Signal Processing 87 (2007) 2659–2672
Delay-dependent robust H
1
filtering for uncertain 2-D
state-delayed systems
Shyh-Feng Chen
a, I-Kong Fong
b,aDepartment of Electrical Engineering, China Institute of Technology, Taipei, Taiwan 11581, ROC bDepartment of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, ROC
Received 24 November 2006; received in revised form 13 April 2007; accepted 27 April 2007 Available online 6 May 2007
Abstract
For uncertain 2-D state-delayed systems in the Fornasini–Marchesini second model, this paper discusses the robust H1 filtering problem. Both filter analysis and synthesis problems are considered. Firstly, a stability condition and an H1-norm performance condition are derived. Then a set of delay-dependent sufficient conditions for the existence of desired robust H1 filters is expressed in terms of linear matrix inequalities. A numerical example is given in the last to show the application of the proposed filter design method.
r2007 Elsevier B.V. All rights reserved.
Keywords: Robust H1filter; 2-D systems; Time-delay systems; Delay-dependence; Linear matrix inequality (LMI)
1. Introduction
The filtering problem is one of the fundamental problems in various system engineering applica-tions, especially in fields of signal processing and automatic control. In these applications, it is usually necessary to estimate the state variables from the system measurement data. One of the most popular filter design approaches is the H1filtering method,
which can guarantee a prescribed noise attenuation over the entire frequency range for the estimation error. In many practical physical systems, however, parameter uncertainties may appear in system models. To handle problems with modeling un-certainties, the robust H1filtering methods for
one-dimensional (1-D) systems have been proposed in the literature[1–3].
Recently, the discrete two-dimensional (2-D) systems, which are physical systems with dynamics depending on two independent integer variables i and j, have attracted increasing attentions due to its theoretical as well as application importance in the fields such as multi-dimensional digital filtering, linear image processing, signal processing, process control, and so on[4–6]. In recent years, the linear matrix inequality (LMI)-based methods [7,8]for 2-D systems have been widely adopted and many results have been obtained [4,9–12]. Among these results, the H1filter is proposed in[4,9],
stabiliza-tion and H1 control problems are discussed in
[4,10,12], and the mixed H2=H1 filtering for 2-D
systems with polytopic uncertainties is reported in
[11]. It is worth noting that most of the researches
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regarding this topic only deal with 2-D systems without delays.
In practical 2-D systems, there are many exam-ples containing inherent delays, such as discretiza-tion time in discrete models describing delayed lattice differential equation [13] and partial differ-ence equations [14,15]. The delay effects are often adverse and need to be treated properly. For 1-D state-delayed systems, there have been many works on various problems. See, e.g., [16–21], and the references cited therein. Recently, research results about the stability and control problems of un-certain 2-D discrete state-delayed systems is re-ported in[22], and results about the mixed H2=H1
filter design problem by using a parameter-depen-dent Lyapunov function approach is first proposed in [14]. These works are, however, based on the independent approach. In general, the delay-dependent results are less conservative than the delay-independent counterparts, especially when it is known beforehand that the delays involved are small. Therefore, it is natural to try to derive similar results on the same problems of 2-D systems with state delays.
In this paper, a delay-dependent approach to robust H1 filtering will be proposed for polytopic
2-D state-delayed systems described by the For-nasini–Marchesini second model. In the considered systems, it is assumed that delays appear in both the horizontal and vertical directions. The purpose of the problem under investigation is to design a 2-D filter such that, for all admissible uncertainties, the filtering error dynamics is asymptotically stable, and a prescribed H1-norm performance level is
achieved, within specific delay ranges in both horizontal and vertical directions. Effective methods to solve the robust H1filtering problem by using a
parameter-dependent Lyapunov function [23,24]
will be derived. Different from the quadratic stability framework [21], the use of a parameter-dependent Lyapunov function allows different Lyapunov matrices to be set for different parts of the entire polytope domain, and produces less conservative design results.
The notation used throughout the paper is quite standard. Z is the set of nonnegative integers, Rn is the n-dimensional Euclidean space, and Rnm is the set of n m real matrices. PT stands for the transpose of a matrix P, and P40 ðo0Þ means that the symmetric matrix P is positive (negative) definite. The boldface characters represent matrix variables, and is the Kronecker product. In
symmetric block matrices,%is used as an ellipsis for the terms that are implied by symmetry, and diagf g for block-diagonal matrices. The ‘2 norm
of a 2-D signal wði; jÞ is defined and denoted by kwk2¼ ½P1i;j¼0kwði; jÞk
21=2, where k k is the
Eu-clidean vector norm. A 2-D signal w 2 ‘2 if it has a
bounded ‘2 norm. Finally, for any given M40
kwði; jÞk2M means wði; jÞTMwði; jÞ.
2. Problem formulation
Consider the 2-D state-delayed polytopic system described by the Fornasini–Marchesini second model [5,25] xði þ 1; j þ 1Þ ¼ AðaÞ xði þ 1; jÞ xði; j þ 1Þ " # þAdðaÞ xði þ 1; j d1Þ xði d2; j þ 1Þ " # þBðaÞ wði þ 1; jÞ wði; j þ 1Þ " # ,
yði; jÞ ¼ CðaÞxði; jÞ þ DðaÞwði; jÞ,
zði; jÞ ¼ LðaÞxði; jÞ, ð1Þ where x 2 Rn is the state vector, w 2 Rm is the disturbance input vector, y 2 Rp is the measured output vector, z 2 Rq is the signal vector to be estimated, and d1 and d2 are positive integers
denoting delays along vertical and horizontal directions, respectively. The matrices
AðaÞ ¼ ½A1ðaÞA2ðaÞ,
AdðaÞ ¼ ½Ad1ðaÞAd2ðaÞ,
BðaÞ ¼ ½B1ðaÞB2ðaÞ, ð2Þ
CðaÞ, DðaÞ and LðaÞ are assumed to be constant and unknown (uncertain), but belonging to a convex compact set of polytopic type, namely
A1ðaÞ B1ðaÞ Ad1ðaÞ
A2ðaÞ B2ðaÞ Ad2ðaÞ
CðaÞ DðaÞ 0 LðaÞ 0 0 2 6 6 6 6 4 3 7 7 7 7 5¼ Xt j¼1 aj AðjÞ1 BðjÞ1 AðjÞd1 AðjÞ2 BðjÞ2 AðjÞd2 CðjÞ DðjÞ 0 LðjÞ 0 0 2 6 6 6 6 4 3 7 7 7 7 5, (3) where a ¼ ½a1 atTis unknown in the unit simplex
½a1 atT: Xt j¼1 aj¼1; ajX0 ( ) . (4)
The boundary conditions are defined by
fxði; jÞ ¼ sij; 8 i 2 Z and j ¼ d1; d1þ1;. . . ; 0g,
fxði; jÞ ¼ tij; 8 j 2 Z and i ¼ d2; d2þ1;. . . ; 0g,
s00¼t00, ð5Þ
where sij and tij are given vectors.
