inequality:
V xt < e−αt−t0 V xt0 ,
3.7
where Σ A Ad T P P A Ad α 1P R, Φ −τρ1 1 ε1 η1 ρ2 −1 I.
t
Proof. Let V1 xt xT tP xt, V 2 xt ẋT se−αt−s Rẋsds, V3 xt
t−dt
0 t
ẋT re−αt−r Qẋrdr dθ.
−d tθ
t
Notice that xt − xt − dt t−dt ẋrdr, 3.1 can be written as
ẋt A Ad xt fxt, t − Ad
t
3.8
ẋrdr.
t−dt
Along the trajectory of system 3.1, the time derivative of V1 xt is given by
V̇1 xt 2xT tP A Ad xt fxt, t −
t
2xT tP Ad ẋrdr.
3.9
t−dt
Let a Ad ẋr, b P xt, from Lemma 2.5, we can obtain
−2xT tP Ad ẋr ≤ ẋT rATd XAd ẋr 2ẋT rATd XMP xt
xT tP MT X I X −1 XM IP xt.
3.10
Substituting 3.10 into 3.9 leads to
V̇1 xt ≤ xT t A Ad T P P A Ad dtP MT X I X −1 MX IP xt
x tP fxt, t f xt, tP xt 2x tP M XAd
T
T
T
T
t
ẋrdr
t−dt
t
t−dt
ẋT rATd XAd ẋrdr
≤ xT t A Ad T P P A Ad τP MT X I X −1 XM IP xt
2x tP fxt, t 2x tP M XAd
T
T
T
t
t−dt
ẋrdr t
t−dt
ẋT rATd XAd ẋrdr.
3.11
8
Mathematical Problems in Engineering
Differentiating V2 xt and V3 xt along the trajectory of system 3.1, we have
V̇2 xt ≤ −α
t
e−αt−s ẋT sRẋsds xT tRxt
t−dt
− 1 − τe−αd xT t − dtRxt − dt,
0 t
t
T
T
−αt−r
V̇3 xt dẋ tQẋt −
ẋ re
Qẋrdr − α
ẋT re−αt−r Qẋrdr dθ
t−d
≤ dẋT tQẋt − e−αd
−d
t
ẋT rQẋrdr − α
tθ
0 t
t−dt
−d
ẋT re−αt−r Qẋrdr dθ
tθ
dxT t − dtATd QAd xt − dt
dx tA QAxt 2dx t −
t
0 t
−αd
T
−e
ẋ rQẋrdr − α
ẋT re−αt−r Qẋrdr dθ
T
T
T
dtATd QAxt
−d
t−dt
tθ
d 2xT tAT Qfxt, t 2xT t − dtATd Qfxt, t f T xt, tQfxt, t .
3.12
By Lemma 2.6, we have
2xT tP fxt ≤ xT tP xt f T xtP fxt,
2xT tAT Qfxt, t ≤ ε1−1 xT tAT QAxt ε1 f T xt, tQfxt, t,
2xT t − dtATd Qfxt, t ≤ η1−1 xT t − dtATd QAd xt − dt η1 f T xt, tQfxt, t.
3.13
Therefore
V̇ xt αV xt
V̇1 xt V̇2 xt αV xt
≤ xT t A Ad T P P A Ad dP MT X I X −1 XM IP
α 1P d 1 ε1−1 AT QA xt
2dxT t − dtATd QAxt d 1 η1−1 xT t − dtATd QAd xt − dt
2x tP M XAd
T
T
t
ẋrdr
t−dt
t
t−dt
ẋT r ATd XAd − e−αd Q ẋrdr d 1 ε1 η1 f T xt, t
× Qfxt, t f T xtP fxt
Mathematical Problems in Engineering
≤ xT t A Ad T P P A Ad dP MT X I X −1 XM IP
9
T
T
α 1P d 1 ε1−1 AT QA d 1 ε1 η1 U QU U P U xt
2dxT t − dtATd QAxt d 1 η1−1 xT t − dtATd QAd xt − dt
2x tP M XAd
T
T
t
ẋrdr t
t−dt
t−dt
ẋT r ATd XAd − e−αd Q ẋrdr.
