bbb

Exponential Delay Dependent
Stabilization for Time-Varying Delay
Systems With Saturating
Actuator
Pin-Lin Liu
Associate Professor
Department of Electrical Engineering,
Chienkuo Technology University,
Changhua,
500 Taiwan, R.O.C.
e-mail: [email protected]
This paper deals with the stabilization criteria for a class of timevarying delay systems with saturating actuator. Based on the
Lyapunov–Krasovskii functional combining with linear matrix inequality techniques and Leibniz–Newton formula, delaydependent stabilization criteria are derived using a state feedback
controller. We also consider efficient convex optimization algorithms to the time-varying delay system with saturating actuator
case: the maximal bound on the time delay such that the prescribed level of operation range and imposed exponential stability
requirements are still preserved. The value of the time-delay as
well as its rate of change are taken into account in the design
method presented and further permit us to reduce the conservativeness of the approach. The results have been illustrated by
given numerical examples. These results are shown to be less
conservative than those reported in the literature.
关DOI: 10.1115/1.4002713兴
Keywords: Leibniz–Newton formula, linear matrix inequality,
time delay, delay-dependence
1
Introduction
Both time-delay and saturating controls are commonly encountered in various engineering systems and are frequently a source
of instability. Stability analysis and synthesis of time-delay systems are important issues addressed by many authors, and some
mature methods have been widely used to deal with these problems 关1–22兴. Many methods to check the stability of time-delay
systems 关1,7,8,13,16,22兴 or linear systems with saturating controls
have been proposed 关2–6,9–12,15,17–21兴.Nonlinear systems with
time-delay constitute basic mathematical models of real phenomena, for instance, in circuits theory, economics, and mechanics.
Not only dynamical systems with time-delay are common in
chemical processes and long transmission lines in pneumatic, hydraulic, or rolling mill systems, but computer controlled systems
requiring numerical computation also have time delays in control
loops. The presence of time delays in control loops usually degrades system performance and complicates the analysis and design of feedback controllers. Actuator saturation and time delays
are often observed together in control systems. To deal with both
problems effectively, appropriate design methods are required. Up
to now, only a few methods were reported to deal with these
problems simultaneously. Cao et al. 关2兴 considered the design of
the antiwindup gain for further enlargement of the closed-loop
stability region. Unlike in the design of feedback gain by Hu et al.
关6兴, the design of antiwindup gain by Cao et al. 关2兴 cannot be
formulated into a linear matrix inequality 共LMI兲 optimization
problem. Chen and Wang 关3兴 studied the stabilization problem of
saturating a time-delay system with state feedback and sampledstate feedback and they derived several sufficient conditions to
ensure the system stability in terms of norm inequalities. Chou et
al. 关4兴 exploited a sufficient condition to stabilize a linear uncertain time delay system containing input saturation. The problem of
robust stabilization of uncertain time delay systems containing a
saturating actuator was addressed by Niculescu et al. 关14兴 by a
high gain approach. Oucheriah 关15兴 considered a method to synthesize a globally stabilizing state feedback controller by means of
an asymptotic observer for time-delay systems. In Ref. 关18兴, a
dynamic antiwindup method was presented for the systems with
input delay and saturation. In Ref. 关13兴, a LMI-based approach is
proposed to analyze the stability and domain of attraction for systems with exponential stability.