In this paper, the basic objective is to find a filter of the form
xfði þ 1; j þ 1Þ ¼ Af1xfði þ 1; jÞ þ Af2xfði; j þ 1Þ
þBf1yði þ 1; jÞ þ Bf2yði; j þ 1Þ,
zfði; jÞ ¼ Lfxfði; jÞ (6)
to estimate the signal z from the measurement history of y, where xfði; jÞ 2 Rnis the state vector of
the filter, zfði; jÞ 2 Rq is the estimation of zði; jÞ, and
Af1, Af2, Bf1, Bf2 and Lf are filter parameter
matrices to be determined. Define the augmented state vector ^xði; jÞ ¼ ½xTði; jÞ xTfði; jÞT and the filter-ing error output signal ^zði; jÞ ¼ zði; jÞ zfði; jÞ. Then
the error dynamics equations are ^ xði þ 1; j þ 1Þ ¼ ^AðaÞ ^ xði þ 1; jÞ ^ xði; j þ 1Þ " # þ ^AdðaÞ ~J ^ xði þ 1; j d1Þ ^ xði d2; j þ 1Þ " # þ ^BðaÞ wði þ 1; jÞ wði; j þ 1Þ " # ,
^zði; jÞ ¼ ^LðaÞ ^xði; jÞ, (7) where ^ AðaÞ ¼ A1ðaÞ 0 A2ðaÞ 0 Bf1CðaÞ Af1 Bf2CðaÞ Af2 " # , ^ AdðaÞ ¼ Ad1ðaÞ Ad2ðaÞ 0 0 " # , ^ BðaÞ ¼ B1ðaÞ B2ðaÞ Bf1DðaÞ Bf2DðaÞ " # ; LðaÞ ¼ ½LðaÞ L^ f, ~ J ¼ ½JT1 JT2T; J1 ¼ ½J 0; J2¼ ½0 J, J ¼ ½In 0. ð8Þ
The boundary conditions of the error dynamics equations are defined by
fxði; jÞ ¼ ^s^ ij; 8 i 2 Z and j ¼ d1; d1þ1;. . . ; 0g,
fxði; jÞ ¼ ^t^ ij; 8 j 2 Z and i ¼ d2; d2þ1;. . . ; 0g,
^s00¼ ^t00, ð9Þ
where ^sij and ^tij are vectors determined by (5) and
boundary conditions set in the filter (6).
Throughout this paper, the following definitions apply.
Definition 1. The 2-D state-delayed system (7) is asymptotically stable if limr!1^wr¼0 for w ¼ 0 and
all bounded boundary conditions in (9), where ^wr¼supfk ^xði; jÞk : i þ j ¼ r; i; jX1g. (10) Definition 2. The H1-norm of the 2-D
state-delayed system (7) is defined as sup w k^zk2 kwk2: w 2 ‘2; kwk2a0; ^sij¼ ^tij¼0 8 i; j in ð9Þ . (11) By the above definition, the H1-norm of the 2-D
delay system (7) is less than or equal to g if and only if the H1-norm constraint k ^zk2pg2kwk2is satisfied
for all w 2 ‘2, and ^sij¼ ^tij¼0 with i, j in (9).
The robust H1filtering problem addressed in this
paper is as follows. Given positive integers ¯d1, ¯d2
and a real number g40, find a filter (6) such that the filtering error dynamics (7) is asymptotically stable and satisfies the H1-norm constraint for all
admissible uncertainties and delay di2 ½0; ¯di,
i ¼ 1; 2.
3. Stability andH1-norm performance analysis
In this section, the stability and H1-norm
performance analysis for 2-D state-delayed systems will be carried out. First, a delay-dependent sufficient condition for the asymptotic stability of 2-D state-delayed systems is presented.
3.1. Stability analysis
Consider the 2-D state-delayed system descri-bed by xði þ 1; j þ 1Þ ¼ A xði þ 1; jÞ xði; j þ 1Þ " # þAd xði þ 1; j d1Þ xði d2; j þ 1Þ " # , ð12Þ where A ¼ ½A1 A2 2Rn2n, Ad ¼ ½Ad1 Ad2 2Rn2n
are system matrices, x 2 Rn is the state vector, and d1; d2 are positive integers denoting delay sizes along
vertical and horizontal directions, respectively. In the following theorem, a sufficient delay-dependent
condition will be given that guarantees the asymptotic stability of system (12).
Theorem 3. Given positive integers ¯d1and ¯d2, if there
exist matrices P 2 R2n2n, Q 2 R2n2n, Ml 2Rnn,
Fl2R2n2n and Hl2R2nn, l ¼ 1; 2, such that
U % % HT Q % K ^A K ^Ad K 2 6 6 4 3 7 7 5o0, ð13Þ Fl % HTl Ml " # 40; l ¼ 1; 2, ð14Þ P ¼ P1 % PT3 P2 " # 40; P3¼PT3X0, ð15Þ Q ¼ Q1 % QT3 Q2 " # 40; Q3 ¼QT3X0, ð16Þ where U ¼ P ¯d1F1 ¯d2F2H HT, K ¼ diagfR; ¯d1M1; ¯d2M2g, R ¼ P1þ2P3þP2þQ1þ2Q3þQ2, ^ A ¼ A A A " # I2n 2 6 6 4 3 7 7 5; A^d ¼ Ad Ad Ad 2 6 6 4 3 7 7 5, H ¼ ½H1 H2, ð17Þ
then system (12) is asymptotically stable for any delay di2 ½0; ¯di, i ¼ 1; 2.
Proof. For notational simplicity, let
~ Z ¼ z2In z1In " # ; Z~d ¼ zd1þ1 2 In zd2þ1 1 In " # . (18)
Suppose the condition (13)–(16) is satisfied but (12) is unstable. Then by[5,14]
detðInz2A1z1A2z2d1þ1Ad1zd12þ1Ad2Þ ¼0
(19) for some ðz1; z2Þ 2U ¼ fðz1; z2Þjjz1jp1; jz2jp1g and
there exists a nonzero vector v such that v ¼ ~A ~ Z ~ Zd " # v, (20)
where ~A ¼ ½A Ad. The inequalities in (14) imply
that v ffiffiffiffiffi ¯d1 p ~ Z 1 ffiffiffiffiffi ¯d1 p ðz2zd21þ1ÞIn 2 6 6 4 3 7 7 5 F1 % HT1 M1 " # ffiffiffiffiffi ¯d1 p ~ Z 1 ffiffiffiffiffi ¯d1 p ðz2zd21þ1ÞIn 2 6 6 4 3 7 7 5vX0, ð21Þ v ffiffiffiffiffi ¯d2 p ~ Z 1 ffiffiffiffiffi ¯d2 p ðz1zd12þ1ÞIn 2 6 6 4 3 7 7 5 F2 % HT2 M2 " # ffiffiffiffiffi ¯d2 p ~ Z 1 ffiffiffiffiffi ¯d2 p ðz1zd12þ1ÞIn 2 6 6 4 3 7 7 5vX0, ð22Þ
where denotes the complex conjugate transpose.
