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Robust H∞ Filter Design with Filter Pole Constraints via the π-Sharing Theory

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Robust H

N

filter design with filter pole constraints

via p-sharing theory

C.-M. Lee and I-K. Fong

Abstract: The robust HNfiltering problem subject to pole-placement constraints for continuous-time systems with the polytopic type uncertainties is considered. Different from those considered in the literature, the regional pole-placement constraints considered here focus on the filter dynamics. To solve the problem, the p-sharing theory is extended to offer a stability criterion that covers the bounded real lemma as a special case, and the linear matrix inequality approach is adopted to develop filter design methods based on the convex optimisation procedure. One numerical example is given to illustrate the proposed methods.

1 Introduction

The filter design problem for dynamic systems is important in many engineering applications. The filter design techni-ques therefore have received a large amount of attention in the literature, for both theoretical and practical aspects. In recent years, the convex optimisation based filter design methods under the linear matrix inequality (LMI)

frame-work [1] are very popular because of the efficient

computation algorithms that are available. Also, the consideration of system poles has become an important issue in the filter design problem [2–4], just as in the feedback control problems [5–7]. There are some studies trying to consider the poles in the robust filtering problem using the LMI approach [8–11]. However, in these works there is a common assumption. In order to conveniently impose the pole-placement constraints of the filter, the poles of the entire filtering error dynamics are required to lie inside some desired regions. Thus poles of the system for which signals are to be estimated must also be assumed to lie inside the desired regions. In general, such an assumption not only limits its applicable domain, but also often results in more conservative designs. Unlike in the feedback control problems, for which it is natural and feasible to require poles of the overall system be placed into a certain region, in the filtering problem the only poles that need to be placed are those of the filters. Thus it is more reasonable to merely assume that the system for which signals are to be estimated is stable, i.e., with all poles located in the open left half of the complex plane [12]. With this in mind, this paper considers the robust filtering problem subject to the D-stability[5]constraint for the filter, which enables the filter designers to shape the filter characteristics in a flexible fashion[13, 14].

The concept of energy storage and dissipation in the circuit theory is well inherited in system theory[15], and is applied fruitfully to many control and engineering problems,

such as the stability analysis problems [16–18], the design

and synthesis problems for controllers and filters

[19–21], and so on. Along this line, the p-sharing theory

[22] is an extension of the concepts of passivity [23] and dissipativity [24]. It uses the so-called p-coefficients to describe the ‘energy storage and dissipation’ of systems, and the so-called p-stability[22]is able to deal with both the Lyapunov stability and input-output stability simulta-neously. After proper development[25], the p-sharing theory is shown to be applicable to MIMO systems within a convenient LMI framework. Successful application of the p-sharing theory to the controller design problem can be found in[26]. However, the p-sharing theory in[25, 26]is limited to square systems. Here, an extended p-sharing theory will be developed under the LMI framework, and applied to the above filter design problem, which involves non-square systems. As can be seen subsequently, the p-stability from the extended p-sharing theory covers the bounded real lemma[1, 27]as a special case.

Some notations to be adopted are introduced first. The inequality X  0 means that the matrix X is symmetric and

positive semi-definite, and X  Y means X  Y  0.

Similar definitions apply to symmetric positive/negative definite matrices. Let x(t) and UðtÞ, respectively, be any real vector and symmetric matrix functions of the continuous-time continuous-time index t. Then ðUÞjxðtÞj2¼ xTðtÞUðtÞxðtÞ and

ðUÞkxk2T ¼

RT

0 xTðtÞUðtÞxðtÞdt, where T is a nonnegative

constant. If U¼ I, the identity matrix, then it is omitted from the notations. Finally, for any matrix Z, kZk represents its induced two-norm, and for any symmetric matrix X, lmaxðXÞ and lminðXÞ denote its maximal and

minimal eigenvalues, respectively.

