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1.
A delay decomposition approach to robust stability analysis of uncertain
systems with time-varying delay
Pin-Lin Liu n
Department of Automation Engineering Institute of Mechatronoptic System, Chienkuo Technology University, Changhua 500, Taiwan, ROC
a r t i c l e i n f o
Article history:
Received 5 April 2012
Received in revised form
16 June 2012
Accepted 3 July 2012
Available online 25 July 2012
Keywords:
Lyapunov–Krasovskii’s functional
Stability
Time-varying delay
Linear matrix inequalities (LMIs)
Maximum allowable delay bound (MADB)
a b s t r a c t
This paper is concerned with delay-dependent robust stability for uncertain systems with time-varying
delays. The proposed method employs a suitable Lyapunov–Krasovskii’s functional for new augmented
system. Then, based on the Lyapunov method, a delay-dependent robust criterion is devised by taking
the relationship between the terms in the Leibniz–Newton formula into account. By developing a delay
decomposition approach, the information of the delayed plant states can be taken into full considera-
tion, and new delay-dependent sufﬁcient stability criteria are obtained in terms of linear matrix
inequalities (LMIs) which can be easily solved by various optimization algorithms. Numerical examples
are included to show that the proposed method is effective and can provide less conservative results.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Time delay is one of the instability sources for dynamical
systems, and is a common phenomenon in many industrial and
engineering systems such as those in communication networks,
manufacturing, and biology. Since system stability is an essential
requirement in many applications, much effort has been made to
investigate stability criteria for various time-delay systems during
the last two decades. For details, see the works [1–24] and
references therein.
Recently, some researchers have paid attention to the issue
concerning delay-dependent stability analysis [2–24]. The devel-
opment of the technologies for delay-dependent stability analysis
has been focusing on effective reduction of the conservation of the
stability conditions. The main aim is to derive a maximum
allowable delay bound (MADB) of the time-delay such that the
time delays’ system is asymptotically stable for any delay size less
than the MADB. Accordingly, the obtained MADB becomes a key
performance index to measure the conservatism of a delay-
dependent stability condition.
On the other hand, many uncertain factors exist in practical
systems. Uncertainty in a control system may be attributed to
modeling errors, measurement errors, parameter variations and a
linearization approximation [9]. Therefore, study on the robust
stability of uncertain systems with time-varying delay becomes
signiﬁcantly important. The results obtained will be very useful to
further research for robust stability of uncertain systems with
time-varying delay [4,6,8,9–16,18–20,23,24] and references
therein.
The development of the technologies for delay-dependent
stability analysis has been focusing on effective reduction of the
conservation of the stability conditions. Two effective technolo-
gies have been widely accepted: the bounding technology [4], and
the model transformation technology [2]. The former is used to
evaluate the bounds of some cross terms arising from the analysis
of the delay-dependent problem; while the latter is employed to
transform the original delay models into simpler ones so that the
stability analysis becomes easier.
However, it is also known that the bounding technology and the
model transformation technique are the main source of conserva-
tion. To further improve the performance of delay-dependent
stability criteria, there is a need to avoid this source of conservation.
Therefore, much effort has been devoted recently to the develop-
ment of the free weighting matrices method, in which neither the
bounding technology nor model transformation is employed
[3,11,19–21]. Generally speaking, the free weighting matrices appear
in two forms: the one with a null summing term added to the
Lyapunov functional derivative [19–21], and the one with free
matrices item added to the Lyapunov functional combined with
the descriptor model transformation [3,11]. However, this approach
introduced some slack variables apart from the matrix variables
appearing in Lyapunov–Krasovskii functionals (LKFs). When the
upper bound of delay derivative may be larger than or equal to 1,
Zhu and Yang [23,24] used a delay decomposition approach, and
new stability results were derived. Compared with [6], the stability
results in [23,24] are simpler and less conservative.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.07.001
n
Tel.: þ886 47 111155; fax: þ886 47 111129.
E-mail address: lpl@cc.ctu.edu.tw
ISA Transactions 51 (2012) 694–701
2.
Motivated by the above discussions, we propose new robust
stability criteria for uncertain systems with time-varying delays.
Based on the Lyapunov function method, a novel robust delay-
dependent criterion, which is less conservative than delay- inde-
pendent one when the size of delays is small, is established in
terms of LMIs which can be solved efﬁciently by using the
optimization algorithms. In order to derive less conservative
results, a new integral inequality approach (IIA) which utilizes
free weighting matrices is proposed. Also, the model transforma-
tion technique, which leads to an additional dynamics, is not used
in this work. Finally, ﬁve numerical examples are shown to
support that our results are less conservative than those of the
existing ones.
2. Problem statement
Consider the following uncertain system with a time-varying
state delay:
_xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ ð1aÞ
xðtÞ ¼ fðtÞ, tA½Àh,0, ð1bÞ
where xðtÞARn
is the state vector, A and B are constant matrices
with appropriate dimensions, f(t) is a smooth vector-valued
initial function, h(t) is a time varying delay in the state, and h is
an upper bound on the delay h(t). The uncertainties are assumed
to be the form:
DAðtÞ DBðtÞ
Â Ã
¼ DFðtÞ Ea Eb½ , ð2Þ
where D, Ea and Eb are constant matrices with appropriate
dimensions, and F(t) is an unknown, real, and possibly time-
varying matrix with Lebesgue–measurable elements satisfying
FT
ðtÞFðtÞrI, 8t: ð3Þ
As in [2], we consider two different cases for time varying
delays:
Case I. h(t) is a differentiable function, satisfying for all tZ0:
0rhðtÞrh and _hðtÞ
rhd: ð4Þ
Case II. h(t) is not differentiable or the upper bound of the
derivative of h(t) is unknown, and h(t) satisﬁes
0rhðtÞrh ð5Þ
where h and hdare some positive constants.
