Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 969674, 16 pages
doi:10.1155/2011/969674
Research Article
Novel Stability Criteria of Nonlinear Uncertain
Systems with Time-Varying Delay
Yali Dong,1 Shengwei Mei,2 and Xueli Wang1
1
2
School of Science, Tianjin Polytechnic University, Tianjin 300160, China
Department of Electrical Engineering, Tsinghua University, Beijing 100084, China
Correspondence should be addressed to Yali Dong, dongyl@vip.sina.com
Received 31 January 2011; Revised 8 May 2011; Accepted 9 May 2011
Academic Editor: Irena Rachůnková
Copyright q 2011 Yali Dong 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.
The problem of robust exponential stabilization for dynamical nonlinear systems with uncertainties and time-varying delay is considered in the paper. By constructing the proposed LyapunovKrasovskii functional approach, continuous state feedback controllers are put forward, and the
criteria which guarantee the exponential stabilization of the nonlinear systems with uncertainties
and time-varying delay are established in terms of solutions to the standard Riccati differential
equations. Furthermore, based on the Lyapunov method and the linear matrix inequality
approach, the sufficient conditions of exponential stability for a class of uncertain systems with
time-varying delays and nonlinear perturbations are derived. Finally, two numerical examples are
given to demonstrate the validity of the results.
1. Introduction
The stability problem and stabilization problem of time-delay systems are important problems not yet completely solved and continuously investigated by many people. For control
systems with delayed state, existing stability criteria can be classified into two categories,
that is, delay-independent ones and delay-dependent ones, and delay-independent ones
are usually more conservative than the delay-dependent ones. On the other hand, it is
unavoidable to include uncertain parameters and perturbations in practical control systems
due to modeling errors, measurement errors, approximations, and so on. Therefore, the
robust stabilization problem of uncertain dynamical systems with time-varying delays has
attracted considerable attention of many researchers in recent years, and most of these
papers have always adopted linear matrix equalities to guarantee the exponential stability
of dynamical systems by employing Leibniz-Newton formula or different transformations.
2
Abstract and Applied Analysis
For instance, by employing a descriptor model transformation and a decomposition technique of the delay term matrix, the robust stability of uncertain linear systems with a single
time-varying delay and nonlinear perturbations is investigated in 1. Without introducing
any free weighting matrices, in 2, new delay-dependent stability criteria for time-delay
systems have been derived by introducing a new type of Lyapunov functional which contains
some triple-integral terms and fully uses the information on the lower bound of the delay.
In 3, the exponential stability of linear distributed parameter systems with time-varying
delays is studied through Linear Operator Inequalities which are reduced to standard
Linear Matrix Inequalities. Based on the Lyapunov-Krasovskii functional approach, in 4,
a stability criterion of uncertain linear systems with interval time-varying delay is derived
by introducing some relaxation matrices that can be used to reduce the conservatism of the
criteria. In terms of linear matrix inequality, 5 proposes a new delay-dependent stability
criterion for dynamic systems with time-varying delays and nonlinear perturbations. In 6,
a class of linear systems with unknown norm-bounded uncertainties and time-varying delays
is investigated with Leibniz-Newton formula and linear matrix inequalities, which allow
computing simultaneously the two bounds that characterize the exponential stability rate
of the solution. In 7, the system is transformed into another one by the transformation of
zt eαt xt, and then delay-dependent stability criteria have been derived in terms of
a matrix inequality LMI which can be easily solved using efficient convex optimization
algorithms. In 8, by constructing a suitable augmented Lyapunov’s functional and utilizing
free weight matrices, the criterion for stabilization of uncertain dynamic systems with timevarying delays is established in terms of linear matrix inequalities. Using the method of
the matrix equality, 9 has presented some improved stability criteria to guarantee that the
uncertain systems with two successive delay components are robustly, asymptotically stable.
In 10, 11, the stabilization of uncertain dynamic systems is considered. In 12, the stability
analysis problem of linear neutral delay differential systems with multiple time delays is
investigated. Using the Lyapunov method, some sufficient conditions are presented for the
asymptotic stability of systems. In 13, the problem of robust stabilization of a class of
nonlinear dynamical systems with delayed perturbations is considered. Based on the stability
of the nominal systems, a new stabilizing control law for exponential stability of the system
is designed using Lyapunov stability theory.