From (21)–(22), one gets v ¯d 1Z~F1Z þ~ 1 ¯d1 ðz2zd21þ1Þ M1ðz2zd21þ1Þ þ2 ~ZH1ðz2zd21þ1Þ vX0, ð23Þ v ¯d 2Z~F2Z þ~ 1 ¯d2 ðz1zd12þ1Þ M 2ðz1zd12þ1Þ þ2 ~ZH2ðz1zd12þ1Þ vX0. ð24Þ
Also, the inequality (13) implies that
v ~ Z ~ Zd " # U % HT Q " # þ ~ATR ~A þ ¯d1A~T1M1A~1 ( þ ¯d2A~T2M2A~2 ~ Z ~ Zd " # vp0, ð25Þ
where A~1¼ ½A1In A2 Ad1 Ad2 and A~2¼
½A1 A2In Ad1 Ad2. From (25) and (12), the
following inequality: v½ ~ZP ~Z ~Z dQ ~Zdþ ¯d1Z~F1Z þ ¯d~ 2Z~F2Z~ þ ~ZH ~Z þ ~ZHTZ 2 ~~ ZH ~Z dv þv½R þ ¯d1M1jð1 z2Þj2 þ ¯d2M2jð1 z1Þj2vp0 ð26Þ
can be deduced, which, together with (23)–(24), give v ~ZP ~Z ~ZdQ ~Zd 1 ¯d1 M1jðz2zd21þ1Þj2 1 ¯d2 M2jðz1zd12þ1Þj2 v þ v½R þ ¯d 1M1jð1 z2Þj2 þ ¯d2M2jð1 z1Þj2vp0. ð27Þ
It follows from (15) and (16) that ðz1z2ÞP3ðz1z2ÞX0, ðzd2þ1 1 z d1þ1 2 Þ Q 3ðzd12þ1z d1þ1 2 ÞX0, ð28Þ
which imply, respectively,
z2P3z1z1P3z2XP3jz2j2P3jz1j2, ðzd1þ1 2 Þ Q 3ðzd12þ1Þ ðz d2þ1 1 Þ Q 3ðzd21þ1Þ XQ3jzd1þ1 2 j 2Q 3jzd12þ1j 2, ð29Þ
and one has ~ZP ~Z ¼ P 3jz2j2P3jz1j2z2P3z1z1P3z2 X ðP1þP3Þjz2j2 ðP2þP3Þjz1j2, ~Z dQ ~Zd¼ Q1jzd21þ1j 2 Q2jzd12þ1j 2 ðzd1þ1 2 Þ Q3ðz d2þ1 1 Þ ðz d2þ1 1 Þ Q3ðz d1þ1 2 Þ X ðQ1þQ3Þjzd21þ1j 2 ðQ2þQ3Þjzd12þ1j 2. ð30Þ Hence, (27) implies v ðP1þP3Þð1 jz2j2Þ þ ðP2þP3Þð1 jz1j2Þ ( þ ðQ1þQ3Þð1 jzd21þ1j 2Þ þ ðQ 2þQ3Þð1 jzd12þ1j 2Þ þ ¯d1M1 jð1 z2Þj2 jz2j2 ¯d2 1 jð1 zd1 2 Þj 2 " # þ ¯d2M2 jð1 z1Þj2 jz1j2 ¯d2 2 jð1 zd2 1 Þj 2 " #) vp0. ð31Þ Since P3X0, Q3X0, Pk40, Qk40, Mk40, k ¼ 1; 2,
and jz2j2= ¯d21p1; jz1j2= ¯d22p1 for ðz1; z2Þ 2U, the
left-hand side of (31) are positive except when v ¼ 0. This leads to a contradiction and system (12) must be asymptotically stable. This completes the proof of Theorem 3. &
Remark 4. Theorem 3 offers a new delay-dependent stability condition for 2-D state-delayed systems.
By setting F1¼ nI2n ¯d1 ; M1¼ nIn ¯d1 , F2¼ nI2n ¯d2 ; M2¼ nIn ¯d2 , H ¼ 0; P3¼Q3¼0, ð32Þ
in Theorem 3, the delay-dependent conditions (13)–(16) reduce to the delay-independent condi-tions in[14]when nX0 approaches zero.
3.2. Robust H1-norm performance analysis
In the following theorem, a delay-dependent H1
-norm performance criterion is established.
Theorem 5. Given positive integers ¯d1, ¯d2 and a real
number g40, system (7) is asymptotically stable and its H1-norm is less than or equal to g if di2 ½0; ¯di,
i ¼ 1; 2, and there exist matrices 0oPðaÞ ¼
P1ðaÞ PT 3ðaÞ % P2ðaÞ h i 2R4n4n, QðaÞ ¼ Q1ðaÞ QT 3ðaÞ % Q2ðaÞ h i 2R2n2n, MlðaÞ 2 Rnn, NlðaÞ 2 Rnn, FlðaÞ 2 R4n4n,GlðaÞ 2
R2mn, H
lðaÞ 2 R4nn and KlðaÞ 2 R2m2m; l ¼ 1; 2,
such that Q3ðaÞ ¼ QT3ðaÞX0, ð33Þ KlðaÞ % GTlðaÞ NlðaÞ " # 40; l ¼ 1; 2, ð34Þ P3ðaÞ ¼ PT3ðaÞX0, ð35Þ FlðaÞ % HTlðaÞ MlðaÞ " # 40; l ¼ 1; 2, ð36Þ UðaÞ % % % % HTðaÞ QðaÞ % % %
GðaÞ ~J GðaÞ WðaÞ % % AðaÞ AdðaÞ BðaÞ X1ðaÞ % ^ LðaÞ I2 0 0 0 gI2q 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 o0, ð37Þ for all a in the unit simplex (4), where
UðaÞ ¼PðaÞ ¯d1F1ðaÞ ¯d2F2ðaÞ HðaÞ ~J ~JTHTðaÞ, WðaÞ ¼ gI2m ¯d1K1ðaÞ ¯d2K2ðaÞ,
XðaÞ ¼ diagfRðaÞ; ¯d1½M1ðaÞ þ N1ðaÞ, ¯d2½M2ðaÞ þ N2ðaÞg,
RðaÞ ¼ P1ðaÞ þ 2P3ðaÞ þ P2ðaÞ
þJT½Q1ðaÞ þ 2Q3ðaÞ þ Q2ðaÞJ, HðaÞ ¼ ½H1ðaÞ H2ðaÞ,
GðaÞ ¼ ½G1ðaÞ G2ðaÞ,
AðaÞ ¼ ^ AðaÞ J ^AðaÞ J1 J ^AðaÞ J2 2 6 6 4 3 7 7 5; AdðaÞ ¼ ^ AdðaÞ J ^AdðaÞ J ^AdðaÞ 2 6 6 4 3 7 7 5, BðaÞ ¼ ^ BðaÞ J ^BðaÞ J ^BðaÞ 2 6 6 4 3 7 7 5. ð38Þ
Proof. First, the asymptotic stability of system (7) is established. For all a in the unit simplex (4), (37) implies (13). It follows from Theorem 3 that system (7) is asymptotically stable.