To do this, three fundamental lemmas are reviewed. First,
Lemma 1 induces the integral inequality approach.
Lemma 1. [9] For any positive semi-deﬁnite matrices:
X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0, ð6aÞ
the following integral inequality holds:
À
Z t
tÀhðtÞ
_xT
ðsÞX33 _xðsÞ ds
r
Z t
tÀhðtÞ
xT
ðtÞ xT
ðtÀhðtÞÞ _xT
ðsÞ
h i
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 0
2
6
4
3
7
5
xðtÞ
xðtÀhðtÞÞ
_xðsÞ
2
6
4
3
7
5 ds:
ð6bÞ
Secondary, we introduce the following Schur complement
which is essential in the proofs of our results.
Lemma 2. [1] The following matrix inequality:
QðxÞ SðxÞ
ST
ðxÞ RðxÞ
" #
o0, ð7aÞ
where QðxÞ ¼ QT
ðxÞ, RðxÞ ¼ RT
ðxÞ and SðxÞ depend on afﬁne on x, is
equivalent to
RðxÞo0, ð7bÞ
QðxÞo0, ð7cÞ
and
QðxÞÀSðxÞRÀ1
ðxÞST
ðxÞo0: ð7dÞ
Lemma 3. [1] Given matrices Q ¼ QT
, D, E, and R ¼ RT
40 of
appropriate dimensions,
Q þDFðtÞEþET
FT
ðtÞDT
o0, ð8Þ
for all F satisfying FT
ðtÞFðtÞrH, if and only if there exists some
e40 such that
Q þeDDT
þeÀ1
ET
HEo0: ð9Þ
The purpose of this paper is to ﬁnd new stability criteria, which
are less conservative than the existing results.
Firstly, we consider the nominal from system (1):
_xðtÞ ¼ AxðtÞþBxðtÀhðtÞÞ, tZ0: ð10Þ
For the nominal system (10), we will give a stability condition
by using a delay decomposition approach as follows:
Theorem 1. In Case I, if 0rh(t)rah, for given scalars h(h40),
a(0oao1) and hd, the system described by (10) with (4) is
asymptotically stable if there exist matrices P ¼ PT
40, Qi ¼
QT
i Z0, Ri ¼ RT
i Z0,ði ¼ 1,2,3Þ, and positive semi-deﬁnite matrices:
X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0, Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5Z0,
Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0 such that
O ¼
O11 O12 O13 O14 O15
OT
12 O22 O23 O24 O25
OT
13 OT
23 O33 O34 O35
OT
14 OT
24 OT
34 O44 O45
OT
15 OT
25 OT
35 OT
45 O55
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
o0, ð11Þ
and
R1ÀX33 Z0, R2ÀY33 Z0, R1 þð1ÀhdÞR3ÀZ33 Z0, ð12Þ
where
O11 ¼ AT
PþPAþQ1 þQ3 þahZ11 þZ13 þZT
13,
O12 ¼ PBþahZ12ÀZ13 þZT
23,
O15 ¼ AT
½ahR1 þð1ÀaÞhR2 þahR3,
O22 ¼ Àð1ÀhdÞQ3 þahX11 þX13 þXT
13 þahZ22ÀZ23ÀZT
23,
O23 ¼ ahX12ÀX13 þXT
23,O25 ¼ BT
½ahR1 þð1ÀaÞhR2 þahR3,
O33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT
23 þð1ÀaÞhY11 þY13 þYT
13,
O34 ¼ ð1ÀaÞhY12ÀY13 þYT
23, O44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT
23,
O55 ¼ À½ahR1 þð1ÀaÞhR2 þahR3, O13 ¼ O14 ¼ O24 ¼ O35 ¼ O45 ¼ 0:
P.-L. Liu / ISA Transactions 51 (2012) 694–701 695
3.