2 p-sharing theory and problem formulation

2.1

Multivariable p-sharing theory

First, the continuous-time multivariable p-sharing theory

[25]is extended in this subsection, but restricted to the linear time-invariant (LTI) case. Consider the system Sowith the

state-space model

_xðtÞ ¼ AxðtÞ þ BuðtÞ

yðtÞ ¼ CxðtÞ þ DuðtÞ ð1Þ

E-mail: ikfong@cc.ee.ntu.edu.tw

The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China

rThe Institution of Engineering and Technology 2006 IEE Proceedings online no. 20045104

doi:10.1049/ip-cds:20045104

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where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm

is the input vector, and yðtÞ 2 Rp is the output vector. The system matrices A, B, C, D are of appropriate dimensions. The system Sois said[22, 25]to be p-sharing with respect to the

constant p-coefficientsfS; C; Q; P; Rg, if for all T  0 Z T 0 uTðtÞSyðtÞdt ðCÞjxðT Þj2 ðCÞjxð0Þj2 þ ðQÞkxk2T þ ðPÞkyk 2 T þ ðRÞkuk 2 T ð2Þ where C; Q2 Rnn

are positive semi-definite symmetric, P2 Rppand R2 Rmmare symmetric, and S 2 Rmpis subject to no special constraint. Define the dissipativity matrix[25]of the system Soas

Mdis¼ M11 M12 MT12 M22 " # ð3Þ where M11¼ ATCþ CA þ Q þ CTPC ; M12¼ CB þ CTPD 1 2C T ST; M22¼ DTPDþ R  1 2ðSD þ D TSTÞ

The dissipativity matrix may be used to qualify a set of p-coefficients in terms of its negative semi-definiteness in the next lemma.

Lemma 1 [22, 25]: If Mdis 0, C  0, and Q  0, then

system So in (1) is p-sharing with respect to fS; C;

Q; P; Rg.

Note that the conditions in the above lemma are LMIs with respect tofS; C; Q; P; Rg for the given So.

In the p-sharing theory, the p-stability is defined to include state and input-output stability at the same time. Below is the definition of the p-stability.

Definition 1[22, 25]: The system (1) is p-stable, if there exist g1; . . . ;g42 R such that

kykT  g1kukT þ g2jxð0Þj;

sup

0tT

jxðtÞj  g3kukT þ g4jxð0Þj

for all uðtÞ 2 Rm, xð0Þ 2 Rn, and T  0.

Obviously, the first condition in Definition 1 is for theL2

stability, and the second one implies stability in the sense of Lyapunov when the external input u 0. The next lemma, adapted from [22, 25] for the LTI case, gives a sufficient condition for the p-stability in terms of the p-coefficients. Lemma 2: If the system Soin (1) is p-sharing with respect to

fS; C; Q; P; Rg with C  gI40, R  r0I , and P p0I 40,

then it is p-stable with g1¼ ðs

ffiffiffiffiffiffiffiffi p0d p Þ=p0, g2¼ ffiffiffiffiffiffiffiffiffiffiffiffi g0=p0 p , and g3¼ g4¼ ffiffiffiffiffiffiffi x=g p

, where d¼ j minf0; r0gj, g0¼ lmaxðCÞ,

x¼ maxfg0;dþ ðs0þ ffiffiffiffiffiffiffiffi p0d p Þ=p0; ffiffiffiffiffiffiffiffiffiffiffiffi g0=p0 p g, and s0¼ kSk.

Compared with the original p-sharing theory[22, 25], the extended p-sharing theory usesRT0 uTðtÞSyðtÞdt instead of

RT

0 u

TðtÞyðtÞdt in the left-hand side of (2). Thus the

difference in form is not large, and the proofs for lemmas 1 and 2 can be extended from the corresponding proofs in

[22, 25]in a straightforward fashion. For the sake of brevity, the proofs are omitted here. However, the introduction of S not only makes handling non-square systems (with m6¼ p)

possible, but also lets lemma 2 cover the bounded real

lemma [27] as a special case. For example, the HN

performance specificationkyk2T  Z2kuk

2

T for a given Z40

may be formulated as the p-stability of system So with

fS; C; Q; P; Rg ¼ f0; C; 0; I; Z2Ig.