Next, the H1-norm performance criterion is
considered. For convenience, the following definitions: ~ xði; jÞ ¼ ^ xði þ 1; jÞ ^ xði; j þ 1Þ " # ; x~dði; jÞ ¼ ^ xði þ 1; j d1Þ ^ xði d2; j þ 1Þ " # , ~ wði; jÞ ¼ wTði þ 1; jÞ wTði; j þ 1Þ " # , and
dði þ 1; tÞ ¼ xði þ 1; t þ 1Þ xði þ 1; tÞ ¼ ðJ ^AðaÞ J1Þxði; tÞ~
þJ ^AdðaÞ ~J ~xdði; tÞ þ J ^BðaÞ ~wði; tÞ, ð39Þ
dðt; j þ 1Þ ¼ xðt þ 1; j þ 1Þ xðt; j þ 1Þ ¼ ðJ ^AðaÞ J2Þxðt; jÞ~
þJ ^AdðaÞ ~J ~xdðt; jÞ þ J ^BðaÞ ~wðt; jÞ ð40Þ
are introduced. Then, xði þ 1; jÞ xði þ 1; j d1Þ ¼ Xj1 t¼jd1 dði þ 1; tÞ, ð41Þ xði; j þ 1Þ xði d2; j þ 1Þ ¼ Xi1 t¼id2 dðt; j þ 1Þ ð42Þ for iXd2 and jXd1. The inequalities (34) and (36)
imply that Xj1 t¼jd1 ~ xði; jÞ dði þ 1; tÞ " #T F1ðaÞ % HT1ðaÞ M1ðaÞ " # xði; jÞ~ dði þ 1; tÞ " # X0, ð43Þ Xi1 t¼id2 ~ xði; jÞ dðt; j þ 1Þ " #T F2ðaÞ % HT2ðaÞ M2ðaÞ " # xði; jÞ~ dðt; j þ 1Þ " # X0, ð44Þ Xj1 t¼jd1 ~ wði; jÞ dði þ 1; tÞ " #T K1ðaÞ % GT1ðaÞ N1ðaÞ " # wði; jÞ~ dði þ 1; tÞ " # X0, ð45Þ Xi1 t¼id2 ~ wði; jÞ dðt; j þ 1Þ " #T K2ðaÞ % GT2ðaÞ N2ðaÞ " # wði; jÞ~ dðt; j þ 1Þ " # X0. ð46Þ From (41)–(46), one has
d1kxði; jÞk~ 2F1ðaÞþ
Xj1 t¼jd1
kdði þ 1; tÞk2M1ðaÞþ2½xði þ 1; jÞ
xði þ 1; j d1ÞTHT1ðaÞ ~xði; jÞX0, ð47Þ
d2kxði; jÞk~ 2F2ðaÞþ
Xi1 t¼id2
kdðt; j þ 1Þk2M2ðaÞþ2½xði; j þ 1Þ
xði d2; j þ 1ÞTHT2ðaÞ ~xði; jÞX0, ð48Þ
d1kwði; jÞk~ 2K1ðaÞþ
Xj1 t¼jd1
kdði þ 1; tÞk2N1ðaÞþ2½xði þ 1; jÞ
xði þ 1; j d1ÞTGT1ðaÞ ~wði; jÞX0, ð49Þ
d2kwði; jÞk~ 2K2ðaÞþ
Xi1 t¼id2
kdðt; j þ 1Þk2N2ðaÞþ2½xði; j þ 1Þ
xði d2; j þ 1ÞTGT2ðaÞ ~wði; jÞX0. ð50Þ
By Schur’s complement, (37) implies that
xTði; jÞ
UðaÞ % %
HTðaÞ QðaÞ % GðaÞ ~J GðaÞ WðaÞ 2 6 6 4 3 7 7 5 8 > > < > > : þ ATðaÞ ATdðaÞ BTðaÞ 2 6 6 4 3 7 7
5XðaÞ½AðaÞ AdðaÞ BðaÞ
þ1 g ^ LTðaÞ I2 0 0 2 6 6 4 3 7 7 5½LðaÞ I^ 2 0 0 9 > > = > > ; xði; jÞp0, ð51Þ where xTði; jÞ ¼ ½ ~xTði; jÞ ~J ~xT
dði; jÞ ~wTði; jÞ. Also it
follows from (7) and (39)–(40) that
kxði þ 1; j þ 1Þk^ 2RðaÞþd1kdði þ 1; jÞk2M1ðaÞþN1ðaÞ
þd2kdði; j þ 1Þk2M2ðaÞþN2ðaÞ kxði; jÞk~
2 PðaÞ
þd1kxði; jÞk~ 2F1ðaÞþd2kxði; jÞk~ 2 F2ðaÞ þ kxði; jÞk~ 2 HðaÞ ~Jþ kxði; jÞk~ 2 ~
JTHTðaÞ k ~J ~xdði; jÞk2QðaÞ
x~Tði; jÞHðaÞ ~J ~xdði; jÞ ~xTdði; jÞ ~J
THTðaÞ ~xði; jÞ
þw~Tði; jÞGðaÞ ~J ~xði; jÞ þ ~xTði; jÞ ~JTGTðaÞ ~wði; jÞ w~Tði; jÞGðaÞ ~J ~xdði; jÞ ~xTdði; jÞ ~JTGTðaÞ ~wði; jÞ
þd1kwði; jÞk~ 2K1ðaÞþd2kwði; jÞk~
2 K2ðaÞ þg1 ^zði þ 1; jÞ ^zði; j þ 1Þ " # 2 g wði þ 1; jÞ wði; j þ 1Þ " # 2 p0, ð52Þ which together with (47)–(50) imply
kxði þ 1; j þ 1Þk^ 2RðaÞþd1kdði þ 1; jÞk2M1ðaÞþN1ðaÞ
þd2kdði; j þ 1Þk2M2ðaÞþN2ðaÞ kxði; jÞk~
2 PðaÞ
k ~Jx~dði; jÞk2QðaÞ
Xj1 t¼jd1
kdði þ 1; tÞk2M1ðaÞþN1ðaÞ
X i1 t¼id2 kdðt; j þ 1Þk2M2ðaÞþN2ðaÞ þg1 ^zði þ 1; jÞ ^zði; j þ 1Þ " # 2 g wði þ 1; jÞ wði; j þ 1Þ " # 2 p0. ð53Þ Since for all P3ðaÞX0 and Q3ðaÞX0,
kxði; jÞk~ 2PðaÞX kxði þ 1; jÞk^ 2P
1ðaÞþP3ðaÞ
kxði; j þ 1Þk^ 2P
2ðaÞþP3ðaÞ, ð54Þ
k ~J ~xdði; jÞk2QðaÞX kxði þ 1; j d1Þk 2
Q1ðaÞþQ3ðaÞ
kxði d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ.