Proof. In Case I, a Lyapunov functional can be constructed as
VðtÞ ¼ V1ðtÞþV2ðtÞþV3ðtÞ, ð13Þ
where
V1ðtÞ ¼ xT
ðtÞPxðtÞ,
V2ðtÞ ¼
Z t
tÀah
xT
ðsÞQ1xðsÞdsþ
Z tÀah
tÀh
xT
ðsÞQ2xðsÞds
þ
Z t
tÀhðtÞ
xT
ðsÞQ3xðsÞds,
V3ðtÞ ¼
Z 0
Àah
Z t
t þy
_xT
ðsÞR1 _xðsÞdsdyþ
Z Àah
Àh
Z t
t þy
_xT
ðsÞR2 _xðsÞdsdy
þ
Z 0
ÀhðtÞ
Z t
t þy
_xT
ðsÞR3 _xðsÞdsdy:
Taking the time derivative of V(t) for tA(0,N) along the
trajectory of (10), it yields that:
_V ðtÞ ¼ _V 1ðtÞþ _V 2ðtÞþ _V 3ðtÞ, ð14Þ
where
_V 1ðtÞ ¼ _xT
ðtÞPxðtÞþxT
ðtÞP_xðtÞ
¼ xT
ðtÞðAT
PþPAÞxðtÞþxT
ðtÞPBxðtÀhðtÞÞþxT
ðtÀhðtÞÞBT
PxðtÞ,
ð15Þ
_V 2ðtÞ ¼ xT
ðtÞðQ1 þQ3ÞxðtÞÀxT
ðtÀhðtÞÞð1À_hðtÞÞQ3xðtÀhðtÞÞ
þxT
ðtÀahÞðQ2ÀQ1ÞxðtÀahÞÀxT
ðtÀhÞQ2xðtÀhÞ
rxT
ðtÞðQ1 þQ3ÞxðtÞÀxT
ðtÀhðtÞÞð1ÀhdÞQ3xðtÀhðtÞÞ
þxT
ðtÀahÞðQ2ÀQ1ÞxðtÀahÞÀxT
ðtÀhÞQ2xðtÀhÞ, ð16Þ
and
_V 3ðtÞ ¼ _xT
ðtÞ½ahR1 þð1ÀaÞhR2 þhðtÞR3_xðtÞÀ
Z t
tÀah
_xT
ðsÞR1 _xðsÞds
À
Z tÀah
tÀh
_xT
ðsÞR2 _xðsÞdsÀð1À _hðtÞÞ
Z t
tÀhðtÞ
_xT
ðsÞR3 _xðsÞds
r _xT
ðtÞ½ahR1 þð1ÀaÞhR2 þahR3_xðtÞÀ
Z t
tÀah
_xT
ðsÞR1 _xðsÞds
À
Z tÀah
tÀh
_xT
ðsÞR2 _xðsÞdsÀð1ÀhdÞ
Z t
tÀhðtÞ
_xT
ðsÞR3 _xðsÞds: ð17Þ
Now, we estimate the upper bound of the last three terms in
inequality (17) as follows:
À
Z t
tÀah
_xT
ðsÞR1 _xðsÞdsÀ
Z tÀah
tÀh
_xT
ðsÞR2 _xðsÞds
Àð1ÀhdÞ
Z t
tÀhðtÞ
_xT
ðsÞR3 _xðsÞds
¼ À
Z tÀhðtÞ
tÀah
_xT
ðsÞR1 _xðsÞdsÀ
Z tÀah
tÀh
_xT
ðsÞR2 _xðsÞds
À
Z t
tÀhðtÞ
_xT
ðsÞðR1 þð1ÀhdÞR3Þ_xðsÞds
¼ À
Z tÀhðtÞ
tÀah
_xT
ðsÞðR1ÀX33Þ_xðsÞdsÀ
Z tÀah
tÀh
_xT
ðsÞðR2ÀY33Þ_xðsÞds
À
Z t
tÀhðtÞ
_xT
ðsÞðR1 þð1ÀhdÞR3ÀZ33Þ_xðsÞds
À
Z tÀhðtÞ
tÀah
_xT
ðsÞX33 _xðsÞdsÀ
Z tÀah
tÀh
_xT
ðsÞY33 _xðsÞds
À
Z t
tÀhðtÞ
_xT
ðsÞZ33 _xðsÞds:Þ ð18Þ
From integral inequality approach of Lemma 1 [9], noticing
that R1ÀX33 Z0, R2ÀY33 Z0, and R1 þð1ÀhdÞR3ÀZ33 Z0, it yields
that:
À
Z tÀhðtÞ
tÀah
_xT
ðsÞX33 _xðsÞdsr
Z tÀhðtÞ
tÀah
xT
ðtÀhðtÞÞ xT
ðtÀahÞ _xT
ðsÞ
h i
Â
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 0
2
6
4
3
7
5
xðtÀhðtÞÞ
xðtÀahÞ
_xðsÞ
2
6
4
3
7
5ds
rxT
ðtÀhðtÞÞðahÀhðtÞÞX11xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞðahÀhðtÞÞX12xðtÀahÞ
þxT
ðtÀhðtÞÞX13
Z t
tÀhðtÞ
_xðsÞdsþxT
ðtÀahÞðahÀhðtÞÞXT
12xðtÀhðtÞÞ
þxT
ðtÀahÞðahÀhðtÞÞX22xðtÀahÞþxT
ðtÀahÞX23
Z t
tÀhðtÞ
_xðsÞds
þ
Z tÀhðtÞ
tÀah
_xT
ðsÞdsXT
13xðtÀhðtÞÞþ
Z tÀhðtÞ
tÀah
_xT
ðsÞdsXT
23xðtÀahÞ
rxT
ðtÀhðtÞÞahX11xðtÀhðtÞÞþxT
ðtÀhðtÞÞahX12xðtÀahÞ
þxT
ðtÀhðtÞÞX13
Z t
tÀhðtÞ