Being different from them, this paper chooses the Riccati differential equation to solve
the stabilization of uncertain time-delay systems. The sufficient conditions which guarantee
that the uncertain systems with time-varying delay and nonlinear perturbations are exponentially stabilizable are presented by employing Lyapunov-Krasovskii functional, and the
controllers are constructed.
The rest of this paper is organized as follows. Section 2 presents notations, assumptions, and lemmas which can be used in the proof of theorems. In Section 3, the main results
on the robust exponential stabilization for the uncertain systems with time-varying delay and
nonlinear perturbations are given. Two numerical examples are provided to illustrate the use
of our results in Section 4. Finally, the conclusion follows in Section 5.
2. Preliminaries
For convenience, we now introduce the following notations that will be employed throughout
the paper. The notation Rn is used to denote the n-dimensional space. R denotes the set of all
real nonnegative numbers. Rn×r denotes the space of all n×r matrices. x, y or xT y denotes
Abstract and Applied Analysis
3
the scalar product of two vectors x and y. x denotes the Euclidean vector norm of x. λA
denotes the set of all eigenvalues of A. λmax A is defined by λmax A max{Re λ : λ ∈ λA}.
λmin A is defined by λmin A min{Re λ : λ ∈ λA}. For h ≥ 0, C−h, 0, Rn denotes the
set of all Rn -valued continuous functions mapping −h, 0 into Rn . A matrix A is semipositive
definite A ≥ 0 if Ax, x ≥ 0, for all x ∈ Rn . A is positive definite A > 0 if Ax, x > 0 for
all x /
0. A ≥ B means A − B ≥ 0. Also, xt is defined by xt {xt s, s ∈ −h, 0}, and xt is
defined by xt sups∈−h,0 xt s.
Now, let us consider a class of uncertain systems with time-varying delay and
nonlinear perturbations of the form
ẋt At ΔAυ, txt A1 t ΔA1 ξ, txt − ht
Bt ΔBς, tut ft, xt, xt − ht, ut,
xt φt,
2.1
t ∈ −h, 0, h ≥ 0,
where xt ∈ Rn is the state, ut ∈ Rm is the control function, At, A1 t, Bt are continuous
matrixes of appropriate dimensions on R , and ΔAυ, t, ΔA1 ξ, t, ΔBς, t represent the
system uncertainties and are assumed to be continuous in all their arguments. Moreover,
υ, ξ, ς ∈ ψ is the uncertain vector, and ψ ⊂ RL is a compact set. In addition, perturbation
ft, xt, xt − ht, ut : R × Rn × Rn × Rm → Rn is a given nonlinear function satisfying
ft, 0, 0, 0 0 for all t ≥ 0, and there exist scalars a, b, d > 0 such that
ft, xt, xt − ht, ut ≤ axt bxt − ht dut,
2.2
for all t, xt, xt − ht, ut ∈ R × Rn × Rn × Rm . The initial function φt ∈ C−h, 0, Rn ,
h > 0, has its norm φ sups∈−h,0 φs. The delay ht is a continuous function satisfying
H1 0 ≤ ht ≤ h, ḣt ≤ δ < 1, for all t ≥ 0.
The purpose of this paper is to design a state feedback controller ut Ktxt such
that the closed-loop system of 2.1
ẋt At ΔAυ, t BtKt ΔBς, tKtxt
A1 t ΔA1 ξ, txt − ht ft, xt, xt − ht, ut,
xt φt,
2.3
t ∈ −h, 0, h ≥ 0,
is robustly α-exponentially stable, that is, every solution xt, φ of the system satisfies
∃N > 0, α > 0,
x t, φ ≤ N φe−αt ,
∀t ∈ R ,
2.4
for all the uncertainties ΔAυ, t, ΔA1 ξ, t, ΔBς, t.
Before proposing our theorems, we introduce for 2.1 the following standard assumptions and lemmas that will be needed for deriving the main results.
4
Abstract and Applied Analysis
Assumption 1. For all x, t ∈ Rn × R there exist continuous matrix functions Hυ, t, H1 ξ, t,
Eς, t of appropriate dimensions such that
ΔAυ, t BtHυ, t,
ΔA1 ξ, t BtH1 ξ, t,
ΔBς, t BtEς, t.