ð55Þ Eq. (53) implies
kxði þ 1; j þ 1Þk^ 2RðaÞ kxði þ 1; jÞk^ 2P1ðaÞþP3ðaÞ kxði; j þ 1Þk^ 2P2ðaÞþP3ðaÞ
kxði þ 1; j d1Þk2Q1ðaÞþQ3ðaÞ
kxði d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ
þd1kdði þ 1; jÞk2M1ðaÞþN1ðaÞ
þd2kdði; j þ 1Þk2M2ðaÞþN2ðaÞ
X
j1 t¼jd1
kdði þ 1; tÞk2M1ðaÞþN1ðaÞ
X i1 t¼id2 kdðt; j þ 1Þk2M2ðaÞþN2ðaÞ þg1 ^zði þ 1; jÞ ^zði; j þ 1Þ " # 2 g wði þ 1; jÞ wði; j þ 1Þ " # 2 p0. ð56Þ Thus, for any positive integers l1 and l2,
X l21 i¼0 X l11 j¼0 kxði þ 1; j þ 1Þk^ 2RðaÞ 8 < : kxði þ 1; jÞk^ 2
P1ðaÞþP3ðaÞ kxði; j þ 1Þk^
2
P2ðaÞþP3ðaÞ
kxði þ 1; j d1Þk2Q1ðaÞþQ3ðaÞ
kxði d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ
þd1kdði þ 1; jÞk2M1ðaÞþN1ðaÞ
þd2kdði; j þ 1Þk2M2ðaÞþN2ðaÞ
X j1 t¼jd1 kdði þ 1; tÞk2M 1ðaÞþN1ðaÞ X i1 t¼id2 kdðt; j þ 1Þk2M2ðaÞþN2ðaÞ þg1 ^zði þ 1; jÞ ^zði; j þ 1Þ " # 2 g wði þ 1; jÞ wði; j þ 1Þ " # 29 = ;p0, ð57Þ and X l21 i¼0
kx^Tði þ 1; l1Þk2P1ðaÞþP3ðaÞ
(
þX
d1
j¼0
kxði þ 1; l1þj d1Þk2Q1ðaÞþQ3ðaÞ
þX
d11
j¼0
ðd1jÞkdði þ 1; l11 jÞk2M1ðaÞþN1ðaÞ
kx^Tði þ 1; 0Þk2P1ðaÞþP3ðaÞ
X
d1
j¼0
kxði þ 1; j d1Þk2Q1ðaÞþQ3ðaÞ
X
d11
j¼0
ðd1jÞkdði þ 1; 1 jÞk2M1ðaÞþN1ðaÞ
þX l11 j¼0 kx^Tðl 2; j þ 1Þk2P2ðaÞþP3ðaÞ ( þX d2 i¼0 kxðl2þi d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ þX d21 i¼0
ðd2iÞkdðl21 i; j þ 1Þk2M2ðaÞþN2ðaÞ
kx^Tð0; j þ 1Þk2P2ðaÞþP3ðaÞ
X
d2
i¼0
kxði d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ
X
d21
i¼0
ðd2iÞkdð1 i; j þ 1Þk2M2ðaÞþN2ðaÞ
) þg1 X l2 i¼0 X l11 j¼0 k^zði; jÞk2X l11 j¼0 k^zð0; jÞk2 ( þX l21 i¼0 Xl1 j¼0 k^zði; jÞk2X l21 i¼0 k^zði; 0Þk2 ) pg X l2 i¼0 X l11 j¼0 kwði; jÞk2X l11 j¼0 kwð0; jÞk2 ( þX l21 i¼0 Xl1 j¼0 kwði; jÞk2X l21 i¼0 kwði; 0Þk2 ) . ð58Þ
When the boundary condition is ^sij¼ ^tij ¼0 with i, j
in (9), the inequality (58) gives X
l21
i¼0
kx^Tði þ 1; l1Þk2P1ðaÞþP3ðaÞ
(
þX
d1
j¼0
kxði þ 1; l1þj d1Þk2Q1ðaÞþQ3ðaÞ
þX
d11
j¼0
ðd1jÞkdði þ 1; l11 jÞk2M1ðaÞþN1ðaÞ
) þX l11 j¼0 kx^Tðl2; j þ 1Þk2P2ðaÞþP3ðaÞ ( þX d2 i¼0 kxðl2þi d2; j þ 1Þk2Q2ðaÞþQ3ðaÞ þX d21 i¼0
ðd2iÞkdðl21 i; j þ 1Þk2M2ðaÞþN2ðaÞ
) þg1 X l2 i¼0 X l11 j¼0 k^zði; jÞk2þX l21 i¼0 Xl1 j¼0 k^zði; jÞk2 ( ) pg X l2 i¼0 X l11 j¼0 kwði; jÞk2X l11 j¼0 kwð0; jÞk2 ( þX l21 i¼0 Xl1 j¼0 kwði; jÞk2X l21 i¼0 kwði; 0Þk2 ) . ð59Þ
Since P3ðaÞX0, Q3ðaÞX0, PlðaÞ40, QlðaÞ40,
MlðaÞ40 and NlðaÞ40, l ¼ 1; 2, the inequality (59)
implies 2g1X 1 i¼0 X1 j¼0 k^zði; jÞk2p2gX 1 i¼0 X1 j¼0 kwði; jÞk2 (60)
when l1 ! 1and l2! 1. Hence, the H1-norm of
system (7) is no greater than g. This completes the proof. &
Remark 6. Theorem 5 provides a new delay-dependent bounded real lemma for 2-D state-delayed systems. Similar to the case of Remark 4, when nX0 approaches zero Theorem 5 with
F1¼ nI4n ¯d1 ; M1¼ nIn ¯d1 ; K1 ¼ nI2m ¯d1 ; N1¼ nIn ¯d1 , F2¼ nI4n ¯d2 ; M2¼ nIn ¯d2 ; K2 ¼ nI2m ¯d2 ; N2¼ nIn ¯d2 , H ¼ 0; P3¼Q3¼0,
reduces to the delay-independent conditions in Theo-rem 2 of [14] with no uncertainty. In another case, when the 2-D system has no state-delays, i.e., Ad1¼
Ad2¼0, Theorem 5 reduces to Theorem 1 of[11].
