Time-Varying Delay Systems Stabilization

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 time-
varying delay systems with saturating actuator. Based on the
Lyapunov–Krasovskii functional combining with linear matrix in-
equality techniques and Leibniz–Newton formula, delay-
dependent stabilization criteria are derived using a state feedback
controller. We also consider efficient convex optimization algo-
rithms to the time-varying delay system with saturating actuator
case: the maximal bound on the time delay such that the pre-
scribed 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 conserva-
tiveness 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 encoun-
tered in various engineering systems and are frequently a source
of instability. Stability analysis and synthesis of time-delay sys-
tems are important issues addressed by many authors, and some
mature methods have been widely used to deal with these prob-
lems 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 phenom-
ena, 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, hy-
draulic, 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 de-
grades system performance and complicates the analysis and de-
sign 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 sampled-
state 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 uncer-
tain 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 syn-
thesize 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 sys-
tems 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 sta-
bility 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 con-
servative. Therefore, recently, a new stability criterion based on
the LMI techniques was proposed 关2,19,20兴. However, the stabil-
ity analyses of the operators are still based on matrix norm ma-
nipulations, which may lead to conservative results. Considering a
Lyapunov–Krasovskii functional, a synthesis technique based
upon LMIs was used to determine simultaneously a robust stabi-
lizing 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 hto keep the stabil-
ilization 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 tech-
niques and the Leibniz–Newton formula, delay-dependent stabili-
zation 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 hvaries with the chosen decay rate
␤ . The results have been illustrated by the given numerical ex-
amples. These results are shown to be less conservative than those
reported in the literature.
2 Main Result
Consider the following time-varying delay systems with a satu-
rating actuator described by
x
˙共t兲=A0x共t兲+A1x共t−h共t兲兲 +Bsat共u共t兲兲 共1a兲
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.
Journal of Dynamic Systems, Measurement, and Control JANUARY 2011, Vol. 133 / 014502-1
Copyright © 2011 by ASME
Downloaded 20 Dec 2010 to 140.120.80.11. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

x共t兲= ␾ 共t兲,∀t苸关−h,0兴共1b兲
where x共t兲苸Rnis the state vector, u共t兲苸Rmis the control input
vector, and xtis the state at time tdenoted by xt共s兲ªx共t+s兲.A0,
A1, and Bare known constant matrices with appropriate dimen-
sions.
␾ 共t兲is a smooth vector-valued initial function.
Time-delay h共t兲is a time-varying continuous function that sat-
isfies
0ⱕh共t兲ⱕh,h
˙共t兲ⱕhd⬍1共2兲
where hand hdare constants.
The saturating function is defined as follows:
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共3兲
The operation of sat共ui共t兲兲 is linear for −UiⱕuiⱕUias
sat共ui共t兲兲 =
冦
−Uiif ui⬍−Ui⬍0
uiif − UiⱕuiⱕUi
Uiif ui⬎Ui⬎0
冧
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 memory-
less 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⬵0of the
functional differential equation associated to the closed-loop sys-
tem 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 writ-
ten in an equivalent form
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 共5兲
where D共
␣ 共x兲兲 is a diagonal matrix for which the diagonal ele-
ments
␣
i共x兲satisfy for i=1,2,...,m,
␣
i共x兲=
冦
−Ui
共Kx兲i
if 共Kx兲i⬍−Ui⬍0
1if − Uiⱕ共Kx兲iⱕUi
Ui
共Kx兲i
if 共Kx兲i⬎Ui⬎0
冧
共6兲
and therefore,
0ⱕ
␣
i共x兲ⱕ1共7兲
From Eqs. 共1a兲,共1b兲, and 共3兲–共7兲, we can rewrite the time-
varying delay system with saturating actuator as follows:
x
˙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 hand 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 re-
viewed.
LEMMA 1 关13兴.If there exist symmetric positive-definite matrix
X33 ⬎0and arbitrary matrices X11 ,X
12 ,X 13 ,X 22 , and X23 such
that
X=
冤
X11 X12 X13
X12
TX22 X23
X13
TX23 TX33
冥
ⱖ0共9兲
then we obtain
−
冕
t−h共t兲
t
x
˙T共s兲X33x ˙共s兲ds ⱕ
冕
t−h共t兲
t
关xT共t兲xT共t−h共t兲兲 x
˙T共s兲兴
⫻
冤
X11 X12 X13
X12
TX22 X23
X13
TX23 T0
冥冤
x共t兲
x共t−h共t兲兲
x
˙共s兲
冥
ds
共10兲
LEMMA 2 关1兴.The following matrix inequality
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
x
˙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 theo-
rem.