2.5
Remark 2.1. Assumption 1 defines the matching condition about the uncertainties and is a
rather standard assumption for robust control problem see, e.g., 10, 14–16.
Lemma 2.2 see 15. For any real vectors a, b and any matrix Q > 0 with appropriate dimensions,
it follows that
2aT b ≤ aT Qa bT Q−1 b.
2.6
Lemma 2.3 see 11. Given constant symmetric matrices S1 , S2 , S3 , and S1 ST1 < 0, S3 ST3 >
T
0, then S1 S2 S−1
3 S2 < 0 if and only if
S1 S2
ST2 −S3
2.7
< 0.
3. Main Results
In this section, we will present our main results on the robust exponential stabilization of
system 2.1.
Given positive numbers α, β, εi , i 1, 2, 3, 4, we set
ω ε1 ε3 ε4 ε2 h,
Pβ t P t βI,
ρξ t maxH1 ξ, t,
ξ
μt min
ς
At At αI,
ρυ t maxHυ, t,
υ
1
λmin Eς, t ET ς, t ,
2
p supP t,
t∈R
4ρξ2 t
4
1
T
−1 2
Rt − −2αh
A1 tA1 t − ε3 ρυ t − 1 BtBT t
2 ε1 e
ε1 e−2αh 1 − δ
1 − δ
4 1 μt
−
ε4−1 a2
2b2
2
d I.
ε1 e−2αh 1 − δ
3.1
We need the following assumption.
Assumption 2. For any t > 0, μt > −1.
Abstract and Applied Analysis
5
Theorem 3.1. Suppose that condition (H1) and Assumptions 1-2 hold. If there exist positive numbers
α, β, εi , i 1, 2, 3, 4, and a symmetric positive semidefinite matrix function P t satisfying the
following Riccati differential equation:
T
Ṗ t A tPβ t Pβ tAt − Pβ tRtPβ t ωI 0,
3.2
then system 2.1 is robustly α-exponentially stabilizable with feedback control
1
BT tPβ txt.
ut − 2 1 μt
3.3
Moreover, the solution xt, φ satisfies the condition
x t, φ ≤
p β hε1 1/2h2 ε2 φe−αt ,
β
t ≥ 0.
3.4
Proof. Let ut Ktxt, where
1
Kt − BT tPβ t,
2 1 μt
t ≥ 0.
3.5
For the closed-loop system 2.3 of system 2.1, we consider the following Lyapunov-Krasovskii functional:
V t, xt V1 V2 V3 ,
3.6
where
V1 P txt, xt βxt, xt,
t
V2 ε1
e2αs−t xs2 ds,
t−ht
V3 ε2
0 t
−h
3.7
e2αs−t xs2 ds dτ.
tτ−htτ
The time derivative of V along the trajectory of 2.3 is given by
V̇ t, xt V̇1 V̇2 V̇3 ,
3.8
6
Abstract and Applied Analysis
where
V̇1 Ṗ txt, xt 2 Pβ tẋt, xt
Ṗ t AT tPβ t Pβ tAt xt, xt 2 Pβ tBtHυ, txt, xt
2 Pβ tA1 txt − ht, xt 2 Pβ tBtH1 ξ, txt − ht, xt
2 Pβ tBtKtxt, xt 2 Pβ tBtEς, tKtxt, xt
2 Pβ tft, xt, xt − ht, ut, xt ,
V̇2 −2αV2 ε1 xt2 − ε1 e−2αht xt − ht2 1 − ḣt
3.9
≤ −2αV2 ε1 xt2 − ε1 e−2αh xt − ht2 1 − δ,
V̇3 −2αV3 ε2
0 xt2 − e−2ατ−htτ xt τ − ht τ2 1 − ḣt τ dτ
−h
2
≤ −2αV3 ε2 hxt − ε2
0
−h
e−2ατ−htτ xt τ − ht τ2 1 − δdτ
≤ −2αV3 ε2 hxt2 .