4. Synthesis of robust filters
The system matrices (8) of the filtering error dynamics (7) can be expressed as
^ AðaÞ ¼X t j¼1 ajA^ðjÞ; A^dðaÞ ¼ Xt j¼1 ajA^ðjÞd , ^ BðaÞ ¼X t j¼1 ajB^ðjÞ; LðaÞ ¼^ Xt j¼1 ajL^ðjÞ, ð61Þ where ^ AðjÞ¼JTAðjÞJ þ I~ TNðI2CðjÞÞ, ^ AðjÞd ¼JTAðjÞd , ^ BðjÞ¼JTBðjÞþITNðI2DðjÞÞ, ^ LðjÞ¼LðjÞJ LfI, AðjÞ¼ ½AðjÞ1 AðjÞ2 ; AðjÞd ¼ ½AðjÞd1 AðjÞd2,
BðjÞ¼ ½BðjÞ1 BðjÞ2 ; I ¼ ½0 In, CðjÞ¼ CðjÞ 0 0 In " # ; DðjÞ¼ D ðjÞ 0 " # , N ¼ ½Bf1 Af1 Bf2 Af2. ð62Þ
Therefore ^AðjÞ, ^BðjÞ and ^LðjÞ are affine functions of the filter system matrix variables N and Lf. This fact
is useful in the subsequent development. To handle the polytopic uncertain system with less conserva-tive design results, the use of a parameter-dependent Lyapunov function allows different Lyapunov matrices to be set for different parts of the entire polytopic domain. The following lemma is pre-sented as a preparation step.
Lemma 7. Given positive integers ¯d1, ¯d2 and a real
number g40, if there exist matrices 0oPðjÞ¼
PðjÞ1 PðjÞT3 % PðjÞ2 2R4n4n, QðjÞ¼ Q ðjÞ 1 QðjÞT3 % QðjÞ2 2R2n2n, S 2 R2n2n, MðjÞl 2Rnn,NðjÞl 2Rnn, FðjÞl 2R4n4n, GðjÞl 2 R2mn, HðjÞl 2R4nn, KlðjÞ2R2m2m and Vl2Rnn, l ¼ 1; 2, such that QðjÞ3 ¼QðjÞT3 X0; (63) KðjÞl % GðjÞTl NðjÞl 2 4 3 540; l ¼ 1; 2, (64) PðjÞ3 ¼PðjÞT3 X0, (65) FðjÞl % HðjÞTl MðjÞl 2 4 3 540; l ¼ 1; 2, (66)
hold for j ¼ 1; 2; . . . ; t, where UðjÞ¼PðjÞ ¯d1FðjÞ1 ¯d2FðjÞ2 H ðjÞJ ~~ JTHðjÞT, WðjÞ¼gI2m ¯d1KðjÞ1 ¯d2KðjÞ2 , HðjÞ¼ ½HðjÞ1 HðjÞ2 ; GðjÞ¼ ½GðjÞ1 GðjÞ2 , KðjÞ1 ¼ ðS þ STÞ RðjÞ, RðjÞ¼PðjÞ1 þ2PðjÞ3 þPðjÞ2 þJT½QðjÞ1 þ2QðjÞ3 þQðjÞ2 J, KðjÞ2 ¼ ðV1þV1TÞ ¯d1ðMðjÞ1 þN ðjÞ 1 Þ, KðjÞ3 ¼ ðV2þV2TÞ ¯d2ðMðjÞ2 þN ðjÞ 2 Þ, ð68Þ
then PðaÞ¼Ptj¼1ajPðjÞ, QðaÞ¼Ptj¼1ajQðjÞ, FlðaÞ¼
Pt
j¼1ajFðjÞl , GlðaÞ¼Ptj¼1ajGðjÞl , HlðaÞ¼Ptj¼1ajHðjÞl ,
KlðaÞ¼Ptj¼1ajKðjÞl , MlðaÞ¼Ptj¼1ajMðjÞl and NlðaÞ¼
Pt
j¼1ajNðjÞl , l ¼ 1; 2, satisfy (33)–(37) of Theorem 5
for all a in the unit simplex (4).
Proof. Clearly, (63)–(66) imply (33)–(36), respec-tively, for all a in the unit simplex (4). Then, for KlðaÞ ¼Ptj¼1ajKðjÞl , l ¼ 1; 2; 3, and all a in the unit
simplex (4), (67) guarantees
Notice that (67) also ensures the nonsingularity of
UðjÞ % % % % % % HðjÞT QðjÞ % % % % % GðjÞJ~ GðjÞ WðjÞ % % % % S ^AðjÞ S ^AðjÞd S ^BðjÞ KðjÞ 1 % % % V1ðJ ^AðjÞJ1Þ V1J ^AðjÞd V1J ^BðjÞ 0 KðjÞ2 % % V2ðJ ^AðjÞJ2Þ V2J ^AðjÞd V2J ^BðjÞ 0 0 KðjÞ3 % ^ LðjÞI2 0 0 0 0 0 gI2q 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 (67) UðaÞ % % % % % % HTðaÞ QðaÞ % % % % %
GðaÞ ~J GðaÞ WðaÞ % % % %
S ^AðaÞ S ^AdðaÞ S ^BðaÞ K1ðaÞ % % % V1½J ^AðaÞ J1 V1J ^AdðaÞ V1J ^BðaÞ 0 K2ðaÞ % % V2½J ^AðaÞ J2 V2J ^AdðaÞ V2J ^BðaÞ 0 0 K3ðaÞ %
^ LðaÞ I2 0 0 0 0 0 gI2q 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 o0. (69)
S, V1 and V2. Thus, performing the congruence
transformation diagfI4n; I2n; I2m; ST; VT1 ; V T 2 ; I2qg
to (69) results in
where ^K1ðaÞ ¼ S1K1ðaÞST, ^K2ðaÞ ¼ V11 K2ðaÞVT1
and ^K3ðaÞ ¼ V12 K3ðaÞVT2 . Since
½R1ðaÞ S1RðaÞ½R1ðaÞ S1TX0,
f ¯d11 ½M1ðaÞ þ N1ðaÞ1V11 g ¯d1½M1ðaÞ þ N1ðaÞ
f ¯d11 ½M1ðaÞ þ N1ðaÞ1V11 g TX0,
f ¯d12 ½M2ðaÞ þ N2ðaÞ1V12 g ¯d2½M2ðaÞ þ N2ðaÞ
f ¯d12 ½M2ðaÞ þ N2ðaÞ1V12 g TX0, one has R1ðaÞpS1RðaÞST ðS1þSTÞ, ¯d11 ½M1ðaÞ þ N1ðaÞ1 p¯d1V11 ½M1ðaÞ þ N1ðaÞVT1 ðV 1 1 þV T 1 Þ, ¯d12 ½M2ðaÞ þ N2ðaÞ1 p¯d2V12 ½M2ðaÞ þ N2ðaÞVT2 ðV12 þVT2 Þ.