3.14
Notice that 2xT t − dtGxt − xt − dt −
appropriate dimension, we have
V̇ xt αV xt ≤
t
t−dt
1
dt
ẋrdr 0, where G is any matrix with
t
3.15
ξT t, rZξt, rdr,
t−dt
where ξT t, r xT t xT t−d ẋT r ,
⎤
⎡
T
Ψ − U Φ−1
U
τAT QAd GT
dtW T Ad
1
⎥
⎢
⎥
⎢
Z ⎢ τAT QA G τAT QAd − 1 − τe−αd R − G − GT
⎥,
−dtG
d
d
⎦
⎣
T
T
−1 T
−ατ
−dtG
dt τ Ad SAd − e Q
dtAd W
3.16
where Ψ A Ad T P P AAd dP MT XIX −1 XMIP α1P Rd1ε1−1 AT QA.
Let W XMP, S τX, by Schur complement lemma, 3.5 is equivalent to the
following inequality:
⎡
Ψ
dAT QAd GT
W T Ad
U
T
⎤
⎢
⎥
⎢ T
⎥
⎢dA QA G dAT QAd − 1 − τe−αd − G − GT
⎥
−G
0
⎢ d
⎥
d
⎢
⎥ < 0.
⎢ AT W
T
−2 T
−1 −ατ
−G
d Ad SAd − d e Q 0 ⎥
⎢
⎥
d
⎣
⎦
U
0
0
Φ
3.17
Using diag{I, I, dtI, I} to pre- and post- multiply the left term of 3.17, respectively, we can
obtain Z < 0. Therefore, V̇ xt αV xt < 0.
The proof is completed.
10
Mathematical Problems in Engineering
Lemma 3.2. Consider system 3.1-3.2, for given positive constant β, ρ1 , ρ2 , ε1 , η1 , if there exist
positive definite symmetric matrices P, S, Q, and any matrices G, W with appropriate dimensions,
such that
P ≤ ρ2 I,
Q ≤ ρ1 I,
⎤
⎡
T
Σ
GT
W T Ad
AT Q
0 d WT P
AT Q
0
U
⎥
⎢
⎥
⎢
⎢∗ −G − GT
−G
ATd Q
0
0
0
ATd Q
0 ⎥
⎥
⎢
⎥
⎢
⎢∗
∗
−d−1 e−αd Q
0
ATd S
0
0
0
0 ⎥
⎥
⎢
⎥
⎢
−1
⎥
⎢∗
∗
∗
−d
Q
0
0
0
0
0
⎥
⎢
⎥
⎢
2
⎥<0
⎢∗
∗
∗
∗
−d
S
0
0
0
0
⎥
⎢
⎥
⎢
⎢∗
∗
∗
∗
∗
−S
0
0
0 ⎥
⎥
⎢
⎥
⎢
−1
⎥
⎢∗
∗
∗
∗
∗
∗
−d
ε
Q
0
0
1
⎥
⎢
⎥
⎢
−1
⎥
⎢∗
∗
∗
∗
∗
∗
∗
−d
η
Q
0
1
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗
Φ
3.18
3.19
holds, then for Lyapunov functional candidate
V xt xT tP xt 0 t
−d
ẋT re−αt−r Qẋrdr dθ,
3.20
tθ
along the trajectory of system 3.1, there holds the following inequality
V xt < eβt−t0 V xt0 ,
3.21
where Σ A Ad T P P A Ad − β − 1P .
Proof. Similarly to the proof line of Lemma 3.1, we can obtain Lemma 3.2.
3.2. Stabilizing Controller Design
In this subsection, we will design a stabilizing controller for system 2.12 with delayed
switching.
In our design approach we only require the subsystems to be stable during matched
period, and the subsystems are allowed to be unstable during mismatched period. Under
delayed switching controller ut Kσ t xt, the corresponding closed-loop system is given
by
ẋt Aσt Bσt Kσ t xt Adσt x t − dσt t fσt xt, t,
xt ϕt,
t ∈ t0 − d, t0 .