2.2

A robust filtering problem

Consider the deconvolution filtering system in Fig. 1. The source signal sðtÞ 2 Rps is assumed to be generated by the

signal model SS: _xsðtÞ ¼ AsxsðtÞ þ BswðtÞ sðtÞ ¼ CsxsðtÞ þ DswðtÞ ( ð4Þ where xsðtÞ 2 Rns is the model state vector, wðtÞ 2 Rms is

the driving signal vector with each element inL2½0; 1Þ, and

As, Bs, Cs, Dsare known constant matrices of appropriate dimensions. The source signals are transmitted through a channel system with an uncertain characteristic modelled by

SC:

_xcðtÞ ¼ AcxcðtÞ þ BcsðtÞ

zcðtÞ ¼ CcxcðtÞ þ DcsðtÞ

(

ð5Þ where xcðtÞ 2 Rnc and zcðtÞ 2 Rpc are the channel state

and output signal vectors, respectively. The channel system matrices fAc;Bc;Cc;Dcg are only known to belong to a

polytopic set Dc ðAc;Bc;Cc;DcÞ ¼ Xl i¼1 tiðAci;Bci;Cci;DciÞ ( ) ð6Þ where all uncertain parameters ti, i¼ 1; 2; . . . ; l, are non-negative and satisfyPli¼1 ti¼ 1, and all vertices fAci;Bci;

Cci;Dcig of the polytope are known and designated by

i¼ 1; 2; . . . ; l.

At the receiving end, the measured signal vector yðtÞ 2 Rpcis equal to z

cðtÞ þ vðtÞ, where v(t) is an energy-bounded

channel noise vector. To integrate, one can combine the signal and channel models as

S : _xðtÞ ¼ AxðtÞ þ BueðtÞ yðtÞ ¼ CxðtÞ þ DueðtÞ sðtÞ ¼ LxðtÞ þ JueðtÞ 8 > > < > > : ð7Þ where xT¼½xT s xTc, uTe¼½wT vT, L¼½Cs 0, and J ¼½Ds 0,

and define the polytopic set Dz¼ ðA; B; C; DÞ ¼ Xl i¼1 tiðAi;Bi;Ci;DiÞ ( ) ð8Þ where Ai¼ As 0 BciCs Aci   ; Bi¼ Bs 0 BciDs 0   ; Ci¼ D½ ciCs Cci; Di¼ D½ ciDs I ð9Þ To optimally recover the source signals sðtÞ, the signal vector yðtÞ is deconvoluted by a filter with order

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nf ¼ nsþ nc: SF: _xfðtÞ ¼ AfxfðtÞ þ BfyðtÞ sfðtÞ ¼ CfxfðtÞ þ DfyðtÞ  ð10Þ where xfðtÞ 2 Rnf is the filter state vector, sfðtÞ 2 Rps is

the filter output vector, and Af;Bf;Cf;Df are filter

system matrices to be designed. Although choosing a fixed filter under the uncertain channel environment may cause some conservativeness than choosing parameter-dependent filters[28, 29], there are also advantages. A fixed filter does not require the knowledge of exact parameter values of ti; i¼ 1; 2; . . . ; l, and is easier to implement.

Define the filtering error as eðtÞ ¼ sðtÞ  sfðtÞ, which

satisfies Se: _xeðtÞ ¼ AexeðtÞ þ BeueðtÞ eðtÞ ¼ CexeðtÞ þ DeueðtÞ  ð11Þ with xT

eðtÞ ¼ ½xTðtÞ xTfðtÞ. The polytopic set De is defined

as De¼ ðAe;Be;Ce;DeÞ ¼ Xl i¼1 tiðAei;Bei;Cei;DeiÞ ( ) ð12Þ where Aei¼ Ai 0 BfCi Af   ; Bei¼ Bi BfDi   Cei¼ L  DfCi Cf   ; Dei¼ J  DfDi ð13Þ

The purpose of this paper is to design a filter SFwith the

three desired properties below.