Thus (67) implies (37) for all a in the unit simplex (4). &
It should be noted that (67) is not an LMI in the matrix variables S, Af1, Af2, Bf1 and Bf2, but can be
converted into one via proper transformations. This will be done in the next theorem, and an LMI based method will be developed for designing a filter (6) such that the H1-norm constraint is satisfied.
Theorem 8. Consider system (1). Given positive integers ¯d1, ¯d2 and a real number g40, if there exist
matrices S ¼^ S^1 ^ S2 ^ S3 ^ S3 h i 2R2n2n, 0o ^PðjÞ¼ ^ PðjÞ1 % ^ PðjÞT3 ^ PðjÞ2 2R4n4n, QðjÞ¼ Q ðjÞ 1 QðjÞT3 % QðjÞ2 2R2n2n, ^N 2 Rnð2nþ2mÞ, L^f 2Rqn, MlðjÞ2Rnn, NðjÞl 2Rnn, ^ FðjÞl 2R4n4n, GðjÞl 2R2mn, H^ðjÞl 2R4nn, KðjÞl 2 R2m2m and Vl 2Rnn, l ¼ 1; 2, such that (63)–(64)
and the following LMIs:
^ PðjÞ3 ¼ ^PðjÞT3 X0, ð71Þ ^ FðjÞl % ^ HðjÞTl MðjÞl 2 4 3 540, ð72Þ H11 % % % % % % ^HðjÞT QðjÞ % % % % % GðjÞJ~ GðjÞ H33 % % % % H41 H42 H43 H44 % % % H51 V1AðjÞd V1BðjÞ 0 H55 % % H61 V2AðjÞd V2BðjÞ 0 0 H66 % H71 0 0 0 0 0 gI2q 2 6 6 6 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 7 7 7 5 o0 ð73Þ hold for j ¼ 1; 2; . . . ; t, where
H11¼ ^PðjÞ ¯d1F^ðjÞ1 ¯d2F^ðjÞ2 ^H ðjÞJ ~~ JTH^ðjÞT, H33¼gI2m ¯d1KðjÞ1 ¯d2KðjÞ2 , H41¼ ^SJTAðjÞJ þ P~ TNðI^ 2CðjÞÞ, H42¼ ^SJTAðjÞd , H43¼ ^SJTBðjÞþPTNðI^ 2DðjÞÞ, H44 ¼ ^S þ ^ST ^P1ðjÞ2 ^PðjÞ3 ^PðjÞ2 JTðQðjÞ1 þ2QðjÞ3 þQðjÞ2 ÞJ, H51¼V1ðAðjÞJ J~ 1Þ, H55¼V1þV1T ¯d1ðMðjÞ1 þN ðjÞ 1 Þ, H61¼V2ðAðjÞJ J~ 2Þ, UðaÞ % % % % % % HTðaÞ QðaÞ % % % % %
GðaÞ ~J GðaÞ WðaÞ % % % %
^
AðaÞ A^dðaÞ BðaÞ^ ^K1ðaÞ % % % J ^AðaÞ J1 J ^AdðaÞ J ^BðaÞ 0 ^K2ðaÞ % % J ^AðaÞ J2 J ^AdðaÞ J ^BðaÞ 0 0 ^K3ðaÞ %
^ LðaÞ I2 0 0 0 0 0 gI2q 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 o0, (70)
H66¼V2þV2T ¯d2ðMðjÞ2 þN ðjÞ 2 Þ, H71¼I2 ðLðjÞJÞ I2 ð ^LfIÞ, ^ HðjÞ¼ ½ ^HðjÞ1 H^ðjÞ2 , GðjÞ¼ ½GðjÞ1 GðjÞ2 , P ¼ ½In In, (74)
then the H1 filtering problem is solvable for
di2 ½0; ¯di, i ¼ 1; 2, and the filter system matrices
for (6) can be obtained from any feasible ^S and ^N ¼ ½ ^Bf1 A^f1 B^f2 A^f2 as N ¼ ½ ^Bf1 A^f1^S 1 3 B^f2 A^f2S^ 1 3 in (62) and Lf ¼ ^LfS^ 1 3 .
Proof. In (73), H44o0 ensures that ^S is
non-singular, which implies that ^S3 is nonsingular.
Define U ¼ In 0
0 S^T3 " #
, (75)
eU ¼ I2U, and change the variables as
PðjÞ¼ eU1P^ðjÞeUT, S ¼ U1SU^ T ¼ ^ S1 In ^ST 3 ^S2 S^T3 2 4 3 5, FðjÞl ¼ eU1F^ðjÞl eUT; l ¼ 1; 2, HðjÞl ¼ eU1H^ðjÞl ; l ¼ 1; 2, N ¼ ^N ~UT; Lf ¼ ^Lf^S13 . ð76Þ
Then, it is easy to check that JU ¼ JUT¼J, UTDðjÞ¼DðjÞ, UTCðjÞ¼CðjÞUT, USIT¼PT, ^ S3I ¼ IUT, US ^AðjÞeUT¼ ^SJTAðjÞJ þ P~ TNðI^ 2CðjÞÞ, US ^AðjÞd ¼ ^SJTAðjÞd , US ^BðjÞ¼ ^SJTBðjÞþPTNðI^ 2DðjÞÞ, V1ðJ ^AðjÞJ1ÞeUT¼V1ðAðjÞJ J~ 1Þ, V2ðJ ^AðjÞJ2ÞeUT¼V2ðAðjÞJ J~ 2Þ, ^ LðjÞUT¼LðjÞJ ^LfI. ð77Þ
Applying the congruence transformations UT, diagfeUT; Ing and diagfeUT; I2n; I2m; UT; In; In; I2qg
to (71), (72) and (73), respectively, yield (65), (66) and (67). Thus the proof is completed. &
Remark 9. With the variables S, V^ 1 and V2
independent of the index j, Theorem 8 provides an effective method to solve the robust H1 filtering
problems involving parameter-dependent system matrices. Recently, a new approach is proposed in
[1], in which all Lyapunov matrix variables are set to be parameter-dependent. This new methods in[1]
may offer less conservative results, especially for 1-D polytopic systems. However, the filtering synth-esis method in [1] for 2-D polytopic systems will involve matrices of quite large dimensions, and the number of LMIs increases faster as the number of vertices in the polytope domain increases.