3.22
3.23
Mathematical Problems in Engineering
11
Let T t0 , t denote the total mismatched period during t0 , t, T − t0 , t denote the total
matched period during t0 , t, then we have the following result.
Theorem 3.3. Consider system 2.12, for given positive constants α, β, ε1 , η1 , ε2 , η2 , ρ1 , ρ2 , ρ3 , ρ4 ,
if there exist positive definite symmetric matrices Xi , Zi , Si , Pij , Qij , Sij , and any matrices Yi , Gij , Wij
with appropriate dimensions, such that, for i, j ∈ N, i /
j,
Zi ≥ ρ1−1 I,
Xi ≥ ρ2−1 I,
Qij ≤ ρ3 I,
3.24
Pij ≤ ρ4 I,
⎤
⎡
Πi Xi
Adi Zi
Ξi
0
2dSi
Ξi
0
Xi UiT
⎥
⎢
⎢ ∗ −2Zi
−Zi
Zi ATdi
0
0
0
Zi ATdi
0 ⎥
⎥
⎢
⎥
⎢
T
−1 −αd
⎥
⎢∗
∗
−d
e
Z
0
Z
A
0
0
0
0
i
i di
⎥
⎢
⎥
⎢
−1
⎢∗
∗
∗
−d Zi
0
0
0
0
0 ⎥
⎥
⎢
⎥
⎢
2
⎥ < 0,
⎢∗
∗
∗
∗
−d
S
0
0
0
0
i
⎥
⎢
⎥
⎢
⎥
⎢∗
∗
∗
∗
∗
−S
0
0
0
i
⎥
⎢
⎥
⎢
−1
⎢∗
∗
∗
∗
∗
∗ −d ε1 Zi
0
0 ⎥
⎥
⎢
⎥
⎢
⎢∗
−1
∗
∗
∗
∗
∗
∗
−d η1 Zi
0 ⎥
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗
Φ1
⎡
Λij
Ξij
GTij
W Tij Adj
Ξij
0 d WijT Pij
0
⎢
⎢
T
Gij
−Gij −GTij −Gij
ATdj Qij
0
0
0
Adj Qij
⎢
⎢
⎢
T
T
T
−1
⎢ Adj Wij
−Gij −d Qij
0
Adj Sij
0
0
0
⎢
⎢
⎢
ΞTij
Qij Adj
0
−d−1 Qij
0
0
0
0
⎢
⎢
⎢
2
0
0
Sij Adj
0
−d Sij
0
0
0
⎢
⎢
⎢
0
0
0
0
0
0
−Sij
⎢τ Wij Pij
⎢
⎢
⎢
ΞTij
0
0
0
0
0
−d−1 ε2 Qij
0
⎢
⎢
−1
⎢
0
Qij Adj
0
0
0
0
0
−d η2 Qij
⎣
Uj
0
0
0
0
0
0
0
3.25
UTj
⎤
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥<0
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎦
Φ2
3.26
holds, then under the switching controller ut Kσ t xt, Ki Yi Xi−1 , and the following average
dwell-time scheme:
ln μ1 μ2
T − t0 , t β λ∗
∗
≥
inf ,
τ
>
τ
,
3.27
a
a
t>t0 T t0 , t
α − λ∗
λ∗
the corresponding closed-loop system is exponentially stable, where 0 < λ∗ < α, μ1 , μ2 ≥ 1 satisfying
Xi−1 < μ1 Pij , Pij < μ2 Xi−1 , Zi−1 < μ1 Qij , Qij < μ2 Zi−1 , Φ1 −dρ1 1 ε1 η1 ρ2 −1 I, Ξi Xi ATi YiT BiT , Πi Ai Adi Xi Xi Ai Adi T α 1Xi Bi Yi YiT BiT Ri , Φ2 −dρ3 1 ε2 η2 ρ4 −1 I, Ξij Aj Bj Yi Xi−1 Qij , Λij Aj Adj T Pij Pij Aj Adj −
β − 1Pij Pij Bj Yi Xi−1 Xi−1 YiT BjT Pij .