DP-1: The filtering error dynamics Se is bounded stable

[30], i.e., there exists a constant b 0 such that jxeðtÞj  b

for all t 0, no matter what initial condition xeð0Þ and

input ueðÞ 2 L2½0; 1Þ are.

DP-2: When xeð0Þ ¼ 0, the filter has the H1 performance

keðtÞk2T  m2kueðtÞk2T ð14Þ

for some scalar m40 and all ue6¼ 0, as T ! 1.

DP-3: Poles of the filter must be constrained in the following region(s)[5, 31]of the x–y plane:

 PC-1: Disc region PDðc0; r0Þ centred at the point ðc0;0Þ

with radius r0.

 PC-2: Vertical strip PVðr1; r2Þ lying between the lines x¼ r1 and x¼ r2 on the x–y plane, where r1or22 R.

 PC-3: Left conic sector PLðcl;ylÞ with the apex at the

pointðcl;0Þ and inner angle yl, where 0 yl p.

 PC-4: Right conic sector PRðcr;yrÞ with the apex at the

point ðcr;0Þ and the inner angle yr, where cr 0 and

0 yr p.

From the above subsection, it is easy to see that both DP-1 and DP-2 may be implied by the p-stability. More specifically, if Se is p-sharing with respect to the

p-coefficients fSe;Ce;Qe;Pe;Reg satisfying the matrix

inequalities ATeiCeþ CeAeiþ Qe CeBei 1 2C T eiS T e C T ei BTeiCe 1 2SeCei Re 1 2ðSeDeiþ D T eiS T eÞ D T ei Cei Dei P1e 2 6 6 6 6 4 3 7 7 7 7 5 0 ð15Þ Ce40; Qe 0 ð16Þ Pe  1 2S T e 1 2Se Re 2 6 4 3 7 5  I0 m02I   ð17Þ

for all i¼ 1; 2; . . . ; l, then not only the H1 performance

specification (14), but also the bounded stability of Se is

ensured, since from lemma 2

keðtÞkT  mkueðtÞkT þ g2jxeð0Þj ð18Þ

for some g240.

The desired property DP-3 is useful for shaping the filter characteristics [13]. The corresponding conditions [5, 31]

are described below, which can be applied individually or together.

 PC-1: Existence of a matrix Cf40 such that

ðAf c0IÞTCfðAf c0IÞ  r20Cfo 0 ð19Þ

 PC-2: Existence of a matrix Cf40 such that

ðAf r2IÞTCf þ CfðAf r2IÞo 0

ðr1I AfÞTCf þ Cfðr1I AfÞo 0

ð20Þ  PC-3: Existence of a matrix Cf40 such that

sinyl

2½ðAf cl

T

Cfþ CfðAf clIÞ

cosyl

2½CfðAf clIÞ  ðAf cl

T Cf 2 6 6 6 4 cosyl 2½ðAf clIÞ T Cf CfðAf clIÞ sinyl 2½ðAf clIÞ T Cfþ CfðAf clIÞ 3 7 7 7 5o 0 ð21Þ

 PC-4: Existence of a matrix Cf40 such that

sinyr 2½ðcrI AfÞ T Cf þ CfðcrI AfÞ cosyr 2½CfðcrI AfÞ  ðcrI AfÞ T Cf 2 6 6 6 4 cosyr 2½ðcrI AfÞ T Cf CfðcrI AfÞ sinyr 2½ðcrI AfÞ T Cfþ CfðcrI AfÞ 3 7 7 7 5o 0 ð22Þ

It is noted that there are other regions and conditions for the pole placement constraint, such as those presented in

[5, 32]. Basically it is also possible to accommodate these conditions in this paper, but the details are omitted for the sake of brevity.