Recently, increasing attention has been paid to the study of the
stability of systems with both time-delay and saturating actuator
because of its practical usefulness 关2–6,9–12,15,17–20兴. The stability problem of time-delay systems with a saturating actuator
has been proposed that are based on the matrix norm or matrix
measure 关3–5,9–12,14,17,18,21兴. Unfortunately, matrix norm and
matrix measure operations usually render the criteria more conservative. Therefore, recently, a new stability criterion based on
the LMI techniques was proposed 关2,19,20兴. However, the stability analyses of the operators are still based on matrix norm manipulations, which may lead to conservative results. Considering a
Lyapunov–Krasovskii functional, a synthesis technique based
upon LMIs was used to determine simultaneously a robust stabilizing state feedback and a set of admissible conditions from
which the resulting trajectories are asymptotically stable when the
saturation effectively occurs. This work concerns both the design
of stabilizing controllers and the determination of the associated
domains of safe initial conditions for linear systems with state
delay and saturating controls. The method used is based on the
Lyapunov–Krasovskii approach 关2,19兴. The synthesis of both a
suitable gain matrix and an associated set of initial conditions is
carried out by LMIs 关1兴. A convex optimization problem is then
proposed in order to maximize the size of the set of admissible
initial conditions 关13兴. However, in the control of a time-varying
delay system with saturating actuator, it is usually desirable to
design a controller, which not only robustly stabilizes the system
but also estimates the bound of delay time h to keep the stabililization of the system. Furthermore, the results are somewhat
conservative, especially in situations where delays are small; there
is room for investigation.
This paper deals with the robust stabilization criteria for a class
of time-varying delay systems with a saturating actuator. Based on
the Lyapunov–Krasovskii functional combining with LMI techniques and the Leibniz–Newton formula, delay-dependent stabilization criteria are derived using a state feedback controller. We
also consider an optimization particular to the time-varying delay
system with a saturating actuator case: the maximal bound on the
time delay such that the prescribed level of operation range and
imposed exponential stability requirements are still preserved. The
designed controller is dependent on the time delay and its rate of
change. From the illustrated examples, if the delay time lengthens,
the decay rate becomes conservative. We claim that the sharpness
of the upper bound delay time h varies with the chosen decay rate
␤. The results have been illustrated by the given numerical examples. These results are shown to be less conservative than those
reported in the literature.
2
Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received
April 13, 2009; final manuscript received February 23, 2010; published online
November 23, 2010. Assoc. Editor: Guoming George Zhu.
Main Result
Consider the following time-varying delay systems with a saturating actuator described by
ẋ共t兲 = A0x共t兲 + A1x共t − h共t兲兲 + B sat共u共t兲兲
Journal of Dynamic Systems, Measurement, and Control
Copyright © 2011 by ASME
共1a兲
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x共t兲 = ␾共t兲,
∀ t 苸 关− h,0兴
共1b兲
−
where x共t兲 苸 Rn is the state vector, u共t兲 苸 Rm is the control input
vector, and xt is the state at time t denoted by xt共s兲 ª x共t + s兲. A0,
A1, and B are known constant matrices with appropriate dimensions. ␾共t兲 is a smooth vector-valued initial function.
Time-delay h共t兲 is a time-varying continuous function that satisfies
0 ⱕ h共t兲 ⱕ h,
ḣ共t兲 ⱕ hd ⬍ 1
冕
t
ẋT共s兲X33ẋ共s兲ds ⱕ
t−h共t兲
The operation of sat共ui共t兲兲 is linear for −Ui ⱕ ui ⱕ Ui as
sat共ui共t兲兲 =
冦
− Ui
if ui ⬍ − Ui ⬍ 0
ui
if − Ui ⱕ ui ⱕ Ui
Ui
if ui ⬎ Ui ⬎ 0
冧
冤
LEMMA 2
sat共Kx共t兲兲 = D共␣共x兲兲Kx共t兲,
D共␣共x兲兲 苸 Rmxn
␣i共x兲 =
and therefore,
冦
Ui
if 共Kx兲i ⬍ − Ui ⬍ 0
共Kx兲i
1
if − Ui ⱕ 共Kx兲i ⱕ Ui
Ui
共Kx兲i
if 共Kx兲i ⬎ Ui ⬎ 0
冧
XT13 XT23
0
冥冤
x共t兲
冥
x共t − h共t兲兲 ds
ẋ共s兲
共10兲
冋
Q共x兲 S共x兲
ST共x兲 R共x兲
册
⬍0
共11兲
where Q共x兲 = QT共x兲 , R共x兲 = RT共x兲 , and S共x兲 depend on affine on x
is equivalent to
R共x兲 ⬍ 0
共12a兲
Q共x兲 ⬍ 0
共12b兲
Q共x兲 − S共x兲R−1共x兲ST共x兲 ⬍ 0
共12c兲
The nominal unforced time-varying delay saturating actuator
system 共Eq. 共1兲兲 can be written as
ẋ共t兲 = A0x共t兲 + A1x共t − h共t兲兲
共13兲
Now, we describe our method for determining the stabilization
of time-varying delay systems 共Eq. 共13兲兲 in the following theorem.