_xðsÞdsþxT
ðtÀahÞahXT
12xðtÀhðtÞÞ
þxT
ðtÀahÞahX22xðtÀahÞþxT
ðtÀahÞX23
Z t
tÀhðtÞ
_xðsÞds
þ
Z tÀhðtÞ
tÀah
_xT
ðsÞdsXT
13xðtÀhðtÞÞþ
Z tÀhðtÞ
tÀah
_xT
ðsÞdsXT
23xðtÀahÞ
¼ xT
ðtÀhðtÞÞ½ahX11 þXT
13 þX13xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞ½ahX12ÀX13 þXT
23xðtÀahÞ
þxT
ðtÀahÞ½ahXT
12ÀXT
13 þX23xðtÀhðtÞÞ
þxT
ðtÀahÞ½ahX22ÀX23ÀXT
23xðtÀahÞ: ð19Þ
Similarly, we obtain
À
Z tÀah
tÀh
_xT
ðsÞY33 _xðsÞdsrxT
ðtÀahÞ½ð1ÀaÞhY11 þYT
13 þY13xðtÀahÞ
þxT
ðtÀahÞ½ð1ÀaÞhY12ÀY13 þYT
23xðtÀhÞ
þxT
ðtÀhÞ½ð1ÀaÞhYT
12ÀYT
13 þY23xðtÀahÞ
þxT
ðtÀhÞ½ð1ÀaÞhY22ÀY23ÀYT
23xðtÀhÞ, ð20Þ
and
À
Z t
tÀhðtÞ
_xT
ðsÞZ33 _xðsÞds
rxT
ðtÞ½hðtÞZ11 þZT
13 þZ13xðtÞþxT
ðtÞ½hðtÞZ12ÀZ13 þZT
23xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞ½hðtÞhZT
12ÀZT
13 þZ23xðtÞ
þxT
ðtÀhðtÞÞ½hðtÞZ22ÀZ23ÀZT
23xðtÀhðtÞÞ
rxT
ðtÞ½ahZ11 þZT
13 þZ13xðtÞþxT
ðtÞ½ahZ12ÀZ13 þZT
23xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞ½ahZT
12ÀZT
13 þZ23xðtÞ
þxT
ðtÀhðtÞÞ½ahZ22ÀZ23ÀZT
23xðtÀhðtÞÞ: ð21Þ
The operator for the term _xT
ðtÞ½ahR1 þð1ÀaÞhR2 þahR3_xðtÞ is as
follows:
_xT
ðtÞ½ahR1 þð1ÀaÞhR2 þahR3_xðtÞ
¼ ½AxðtÞþBxðtÀhðtÞÞT
½ahR1 þð1ÀaÞhR2 þahR3½AxðtÞþBxðtÀhðtÞÞ
¼ xT
ðtÞAT
½ahR1 þð1ÀaÞhR2 þahR3AxðtÞ
þxT
ðtÞAT
½ahR1 þð1ÀaÞhR2 þahR3BxðtÀhðtÞÞ
þxT
ðtÀhðtÞÞBT
½ahR1 þð1ÀaÞhR2 þahR3AxðtÞ
þxT
ðtÀhðtÞÞBT
½ahR1 þð1ÀaÞhR2 þahR3BxðtÀhðtÞÞ: ð22Þ
Combining (13)–(22), it yields:
_V ðtÞrxT
ðtÞXxðtÞÀ
Z tÀhðtÞ
tÀah
_xT
ðsÞðR1ÀX33Þ_xðsÞds
À
Z tÀah
tÀh
_xT
ðsÞðR2ÀY33Þ_xðsÞds
P.-L. Liu / ISA Transactions 51 (2012) 694–701696
4.
À
Z t
tÀhðtÞ
_xT
ðsÞðR1 þð1ÀhdÞR3ÀZ33Þ_xðsÞds, ð23Þ
where xT
ðtÞ ¼ xT
ðtÞ xT
ðtÀhðtÞÞ xT
ðtÀahÞ xT
ðtÀhÞ
h i
and
X ¼
X11 X12 X13 X14
XT
12 X22 X23 X24
XT
13 XT
23 X33 X34
XT
14 XT
24 XT
34 X44
2
6
6
6
6
4
3
7
7
7
7
5
with
X11 ¼ AT
PþPAþQ1 þQ3 þahZ11 þZ13 þZT
13
þAT
½ahR1 þð1ÀaÞhR2 þahR3A,
X12 ¼ PBþahZ12ÀZ13 þZT
23 þAT
½ahR1 þð1ÀaÞhR2 þahR3B,
X22 ¼ Àð1ÀhdÞQ3 þhaX11 þX13 þXT
13 þahZ22ÀZ23
ÀZT
23 þBT
½ahR1 þð1ÀaÞhR2 þahR3B,
X23 ¼ ahX12ÀX13 þXT
23,
X33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT
23 þð1ÀaÞhY11 þY13 þYT
13,
X34 ¼ ð1ÀaÞhY12ÀY13 þYT
23,
X44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT
23,
X13 ¼ X14 ¼ X24 ¼ 0:
From nominal system (10) and the Schur complement of
Lemma 2, it is easy to see that _V ðxtÞo0 holds if: R1ÀX33
Z0, R2ÀY33 Z0, R1 þð1ÀhdÞR3ÀZ33 Z0, and 0rh(t)rah.