3 Main results

3.1

Robust H

N

filter design

To start developing the filter design method, the p-stability conditions for the filtering error dynamics Seare utilised in

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Theorem 1: If there exist feasible solutions m2, U, W, P1e ,

Re, X, Y, M, N, Z, and Df to the following matrix

inequalities

ATiUþUAiþW ðXAiþZCiþMÞTþUAiþW

XAiþZCiþM þATiUþW XAiþZCiþðXAiþZCiÞTþY

BTiU1 2SeðLDfCiNÞ ðXBiþZDiÞ T 1 2SeðLDfCiÞ LDfCiN LDfCi 2 6 6 6 6 6 6 6 6 4 UBi 1 2ðLDfCiNÞ T STe ðLDfCNÞT XBiþZDi 1 2ðLDfCÞ T STe ðLDfCiÞT Re 1 2½SeðJ DfDiÞþðJ DfDiÞ T STe ðJ DfDiÞT JDfDi P1e 3 7 7 7 7 7 7 7 7 7 7 7 5 0 ð23Þ U U U X " # 40; Y  W  0; m240 ð24Þ P1e 1 2P 1 e S T e P 1 e 1 2SeP 1 e m2I Re 0 P1e 0 I 2 6 6 6 6 6 4 3 7 7 7 7 7 5  0 ð25Þ

for all i¼ 1; . . . ; l, then the filter SFwith the gain matrices

Af ¼ U1MUT; Cf ¼ NUT;

Bf ¼ U1Z; Df ¼ Df

ð26Þ satisfies the desired properties DP-1 and DP-2 of the filtering problem, where U is any non-singular matrix satisfying UUT¼ X  U.

Proof: By the Schur complement[1]and the first inequality

in (24), U40 and X U40. Thus, I  XU1 is

non-singular and there exist non-non-singular matrices U and V such that I XU1¼ UVT. Let ^ T¼ U 1 I VT 0 " # and T¼ I X 0 UT " # ð27Þ where ^T is non-singular since

^ T1 ¼ 0 V T I U1VT " # Define Ce¼ T ^T 1 ¼ Ce1 Ce0 CTe0 Ce2 " # ¼ X U UT I " #

by letting U¼ UV. Under this arrangement Ce40

because X UUT¼ X þ UVTU¼ U40.

Next, pre- and post-multiply (23) by diagðU1;I ; I ; IÞ at the same time. Then with (13), (26), (27), U¼ UV,

Ce¼ T ^T

1

, and Qe¼

Y ðY  WÞUT

U1ðY  WÞ U1ðY  WÞUT

2 4

3

5 ð28Þ

the result can be re-written as U11 U12 T^ T CTei U12T U22 DTei CeiT^ Dei P1e 2 6 6 6 4 3 7 7 7 5 0 ð29Þ where U11¼ ^T T ATeiCeT^þ ^T T CeAeiT^þ ^T T QeT^ U12¼ ^T T CeBei 1 2 ^ TTCTeiSTe U22¼ Re1 2ðSeDeiþ D T eiSTeÞ

Note that Qe 0 by (24). Now, pre- and post-multiplying

(29), respectively, by diagð ^TT;I ; IÞ and diagð ^T1;I ; IÞ results in (15). Therefore from lemma 2 the error dynamics Sein (11) is p-stable with respect to the p-coefficientsfSe;Ce;

Qe;Pe;Reg found in the proof.

Finally, with m240, (25) may be re-written as (17) using

the Schur complement. Hence the H1 performance

constraint in (14) is satisfied.

It is worth noting that, the result in[10]may be regarded as a special case of what is obtained here with fSe;Ce;

Qe;Pe;Reg ¼ f0; Ce; 0;I ;m2Ig.