THEOREM 1. Given the scalars h ⬎ 0 and hd ⬎ 0 , the nominal
unforced time-varying delay system (Eq. (13)) is asymptotically
stable if there exist symmetric positive-definite matrices P ⬎ 0 ,
Q ⬎ 0 , R ⬎ 0 , and X ⱖ 0 and a semipositive-definite matrix
冤
共5兲
X11 X12 X13
冥
X = XT12 X22 X23 ⱖ 0
where D共␣共x兲兲 is a diagonal matrix for which the diagonal elements ␣i共x兲 satisfy for i = 1 , 2 , . . . , m,
−
X11 X12 X13
关1兴. The following matrix inequality
共4兲
Throughout this paper, we will use the following concept of
stabilization for the time delay system with a saturating actuator
共Eq. 共1兲兲.
DEFINITION 1. The time-varying delay system with a saturating
actuator (Eq. 共1兲) is said to be stable in closed-loop via memoryless state feedback control law if there exists a control law u共t兲
= Kx共t兲 , K 苸 Rmxn such that the trivial solution x共t兲 ⬵ 0 of the
functional differential equation associated to the closed-loop system is uniformly asymptotically stable.
In order to develop our result by considering a state feedback
control law u共t兲 = Kx共t兲, the saturating term sat共Kx共t兲兲 can be written in an equivalent form
关xT共t兲 xT共t − h共t兲兲 ẋT共s兲 兴
t−h共t兲
where h and hd are constants.
The saturating function is defined as follows:
共3兲
t
⫻ XT12 X22 X23
共2兲
sat共u共t兲兲 = 关sat共u1共t兲兲,sat共u2共t兲兲, . . . ,sat共um共t兲兲兴T
冕
XT13 XT23 X33
such that the following LMIs hold:
冤
⌽11 ⌽12 ⌽13
冥
⌽ = ⌽T12 ⌽22 ⌽23 ⬍ 0
共6兲
⌽T13
⌽33
⌽T23
R − X33 ⱖ 0
共14a兲
共14b兲
where
0 ⱕ ␣i共x兲 ⱕ 1
共7兲
⌽11 = AT0 P + PA0 + Q + 共1 − hd兲共X13 + XT13 + hX11兲
From Eqs. 共1a兲, 共1b兲, and 共3兲–共7兲, we can rewrite the timevarying delay system with saturating actuator as follows:
ẋ共t兲 = AFx共t兲 + A1x共t − h共t兲兲
共8兲
where AF = A0 + BD共␣共x兲兲K.
For the above system 共Eq. 共8兲兲, the main objective is to find the
range of h and guarantee the stabilization for the time-varying
delay system with a saturating actuator 共Eq. 共8兲兲. When the time
delay is unknown, how long can time delay be tolerated to keep
the system stable? To do this, two fundamental lemmas are reviewed.
LEMMA 1 关13兴. If there exist symmetric positive-definite matrix
X33 ⬎ 0 and arbitrary matrices X11 , X12 , X13 , X22 , and X23 such
that
冤
X11 X12 X13
冥
X = XT12 X22 X23 ⱖ 0
XT13 XT23 X33
then we obtain
014502-2 / Vol. 133, JANUARY 2011
⌽12 = PA1 + 共1 − hd兲共− X13 + XT23 + hX12兲
⌽13 = hAT0 R
⌽23 = hAT1 R
⌽22 = 共1 − hd兲共− Q − X23 − XT23 + hX22兲
⌽33 = − hR
Proof. Select the following Lyapunov–Krasovskii functional to
be
V共xt兲 = xT共t兲Px共t兲 +
共9兲
冕 冕
0
+
−h共t兲
冕
t
xT共s兲Qx共s兲ds
t−h共t兲
t
ẋT共s兲Rẋ共s兲dsd␪
共15兲
t+␪
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Calculating the derivative of Eq. 共15兲 with respect to t along the
trajectory of the nominal unforced time-varying delay system 共Eq.