T
12
Mathematical Problems in Engineering
Proof. Suppose that the ith subsystem is activated at the switching instant tk−1 , the jth
subsystem is activated at the switching instant tk , then the corresponding switchings of the
controller occur at the switching instant tk−1 Δk−1 and tk Δk , respectively.
When t ∈ tk−1 Δk−1 , tk , system 3.22 can be written as
ẋt AKi xt Adi xt − di t fi xt, t,
3.28
where AKi Ai Bi Ki .
Consider Lyapunov functional candidate as follows:
Vi xt x tPi xt T
t
ẋ se
T
−αt−s
Ri ẋsds t−dt
0 t
−d
ẋT re−αt−r Qi ẋrdr dθ.
tθ
3.29
For given positive constants α, ρ1 , ρ2 , ε1 , η1 , if there exist positive definite symmetric
i , and any matrices Gi , Wi with appropriate dimensions such that the
matrices Pi , S
i , Qi , R
follows matrix inequalities 3.30-3.31 hold, then from Lemma 3.1 we have V̇i xt αVi xt < 0:
Qi ≤ ρ1 I,
Pi ≤ ρ2 I,
⎡
Θi
GTi
WiT Adi
ATKi Qi
0 d WiT Pi
0
ATKi Qi
⎢
T
T
T
⎢ ∗ −1−τe−αd R
i −Gi −G
−Gi
Adi Qi
0
0
0
Adi Qi
⎢
i
⎢
⎢∗
0
ATdi S
i
0
0
0
∗
−d−1 e−αd Qi
⎢
⎢
⎢∗
−1
∗
∗
−d Qi
0
0
0
0
⎢
⎢
⎢
⎢∗
∗
∗
∗
−d2 S
i
0
0
0
⎢
⎢
⎢∗
∗
∗
∗
∗
−S
i
0
0
⎢
⎢
−1
⎢∗
∗
∗
∗
∗
∗
−d ε1 Qi
0
⎢
⎢
−1
⎢∗
∗
∗
∗
∗
∗
∗
−d η1 Qi
⎣
∗
∗
∗
∗
∗
∗
∗
∗
3.30
⎤
UiT
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎦
Φ1
< 0,
3.31
where
i
Θi AKi Adi T Pi Pi AKi Adi α 1Pi R
−1
Φ1 − dρ1 1 ε1 η1 ρ2 I.
3.32
−1
−1
−1
Using diag{Pi−1 , Qi−1 , Qi−1 , Qi−1 , S−1
i , Si , Qi , Qi , I} to pre- and postmultiply the left term of
3.31, and denoting Pi−1 Xi , Ki Pi−1 Yi , Qi−1 Zi , S
−1
i Si , Wi Pi , Gi Qi , Ri 0, we
have that 3.31 is equivalent to 3.25.
Mathematical Problems in Engineering
13
Therefore, we have
i
Vi xt < e−αt−t0 Vi x ti0 ,
3.33
where ti0 represents the initial value of the ith subsystem.
When t ∈ tk , tk Δk , system 3.22 can be written as
ẋt AKij xt Adj x t − dj t fj xt, t,
3.34
where AKij Aj Bj Ki .
Consider Lyapunov functional candidate for system 3.34 as follows:
Vij xt x tPij xt T
0 t
−τ
3.35
ẋT reβt−r Qij ẋrdr dθ.
tθ
For given positive constants β, ρ3 , ρ4 , ε2 , η2 , if there exist positive definite symmetric matrices
Pij , Sij , Qij , and any matrices Gij , Wij with appropriate dimensions such that the follows
matrix inequalities 3.36-3.37 hold, then from Lemma 3.2 we have V̇ij xt−βVij xt < 0:
Qij ≤ ρ3 I,
⎡
Θij
GTij
W Tij Adj ATKij Qij
0
⎢
⎢
Gij
−Gij − GTij −Gij ATdj Qij
0
⎢
⎢
⎢
T
T
T
−1
⎢ Adj Wij
−Gi
−τ Qij
0
Adj Sij
⎢
⎢
⎢ Qij AKij
Qij Adj
0
−τ −1 Qij
0
⎢
⎢
2
⎢
0
0
Sij Adj
0
−τ Sij
⎢
⎢
⎢τ Wij Pij
0
0
0
0
⎢
⎢
⎢ Qij AKij
0
0
0
0
⎢
⎢
⎢
0
Qij Adj
0
0
0
⎣
Uj
0
0
0
0
Pij ≤ ρ4 I,
ATKij Qij
τ WijT P ij
0
0
0
ATdj Qij
0
0
0
0
0
0
0
0
0
−Sij
0
0
0
−τ −1 ε2 Qij
0
−1
0
0
−τ η2 Qij
0
0
0
UTj
3.36
⎤
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
< 0,
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎦
Φ2
3.37
where Θij AKij Adj T Pij Pij AKij Adj − β − 1Pij , Φ2 −dρ3 1 ε2 η2 ρ4 −1 I.