THEOREM 1.Given the scalars h⬎0and 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 Xⱖ0and a semipositive-definite matrix
X=
冤
X11 X12 X13
X12
TX22 X23
X13
TX23 TX33
冥
ⱖ0
such that the following LMIs hold:
⌽=
冤
⌽11 ⌽12 ⌽13
⌽12
T⌽22 ⌽23
⌽13
T⌽23 T⌽33
冥
⬍0共14a兲
R−X33 ⱖ0共14b兲
where
⌽11 =A0
TP+PA0+Q+共1−hd兲共X13 +X13 T+hX11兲
⌽12 =PA1+共1−hd兲共−X13 +X23
T+hX12兲
⌽13 =hA0
TR
⌽23 =hA1
TR
⌽22 =共1−hd兲共−Q−X23 −X23
T+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+
冕
t−h共t兲
t
xT共s兲Qx共s兲ds
+
冕
−h共t兲
0
冕
t+ ␪
t
x
˙T共s兲Rx ˙共s兲dsd
␪ 共15兲
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Calculating the derivative of Eq. 共15兲with respect to talong the
trajectory of the nominal unforced time-varying delay system 共Eq.
共13兲兲 yields
V
˙共xt兲=xT共t兲共A0
TP+PA0兲x共t兲+xT共t兲PA1x共t−h共t兲兲 +xT共t
−h共t兲兲A1
TPx共t兲+xT共t兲Qx共t兲−xT共t−h共t兲兲共1−h
˙共t兲兲Qx共t
−h共t兲兲 +x
˙T共t兲hRx
˙共t兲−共1−h
˙共t兲兲
冕
t−h
t
x
˙T共s兲Rx ˙共s兲ds =xT共t兲
⫻共A0
TP+PA0+Q兲x共t兲+xT共t兲PA1x共t−h共t兲兲 +xT共t
−h共t兲兲A1
TPx共t兲−xT共t−h共t兲兲共1−hd兲Qx共t−h共t兲兲
+x
˙T共t兲hRx
˙共t兲+共1−hd兲
冕
t−h共t兲
t
x
˙T共s兲共X33 −R兲x ˙共s兲ds −共1
−hd兲
冕
t−h共t兲
t
x
˙T共s兲X33x ˙共s兲ds 共16兲
Using the Leibniz–Newton formula x共t兲−x共t−h共t兲兲
=兰t−h共t兲
tx
˙共s兲ds and Lemma 1, we obtain
−共1−hd兲
冕
t−h共t兲
t
x
˙T共s兲X33x ˙共s兲ds ⱕ共1−hd兲
冕
t−h共t兲
t
关xT共t兲xT共t−h共t兲兲 x
˙T共s兲兴
冤
X11 X12 X13
X12
TX22 X23
X13
TX23 T0
冥冤
x共t兲
x共t−h共t兲兲
x
˙共s兲
冥
ds ⱕ共1−hd兲
⫻
再
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
冕
t−h共t兲
t
x
˙共s兲ds +xT共t−h共t兲兲hX12
Tx共t兲+xT共共t
−h共t兲兲hX22x共t−h共t兲兲兲+xT共t−h共t兲兲X23
冕
t−h共t兲
t
x
˙共s兲ds +
冕
t−h共t兲
t
x
˙T共s兲dsX13
Tx共t兲+
冕
t−h共t兲
t
x
˙T共s兲dsX23
Tx共t
−h共t兲兲
冎
=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hX12
Tx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TX13
Tx共t兲+关x共t兲−x共t−h共t兲兲兴TX23 Tx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X13
Tx共t兲+xT共t−h共t兲兲hX12 Tx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X13
Tx共t兲−xT共t兲X13 Tx共t−h共t兲兲 +xT共t兲X23 Tx共t−h共t兲兲
−xT共t−h共t兲兲X23
Tx共t−h共t兲兲其 =共1−hd兲兵xT共t兲关hX11 +X13 T+X13兴x共t兲+xT共t兲关hX12 −X13 +X23 T兴x共t
−h共t兲兲 +xT共t−h共t兲兲关hX12
T−X13 T+X23兴x共t兲+xT共t−h兲关hX22 −X23 −X23 T兴x共t−h共t兲兲其 共17兲
Substituting the above Eq. 共17兲into Eq. 共16兲yields the follow-
ing equation:
V
˙共xt兲⬍
␰
T共t兲⌶
␰ 共t兲−共1−hd兲
冕
t−h共t兲
t
x
˙T共s兲共R−X33兲x ˙共s兲ds
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
⌶12
T⌶22
册
with
⌶11 =A0
TP+PA0+Q+共1−hd兲共X13 +X13 T+hX11兲+hA0 TRA0
⌶12 =PA1+共1−hd兲共−X13
T+X23 +hX12兲+hA0 TRA1
⌶22 =共1−hd兲共−Q−X23 −X23
T+hX22兲+hA1 TRA1
Finally, using the Schur complements of Lemma 2, with some