共13兲兲 yields
V̇共xt兲 = x
T
−
共t兲共AT0 P
h共t兲兲AT1 Px共t兲
T
+ xT共t兲Qx共t兲 − xT共t − h共t兲兲共1 − ḣ共t兲兲Qx共t
冕
ẋT共s兲Rẋ共s兲ds = xT共t兲
⫻共AT0 P + PA0 + Q兲x共t兲 + xT共t兲PA1x共t − h共t兲兲 + xT共t
t
ẋ 共s兲X33ẋ共s兲ds ⱕ 共1 − hd兲
T
t−h共t兲
− h d兲
t
t−h
冕
+ ẋT共t兲hRẋ共t兲 + 共1 − hd兲
+ PA0兲x共t兲 + x 共t兲PA1x共t − h共t兲兲 + x 共t
T
− h共t兲兲 + ẋT共t兲hRẋ共t兲 − 共1 − ḣ共t兲兲
− 共1 − hd兲
− h共t兲兲AT1 Px共t兲 − xT共t − h共t兲兲共1 − hd兲Qx共t − h共t兲兲
冕
再
T
T
− h共t兲兲hX22x共t − h共t兲兲兲 + xT共t − h共t兲兲X23
冎
t
ẋT共s兲X33ẋ共s兲ds
共16兲
Using
the
Leibniz–Newton
formula
t
ẋ共s兲ds and Lemma 1, we obtain
= 兰t−h共t兲
冤
X11 X12 X13
⫻ xT共t兲hX11x共t兲 + xT共t兲hX12x共t − h共t兲兲 + xT共t兲X13
− h共t兲兲
ẋT共s兲共X33 − R兲ẋ共s兲ds − 共1
t−h共t兲
关x 共t兲 x 共t − h共t兲兲 ẋ 共s兲 兴 XT12 X22 X23
XT13 XT23 0
T
t
t−h共t兲
t
t−h共t兲
冕
冕
冕
冕
冥冤
x共t兲
冥
x共t − h共t兲兲 ds ⱕ 共1 − hd兲
ẋ共s兲
t
ẋ共s兲ds + xT共t − h共t兲兲hXT12x共t兲 + xT共共t
t−h共t兲
t
t−h共t兲
x共t兲 − x共t − h共t兲兲
ẋ共s兲ds +
冕
t
ẋT共s兲dsXT13x共t兲 +
t−h共t兲
冕
t
ẋT共s兲dsXT23x共t
t−h共t兲
= 共1 − hd兲兵xT共t兲hX11x共t兲 + xT共t兲hX12x共t − h共t兲兲 + xT共t兲X13关x共t兲 − x共t − h共t兲兲兴 + xT共t
− h共t兲兲hXT12x共t兲 + xT共t − h共t兲兲hX22x共t − h共t兲兲 + xT共t − h共t兲兲X23关x共t兲 − x共t − h共t兲兲兴 + 关x共t兲 − x共t
− h共t兲兲兴TXT13x共t兲 + 关x共t兲 − x共t − h共t兲兲兴TXT23x共t − h共t兲兲其 = 共1 − hd兲兵xT共t兲hX11x共t兲 + xT共t兲hX12x共t − h共t兲兲
+ xT共t兲X13x共t兲 − xT共t − h共t兲兲XT13x共t兲 + xT共t − h共t兲兲hXT12x共t兲 + xT共t − h共t兲兲hX22x共t − h共t兲兲 + xT共t
− h共t兲兲X23x共t兲 − xT共t − h共t兲兲X23x共t − h共t兲兲 + xT共t兲XT13x共t兲 − xT共t兲XT13x共t − h共t兲兲 + xT共t兲XT23x共t − h共t兲兲
− xT共t − h共t兲兲XT23x共t − h共t兲兲其 = 共1 − hd兲兵xT共t兲关hX11 + XT13 + X13兴x共t兲 + xT共t兲关hX12 − X13 + XT23兴x共t
− h共t兲兲 + xT共t − h共t兲兲关hXT12 − XT13 + X23兴x共t兲 + xT共t − h兲关hX22 − X23 − XT23兴x共t − h共t兲兲其
Substituting the above Eq. 共17兲 into Eq. 共16兲 yields the following equation:
V̇共xt兲 ⬍ ␰T共t兲⌶␰共t兲 − 共1 − hd兲
冕
t
ẋT共s兲共R − X33兲ẋ共s兲ds
t−h共t兲
共18兲
where
␰T共t兲 = 关xT共t兲 xT共t − h共t兲兲 兴
and
⌶=
冋
⌶11 ⌶12
⌶T12 ⌶22
册
with
⌶11 = AT0 P + PA0 + Q + 共1 − hd兲共X13 + XT13 + hX11兲 + hAT0 RA0
⌶12 = PA1 + 共1 − hd兲共− XT13 + X23 + hX12兲 + hAT0 RA1
⬍ 0 and R − X33 ⱖ 0, if and only if Eq. 共14兲 holds. Therefore, the
nominal unforced time-varying delay system 共Eq. 共13兲兲 is asymptotically stable. This completes the proof.