Denoting Ki Yi Xi−1 , we know that 3.37 is equivalent to 3.26. Thus, we have
ij ij
Vij xt < eβt−t0 Vij x t0 ,
ij
where t0 represents the initial value of the jth subsystem.
3.38
14
Mathematical Problems in Engineering
Let t0 , t1 , . . . , tk denote the switching instants in t0 , t, from 3.33 and 3.38, for t ≥
tk Δk , we have
V t < e−αt−tk −Δk V tk Δk k
< μ1 μ2 e−αt−tk −Δk tk −tk−1 −Δk−1 ···t2 −t1 −Δ1 t1 −t0 −Δ0 βΔk Δk−1 ···Δ1 Δ0 V t0 3.39
k
−
μ1 μ2 e−αT t0 ,tβT t0 ,t V t0 .
By Definition 2.4, we have
k ≤ N0 t − t0
.
τa
3.40
From 3.27, it follows that
−T − t0 , tλ− T t0 , tλ ≤ −λ∗ t − t0 .
3.41
Substituting 3.40 and 3.41 into 3.39, we have
N t−t0 /τa −λ∗ t−t0 V t < μ1 μ2 0
e
V t0 N
∗
μ1 μ2 0 elnμ1 μ2 /τa −λ t−t0 V t0 .
3.42
Thus
xt <
N /2
b ∗
· μ1 μ2 0 elnμ1 μ2 /τa −λ t−t0 /2 xt0 h ,
a
3.43
where a mini,j∈N,i / j {λmin Xi−1 , λmin Pij }, b maxi,j∈N,i / j{λmax Xi−1 τ 2 /2λmax Zi−1 ,
λmax Pij τ 2 /2λmax Qij }.
The proof is completed.
Remark 3.4. When μ1 μ2 1, we have τa∗ 0, which implies that switching signals can be arbitrary.
3.3. Robust Reliable Controller Design
Now, we are in a position to present sufficient conditions for the existence of robust reliable
controller for system 2.1 with delayed switching.
Mathematical Problems in Engineering
15
Theorem 3.5. Consider system 2.1, for given positive constants α, β, ε1 , η1 , ρ1 , δ1 , ε2 , η2 , ρ2 , δ2 ,
ρ3 , δ3 , ρ4 , δ4 , if there exist positive definite symmetric matrices Xi , Zi , Ri , Pij , Qij , Sij , and any
j,
matrices Yi , Gij , Wij with appropriate dimensions, such that, for i, j ∈ N,i /
Zi ≥ ρ1−1 I,
Xi ≥ ρ2−1 I,
⎡
Ω i ϑi
⎢
⎢ ∗ Ψi
⎣
∗ ∗
⎡
Ωij ϑij
⎢
⎢ ∗ Ψij
⎣
∗ ∗
Φi
Qij ≤ ρ3 I,
⎤
Pij ≤ ρ4 I
⎥
0⎥
⎦ < 0,
3.44
3.45
Γi
Φij
⎤
⎥
Δij ⎥
⎦<0
3.46
Γij