Theorem 2. In Case I, if ahrh(t)rh, for given scalars h(h40),
a(0oao1) andhd, the system described by (10) with (4) is
asymptotically stable if there exist matrices P ¼ PT
40,
Qi ¼ QT
i Z0, Ri ¼ RT
i Z0, ði ¼ 1,2,3Þ, and positive semi-deﬁnite
matrices: X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0, Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5 Z0,
Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0
such that
O ¼
O11 O12 O13 O14 O15
O
T
12 O22 O23 O24 O25
O
T
13 O
T
23 O33 O34 O35
O
T
14 O
T
24 O
T
34 O44 O45
O
T
15 O
T
25 O
T
35 O
T
45 O55
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
o0, ð24Þ
and
R1 þð1ÀhdÞR3ÀX33 Z0, R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, ð25Þ
where
O11 ¼ AT
PþPAþQ1 þQ3 þahX11 þX13 þXT
13,O12 ¼ PB,
O13 ¼ ahX12ÀX13 þXT
23,
O15 ¼ AT
½ahR1 þð1ÀaÞhR2 þhR3,
O22 ¼ Àð1ÀhdÞQ3 þð1ÀaÞhY22ÀY23ÀYT
23 þð1ÀaÞhZ11 þZ13 þZT
13,
O23 ¼ ð1ÀaÞhYT
12ÀYT
13 þY23, O24 ¼ ð1ÀaÞhZ12ÀZ13 þZT
23,
O25 ¼ BT
½ahR1 þð1ÀaÞhR2 þhR3,
O33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT
23 þð1ÀaÞhYT
11 þY13 þYT
13,
O44 ¼ ÀQ2 þð1ÀaÞhZ22ÀZ23ÀZT
23, O55 ¼ À½ahR1 þð1ÀaÞhR2 þhR3,
O14 ¼ O34 ¼ O35 ¼ O45 ¼ 0:
Proof. If ahrh(t)rh, it gets:
À
Z t
tÀah
_xT
ðsÞR1 _xðsÞdsÀ
Z tÀah
tÀh
_xT
ðsÞR2 _xðsÞds
Àð1ÀhdÞ
Z t
tÀhðtÞ
_xT
ðsÞR3 _xðsÞds
¼ À
Z t
tÀah
_xT
ðsÞðR1 þð1ÀhdÞR3Þ_xðsÞds
À
Z tÀah
tÀhðtÞ
_xT
ðsÞðR2 þð1ÀhdÞR3Þ_xðsÞds
À
Z tÀhðtÞ
tÀh
_xT
ðsÞR2 _xðsÞds
¼ À
Z t
tÀah
_xT
ðsÞðR1 þð1ÀhdÞR3ÀX33Þ_xðsÞds
À
Z tÀah
tÀhðtÞ
_xT
ðsÞðR2 þð1ÀhdÞR3ÀY33Þ_xðsÞds
À
Z tÀhðtÞ
tÀh
_xT
ðsÞðR2ÀZ33Þ_xðsÞdsÀ
Z t
tÀah
_xT
ðsÞX33 _xðsÞds
À
Z tÀah
tÀhðtÞ
_xT
ðsÞY33 _xðsÞdsÀ
Z tÀhðtÞ
tÀh
_xT
ðsÞZ33 _xðsÞds: ð26Þ
From integral inequality matrix [9], noticing that R1 þð1ÀhdÞ
R3ÀX33 Z0, R2ÀZ33 Z0, and R2 þð1ÀhdÞR3ÀY33 Z0, it yields:
À
Z t
tÀah
_xT
ðsÞX33 _xðsÞdsrxT
ðtÞ½ahX11 þXT
13 þX13xðtÞ
þxT
ðtÞ½ahX12ÀX13 þXT
23xðtÀahÞ
þxT
ðtÀahÞ½ahXT
12ÀXT
13 þX23xðtÞ
þxT
ðtÀahÞ½ahX22ÀX23ÀXT
23xðtÀahÞ, ð27Þ
À
Z tÀah
tÀhðtÞ
_xT
ðsÞY33 _xðsÞdsrxT
ðtÀahÞ½ð1ÀaÞhY11 þYT
13 þY13xðtÀahÞ
þxT
ðtÀahÞ½ð1ÀaÞhY12ÀY13 þYT
23xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞ½ð1ÀaÞhYT
12ÀYT
13 þY23xðtÀahÞ
þxT
ðtÀhðtÞÞ½ð1ÀaÞhY22ÀY23ÀYT
23xðtÀhðtÞÞ, ð28Þ
À
Z tÀhðtÞ
tÀh
_xT
ðsÞZ33 _xðsÞdsrxT
ðtÀhðtÞÞ½ð1ÀaÞhZ11 þZT
13 þZ13xðtÀhðtÞÞ
þxT
ðtÀhðtÞÞ½ð1ÀaÞhZ12ÀZ13 þZT
23xðtÀhÞ
þxT
ðtÀhÞ½ð1ÀaÞhZT
12ÀZT
13 þZ23xðtÀhðtÞÞ
þxT
ðtÀhÞ½ð1ÀaÞhZ22ÀZ23ÀZT
23xðtÀhÞ: ð29Þ
Combining (13)–(22) and (26)–(29), it yields:
_V ðtÞrxT
ðtÞXxðtÞÀ
Z t
tÀah
_xT
ðsÞ½R1 þð1ÀhdÞR3ÀX33_xðsÞds
À
Z tÀah
tÀhðtÞ
_xT
ðsÞ½R2 þð1ÀhdÞR3ÀY33_xðsÞds
À
Z tÀhðtÞ
tÀh
_xT
ðsÞðR2ÀZ33Þ_xðsÞds, ð30Þ
where X ¼
X11 X12 X13 X14
X
T
12 X22 X23 X24
X
T
13 X
T
23 X33 X34
X
T
14 X
T
24 X
T
34 X44
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
and
X11 ¼ AT
PþPAþQ1 þQ3 þahX11 þX13 þXT
13
þAT
½ahR1 þð1ÀaÞhR2 þhR3A,
X12 ¼ PBþAT
½ahR1 þð1ÀaÞhR2 þhR3B,
X13 ¼ ahX12ÀX13 þXT
23,
X22 ¼ Àð1ÀhdÞQ3 þð1ÀaÞhY22ÀY23ÀYT
23 þð1ÀaÞhZ11 þZ13 þZT
13
þBT
½ahR1 þð1ÀaÞhR2 þhR3B,
P.-L. Liu / ISA Transactions 51 (2012) 694–701 697
5.