According to Theorem 1, we describe our method for determining the stabilization of the time-varying delay system with a saturating actuator 共Eq. 共1兲兲. The main aim of this paper is to develop
delay-dependent conditions for the stabilization of the timevarying delay saturating actuator system 共Eq. 共1兲兲 under the state
feedback control law u共t兲 = Kx共t兲. More specifically, our objective
is to determine bounds for the delay time by using the Lyapunov–Krasovskii functional and LMI methods with the Leibniz–Newton
formula. Theorem 2 gives an LMI-based computational procedure
to determine the state feedback controller. Then, we have the following result.
THEOREM 2. Given the scalars h ⬎ 0 and hd ⬎ 0 , the timevarying delay saturating actuator system (Eq. 共1兲) is asymptotically stabilizable via the memoryless state feedback controller if
there exist symmetric positive-definite matrices W ⬎ 0 , U ⬎ 0 ,
and Z ⬎ 0 , a semipositive-definite matrix
冤
T11 T12 T13
冥
⌶22 = 共1 − hd兲共− Q − X23 − XT23 + hX22兲 + hAT1 RA1
T = TT12 T22 T23 ⱖ 0
Finally, using the Schur complements of Lemma 2, with some
effort we can show that Eq. 共18兲 guarantees V̇共xt兲 ⬍ 0. Condition
TT13 TT23 T33
共14兲 of the present Theorem 1 is satisfied if V̇共xt兲 ⬍ 0, then ⌶
Journal of Dynamic Systems, Measurement, and Control
共17兲
and a matrix Y with appropriate dimensions such that the following holds:
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冤
⍀11 ⍀12 ⍀13
冥
W − T33 ⱖ 0
⍀ = ⍀T12 ⍀22 ⍀23 ⬍ 0
⍀T13 ⍀T23 ⍀33
共19a兲
⌿11 = W共A0 + ␤I兲T + 共A0 + ␤I兲W + BD共␣共x兲兲Y + Y TDT共␣共x兲兲BT
W − T33 ⱖ 0
+ U + 共1 − hd兲共T13 + TT13 + hT11兲
共19b兲
where
⍀11 =
WAT0
⌿12 = A1e␤hW + 共1 − hd兲共− T13 + TT23 + hT12兲
+ A0W + BD共␣共x兲兲Y + Y D 共␣共x兲兲B + U + 共1 − hd兲
T
T
T
⌿13 = h共W共A0 + ␤I兲T + Y TDT共␣共x兲兲BT兲
⫻共T13 + TT13 + hT11兲
⌿22 = 共1 − hd兲共− U − T23 − TT23 + hT22兲
⍀12 = A1W + 共1 − hd兲共− T13 + TT23 + hT12兲
⌿23 = hWAT1 e␤h
⍀13 = h共WAT0 + Y TDT共␣共x兲兲BT兲
⌿33 = − hZ
⍀22 = 共1 − hd兲共− U − T23 − TT23 + hT22兲
⍀23 = hWAT1
⍀33 = − hZ
Then, the time-varying delay system with a saturating actuator
共Eq. 共1兲兲 is asymptotically stable for with allowable time delay h
under the state feedback control law and K = YW−1 is a stabilizing
gain.