hold, then under the reliable switching controller ut Kσt xt, Ki Yi Xi−1 , and the following
average dwell-time scheme:
T − t0 , t β λ∗
≥
,
inf t>t0 T t0 , t
α − λ∗
τa >
τa∗
ln μ1 μ2
,
λ∗
3.47
the corresponding closed-loop system is exponentially stable,where 0 < λ∗ < α,μ1 , μ2 ≥ 1 satisfying
Xi−1 < μ1 Pij , Pij < μ2 Xi−1 , Zi−1 < μ1 Qij ,
Qij < μ2 Zi−1 ,
⎡
⎤
Ωi11 Xi
Adi Zi
⎢
⎥
⎥,
Ωi ⎢
−Zi
⎣ ∗ −2Zi
⎦
∗
∗ −d−1 e−αd Zi
Ωi11 Ai Adi Xi Xi Ai Adi T α 1Xi Bi Yi YiT BiT δ1 Hi HiT δ2 Bi Ji BiT ,
⎤
⎡
Xi ATi YiT BiT
0
2dRi Xi ATi YiT BiT
0
Xi UiT
⎥
⎢
⎥
⎢
Zi ATdi
0
0
0
Zi ATdi
0 ⎥,
ϑi ⎢
⎦
⎣
0
0
0
0
Zi ATdi 0
⎤
⎡
T
T
T
T
Xi E1i E2i T
0 Xi E1i
YiT E2i
Xi E1i
YiT E2i
0 YiT Mi0 Ji1/2
⎥
⎢
⎥
⎢
T
Φi ⎢
⎥,
0
Zi ET2i
Zi E2i
0
0
0
⎦
⎣
T
T
0
0
0
Zi E2i
0
Zi E2i
16
Mathematical Problems in Engineering
Ψi diag −d−1 Zi δ1 Hi HiT δ2 Bi Ji BiT , −d2 Ri δ1 Hi HiT , −Ri ,
−1 −d−1 ε1 Zi δ1 Hi HiT δ2 Bi Ji BiT , −d−1 η1 Zi δ1 Hi HiT , − dρ1 1 ε1 η1 ρ2 I ,
Γi diag{−δ1 I, −δ1 I, −δ1 I, −δ1 I, −δ1 I, −δ2 I},
⎡
⎤
Ωij1
GTij
WijT Adj
⎢
⎥
⎢
⎥
⎢ ∗ −G − GT 2δ ET E
⎥
−Gij
Ωij ⎢
⎥,
ij
2 2j 2j
ij
⎢
⎥
⎣
⎦
T
∗
∗
−τ −1 Qij δ3 E2j
E2j
T
Ωij1 Aj Adj Pij Pij Aj Adj − β − 1 Pij Pij Bj Yi Xi−1 Xi−1 YiT BjT Pij
T
3δ3 E1j
E1j δ4 Pij Bj Ji BjT Pij ,
⎡
⎢
⎢
⎢
ϑij ⎢
⎢
⎣
Aj Bj Yi Xi−1
T
Qij
0
T
τ WijT Pij
Aj Bj Yi Xi−1 Qij
0
ATdj Qij
0
0
0
ATdj Qij
0
ATdj Sij
0
0
0
⎡
⎤
Pij Hj 0 0 0 WijT Hj
⎢
⎥
⎢
⎥
⎥,
Φij ⎢
0
0
0
0
0
⎢
⎥
⎣
⎦
0
0 0 0
0
UjT
⎤
⎥
⎥
⎥
0 ⎥,
⎥
⎦
0
⎤
⎡
0
0
Qij Hj
0
0
⎥
⎢
⎥
⎢
⎢0
0
0
0
S
H
ij j ⎥
⎥
⎢
⎥
⎢
⎥
⎢
⎢0
0
0
0
0 ⎥
⎥
⎢
Δij ⎢
⎥,
⎢0
0
0
Qij Hj
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢0 Qij Hj
0
0
0
⎥
⎢
⎦
⎣
0
0
0
0
0
Ψij diag −d−1 Qij δ4 Qij Bj Ji BjT Pij , −d2 Sij , −Sij , −d−1 ε2 Qij δ4 Qij Bj Ji BjT Qij , −d−1 η2 Qij ,
−1 − dρ3 1 ε2 η2 ρ4 I ,
Γij diag{−δ3 I, −δ3 I, −δ3 I, −δ3 I, −δ3 I, −δ4 I}.
