Research Journal of Applied Sciences and Technology 4(17): 2861-2865, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: November 25, 2011
Accepted: December 16, 2011
Published: September 01, 2012
Non-fragile Passive Filtering for a Class of Sampled-data System with
Long Time-Delay
1, 2
Shi-gang Wang and 1Jun-feng Wu
1
School of Automation, Harbin University of Science and Technology, Harbin, P.R. China
2
School of Mechanical & Electronic Engineering, Heilongjing University, Harbin, P.R. China
Abstract: In this study, the problem of non-fragile passive filtering for a class of sampleded-data system with
long time delay is addressed. The uncertain parameters are supposed to belong to norm-bounded uncertainties.
A direct distribution processing methodology is developed to design a stable linear filter that assures asymptotic
stability and a prescribed passive performance for the filtering error, in spite of the uncertainty and the long
time-delay. The proposed algorithm is given in term of linear matrix inequality, whose feasibility and
effectiveness has been shown by a numberical example.
Keywords: Long time-delay system, LMI, non-fragile passive filtering, sampled-data system
INTRODUCTION
Sampled-data system extensively exists in lots of
industrial processes, such as welding industry, aeronautics
and astronautics, chemical industry, etc., Chen and
Francis (1995), which is characterized by a continuous
control plant and discrete controller. Due to uncertainties
and time-delay frequently appearing in sampled-data
system, which makes the system instable and its
performance deteriorated. Therefore, robust control and
robust filtering have gradually become hot topics of
control field and signal processing (Wu et al., 2001; Wu
et al., 2002; Theodor et al., 1994; Xie et al., 1991; Xie et
al., 1996; Li and Fu, 1997). However, above-mentioned
results are based on the accurate feedback controllers. In
fact, because of the existence of the accuracy problem
parameter drift and other factors, it is shown that
relatively small perturbation of the controller parameters
might destabilize the closed-loop system, even lead to the
performance degradation. Thus, it is necessary to design
a controller which can tolerate some level of controller
parameter variables. This is known as the non-fragile
control problem. To data, this problem of non-fragile
control and fitering has been widely investigated by many
researchers, (Wang, 2011; Wang and Wu, 2011;
Mahmoud, 2004; Yang and Che, 2008; Che and Yang,
2008).
On the other hand, passive theory has been important
effect on stability analysis, which makes the product of
the input and output as the supply of energy rate, in order
to reflect energy decay characteristics in bounded input
limit. In recent years, researches has made lots of works
in passive theory, such as Mahmoud, (1988) and
Zhang et al., (2006).
This study deals with non-fragile passive filtering
problem for sampled- data systems whose delay is longer
than a sampling period. The uncertain parameters are
assumed to belong to a given norm-bounded type. A
methodology for the design of a full- order stable linear
filter that assures asymptotic stability and a prescribed
passive performance for the filtering error system,
irrespective of the uncertainty and long time delay, is
developed by solving a set of LMIs.
PROBLEM FORMULATION
Consider sampled-data system:
⎧ x (t ) = A0 x (t ) (t − τ ) B0ω (t )
⎪ y (t ) = C x (t ) + Dω (t )
⎪
0
0
⎨
z
(
t
)
=
L
x
(
t
)
0
⎪
⎪⎩ x (t ) = x 0 , t ∈ [ − τ , 0]
(1)
where, x(t) , Rn is the state vector and y(t) , Rr is the
measured output, z(t),Rl is the signal to be estimated, T(t)
, Rp is the external disturbance input that belongs to
L2[0,4], A0, A1, B0, C0, D0 and L0 are the constant matrices
of approcaite dimensions. X0 is the initial state vector. J is
time delay, which is uncertain and assumed to be evaluted
between two agjacent sampling periods, namely, (m-1)h#
J # mh, where m $ 1 is a known constant.