Proof. In view of Theorem 1, to prove the asymptotic stability
of the closed-loop system with control u共t兲 = Kx共t兲, it suffices to
show that there exist symmetric, positive-definite matrices P ⬎ 0,
Q ⬎ 0, R ⬎ 0, and X ⱖ 0 such that Eq. 共14兲 remains valid with A0
replaced by A0 + BD共␣共x兲兲K. Pre- and post-multiplying both sides
of Eq. 共14兲 by diag兵P−1 , P−1 , R−1其 and letting W = P−1, K = YW−1,
P−1QP−1 = U, P−1Xij P−1 = Tij 共i , j = 1 , 2 , 3兲, Z = R−1, and
R
关R−1 P−1兴关 −X33 兴 P−1 = W − T33 lead to Eq. 共18兲. Thus, if W, U, Z, Y,
and Tij are a set of feasible solutions to LMI 共Eq. 共19兲兲, then W
= P−1, P−1QP−1 = U, P−1Xij P−1 = Tij, and Y = KW−1, satisfying Eq.
共19兲 with A0 replaced by A0 + BD共␣共x兲兲K. This completes our
proof.
3 Extension to Exponential Stability for Time Delay
Saturating Actuator Systems
Consider that the time-varying delay system with a saturating
actuator 共Eq. 共1兲兲 utilizes the following transformation:
z共t兲 = e␤tx共t兲
共20兲
where ␤ ⬎ 0 is stability degree 共delay decay rate兲 to transform Eq.
共1兲 into
ż共t兲 = A␤z共t兲 + A1␤z共t − h共t兲兲
共21兲
where A␤ = A0 + BD共␣共x兲兲K + ␤I and A1␤ = A1e␤h.
By applying Theorem 2 to system 共21兲, we can immediately
have Theorem 3.
THEOREM 3. Given scalars h ⬎ 0 and hd ⬎ 0 , the time-varying
delay saturating actuator system (Eq. (21)) is exponentially stable
with decay rate ␤ via the memoryless state feedback controller if
there exist symmetric positive-definite matrices W ⬎ 0 , U ⬎ 0 , Z
⬎ 0 , a semipositive-definite matrix
冤
T11 T12 T13
冥
Then, the time delay system 共Eq. 共21兲兲 is exponentially stable
for with allowable time delay h under the state feedback control
law and K = YW−1 is a stabilizing gain.
Proof. Replace A0 and A1 with A0 + BD共␣共x兲兲K + ␤I and A1e␤h,
respectively, in Theorem 2 and let K = YW−1 then the proof of
Theorem 3 follows from Theorem 2.
Remark 1. As in the stabilization problem, the upper bound h,
which ensures that time-delays 共Eq. 共1兲兲 is stabilizable for any h,
can be determined by solving the following quasi-convex optimization problem when the other bound of time delay h is known:
maximize
Z ⬎ 0,
⌿=
冤
⌿11 ⌿12 ⌿13
⌿T12
⌿T13
冥
⌿22 ⌿23 ⬍ 0
⌿T23 ⌿33
4
014502-4 / Vol. 133, JANUARY 2011
U ⬎ 0,
Tⱖ0
and
共23兲
Examples
In this section, two numerical examples are presented to compare with the proposed stabilization method with previous results.