3.48
Mathematical Problems in Engineering
17
Proof. Consider the following inequalities 3.49 and 3.50:
⎤
⎡
di Zi
i
i
Πi Xi
0
2dRi
0
Xi UiT
Ξ
Ξ
A
⎥
⎢
⎥
⎢
T
T
⎥
⎢ ∗ −2Z
−Z
Z
0
0
0
Z
0
A
A
i
i
i di
i di
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
T
0
Zi A
0
0
0
0 ⎥
∗ −d−1 e−αd Zi
⎢∗
di
⎥
⎢
⎥
⎢
⎥
⎢
−1
⎢∗
0
0
0
0
0 ⎥
∗
∗
−d Zi
⎥
⎢
⎥
⎢
⎥
⎢
2
Ti ⎢ ∗
⎥ < 0,
R
0
0
0
0
∗
∗
∗
−d
i
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢∗
0
0
0
∗
∗
∗
∗
−R
i
⎥
⎢
⎥
⎢
⎥
⎢
−1
⎢∗
0
0 ⎥
∗
∗
∗
∗
∗ −d ε1 Zi
⎥
⎢
⎥
⎢
⎥
⎢
−1
⎢∗
0 ⎥
∗
∗
∗
∗
∗
∗
−d η1 Zi
⎥
⎢
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗
Φ1
3.49
⎤
⎡
dj
ij
ij
ij
Λ
GTij
WijT A
0 d WijT Pij
0
UjT
Ξ
Ξ
⎥
⎢
⎥
⎢
⎥
⎢
T Qij
T Qij
⎥
⎢ ∗ −Gij − GTij −Gij
0
0
0
A
0
A
dj
dj
⎥
⎢
⎥
⎢
⎥
⎢
T
−1
⎢∗
Sij
∗
−d Qij
0
A
0
0
0
0 ⎥
⎥
⎢
dj
⎥
⎢
⎥
⎢
⎢∗
−1
0
0
0
0
0 ⎥
∗
∗
−d Qij
⎥
⎢
⎥
⎢
⎥
⎢
2
Tij ⎢
0
0
0
0 ⎥
∗
∗
∗
−d Sij
⎥ < 0,
⎢∗
⎥
⎢
⎥
⎢
⎥
⎢
0
0
0 ⎥
∗
∗
∗
∗
−Sij
⎢∗
⎥
⎢
⎥
⎢
⎥
⎢
⎢∗
0
0 ⎥
∗
∗
∗
∗
∗
−d−1 ε2 Qij
⎥
⎢
⎥
⎢
⎥
⎢
−1
⎥
⎢∗
η
Q
0
∗
∗
∗
∗
∗
∗
−d
2 ij
⎥
⎢
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗
Φ2
3.50
i A
di Xi Xi A
i A
di T T Y T Mi BT , Π
i Xi A
i A
where Φ1 −dρ1 1 ε1 η1 −1 I, Ξ
i
i
i
j Bj Mi Yi X −1 T Qij , Λ
ij A
ij α 1Xi Bi Mi Yi YiT Mi BiT , Φ2 −dρ2 1 ε2 η2 −1 I, Ξ
i
T
j A
dj Pij Pij A
j A
dj − β − 1Pij Pij Bj Mi Yi X −1 X −1 Y T Mi BT Pij .