Discretizing system (1) in one period, we can obtain
the discrete state equation of the sampled-data
system:
Corresponding Author: Shi-gang Wang, School of Automation, Harbin University of Science and Technology, Harbin, P.R. China
2861
Res. J. App. Sci. Eng. Technol., 4(17): 2861-2865, 2012
A f = A f 1 ( I + ∆1 )
⎧ x( k + 1) = G0 x ( k ) + G1 x ( k − m + 1) +
⎪
G2 x( k − m) + H0ω ( k )
⎪⎪
⎨ y ( k ) = C0 x ( k ) + D0ω ( k )
⎪z( k ) = L x ( k )
0
⎪
⎪⎩ x( k ) = x0 , k ≤ 0
B f = B f 1 ( I + ∆1 )
(2)
In order to be convenient to solve the following linear
matrix inequality,
Letting ~
x
= Mx$ ( k )
$
and then x ( k ) = M −1 ~
x (k )
filter transformed:
where,
G0 = e A0h , G1 =
G2 =
∫
h
mh − τ
∫
mh − r
0
(6)
C f = C f 1 ( I + ∆1 )
e A0h dtA1
∫
e A0h dt A1 , H 0 =
h
0
−1 ~
~
⎧
⎪ x ( k + 1) = MA f M x ( k ) + MB f y ( k )
⎨~
−1 ~
⎪Z ( k ) = C f M x ( k )
⎩
e A0h dtB0
Since time-delay J is uncertain, G1 and G2 are uncertain
matrices. Let
(7)
Denote:
A0 = L diag {81, …, 8n} L
-1
⎡ x( k ) ⎤
~
⎥ , e( k ) = z ( k ) − z ( k )
⎣ x ( k )⎦
ξ (k )⎢ ~
where, L is a n×n nonsingular matrix, 81, …, 8n are the
eigenvalues of matrix A0, here, assuming that 81, …, 8n
are unequal to 0, then:
G1 (τ ) = G1 + DF (τ ) E
Then, fitering error system:
~
~
⎧ξ ( k + 1) = G
0ξ ( k ) + Gm−1ξ ( k − m + 1) +
⎪
~
~
⎪
Gmξ ( k − m) + H0ω ( k )
⎨
~
⎪
e( k ) = z ( k ) − ~
z ( k ) = L0ξ ( k )
⎪
⎩
(3)
G2 (τ ) = G2 + DF (τ ) E
where,
G1 = − L diag {1 / λ1 ,...,1 / λn } E
{
where,
}
}
D = L diag e λ1β1 / λ1 ,..., e λnβn / λn E = L−1 A1
{
F (τ ) = diag e λ1 ( mh − τ − β1 ) ,..., e λn ( mh − τ − βn )
8i(mh- J - $i )
Selection of $i make e
0 ⎤ ~
⎡G m − 1 0 ⎤
⎥, Gm−1 = ⎢
0⎥⎦
MA f M − 1 ⎥⎦
⎣0
⎡ G0
~
G0 = ⎢
⎢⎣ MB f C0
G2 = L diag e λ1h / λ1 ,..., e λn h / λn E
{
(8)
⎡G m − 1
~
Gm−1 = ⎢
⎣ 0
}
[
⎡ H0 ⎤
0⎤ ~
,H = ⎢
⎥
0⎥⎦ 0 ⎣ MB f D0 ⎦
L0 = L0 − C f M −1
]
# I. Thus it is clear that:
FT(J)F(J) # I
(4)
Remark 1 If A0 can’t be transformed into a diagonal
matrix or it has j(0 # j# n) eigenvalues equal to 0, a
similar result can be obtained, but G 1 , G 2 , D, F(J), E,
should be changed correspondingly.
The aim of this section is to design a full-order,
linear, time-invariant asymp- totically stable filter for
system (2), The state-space realization of the filter has the
form:
The non-fragile passive filtering problem address in
this section is stated as follows:
Given scalars 0 > 0, find a full-order, linear, timeinvariant, asymptotically stable filter with a state-space
realization of the form (7) for system (2), such that:
C
C
Filtering-error system (8) with T(k) = 0 is
asymptotically stable
The passive performance
∞
2 ∑ eT ( k )ω ( k ) ≥ 0
(9)
K = 0
⎧
⎪ x$ ( k + 1) = A f x$ ( k ) + B f y ( k )
⎨$
⎪z ( k ) = C f x$ ( k )
⎩
(5)
is guranateed under zero-initial conditions for all nonzero
T(k) , l 2[0, 4].
where, Af , Rn×n, Bf , Rn×r, Cf , Rl×n are filter parameters to
be determined.