X23 ¼ ð1ÀaÞhYT
12ÀYT
13 þY23,
X24 ¼ ð1ÀaÞhZ12ÀZ13 þZT
23,
X33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT
23 þð1ÀaÞhYT
11 þY13 þYT
13,
X44 ¼ ÀQ2 þð1ÀaÞhZ22ÀZ23ÀZT
23,
X14 ¼ X34 ¼ 0:
From nominal system (10) and the Schur complement, it is
easy to see that _V ðxtÞo0 holds if: R1 þð1ÀhdÞR3ÀX33 Z0,
R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, and ahrh(t)rh.
Theorem 3. In Case II, for given scalars h(h40), a(0oao1), the
system described by (10) with (5) is asymptotically stable if there
exist matrices P ¼ PT
40, Qi ¼ QT
i Z0, Ri ¼ RT
i Z0, ði ¼ 1,2Þ, and
positive semi-deﬁnite matrices: X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5 Z0,
Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5Z0, Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0 such that
C ¼
C11 C12 C13 C14 C15
CT
12 C22 C23 C24 C25
CT
13 CT
23 C33 C34 C35
CT
14 CT
24 CT
34 C44 C45
CT
15 CT
25 CT
35 CT
45 C55
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
o0, ð31Þ
and
R1ÀX33 Z0, R2ÀY33 Z0, R1ÀZ33 Z0, ð32Þ
where
C11 ¼ AT
PþPAþQ1 þhZ11 þZ13 þZT
13,C12 ¼ PBþhZ12ÀZ13 þZT
23,
C15 ¼ AT
½ahR1 þð1ÀaÞhR2,C22 ¼ ahX11 þX13 þXT
13 þhZ22ÀZ23ÀZT
23,
C23 ¼ ahX12ÀX13 þXT
23,C25 ¼ BT
½ahR1 þð1ÀaÞhR2,
C33 ¼ Q2ÀQ1 þahX22ÀX23ÀXT
23 þð1ÀaÞhY11 þY13 þYT
13,
C34 ¼ ð1ÀaÞhY12ÀY13 þYT
23,C44 ¼ ÀQ2 þð1ÀaÞhY22ÀY23ÀYT
23,
C55 ¼ À½ahR1 þð1ÀaÞhR2,C13 ¼ C14 ¼ C24 ¼ C35 ¼ C45 ¼ 0:
Proof. In Case II, a Lyapunov functional can be chosen as (13) with
Q3 ¼ R3 ¼ 0: Similar to the above analysis, one can get that _V ðtÞo0
holds if Co0. Thus, the proof is completed.
Now, extending Theorems 1–3 to uncertain system (1) with
time-varying delays yields the following Theorems.
Theorem 4. In Case I, if 0rh(t)rah, for given scalars h(h40),
a(0oao1) and hd, the uncertain system described by (1) with (4) is
asymptotically stable if there exist matrices P ¼ PT
40, Qi ¼ QT
i Z0,
Ri ¼ RT
i Z0,ði ¼ 1,2,3Þ, e40, and positive semi-deﬁnite matrices:
X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0, Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5Z0, Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0,
such that:
Oe ¼
O11 þeET
a Ea O12 þeET
a Eb O13 O14 O15 PD
OT
12 þeET
bEa O22 þeET
bEb O23 O24 O25 0
OT
13 OT
23 O33 O34 O35 0
OT
14 OT
24 OT
34 O44 O45 0
OT
15 OT
25 OT
35 OT
45 O55 O56
DT
p 0 0 0 OT
56 ÀeI
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
o0,
ð33Þ
and
R1ÀX33 Z0, R2ÀY33 Z0, R1 þð1ÀhaÞR3ÀZ33 Z0, ð34Þ
where
O56 ¼ DT
ðahR1 þð1ÀaÞhR2 þahR3Þ and Oij,ði,j ¼ 1,2,:::,5; iojr5Þ
are deﬁned in (11).
Proof. Replacing A and B in (11) with AþDFðtÞEa and BþDFðtÞEb,
respectively, we apply Lemma 3 [1] for system (1) which is
equivalent to the following condition:
OþGdFðtÞGe þGT
e FðtÞGT
d o0, ð35Þ
where
Gd ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2 þahR3ÞD
h i
, and
Ge ¼ Ea Eb 0 0 0
Â Ã
:
By Lemma 3 [1], a sufﬁcient condition guaranteeing (11) for
system (1) is that there exists a positive number e40 such that:
OþeÀ1
GT
dGd þeGT
e Ge o0: ð36Þ
Applying the Schur complement of Lemma 2 shows that (36) is
equivalent to (33). This completes the proof.
Theorem 5. In Case I, if ahrh(t)rh, for given scalars
h(h40),a(0oao1) and hd, the uncertain system (1) with (4) is
asymptotically stable if there exist matrices P ¼ PT
40, Qi ¼
QT
i Z0, Ri ¼ RT
i Z0,ði ¼ 1,2,3Þ, e40, and positive semi-deﬁnite
matrices: X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0, Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5Z0,
Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0 such that:
Oe ¼
O11 þeET
aEa O12 þeET
aEb O13 O14 O15 PD
O
T
12 þeET
bEa O22 þeET
bEb O23 O24 O25 0
O
T
13 O
T
23 O33 O34 O35 0
O
T
14 O
T
24 O
T
34 O44 O45 0
O
T
15 O
T
25 O
T
35 O
T
45 O55 O56
DT
p 0 0 0 O
T
56 ÀeI
2
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
5
o0,
ð37Þ
and
R1 þð1ÀhdÞR3ÀX33 Z0, R2 þð1ÀhdÞR3ÀY33 Z0, R2ÀZ33 Z0, ð38Þ
where O56 ¼ DT
ðahR1 þð1ÀaÞhR2 þhR3Þ and Oij,ði,j ¼ 1,2,:::,5;
iojr5Þ are deﬁned in (24).