Remark 10. In Theorem 8, g is a given real number. But the conditions in Theorem 8 are still LMIs when g is regarded as a variable. Thus it is possible to formulate the following LMI optimization problem to find a filter with the smallest H1-norm:
Minimize: g
subject to: ð63Þ2ð64Þ; ð71Þ2ð73Þ ð78Þ with respect to g and the variables stated in Theorem 8.
In the delay-independent case, Theorem 8 may be reduced to following simpler results.
Corollary 11. Consider system (1) and let g40 be a given real number. If there exist matrices
^ S ¼ S^1 ^ S2 ^ S3 ^ S3 h i , ^PðjÞ¼ P^ ðjÞ 1 ^ PðjÞT3 % ^ PðjÞ2 , QðjÞ¼ Q ðjÞ 1 QðjÞT3 % QðjÞ2 , ^N and ^Lf such that the LMI
^PðjÞ % % % % 0 QðjÞ % % % 0 0 gI2m % % H41 H42 H43 H44 % H71 0 0 0 gI2q 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 o0 (79)
hold for j ¼ 1; 2; . . . ; t, where H41, H42, H43, H44and
H71 are defined in Theorem 8, then the H1 filtering
problem is solvable for all delay sizes, and the filter system matrices in (6) can be obtained from any feasible S^ and N ¼ ½ ^^ Bf1 A^f1 B^f2 A^f2 as N ¼
½ ^Bf1 A^f1^S13 B^f2 A^f2S^13 in (62) and Lf ¼ ^LfS^13 .
Remark 12. In Corollary 11, an upper bound of H1-norm can be found from the following LMI
Minimize: g
subject to: ð79Þ ð80Þ
with respect to g and the variables stated in Corollary 11.
Remark 13. When Theorem 8 and Corollary 11 are compared, it is seen that matrices of the same dimensions are involved in (73) and (79), but the number of matrix variables to be determined in Corollary 11 is less than that in Theorem 8.
5. Numerical example
Consider a heat diffusion system along a line described by the partial differential equation
q2
qx2uðx; tÞ ¼ cðaÞ q
qtuðx; tÞ þ f ðx; tÞ þ wðx; tÞ, (81) which has been studied in[14]. In (81), x 2 ½0; ¯x is the spatial variable, t 2 ½0; 1Þ is the time variable, uðx; tÞ is the temperature of the line at x and t, cðaÞ is the thermal diffusivity depending on an uncertain parameter vector a ¼ ½a1 a2T, f ðx; tÞ is the control
input, and wðx; tÞ is the noise input. Suppose cðaÞ ¼ 0:8 0:4a1þ0:32a2 and the system is controlled by
a ‘‘mixed’’ state feedback law f ðx; tÞ ¼ 200uðx; tÞþ 102uðx x0; tÞ, where x0 ¼0:1. By the central and
back difference approximations as in[14], (81) can be converted into the Fornasini–Marchesini second model of the form (1) with
xði; jÞ ¼ ½uði; jÞ 0:5cðaÞuði; j 1Þ þ 0:5uði 1; jÞ T,
yði; jÞ ¼ cðaÞuði; j 1Þ þ uði 1; jÞ, zði; jÞ ¼ uði; jÞ, A1ðaÞ ¼ 0 0 0:5cðaÞ 0 " # ; A2ðaÞ ¼ 0:1cðaÞ 0:2 0:5 0 " # , Ad1 ¼0; Ad2¼ 0:12 0 0 0 " # , B1¼0; B2¼ ½0:01 0T, C ¼ ½0 2; D ¼ 0; L ¼ ½1 0, d1¼0; d2¼1.
The matrices A1ðaÞ and A2ðaÞ can be, respectively,
represented by A1ðaÞ ¼ L0þa1L1þa2L2, ð82Þ A2ðaÞ ¼ P0þa1P1þa2P2, ð83Þ where L0¼ 0 0 0:4 0 " # ; L1¼ 0 0 0:2 0 " # , L2¼ 0 0 0:16 0 " # , P0¼ 0:08 0:2 0:5 0 " # ; P1 ¼ 0:04 0 0 0 " # , P2¼ 0:032 0 0 0 " # .
By solving the optimization problem (78) from Theorem 8 and the optimization problem (80) from Corollary 11, the optimal H1-norm bounds of
0:0184 and 1, respectively, are obtained. The bound of 1 means no feasible solutions exist for the delay-independent case. For the delay-depen-dent case the corresponding filter matrices are
Af1¼ 0:0107 0:5503 0:0063 0:9748 " # 103, Af2¼ 0:0998 1:7315 0:0003 0:0078 " # , Bf1¼ 0:2696 0:9646 " # 104; Bf2¼ 13:9253 0:1860 " # , Lf ¼ ½0:0083 0:0159.
Fig. 1shows the magnitude plot of the filtering error dynamics over grid frequencies in the range of ½p; p for a1 ¼1 and a2¼0. It can be seen that the
maximum magnitude is below the guaranteed H1
-norm bound. This is also true for other checked uncertainties a ¼ ½0 1T, ½0:1 0:9T; . . . ; ½0:9 0:1T.
6. Conclusion
For 2-D state-delayed systems with polytopic uncertainties described by the Fornasini–Marchesini second model, this paper proposes filter synthesis methods under the LMI framework. Based on the delay-dependent H1-norm performance criteria,
effective methods for solving the robust H1filtering
problems with a parameter-dependent Lyapunov function approach are derived. An example is given to illustrate the usage of the proposed methods.
A natural extension of current results would be the consideration of feedback control problems for the 2-D state-delayed systems. However, when
applied to the control problems the conditions in Theorems 3 and 5 will become non-convex bilinear matrix inequalities. To develop a set of new LMI-based delay-dependent conditions for the design of feedback controllers is a research topic yet to be studied.
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful and valuable comments that help improve this paper. This research is supported by the National Science Council of Taiwan under Grant NSC 95-2221-E-002-130-MY3.
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