Lemma 1: For given matrices Q = QT, H and E, with
approciate dimension:
2862
Res. J. App. Sci. Eng. Technol., 4(17): 2861-2865, 2012
Q+HF(k)E+ETFT(k) HT<0
V (ξ ( k )) = ξ T ( k ) Pξ ( k ) +
holds for all F(k) satisfying F (k)F(k)#I if and only if
there exists , >0, such that:
T
T
∑ ξ ( k )Q1ξ ( k )
Denote:
Theorem 1: Consider the filtering error system (7), the
system is asymptotically table and (9) is satisfied under
zero- initial conditions for all nonzero T(k), l 2[0,4]., if
there exist matrices such that the following matrix
inequalities hold:
⎡Ω ΩT ⎤
Ω = ⎢ 11 21 ⎥ < 0
⎢⎣Ω21 Ω22 ⎥⎦
⎡ ξ(k )
⎤
⎢
⎥
ξ = ⎢ ξ ( k − m + 1) ⎥
⎢ ξ ( k − m) ⎥
⎣
⎦
⎡ − P1 + Q1 + Q2
*
* ⎤
⎢
⎥
* ⎥ ξ(k ) +
0
∆ V = ξ T (k )⎢
− Q1
⎢
0
0
− Q2 ⎥⎦
⎣
~
T
⎡ G0 ⎤
⎢~ ⎥
~ ~
~
T
ξ ( k ) ⎢ GmT−1 ⎥ P1 G0 Gm−1 Gm ξ ( k )
⎢ ~T ⎥
⎢⎣ Gm ⎥⎦
(10)
[
where,
Ω 11 =
*
*
⎡Θ 1
⎢Θ
*
Θ
2
⎢ 1
⎢ 0
0
− R1
⎢
⎢ 0
0
0
⎢
$
⎢ L0 − C f L0
0
⎢
SG
SG
SG
⎢ 0
0
m− 1
⎢ Θ
Θ4
Θ5
3
⎣
Ω 22 = diag
*
*
*
*
*
*
*
*
*
− R2
*
*
0
0
*
SGm
Θ6
SH 0 − S
Θ7 − I
0
0
0
0
Θ8
0
0
0
E2 C0 0
0
0
0
0
E2 D0 0
0
0
0
0
0
0
0
0
0
1
(11)
]
Firstly, we consider the fitering error system (8) is
asymotically stable with T (k) = 0 and then according to
LMI (10) , the following matrix inequality is true:
⎡ − P1 + Q1 + Q2
⎢
0
⎢
⎢
0
⎢
~
G0
⎣
*
*
− Q1
*
0
~
Gm− 1
− Q2
~
Gm
⎤
⎥
⎥<0
* ⎥
⎥
− P1 − 1 ⎦
*
*
(12)
By Schur complement, (12) is equivalent to:
0
0
0
0
1
⎤
⎥
⎥
⎥
*
⎥
⎥
*
⎥
⎥
*
⎥
*
⎥
− R0 ⎥⎦
*
*
⎤
⎥
⎥
Θ9 ⎥
⎥
0 ⎥
Θ10 ⎥
⎥
0 ⎥⎦
0
0
{ ε I ,− ε I − ε I ,− ε I ,− ε
0
ξ T ( k )Q1ξ ( k ) +
j = k − m+1
RESULTS AND DISCUSSION
0
0
∑
j = k − m +1
k −1
Q+ , HHT+ , -1 ETE<0
⎡0
⎢
⎢ E3
⎢0
Ω 21 = ⎢
⎢ E2 C0
⎢0
⎢
⎢⎣ E1
k −1
2
⎡ − P1 + Q1 + Q2
*
* ⎤
⎢
⎥
* ⎥+
− Q1
0
⎢
⎢⎣
− Q2 ⎥⎦
0
0
~
⎡ G0T ⎤
⎢ ~T ⎥
~ ~
~
⎢ Gm−1 ⎥ P1 G0 Gm−1 Gm < 0
⎢ ~T ⎥
⎣ Gm ⎦
[
I ,− ε 2 I }
Θ 1 = − S + R1 + R2
Θ 2 = − R0 + R1 + R2
Θ 3 = R0 G0 + B$ f C0 + A$ f
Θ 4 = R0 G0 + B$ f C0
Θ 5 = R0 Gm−1
Θ 6 = RGm
]
According to Lyapunov function, we are known that
the filtering error system is inner-stable.
Secondly, passive performance index is considered to (8).