P.-L. Liu / ISA Transactions 51 (2012) 694–701698
6.
Proof. Replacing A and B in (24) with AþDFðtÞEa and BþDFðtÞEb,
respectively, we apply Lemma 3 [1] for system (1) which is
equivalent to the following condition:
O þGdFðtÞGe þG
T
e FðtÞG
T
d o0, ð39Þ
where
Gd ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2 þhR3ÞD
h i
, and
Ge ¼ Ea Eb 0 0 0
Â Ã
:
By Lemma 3 [1], a sufﬁcient condition guaranteeing (24) for
system (1) is that there exists a positive number e40 such that
O þeÀ1
G
T
dGd þeG
T
e Ge o0: ð40Þ
Applying the Schur complement shows that (40) is equivalent
to (37). This completes the proof.
Theorem 6. In Case II, for given scalars h(h40), a(0oao1), the
uncertain system (1) with (5) is asymptotically stable if there exist
matrices P ¼ PT
40, Qi ¼ QT
i Z0, Ri ¼ RT
i Z0,ði ¼ 1,2Þ, e40, and
positive semi-deﬁnite matrices: X ¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT
23 X33
2
6
4
3
7
5Z0,
Y ¼
Y11 Y12 Y13
YT
12 Y22 Y23
YT
13 YT
23 Y33
2
6
4
3
7
5Z0, Z ¼
Z11 Z12 Z13
ZT
12 Z22 Z23
ZT
13 ZT
23 Z33
2
6
4
3
7
5Z0:
such that
C ¼
C11 þeET
aEa C12 þeET
a Eb C13 C14 C15 PD
CT
12 þeET
bEa C22 þeET
bEb C23 C24 C25 0
CT
13 CT
23 C33 C34 C35 0
CT
14 CT
24 CT
34 C44 C45 0
CT
15 CT
25 CT
35 CT
45 C55 C56
DT
P 0 0 0 CT
56 ÀeI
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
o0
ð41Þ
R1ÀX33 Z0, R2ÀY33 Z0, R1ÀZ33 Z0 ð42Þ
where C56 ¼ DT
½ahR1 þð1ÀaÞhR2 and Cij,ði,j ¼ 1,2,:::,5; iojr5Þ
are deﬁned in (31).
Proof. Replacing A and B in (31) with AþDFðtÞEa and BþDFðtÞEb,
respectively, we apply Lemma 3 [1] for system (1) is equivalent to
the following condition:
CþG
dFðtÞG
e þG
T
e FðtÞG
T
d o0, ð43Þ
where
G
d ¼ PD 0 0 0 ðahR1 þð1ÀaÞhR2ÞD
h i
, and G
e ¼ Ea Eb 0 0 0
Â Ã
:
By Lemma 3 [1], a sufﬁcient condition guaranteeing (31) for
system (1) is that there exists a positive number e40 such that
CþeÀ1G
T
d
G
d þeG
T
e
G
e o0: ð44Þ
Applying the Schur complement of Lemma 2 shows that (44) is
equivalent to (41). This completes the proof.
Remark 1. In the proof of Theorems 1–6, the interval [tÀh,t] is
divided into two subintervals [tÀh,tÀah] and [tÀah,t], the infor-
mation of delayed state x(tÀah) can be taken into account. It is
clear that the Lyapunov function deﬁned in Theorems 1–6 are
more general than the ones in [6,20], etc. Since the delay decom-
position approach is introduced in time delay, it is clear that the
stability results are based on the delay decomposition approach.
When the positions of delay decomposition are varied, the
stability results of proposed criteria are also different. In order
to obtain the optimal delay decomposition sequence, we proposed
an implementation based on optimization methods. The proposed
stability conditions are much less conservative and are more
general than some existing results.
Remark 2. In the previous works except [2,6,7,11,13,14,16,18,19],
the time delay term h(t) was usually estimated as h when
estimating the upper bound of some cross term, this may lead to
increasing conservatism inevitably. In Theorems 1–6, the value of
the upper bound of some cross term is estimated to be more
exactly than the previous methods since h(t) is conﬁned to the
subintervals 0rh(t)rah or ahrh(t)rh. So, such decomposition
method may lead to reduction of conservatism.
Remark 3. In the stability problem, maximum allowable delay
bound (MADB) h which ensures that time-varying delay uncertain
system (1) is asymptotically stable for any h can be determined by
solving the following quasi-convex optimization problem when
the other bound of time-varying delay h is known.
Maximize h
Subjectto ð33Þ ðð37Þ or ð41ÞÞ:
(
ð45Þ
Inequality (45) is a convex optimization problem and can be
obtained efﬁciently using the MATLAB LMI Toolbox.
Five examples will be presented in the following section to
highlight the effectiveness of the proposed method.
3. Illustrative examples
In this section, ﬁve examples are provided to illustrate the
advantages of the proposed stability results.