Applying Lemma 2.2 gives
2 Pβ tA1 txt − ht, xt
ε 1 − δe−2αh
4
1
Pβ tA1 tAT1 tPβ txt, xt xt − ht, xt − ht,
−2αh
4
ε1 1 − δe
2 Pβ tBtH1 ξ, txt − ht, xt
≤
≤
ε1 1 − δe−2αh
xt − ht, xt − ht
4
4
T
T
P
tH
tB
xt
tBtH
ξ,
ξ,
tP
txt,
β
1
β
1
ε1 1 − δe−2αh
4ρξ2 t
ε1 1 − δe−2αh
T
,
≤
P
xt
tBtB
tP
txt,
xt − ht, xt − ht β
β
4
ε1 1 − δe−2αh
2 Pβ tBtHυ, txt, xt
≤ ε3 xt, xt ε3−1 Pβ tBtHυ, tH T υ, tBT tPβ txt, xt
≤ ε3 xt, xt ε3−1 ρυ2 t Pβ tBtBT tPβ txt, xt .
3.10
Abstract and Applied Analysis
7
Using 2.2 and 3.3, we get
2 Pβ tft, xt, xt − ht, ut, xt
≤ 2ft, xt, xt − ht, utPβ txt
≤ 2axtPβ txt 2bxt − htPβ txt 2dutPβ txt
2
≤ ε4−1 a2 Pβ txt ε4 xt2 2b2
Pβ txt2
−2αh
ε1 e
1 − δ
2
ε1 e−2αh 1 − δ
xt − ht2 d2 Pβ txt ut2
2
2
2b2
−1 2
2
ε4 xt, xt
P
ε4 a xt
d
txt,
β
ε1 e−2αh 1 − δ
ε e−2αh 1 − δ
1
1
T
xt − ht2 .
2 Pβ tBtB tPβ txt, xt 2
4 1 μt
3.11
In addition, it is easy to obtain the following:
2 Pβ tBtKtxt, xt 2 Pβ tBtEς, tKtxt, xt
1
1
T
T
x tPβ tBt I Eς, t E ς, t BT tPβ txt
−
1 μt
2
2
1
1
T
≤−
λmin I Eς, t E ς, t BT tPβ txt
1 μt
2
− Pβ tBtBT tPβ txt, xt .
3.12
The last equality is got because of μt minς 1/2λmin Eς, t ET ς, t.
Therefore, we get
V̇ t, xt 2αV t, xt ≤
T
Ṗ t A tPβ t Pβ tAt 1
4ρξ2 t
4
Pβ tA1 tAT1 tPβ t
ε1 1 − δe−2αh
ε3−1 ρυ2 t
− 1 Pβ tBtBT tPβ t
2 ε1 1 − δe−2αh
4 1 μt
2b2
2
−1 2
2
ε4 a d Pβ t ε1 ε3 ε4 ε2 hI xt, xt .
ε1 1 − δe−2αh
3.13
8
Abstract and Applied Analysis
Then introducing 3.2 into 3.13 we can get
V̇ t, xt 2αV t, xt ≤ 0,
∀t ≥ 0.
3.14
It is obvious that
e2αt V̇ t, xt 2αe2αt V t, xt ≤ 0.
3.15
Integrating both sides of 3.15 from 0 to t, we get
V t, xt ≤ V 0, x0 e−2αt ,
∀t ≥ 0.
3.16
On the other hand, from the expression of V t, xt , it is easy to see that
βxt2 ≤ V t, xt ,
∀t ≥ 0.
3.17
In addition, since
2
V1 0, x0 ≤ p β φ ,
V3 0, x0 ε2
V2 0, x0 ε1
0 0
−h
0
−h0
e
2αs
e2αs xs2 ds dτ ≤ ε2
τ−hτ
2
xs ds ≤ ε1
0
−h
2
φ ds ≤ ε1 hφ2 ,
0 0
−h
2
φ ds dτ ≤ 1 ε2 h2 φ2 ,
2
τ−h
3.18
we get V 0, x0 ≤ p β hε1 1/2h2 ε2 φ2 .
Hence we have
x t, φ ≤
p β hε1 1/2h2 ε2 φe−αt ,
β
t ≥ 0.
3.19
So the closed-loop system 2.3 is exponentially stable. This completes the proof.