Example 1. Consider the time-varying delay system with an
actuator saturated at level ⫾1 described as follows:
共24兲
ẋ共t兲 = A0x共t兲 + A1x共t − h共t兲兲 + B sat共u共t兲兲
where
A0 =
冋
−4
0
1
−5
册
,
A1 =
冋
−1
1
0
−1
册 冋 册
,
B=
1 0
0 1
Assume that the operation range ␣i共x兲 is inside the sector 共0.5,
1兲. The problem is to design a state feedback controller to estimate
the delay time h such that the above system is to be asymptotically stable.
Solution. Here, we want to find a state feedback controller that
asymptotically stabilizes a class of time delay systems with a saturating actuator. First, apply the same state feedback controller 关11兴
as hd = 0 and
冋
− 0.1023 − 0.0168
− 0.0168 − 0.0855
册
We check the feasibility of LMIs 共Eq. 共19兲兲, and we can find that
the LMIs are feasible and obtain the solutions of the inequalities
W=
共22a兲
W ⬎ 0,
Inequality 共23兲 is a quasi-convex optimization problem and can
be obtained efficiently using MATLAB LMI Toolbox. Then, the
controller K = YW−1 stabilizes system 共1兲.
To show the usefulness of our result, let us consider the following numerical examples.
K=
and a matrix Y with appropriate dimensions such that the following LMIs hold:
h
subject to Eq. 共19兲 or共22兲:
T = TT12 T22 T23 ⱖ 0
TT13 TT23 T33
共22b兲
where
冋
4.9680 0.7689
0.7689 4.0779
Z=
冋
册
,
U=
79.1165
0.7696
0.7696
78.1970
册
冋
16.3152
0.0344
0.0344
16.1438
册
,
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Table 1 Bound of delay time h for various parameters ␣i„x…
␣i共x兲
h
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
2.0599
2.0371
2.0351
2.0065
1.9978
1.9905
1.9501
1.9203
1.8271
Table 2 Bound of delay time h for various decay rates ␤ and the change in time-varying delay hd „saturated range ␣i„x… = 0.1…
hd
␤
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
T11 =
冋
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
7.3955
5.3596
5.3479
5.3169
5.2599
5.2452
5.2300
3.5644
3.5191
5.5316
5.3199
5.2500
5.1119
3.9510
3.6099
3.1330
3.0900
3.0631
5.3654
5.0165
4.8567
4.0119
3.2110
3.1011
2.6010
2.5401
2.4999
5.3414
4.9901
4.8167
3.1564
3.1101
3.0425
2.5010
2.4203
2.2010
5.0014
4.5991
3.5011
3.0449
2.9868
2.1300
2.0563
2.0401
2.0005
4.9821
4.3331
3.3219
3.0228
2.6878
1.8801
1.8521
1.8411
1.8205
3.5921
3.0111
2.9929
2.9001
2.6799
1.8709
1.8455
1.8391
1.8101
3.4995
2.6005
2.5401
2.1111
1.8660
1.5706
1.5499
1.7001
1.6201
3.0011
2.3534
2.3337
1.8511
1.6896
1.4796
1.5354
1.6001
1.5347
5.5236 0.0692
0.0692 5.4325
T13 =
T22 =
冋
冋
册
T12 =
冋
− 0.0997
− 0.0997
− 12.7109
0.1257 6.7435
冋
册
,
T23 =
53.1382
0.4699
0.4699
52.6460
册
− 5.0487 − 0.0687
− 0.0687 − 4.9472
册
冋
− 12.8052
6.8806 0.1257
T33 =
,
12.3777
0.0863
0.0863
12.2943
册
册
K = YW−1 =
,
冋
− 8.2040
15.9474
6.4402
− 17.8635
册
and h ⬍ 1.9978.