effort we can show that Eq. 共18兲guarantees V
˙共xt兲⬍0. Condition
共14兲of the present Theorem 1 is satisfied if V
˙共xt兲⬍0, then ⌶
⬍0 and R−X33 ⱖ0, if and only if Eq. 共14兲holds. Therefore, the
nominal unforced time-varying delay system 共Eq. 共13兲兲 is asymp-
totically stable. This completes the proof.
According to Theorem 1, we describe our method for determin-
ing the stabilization of the time-varying delay system with a satu-
rating actuator 共Eq. 共1兲兲. The main aim of this paper is to develop
delay-dependent conditions for the stabilization of the time-
varying 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 fol-
lowing result.
THEOREM 2.Given the scalars h⬎0and hd⬎0, the time-
varying delay saturating actuator system (Eq. 共1兲) is asymptoti-
cally 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
T=
冤
T11 T12 T13
T12
TT22 T23
T13
TT23 TT33
冥
ⱖ0
and a matrix Y with appropriate dimensions such that the follow-
ing holds:
Journal of Dynamic Systems, Measurement, and Control JANUARY 2011, Vol. 133 / 014502-3
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⍀=
冤
⍀11 ⍀12 ⍀13
⍀12
T⍀22 ⍀23
⍀13
T⍀23 T⍀33
冥
⬍0共19a兲
W−T33 ⱖ0共19b兲
where
⍀11 =WA0
T+A0W+BD共
␣ 共x兲兲Y+YTDT共 ␣ 共x兲兲BT+U+共1−hd兲
⫻共T13 +T13
T+hT11兲
⍀12 =A1W+共1−hd兲共−T13 +T23
T+hT12兲
⍀13 =h共WA0
T+YTDT共
␣ 共x兲兲BT兲
⍀22 =共1−hd兲共−U−T23 −T23
T+hT22兲
⍀23 =hWA1
T
⍀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 Xⱖ0 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−1XijP−1 =Tij 共i,j=1,2,3兲,Z=R−1, and
关R−1 P−1兴关R
−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−1XijP−1 =Tij, and Y=KW−1, satisfying Eq.
共19兲with A0replaced 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
z
˙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+ ␤ Iand A1
␤
=A1e
␤ h.
By applying Theorem 2 to system 共21兲, we can immediately
have Theorem 3.
THEOREM 3.Given scalars h⬎0and 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
T=
冤
T11 T12 T13
T12
TT22 T23
T13
TT23 TT33
冥
ⱖ0
and a matrix Y with appropriate dimensions such that the follow-
ing LMIs hold:
⌿=
冤
⌿11 ⌿12 ⌿13
⌿12
T⌿22 ⌿23
⌿13
T⌿23 T⌿33
冥
⬍0共22a兲
W−T33 ⱖ0共22b兲
where
⌿11 =W共A0+
␤ I兲T+共A0+ ␤ I兲W+BD共 ␣ 共x兲兲Y+YTDT共 ␣ 共x兲兲BT
+U+共1−hd兲共T13 +T13
T+hT11兲
⌿12 =A1e
␤ hW+共1−hd兲共−T13 +T23
T+hT12兲
⌿13 =h共W共A0+
␤ I兲T+YTDT共 ␣ 共x兲兲BT兲
⌿22 =共1−hd兲共−U−T23 −T23
T+hT22兲
⌿23 =hWA1
Te
␤ h
⌿33 =−hZ
Then, the time delay system 共Eq. 共21兲兲 is exponentially stable
for with allowable time delay hunder the state feedback control
law and K=YW−1 is a stabilizing gain.