Remark 3.2. Note that from the proof of Theorem 3.1, condition 3.2 can be relaxed via the
following matrix inequality:
T
Ṗ t A tPβ t Pβ tAt − Pβ tRtPβ t ωI ≤ 0.
3.20
In addition, we consider a class of uncertain systems with time-varying delays and
simple nonlinear perturbations as follows:
ẋt A ΔAυ, txt A1 ΔA1 ξ, txt − ht
B ΔBς, tu ft, xt, xt − ht,
xt φt,
t ∈ −h, 0, h ≥ 0,
3.21
Abstract and Applied Analysis
9
where A, A1 , B are constant matrices of appropriate dimensions and ΔAυ, t, ΔA1 ξ, t,
ΔBς, t are unknown time-varying uncertain matrices, and there exist scalars a, b, d, l1 , l2 , l3 >
0 such that
ft, xt, xt − ht ≤ axt bxt − ht,
ΔAυ, tΔAT υ, t ≤ l1 I,
∀t, xt ∈ R × Rn ,
ΔA1 ξ, tΔAT1 ξ, t ≤ l2 I,
ΔB ≤ l3 .
3.22
Given positive numbers α, β, εi , i 1, 2, 3, 4, we set
A A αI,
P β P βI,
ω ε1 ε3 ε4 ε2 h,
4
A1 AT1 BBT
ε1 e−2αh 1 − δ
4l2
2b2
T
−1
−1 2
− ε3 l1 ε4 a l3 B I.
ε1 1 − δe−2αh ε1 1 − δe−2αh
R−
3.23
We have the following theorem.
Theorem 3.3. Suppose that condition (H1) holds. If there exist positive numbers α, β, εi , i 1, 2, 3, 4,
and a symmetric positive define matrix X such that the following LMIs hold:
⎛ T ⎝X A AX − R
X
−X
X
X −β−1 I
⎞
X ⎠
≤ 0,
−ω−1 I
3.24
3.25
< 0,
then system 3.21 is robustly exponentially stabilizable with feedback control
1
ut − BT X −1 xt.
2
3.26
Moreover, the solution xt, φ satisfies the condition
x t, φ ≤
λmax P β hε1 1/2h2 ε2 φe−αt ,
β
t ≥ 0,
3.27
where P X −1 − βI and λmax P represents the maximum eigenvalue of P .
Proof. From 3.25, we get that P is a symmetric positive define matrix.
For system 3.21, select a Lyapunov-Krasovskii functional as
V t, xt V1 V2 V3 ,
3.28
10
Abstract and Applied Analysis
where
V1 P xt, xt βxt, xt,
V2 ε1
t
e2αs−t xs2 ds,
t−ht
V3 ε2
0 t
−h
3.29
e2αs−t xs2 ds dτ.
tτ−htτ
The time derivative of V along trajectory of 3.21 is given by
V̇ t, xt V̇1 V̇2 V̇3 ,
3.30
where
V̇1 2 P β ẋt, xt
AT P β P β A xt, xt 2 P β ΔAυ, txt, xt 2 P β A1 xt − ht, xt
2 P β ΔA1 ξ, txt − ht, xt 2 P β ft, xt, xt − ht, xt
2 P β BKxt, xt 2 P β ΔBKxt, xt ,
V̇2 −2αV2 ε1 xt2 − ε1 e−2αht xt − ht2 1 − ḣt
≤ −2αV2 ε1 xt2 − ε1 e−2αh xt − ht2 1 − δ,
0 V̇3 −2αV3 ε2
xt2 − e−2ατ−htτ xt τ − ht τ2 1 − ḣt τ dτ
−h
2
≤ −2αV3 ε2 hxt − ε2
0
−h
e−2ατ−htτ xt τ − ht τ2 1 − δdτ
≤ −2αV3 ε2 hxt2 ,
P β P βI.