Apparently, the matrix inequalities in theorem 1 are not linear, but bilinear with respect to the variables. However, closer examination reveals that these inequalities are linear with respect to m2 and V ¼ fU; W; P1e ;Re;X; Y; M; N ;

Z; Dfg if Se is held constant. Also, these inequalities are

linear with respect to m2and SeifV is held constant. Thus a coordinate-by-coordinate minimisation [33] procedure is proposed in the following for finding a filter SFthat satisfies

the desired properties DP-1 and DP-2, and gives the minimal value of m. Note that convergence of the procedure is guaranteed at least to a local optimum, because after each convex optimisation step m is monotonically non-increasing. Procedure A

Step 0. Initialise Sewith a pre-selected matrix.

Step 1. Fix Seto the value obtained in the previous step, and solve the convex optimisation problem

min

m2;Vm

2 subject toð23Þð25Þ

Step 2. FixV to the set obtained in the previous step, and solve the convex optimisation problem

min

m2;Sem

2 subject toð23Þ; ð25Þ; and m240

Step 3. Repeat steps 1 and 2 until m converges to a local optimum. At convergence, compute the filter gain matrices.

3.2

Regional pole-placement constraints

To deal with the third desired property DP-3, i.e. the regional pole-placement constraints, it is shown first that another set of filter gain matrices

Af ¼ ðU  XÞ1M; Cf ¼ N;

Bf ¼ ðU  XÞ1Z; Df ¼ Df

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may also be used. Consider the transfer function matrix GfðsÞ of the filter SFwith the gain matrices (26) and recall

the relationship UUT¼ X  U. The following derivation GfðsÞ ¼ CfðsI  AfÞ1Bfþ Df

¼ NUT½sI  ðU1MUTÞ1U1Zþ Df

¼ N½UðsI  ðU1MUTÞÞðUTÞ1Zþ Df

¼ N½ðUUTÞ  sI  M1

Zþ Df

¼ N½sI  ðU  XÞ1M1ðU  XÞ1Zþ Df

ð31Þ shows the equivalence of SFwith (30) to SFwith (26) in the

sense that the filter state vector is transformed from xf to

UTxf. Hence the input-output characteristic and stability

of the filter are not affected.

With the new formulas (31) for the filter gain matrices,

and with the choice Cf ¼ X  U in (19)–(22), the

conditions for regional pole-placement constraints may be easily converted to LMIs with respect to the variables U, X, and M:

 PC-1: Existence of U, X, and M such that

r2

0ðU  XÞ c0ðU  XÞ  MT

c0ðU  XÞ  M U X

" #

o0 ð32Þ

 PC-2: Existence of U, X, and M such that M  MTþ 2r2U 2r2Xo0

Mþ MT 2r1Uþ 2r1Xo0

ð33Þ  PC-3: Existence of U, X, and M such that

Fðyl; clÞ cos yl 2ðM  M TÞ cosyl 2ðM T MÞ Fðy l; clÞ 2 6 6 4 3 7 7 5o0 ð34Þ where Fðy; cÞ ¼ siny 2½2cðU  XÞ  M  M T

 PC-4: Existence of U, X, and M such that Fðyr; crÞ cos yr 2ðM T MÞ cosyr 2ðM  M TÞ Fðy r; crÞ 2 6 6 4 3 7 7 5o0 ð35Þ

The constraints in (32)–(35) are much simpler than the ones developed for the entire filtering error dynamics[9, 10], and can be augmented into procedure A to produce a procedure in the following for finding a filter SF satisfying all three

desired properties, and with the minimal value of m. Procedure B

Step 0. Initialise Sewith a pre-selected matrix.

Step 1. Fix Seto the value obtained in the previous step, and solve the convex optimisation problem

min

m2;Vm

2 subject toð23Þð25Þ and any combination

of ð32Þð35Þ

Step 2. FixV to the set obtained in the previous step, and solve the convex optimisation problem

min

m2;S

e

m2 subject toð23Þ; ð25Þ and m240

Step 3. Repeat steps 1 and 2 until m converges to a local optimum. At convergence, compute the filter gain matrices.