A
i
i
Substituting 2.3 and 2.10 into 3.49, we can obtain
Ti T1i T2i ,
i
j
3.51
18
Mathematical Problems in Engineering
where
⎤
⎡
Πi Xi
Adi Zi
Ξi
0
2dRi
Ξi
0
Xi UiT
⎥
⎢
⎥
⎢
T
T
⎥
⎢ ∗ −2Zi
−Z
Z
A
0
0
0
Z
A
0
i
i di
i di
⎥
⎢
⎥
⎢
⎥
⎢
∗ −d−1 e−αd Zi
0
Zi ATdi 0
0
0
0 ⎥
⎢∗
⎥
⎢
⎥
⎢
⎢∗
−1
∗
∗
−d Zi
0
0
0
0
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢
2
⎥,
∗
∗
∗
∗
−d
T1i ⎢
R
0
0
0
0
i
⎥
⎢
⎥
⎢
⎥
⎢
⎢∗
∗
∗
∗
∗
−Ri
0
0
0 ⎥
⎥
⎢
⎥
⎢
⎥
⎢
−1
∗
∗
∗
∗
∗ −d ε1 Zi
0
0 ⎥
⎢∗
⎥
⎢
⎥
⎢
⎢∗
−1
∗
∗
∗
∗
∗
∗
−d η1 Zi
0 ⎥
⎥
⎢
⎦
⎣
∗
∗
∗
∗
∗
∗
∗
∗
Φ1
⎡
Υi
⎢
⎢∗
⎢
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢
T2i ⎢ ∗
⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢
⎢
⎢∗
⎣
∗
0 κi φiT
0 0 φiT
0 0 κTi
0 0 0
∗ 0
0 κTi 0 0
∗ ∗
0
0 0 0
∗ ∗
∗
0 0 0
∗ ∗
∗
∗
∗ ∗
∗
∗ ∗
0
∗ ∗
∗
∗ ∗
∗
∗ ∗
∗
∗ ∗
∗
0 0
⎤
0 0
⎥
κTi 0⎥
⎥
⎥
⎥
0 0⎥
⎥
⎥
0 0⎥
⎥
⎥
⎥
0 0⎥,
⎥
⎥
0 0⎥
⎥
⎥
0 0⎥
⎥
⎥
⎥
0 0⎥
⎦
∗ 0
3.52
3.53
where Πi Ai Adi Xi Xi Ai Adi T α1Xi Bi Mi0 Yi YiT Mi0 BiT , Υi Hi Fi E1i E2i Xi Xi E1i E2i T FiT HiT Bi Mi0 Li Yi YiT Li Mi0 BiT , κi Hi Fi E2i Zi , φi Hi Fi E1i Xi Bi Mi0 Li Yi .
By Lemma 2.7, Lemma 2.8, and Schur complement lemma, we know that 3.49 is
equivalent to 3.45. Similarly, substituting 2.3 into 3.50, we can obtain that 3.50 is
equivalent to 3.46.
From Theorem 3.3, we know that Theorem 3.5 holds.
The proof is completed.
Remark 3.6. In actual operation, the condition 3.27 or 3.47 is difficult to check. Let Δmax
be a known positive scalar that represents the maximum delayed period, a simple condition
of the average dwell time is proposed, that is to say, 3.27 or 3.47 can be reduced to the
following condition:
τa >
τa∗
ln μ1 μ2
β λ∗
max
,
1 Δmax ,
λ∗
α − λ∗
0 < λ∗ < α.
3.54
Mathematical Problems in Engineering
19
Proof of 3.54: According to 3.27, we have
T − t0 , t β λ∗
t − t0 − T t0 , t β λ∗
≥
≥
⇒
T t0 , t α − λ∗
T t0 , t
α − λ∗
⇒ t − t0 ≥
β λ∗ T t0 , t T t0 , t
α − λ∗
⇒ t − t0 ≥
3.55
β λ∗
1 T t0 , t.
α − λ∗
On the other hand
β λ∗
β λ∗
1
T
,
t
≤
1
Nσt t0 , tΔmax
t
0
α − λ∗
α − λ∗
≤
β λ∗
t − t0
1
Δmax .
∗
α−λ
τa∗
3.56
Obviously, if the following inequality 3.57 is satisfied, then we have that 3.55 holds
t − t0 ≥
β λ∗
t − t0
1
Δmax .
∗
α−λ
τa∗
3.57
From 3.57, we have τa∗ ≥ β λ∗ /α − λ∗ 1Δmax .
The proof is completed.
4. Conclusion
In this paper, we have investigated the robust reliable stabilization problem for switched
nonlinear systems with time-varying delays and delayed switching. The average dwell-time
approach is utilized for stability analysis and controller design. Our future work will focus on
extending the proposed design method to H∞ performance analysis for switched nonlinear
time-varying systems with delayed switching.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant
no. 60974027. The authors would like to thank anonymous reviewers for their valuable
comments that greatly improved the quality and presentation of this paper.
20
Mathematical Problems in Engineering
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