3.31
Abstract and Applied Analysis
11
Applying Lemma 2.2 and inequality 3.22 gives
2 P β ΔAυ, txt, xt
≤ ε3−1 P β ΔAυ, tΔAT υ, tP β xt, xt ε3 xt, xt
2
≤ ε3−1 l1 P β xt, xt ε3 xt, xt,
2 P β A1 xt − ht, xt
≤
4
T
P
A
A
P
xt,
xt
β
1
β
1
ε1 1 − δe−2αh
ε1 1 − δe−2αh
xt − ht, xt − ht,
4
2 P β ΔA1 ξ, txt − ht, xt
≤
4
T
P
ΔA
xt,
xt
tΔA
tP
ξ,
ξ,
β
1
β
1
ε1 1 − δe−2αh
ε1 1 − δe−2αh
xt − ht, xt − ht
4
ε 1 − δe−2αh
2
4l2
1
P
xt,
xt
≤
xt − ht, xt − ht,
β
4
ε1 1 − δe−2αh
2 P β ft, xt, xt − ht, xt
≤ 2ft, xt, xt − htP β txt
≤ 2axtP β xt 2bxt − htP β xt
2
≤ ε4−1 a2 P β xt ε4 xt2 2 ε e−2αh 1 − δ
2b2
1
xt
xt − ht2
P
β
2
ε1 e−2αh 1 − δ
2
2b2
ε1 e−2αh 1 − δ
−1 2
ε
ε4 a P
xt,
xt
xt
xt,
xt − ht2 .
4
β
2
ε1 e−2αh 1 − δ
3.32
Noticing
2 P β BKxt, xt − P β BBT P β xt, xt ,
2
2 P β ΔBKxt, xt P β ΔBBT P β xt, xt ≤ l3 BT P β xt ,
3.33
12
Abstract and Applied Analysis
we get
V̇ t, xt 2αV t, xt ≤
T
4
A1 AT1 − BBT P β ε1 ε3 ε4 ε2 hI
ε1 1 − δe−2αh
2
4l2
2b2
T
−1
−1 2
ε3 l1 ε4 a l3 B P β xt, xt .
ε1 1 − δe−2αh ε1 1 − δe−2αh
3.34
A P β P βA P β
Hence we obtain
T
V̇ t, xt 2αV t, xt ≤ xT t A P β P β A − P β R P β ωI xt.
3.35
Noticing P β X −1 , and from 3.24 and using Lemma 2.3, we obtain
−1 T
−1
−1
−1
P β A AP β − R ωP β P β ≤ 0.
3.36
By pre-multiplying and post-multiplying the right-hand side of 3.36 with P β it follows that
T
A P β P β A − P β R P β ωI ≤ 0.
3.37
So we have
V̇ t, xt 2αV t, xt ≤ 0,
∀t ≥ 0.
3.38
It is obvious that
e2αt V̇ t, xt 2αe2αt V t, xt ≤ 0.
3.39
Integrating both sides of 3.15 from 0 to t, we get
V t, xt ≤ V 0, x0 e−2αt ,
∀t ≥ 0.
3.40
On the other hand, from the expression of V t, xt , it is easy to see that
βxt2 ≤ V t, xt ,
∀t ≥ 0.
3.41
Abstract and Applied Analysis
13
In addition, since
2
V1 0, x0 ≤ λmax P β φ ,
2
V2 0, x0 ≤ ε1 hφ ,
V3 0, x0 ≤
2
1
ε2 h2 φ ,
2
3.42
we get V 0, x0 ≤ λmax P β hε1 1/2h2 ε2 φ2 .
Hence we have
x t, φ ≤
λmax P β hε1 1/2h2 ε2 φe−αt ,
β
t ≥ 0.
3.43
So the system 3.21 is robustly exponentially stabilizable. This completes the proof of
Theorem 3.3.
4. Illustrative Examples
In this section, we will give numerical examples to demonstrate the effective of the proposed
methods.