4 A numerical example

In this Section, an example is worked out to illustrate the proposed filter design procedures. Suppose that the system shown in Fig. 1 has the signal model SSwith the following

system matrices As¼ 1:7270 0:3405 1 0   ; Bs¼ 1 0   ; Cs¼ 0½ 1; Ds¼ 0

and the channel model SC with the following system

matrices Ac¼ 2:7731 þ 0:3a2 2:1222 1:2302  0:23a1 1 0:4a2 0 0:35a2 1 0:5a1 2 6 4 3 7 5 BTc ¼ 1 þ a½ 1 a2 0; Cc¼ 0 1½ a1; Dc¼ 1 þ 2a2

Assume the uncertain parameters a12 ½0; 0:5 and

a22 ½0:4; 0:5. An optimal H1 filter SF is designed by

using procedure A implemented with the LMI Toolbox

[31]. The initial value of STe is set to [0 0], and the

corresponding optimal m2is 8.0664 after step 1 is executed once. Then the procedure is executed until convergence, and the final optimal m2 obtained is 8.0653, corresponding to STe ¼ ½0:0010  0:0012. These results are put in the first three rows under the column title ‘Procedure A’ in Table 1, which show the advantage of having the extra matrix variable STe.

Next, by applying procedure B, four regional pole-placement constraints, PC-1 with PDð2; 0:5Þ, PC-2

with PVð2; 0:5Þ, PC-3 with PLð1:5; p=6Þ, and PC-4

with PRð1:5; p=6Þ are separately considered. As in the

above, the initial value of STe is set to [0 0], and step 1 is executed once first. Then the procedure is carried out until it converges. The results are put in the first three rows of Table 1 under the column title ‘Procedure B’. Correspond-ing to these constraints, the poles of the filters and the system, including the signal and channel models at

Table 1: Filter design results from the example

Procedure A Procedure B PD PV PL PR m2 8.0664 8.3682 8.0923 8.0896 8.1446 m2 8.0653 8.3660 8.0914 8.0886 8.1425 Se 0:0010 0:0012   0:0032 0:0003   0:0007 0:0013   0:0022 0:0011   0:0022 0:0020   m2 8.0686 8.3744 8.0934 8.0956 8.1463

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the vertices ofDc, are shown in Fig. 2. Clearly, the poles of

the filters indeed lie inside the desired regions.

Finaly, the proposed method is applied with a special constraint, i.e., with the p-coefficients fSe;Ce;Qe;Pe;Reg

set to f0; Ce; 0;I ;m2Ig in (23)–(25), where Qe¼ 0 is

accomplished by setting Y¼ W ¼ 0. This simulates the solution of the current filtering problem by applying the bounded real lemma [27] originating from the concept of positive realness. The last row in Table 1 shows the more conservative results.

5 Conclusion

The robust H1 filtering problem for linear signal models

and uncertain channels are solved by using the extended p-sharing theory. Regional pole-placement constraints on the filter, rather than the entire filtering error dynamics, are also considered and formulated as LMIs that can be augmented to the robust H1 filter design procedure. An

example is worked out to illustrate the effectiveness of the proposed method, in addition to the improvement in reducing the conservativeness from the use of the bounded real lemma.

6 Acknowledgment

This research is supported by the National Science Council of the Republic of China under Grant NSC 92-2213-E-002-040.

7 References

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3 Tseng, C.C., and Pei, S.C.: ‘Stable IIR notch filter design with optimal pole placement’, IEEE Trans. Signal Process., 2001, 49, pp. 2673–2681

4 Teng, J., Fidler, J.K., and Zheng, L.: ‘Pole placement algorithm for equiripple bandstop filter design’, IEE Proc., Circuits Devices Syst., 1994, 149, pp. 325–327

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Fig. 2 Poles of the robust filter, marked by J, and poles of the system with signal and channel models at the vertices ofDc, marked

by, obtained from procedure B with respect to four regional pole-placement constraints

數據

Table 1: Filter design results from the example
Fig. 2 Poles of the robust filter, marked by J, and poles of the system with signal and channel models at the vertices of D c , marked by , obtained from procedure B with respect to four regional  pole-placement constraints

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