Letting:
∆ V − 2e T ( k )ω ( k ) =
Θ 7 = R0 H 0 + B$ f D0
⎡ − P1 + Q1 + Q2
⎢
0
⎢
⎢
0
⎢
~
L0
⎣
Θ 8 = − ε 0 H 3T C$ Tf
Θ 9 = − ε1 H 2T B$ Tf
Θ 10 = ε 2 H1T A$ Tf
Proof. Select the Lyapunov function candidate:
(13)
*
*
− Q1
*
0
− Q2
0
0
*⎤
*⎥⎥
+
*⎥
⎥
0⎦
~T ⎤
⎡G
0
⎢~
⎥
⎢ GYm−1 ⎥
~ ~
~ ~
⎢ ~ T ⎥ P1 G0 Gm− 1 Gm H 0
⎢ Gm ⎥
⎢ ~T ⎥
⎣ Hm ⎦
Applying Schur Complement:
2863
[
]
(15)
Res. J. App. Sci. Eng. Technol., 4(17): 2861-2865, 2012
⎡− P1 + Q1 + Q2
⎢
0
⎢
⎢
0
⎢
~
L
⎢
0
~
⎢
PG
1 0
⎣
*
− Q1
0
*
*
− Q2
*
*
*
0
~
PG
1 m−1
0
~
PG
1 m
0
~
H0
⎤
⎥
⎥
⎥
⎥
* ⎥
− P1 ⎥⎦
*
*
*
θ$1 = − S + R1 + R2
θ$2 = − R0 + R1 + R2
(16)
θ$3 = L0 − C f
θ$4 = R0G0 + ( S − R0 ) B f C0 + ( S − R0 ) A f
θ$5 = R0G0 + ( S − R0 ) B f C0
θ$6 = R0 H0 + ( S − R0 ) B f D0
Setting
⎡R
P1 ⎢ 0T
⎣ X 12
X 12 ⎤ −1 ⎡ S
⎥ , P1 ⎢ T
X 22 ⎦
⎢⎣ Y12
⎡S −1
T1 = ⎢ T
⎢⎣ Y12
M = Y12T S
X 12 M = S − R0
Y12 ⎤
⎥
Y22 ⎥⎦
−1
A f = A f 1 ( I + ∆1 )
⎡ R1 0⎤
⎡ R2
I⎤
⎥ , Q1 = ⎢
⎥ , Q2 = ⎢
0⎥⎦
⎣ 0 0⎦
⎣0
Bf = Bf 1( I + ∆2 )
0⎤
⎥
0⎦
Cf = C f 1( I + ∆3)
(18) is separated from definite part to uncertain part:
Choose congruent transformation diag {I, I, I, T1}, Preand Post-multiplying both sides of ineuqalities (16), one
obtains the following inequality:
*
⎡ θ1
⎢ θ
θ
3
⎢ 2
⎢ 0
0
⎢
0
⎢ 0
⎢ θ4
L0
⎢
−1
G0
⎢G0 S
⎢ θ
θ6
⎣ 5
*
*
*
*
*
*
*
*
− R1
*
*
*
0
− R1
*
*
0
0
0
*
Gm−1
Gm
H0
θ7
θ8
θ9
−1
−S
−I
* ⎤
* ⎥⎥
* ⎥
⎥
* ⎥
* ⎥
⎥
* ⎥
− R0 ⎥⎦
⎡ θ$1
⎢ $
⎢ θ1
⎢ 0
⎢
⎢ 0
⎢Π
⎢ 1
⎢ SG0
⎢Π
⎣ 2
(17)
*
*
*
*
2
*
*
*
*
0
0
− R1
0
*
− R2
*
*
*
*
L0
0
0
0
*
SG0
SGm−1
SGm
SH 0
−S
Π3
R0 Gm−1
R0 Gm
Π4
−I
* ⎤
⎥
* ⎥
* ⎥
⎥
* ⎥
* ⎥
⎥
* ⎥
− R0 ⎥⎦
+ Θ 1 F (τ )Θ 2 + Θ T2 F T (τ )Θ T2 + Θ 3 F (τ )Θ 4 +
by Lemma 2:
where,
⎡ θ$1
⎢ $
⎢ θ1
⎢ 0
⎢
⎢ 0
⎢Π
⎢ 1
⎢ SG0
⎢
⎣ Π2
θ1 = − S + S −1 R1S −1 + S −1 R2 S −1
−1
θ2 = − I + R1S + R2 S
θ3 = − R0 + R1 + R2
−1
−1
−1 T
12
θ4 = − L0 S − C f M Y
θ5 = R0G0 S −1 + X 12 MB f C0 S −1 + X 12 MA f M −1Y12T
θ6 = R0G0 + X 12 MB f C0
θ7 = R0Gm−1
θ8 = R0Gm
θ9 = R0 H0 + X 12 MB f D0
⎡ θ$1
⎢ $
⎢ θ1
⎢ 0
⎢
θ$1 = − S + R1 + R2 ⎢ 0
⎢ θ$
⎢ 3
⎢ SG0
⎢ $
⎣ θ4
0
0
L0
SG0
*
*
− R1
0
0
SGm−1
*
*
*
− R2
0
SGm
*
*
*
*
0
SH0
*
*
*
*
*
−S
Π3
R0Gm−1
R0Gm
Π4
−I
ε 2Θ5Θ5T + ε 2−1Θ 6Θ 6T
*
θ$
*
*
*
*
2
*
*
*
*
0
0
− R1
0
*
− R2
*
*
*
*
L0
0
0
0
*
SG0
θ$
SGm−1
SGm
R0 G m
SH 0
θ$
−S
R0 G m − 1
5
*
θ$2
ε 0Θ1Θ1T + ε 0−1Θ T2 Θ 2 + ε1Θ 3Θ T3 + ε1−1Θ 4 Θ T4 +