Example 4.1. Consider system 2.1 with ht cos2 0.25t and
⎞
3
cos
tx
sin
tx
−
ht
t
t
1
2
⎟
⎜
4
ft, xt, xt − ht, ut ⎝ 2
⎠,
1
3
sin tx2 t cos tut
2
4
⎛1
⎛
⎞
17
4 − sin t
− 12e −
− 4 cos t −
− 2 cos t − 8e − 4
−6e
−4e
⎜
⎟
2 cos t 4
2
At ⎝
17 ⎠,
4 − sin t
− 12e −
−4e − 2 cos t − 8e − 4
−6e − 4 cos t −
2 cos t 4
2
−0.5
0
1
1 e
,
Bt ,
A1 t −0.5
2
0 e
1
ΔAυ, t υt 0
υt 0
,
ΔA1 ξ, t 0 2ξt
,
0 2ξt
ΔBς, t ςt
ςt
,
4.1
where υt η1 , ξt η2 , ςt η3 with |η1 | ≤ 2, |η2 | ≤ 0.5, |η3 | ≤ 0.5. It is easy to obtain that
ft, xt, xt − ht, ut ≤ xt xt − ht ut,
4.2
that is, a b d 1. We can get h 1, δ 0.5. ρυ t 2, ρξ t 1, μt −0.5, ω 4. For
the positive numbers ε1 ε2 ε3 ε4 1, β 1, α 0.5, we can verify that all the conditions
14
Abstract and Applied Analysis
of Theorem 3.1 are satisfied, and the solution of the Riccati differential equation 3.2 is given
by
P t cos t 1
0
0
4.3
.
cos t 1
Therefore, the system is robustly α-exponentially stabilizable with feedback control
ut −cos t 1x1 t − cos t 1x2 t.
4.4
Noting p supt∈R P t 2, the solution of the system satisfies
x t, φ ≤ 2.1213φe−0.5t ,
t ∈ R .
4.5
Remark 4.2. The systems in the examples that are dealt by 17 do not contain uncertainty
in the linear part on state and control, and those systems are robustly stabilizable. But, our
Example 4.1 contains uncertainty in the linear part on state and control, and the system in
Example 4.1 is robustly α-exponentially stabilizable. Using the similar approach in 17, our
results can be extended to the systems with multiple delays.
Example 4.3. Consider system 3.21 with ht 0.5 and
A
−25.5
0
1
−25.5
l1 1,
1
A1 2
,
e−0.5
0
0
e−0.5
√
l3 2,
l2 1,
1
B
,
1
,
a 1,
4.6
b 1.
Taking ε1 ε2 ε3 ε4 1, β 1, α 0.5, we can get
h 1,
δ 0.5,
A A αI −25
0
−25
1
,
R −2I BBT − 1 1 8e 4e 2I −6 12eI ω 4,
1 1
1 1
4.7
.
We can verify that all the conditions of Theorem 3.3 and LMI 3.24 and 3.25 are satisfied with
P
0.11111
0
0
0.11111
.
4.8
By Theorem 3.3, the system is exponentially stabilizable with feedback control
ut −0.555556x1t − 0.555556x2t,
4.9
Abstract and Applied Analysis
15
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
0.5
1
Time (s)
1.5
2
x1
x2
Figure 1: The state x1 and x2 of the closed-loop system in Example 4.3.
and the solution of the system satisfies
x t, φ ≤ 1.61589φe−0.5t ,
t ≥ 0.
4.10
For
⎞
3
cos
tx
sin
tx
−
0.5
t
t
1
2
⎟
⎜
4
ft, xt, xt − ht ⎝ 2
⎠,
1
sin tx2 t
2
0.5
ΔAυ, t 0.5I,
ΔA1 ξ, t 0.5I,
ΔBς, t ,
0.5
⎛1
4.11
and the initial condition
#
$T
φt −0.5 0.5 ,
∀ − 0.5 ≤ t < 0,
4.12
the simulation result is shown in Figure 1. It is seen from Figure 1 that the closed-loop system
is exponentially stable.
5. Conclusions
The problem of robust stabilization for a class of dynamical nonlinear systems with uncertainties and time-varying delays has been considered. On condition that the derivative of timevarying delays has restriction, a novel stability criterion which can guarantee the exponential
16
Abstract and Applied Analysis
stabilization of the uncertain systems with time-varying delay and nonlinear perturbations
has been established by using the Riccati differential equation. The continuous state feedback
controller has been proposed. Furthermore, based on the Lyapunov method, a linear matrix
inequality approach to robust exponentially stabilization for a class of uncertain systems
with time-varying delays and nonlinear perturbations via linear memoryless state feedback
has been proposed. Finally, two illustrative examples have been given to demonstrate the
utilization of the results.
Acknowledgments
This work was supported by the National Nature Science Foundation of China under Grant
no. 50977047 and the Natural Science Foundation of Tianjin under Grant no. 11JCYBJC06800.
References
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