Example 1. Consider the uncertain system with time varying
delay as follows:
_xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð46Þ
where
A ¼
À1:2 0:1
À0:1 À1
!
, B ¼
À0:6 0:7
À1 À0:8
!
, D ¼ I, Ea ¼ Eb ¼ diag 0:1,0:1:
Now, our problem is to estimate the bound of delay time h to
keep the stability of system.
Solution: when hd ¼ 0:5, using the stability criteria in Parlakci
[12], Qian et al. [16] and Theorem 4 of this paper, the calculated
maximum allowable delay bound (MADB) for the time delay are
h¼1.0097 and h¼1.3618, and h¼2.2241, respectively. So the
proposed method in this paper yields a less conservative result
than that given in Parlakci [12] and Qian et al. [16].
Example 2. Consider the uncertain system with time varying
delay as follows:
_xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð47Þ
where
A ¼
À2 0
0 À1
!
,B ¼
À1 0
À1 À1
!
, D ¼ I, Ea ¼ diag 1:6,0:05f g,
Eb ¼ diag 0:1, 0:3f g:
Solution: for hd ¼ 0:1ða ¼ 0:5Þ, by Theorem 4, we can obtain the
maximum upper bound on the allowable size to be h¼2.2143.
However, applying criteria in [22,15,18], the maximum value of h
for the above system is 0.92, 1.1072 and 1.1075. We also apply
Theorems 4 and 5 to calculate the maximum allowable value for
different hd Table 1 shows the comparison of our results with
those in [4,8,10,16,19,21]. This example demonstrates that our
robust stability condition gives a less conservative result. Hence, it
P.-L. Liu / ISA Transactions 51 (2012) 694–701 699
7.
is obvious that the results obtained from our simple method are
less conservative than those obtained from the existing methods.
Example 3. Consider the uncertain system with time varying
delay as follows:
_xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð48Þ
where A ¼
À0:5 À2
1 À1
!
, B ¼
À0:5 À1
0 0:6
!
, Ea ¼ Eb ¼ diag 0:2,0:2f g,
D ¼ I:
Solution: for comparison, the Table 2 also lists the maximum
allowable delay bound (MADB) h obtained from the criteria
[2,6,7,11,13,14,16,18,19]. It is clear that Theorems 4 and 5 give
much better results than those obtained by [2,6,7,11,13,14,16,
18,19]. It is illustrated that the proposed robust stability criteria
are effective in comparison to earlier and newly published results
existing in the literature.
Example 4. Consider the uncertain system with time varying
delay as follows:
_xðtÞ ¼ ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðtÀhðtÞÞ, ð49Þ
where
A ¼
À0:6 À2:3
0:8 À1:2
!
, B ¼
À0:9 0:6
0:2 0:1
!
, D ¼ diag l,l
È É
,Ea ¼ Eb ¼ I:
Solution: when h is a constant, by the proposed method of
Theorem 6 in this paper and those in [11,13,16,20], for different
values of l,we can get the maximum allowable delay bound
(MADB) h and the results are listed in Table 3, which indicate
that our result is less conservative than those in [11,13,16,20].
When l¼0.3, Table 4 gives the comparisons of the maximum
allowed delay h for various hd It can be seen that the robust
stability condition in this paper is less conservative than the one in
[6,23].
Example 5. Consider the system with time varying delay as
follows:
_xðtÞ ¼ AxðtÞþBxðtÀhðtÞÞ, ð50Þ
where A ¼
0 À0:12þ12r
1 À0:465Àr
" #
, B ¼
À0:1 À0:35
0 0:3
!
:
Solution: we let r¼0.035 as [5] did. From Table 5, we could
easily ﬁnd that the results proposed in this note are better than
those of [5,6,17]. The conclusion we draw is better than [5,6,17]
when hd is small.
4. Conclusions
In this paper, we have proposed new robust stability criteria for
uncertain systems with time-varying delays. By developing a
delay decomposition approach, the information of the delayed
plant states can be taken into full consideration, and new delay-
dependent sufﬁcient robust stability criteria are obtained in terms
of linear matrix inequalities (LMIs), which can be easily solved by
various optimization algorithms. Since the delay terms are con-
cerned more exactly, less conservative results are presented.
Moreover, the restriction on the change rate of time-varying
delays is relaxed in the proposed criteria. The proposed criteria
are computationally attractive, and it provides less conservative
results than the existing results. Numerical examples are given to
illustrate the effectiveness of our theoretical results.
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Han [4] 1.03 0.82 0.61 0.40 0.18
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Zhu and Yang [23] (Corollary 2) 0.6946 0.6913 0.6896
Zhu and Yang [23] (Corollary 3) 1.5633 1.5632 1.5630
Theorem 4 (a¼0.3) 2.2899 2.2898 2.2898
Table 5
Calculation results for Example 5.
hd 0 0.1 0.3 0.9 any hd
[5] (Lemma 1) 0.89 0.87 0.84 0.84 0.84
[6] (Theorem 1) 0.89 0.88 0.87 0.87 0.87
Song and Wang [17] 1.67 1.45 0.89 0.84 0.84
Theorem 1 (a¼0.4) 2.2214 2.1545 1.8459 1.5699 1.3936
Theorem 2 (a¼0.4) 2.2412 2.2014 2.1749 2.1235 1.3936
P.-L. Liu / ISA Transactions 51 (2012) 694–701700
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P.-L. Liu / ISA Transactions 51 (2012) 694–701 701
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