Choose congruent transformation diga{S ,I, I, I, I, S, I,},
pre- and Post- multiplying both sides of ineuqalities (17),
one obtains the following inequality:
where,
*
θ$
6
−I
* ⎤
⎥
* ⎥
* ⎥
⎥
* ⎥
* ⎥
⎥
* ⎥
⎥
− R0 ⎦
where,
[
]
$ )T 0 0
Θ 1 = 0 0 0 0 − (CH
3
[
]
Θ 2 = E3 0 0 0 0 0 0
(
Θ 3 = ⎡⎢ 0 0 0 0 0 0 B$ f H2
⎣
[
) ⎤⎥⎦
T
Θ 4 = E2 C0 E2 C0 0 0 E2 D0 0 0
[
(18)
Θ 5 = 0 0 0 0 0 0 A$ f H1
[
]
]
]
Θ 6 = E1 0 0 0 0 0 0
by Lemma1, we can attain Themema1.
2864
⎤
⎥
⎥
⎥
⎥
⎥+
⎥
⎥
⎥
⎥
− R0 ⎦
*
*
*
*
*
*
(19)
Res. J. App. Sci. Eng. Technol., 4(17): 2861-2865, 2012
Simulation results: We consider the system (1) with:
REFERENCES
.⎤
⎡ 0.5 − 35
⎡− 1 0.5⎤
,
,A =
A0 = ⎢
. ⎥⎦
.
0.6 ⎥⎦ 1 ⎢⎣ 0 01
⎣− 12
.⎤
⎡− 55
B0 = ⎢
⎥ C 0 = [ − 3 0.2], D0 = 0.5, L0 = [ − 2 1]
⎣ 1 ⎦
Letting h = 0.1, m = 2 and discretizing system (1), a new
state equation is attained with corresponding parameter:
.
− 0.3724⎤
⎡ 10735
⎡ − 0.5862⎤
G0 = ⎢
⎥ , H0 = ⎢ 01381
⎥
.
10841
.
⎣ − 01277
⎦
⎣ .
⎦
− 0.2337⎤
.
0.0498⎤
.
⎡ − 01033
⎡ 01011
G1 = ⎢
⎥ , G2 = ⎢0.6279 − 0.3188⎥
⎣ 0.0062 0.0073⎦
⎣
⎦
⎡0.4982 0.4277 ⎤
⎡ 0.5899 − 0.3933⎤
D=⎢
E
=
,
⎥
⎢ − 0.5689 01849
⎥
.
⎦
⎣0.2847 − 0.2566⎦
⎣
Apply MATLAB LMI Toolbox to solve and then,
filter parameter is given, as follows:
.
⎡ 3.0894 − 18758
⎤
Af = ⎢
− 0.0562⎥⎦
.
⎣− 15262
B f = [2.9608 − 12716
.
]
T
C f = [ − 19827
0.9591]
.
By numerical experiment, fitering effect of non-fragile
passive filter is better than regular filter obviously.
CONCLUSION
The problem of the non-fragile passive filtering for a
class of sampled-data system with long time-delay has
been investigated. Using direct solvable approach,
dimension of state variable is deceased, a novel filter is
established and sufficient condition for the passivity of
the combined system is derived via linear matrix
inequality (LMI). Finally, a simulation example is
presented to show the validity and advantages of the
proposed method.
ACKNOWLEDGMENT
This study is partially supported by the National
Natural Science Foundation of China (50975068),
Doctoral Fund of Ministry of Education of China
(20060214004), Heilongjiang Province Postdoctoral
Science Foundation (LBHQ06050), Science and
Technology Research Foudation of Heilongjiang
Education Department (12521437).
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