Proof. Replace A0and A1with A0+BD共
␣ 共x兲兲K+ ␤ Iand 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 optimi-
zation problem when the other bound of time delay his known:
maximize h
subject to Eq. 共19兲or共22兲:W⬎0, U⬎0,
Z⬎0, and Tⱖ0共23兲
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 follow-
ing numerical examples.
4 Examples
In this section, two numerical examples are presented to com-
pare 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:
x
˙共t兲=A0x共t兲+A1x共t−h共t兲兲 +Bsat共u共t兲兲 共24兲
where
A0=
冋
−4 0
1−5
册
,A1=
冋
−1 1
0−1
册
,B=
冋
10
01
册
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 hsuch that the above system is to be asymptoti-
cally stable.
Solution. Here, we want to find a state feedback controller that
asymptotically stabilizes a class of time delay systems with a satu-
rating actuator. First, apply the same state feedback controller 关11兴
as hd=0 and
K=
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=
冋
4.9680 0.7689
0.7689 4.0779
册
,U=
冋
16.3152 0.0344
0.0344 16.1438
册
,
Z=
冋
79.1165 0.7696
0.7696 78.1970
册
014502-4 / Vol. 133, JANUARY 2011 Transactions of the ASME
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T11 =
冋
5.5236 0.0692
0.0692 5.4325
册
,T12 =
冋
− 5.0487 − 0.0687
− 0.0687 − 4.9472
册
,
T13 =
冋
− 12.8052 − 0.0997
− 0.0997 − 12.7109
册
T22 =
冋
6.8806 0.1257
0.1257 6.7435
册
,T23 =
冋
12.3777 0.0863
0.0863 12.2943
册
,
T33 =
冋
53.1382 0.4699
0.4699 52.6460
册
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 simu-
lation 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 up-
per bound hfor which the system is stabilized by the correspond-
ing state feedback,
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 accu-
racy of 0.01兲by taking
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 in-
creases 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 hvaries 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,0ⱕ
␣
i共x兲ⱕ1 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
x
˙共t兲=A0x共t兲+A1x共t−h共t兲兲 +Bsat共u共t兲兲 共25兲
where
A0=
冋
−2 0
1−3
册
,A1=
冋
−1 0
− 0.8 − 1
册
,B=
冋
12
−1 4
册
The problem is to design a state feedback controller to estimate
the delay time hsuch that the system 共Eq. 共25兲兲 is to be asymp-
totically stable.
Solution. First, apply the same memoryless state feedback con-
troller 关17兴as K=关−0.0212 0.0272
−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 stabil-
ity 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 up-
per bound is hfor 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
Table 1 Bound of delay time hfor various parameters ␣
i„x…
␣
i共x兲0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
h2.0599 2.0371 2.0351 2.0065 1.9978 1.9905 1.9501 1.9203 1.8271
Table 2 Bound of delay time hfor 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
0.1 7.3955 5.5316 5.3654 5.3414 5.0014 4.9821 3.5921 3.4995 3.0011
0.2 5.3596 5.3199 5.0165 4.9901 4.5991 4.3331 3.0111 2.6005 2.3534
0.3 5.3479 5.2500 4.8567 4.8167 3.5011 3.3219 2.9929 2.5401 2.3337
0.4 5.3169 5.1119 4.0119 3.1564 3.0449 3.0228 2.9001 2.1111 1.8511
0.5 5.2599 3.9510 3.2110 3.1101 2.9868 2.6878 2.6799 1.8660 1.6896
0.6 5.2452 3.6099 3.1011 3.0425 2.1300 1.8801 1.8709 1.5706 1.4796
0.7 5.2300 3.1330 2.6010 2.5010 2.0563 1.8521 1.8455 1.5499 1.5354
0.8 3.5644 3.0900 2.5401 2.4203 2.0401 1.8411 1.8391 1.7001 1.6001
0.9 3.5191 3.0631 2.4999 2.2010 2.0005 1.8205 1.8101 1.6201 1.5347
Fig. 1 The simulation of the example 1 for h=3.8 sec
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