Furthermore, using the LMI Toolbox in MATLAB 共with an accuracy of 0.01兲 by taking
,
h ⬍ 3.8799
An upper bound given by Liu and Su 关11兴 is h ⬍ 0.405. Hence,
for this example, the robust stability criterion of this paper is less
conservative than the existing result of Liu and Su 关11兴. The simulation of the above closed system for h = 3.8 is depicted in Fig. 1.
On the other hand, for the case of hd = 0.9, solving the following
quasi-convex optimization problem 共Eq. 共24兲兲, the maximum upper bound h for which the system is stabilized by the corresponding state feedback,
K=
冋
− 8.2040
15.9474
6.4402
− 17.8635
册
and 0 ⬍ ␣i共x兲 ⬍ 1, the variations of ␣i共x兲 in bound of delay time h
are shown in Table 1. As Table 1 indicates, the delay time increases when the parameter ␣i共x兲 decreases from 0.9 to 0.1. For
reasons mentioned above, if the operation range lengthens, the
delay time becomes less conservative. We claim that the sharpness
of the upper bound delay time h varies with the chosen operation
range of ␣i共x兲.
Remark 2. The operational range of the saturation nonlinearity
is inside the sector 关␣i共x兲 , 1兴, 0 ⱕ ␣i共x兲 ⱕ 1 in actual system and
the results can be improved considerably.
Example 2. Consider the time-varying delay system with an
actuator saturated at level ⫾1 of the form
共25兲
ẋ共t兲 = A0x共t兲 + A1x共t − h共t兲兲 + B sat共u共t兲兲
where
A0 =
Fig. 1 The simulation of the example 1 for h = 3.8 sec
Journal of Dynamic Systems, Measurement, and Control
冋
−2
0
1
−3
册
,
A1 =
冋
−1
0
− 0.8 − 1
册 冋 册
,
B=
1
2
−1 4
The problem is to design a state feedback controller to estimate
the delay time h such that the system 共Eq. 共25兲兲 is to be asymptotically stable.
Solution. First, apply the same memoryless state feedback con−0.0212 0.0272
troller 关17兴 as K = 关 −0.0323 −0.1060 兴. We calculate that the solutions
of the LMIs given in Eq. 共19兲 are feasible and h ⬍ 4.3949. An
upper bound given by Refs. 关14,17兴 are h ⬍ 0.3819 and h
⬍ 0.6153, respectively. Hence, for this example, the robust stability criterion of this paper is less conservative than the existing
results of Refs. 关14,17兴.
Next, when ␣i共x兲 = 0.5, hd = 0.1, and ␤ = 0, solving the following
quasi-convex optimization problem 共Eq. 共24兲兲, the maximum upper bound is h for which the system is h ⬍ 7.6244. Therefore, we
can get the stabilizing state feedback controller for the system
共Eq. 共25兲兲, which is
JANUARY 2011, Vol. 133 / 014502-5
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K=
冋
− 344.0687 − 0.4266
− 201.2766 − 0.0057
册
Moreover, by taking the decay rate ␤ and the change of timevarying delay, hd obtained from Theorem 3 is shown in Table 2.
From the results of Table 2, if the decay rate ␤ or the change of
time-varying delay hd increases the delay, then time length decreases. We claim that the sharpness of the upper bound of the
delay time h varies with the chosen decay ␤ or the change of
time-varying delay hd.
5
Conclusion
This paper deals with the problem of robust stabilization criteria for a class of time-varying delay saturating actuator systems
via the Lyapunov–Krasovskii functional combined with LMI techniques; simple and improved delay-dependent stabilities are proposed. The results have been illustrated by the given numerical
examples. A saturating control law is designed and a region is
specified in which the stability of the closed-loop system is ensured. From the obtained results, if the decay rate ␤ lengthens,
then the delay time becomes conservative. However, if the saturated range ␣i共x兲 lengthens, then the delay time becomes less
conservative. It appears that the upper bound delay time h is more
dependent on the decay rate ␤ in the operational range of the
saturation nonlinearity ␣i共x兲.
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