Multi-rate Filter 1

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0165-1684/$ - se

doi:10.1016/j.sig

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Signal Processing 86 (2006) 1061–1075

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A simple design method of reduced-order filters and itsapplications to multirate filter bank design$

Zhisheng Duana,�,1, Jingxin Zhangb, Cishen Zhangc, Edoardo Moscad

aState Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, Peking University,

Beijing 100871, PR ChinabDepartment of Electrical and Computer Systems Engineering, Monash University, Claydon, Vic 3800, Australia

cSchool of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, SingaporedDipartimento di Sistemi e Informatica, Universita di Firenze, Italy

Received 23 December 2004; accepted 25 July 2005

Available online 6 September 2005

Abstract

Based on linear matrix inequality (LMI) technique, a new design method is proposed for the reduced-order filters of

continuous and discrete time linear systems. The method is derived from decomposing the key matrix in LMIs which

determines the order of designed filters. Different from the existing methods, the proposed method first minimizes the

upper bound of the key matrix and then eliminates its near-zero eigenvalues, which results in a simpler, more direct and

reliable design procedure. The proposed method can be used to design H2 and H1 reduced-order filters and multirate filter

banks. Its effectiveness is illustrated by several examples.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Linear matrix inequality (LMI); Reduced-order filter; Multirate filter banks

1. Introduction

Common filter/controller design methods such as H2 and H1 optimal design often, if not always, yield highorder filters/controllers, especially in the design of multirate filter banks, see [1–4] and references therein forvarious filtering and control techniques. Since a high order filter/controller tends to incur implementationdifficulties, the effective design methods for reduced order filter/controller are obviously important from apractical point of view.

Over the past few decades, there has been some progress in developing design algorithms for the optimalreduced-order filters/controllers. For example, fixed-order controller and model reduction problems have been

e front matter r 2005 Elsevier B.V. All rights reserved.

pro.2005.07.029

orted by the National Science Foundation of China under grant 60204007, 60334030 and Australian Research Council

P03430457.

ing author. Tel.: +861062765037; fax: +861062764044.

esses: [email protected] (Z. Duan), [email protected] (J. Zhang), [email protected] (C. Zhang),

.it (E. Mosca).

leave at Department of Electrical and Computer Systems Engineering, Monash University, Australia.

www.elsevier.com/locate/sigpro

ARTICLE IN PRESSZ. Duan et al. / Signal Processing 86 (2006) 1061–10751062

studied extensively, see [5–9] and references therein. Also some algorithms for reduced-order filter design weredeveloped in [10–14]. Although reduced-order filters and controllers have been extensively studied, the designproblem is far from being solved. The key difficulty in LMI based reduced-order design methods [15,16] stemsfrom the feasibility of rank-constrained LMI. Because of the non-convexity, it is extremely hard to find theglobally optimal solutions to the generic design problems of reduced-order filters and controllers.

Multirate filter banks (FBs) have found a wide range of applications in signal processing andtelecommunications and have aroused the interest of control community in recent years, see [1,2,17–19]. In[1], the design problem of FBs is cast into that of optimal model matching. The advantages of optimal modelmatching approach are that the design problems of FBs can be handled in a unified framework as in robustcontrol and filtering, and that there are reliable algorithms and software for globally optimal solutions. Themain limitation of this approach is that the structural features of FBs are ignored, which generally results inthe FBs with very high-order. To make this approach practically applicable, optimal model matching withreduced-order FBs is required.

This paper is devoted to developing a simple LMI-based design method for the reduced-order filters andFBs. Compared with the reduced-order control problem, the reduced-order filtering problem is a little easier.In the LMI formulae, the key matrix which determines the order of filters can be written out linearly. Based onthis simplicity, we have developed a simple method for the design of reduced-order filters.

This paper is inspired by [14]. But we will present a much simpler algorithm for the search of the positivesemi-definite matrix which determines the order of the designed filters. The rest of this paper is organized asfollows. In Section 2, a simple design method is given for H2 and H1 reduced order filters of continuous-timelinear systems. In Section 3, the proposed method in Section 2 is generalized to discrete-time systems. Section 4applies the result of Section 3 to the design of multirate filter banks. In Section 5, several examples are given toillustrate the proposed design method and to compare it with the results of [12,14,17]. The comparison showsthat the method proposed in this paper gives better results in most cases. The last section concludes the paper.

2. A simple H2 and H1 reduced-order filtering method for continuous-time linear systems

The standard filtering problem is formulated as follows. Consider the following linear stable system:

_x ¼ Axþ Bw1,

y ¼ CxþDw2,

z ¼ Lx, (1)

where x 2 Rn is the system state vector, y 2 Rq is the measurement, z 2 Rp is the signal to be estimated,w1 2 Rm1 is the input noise and w2 2 Rm2 is the measurement noise. w1 and w2 are assumed to be mutuallyindependent zero-mean Gaussian white noises with unit power spectrum density matrix. The canonicalfiltering problem is to find a stable filter of the form:

_xF ¼ AFxF þ BFy,

zF ¼ LFxF, (2)

which minimizes

E½ðz� zFÞTðz� zFÞ�,

where E½z� denotes mathematical expectation of the random variable z. The filter (2) is called a full-order filterwhen xF 2 Rn, or a reduced-order filter when xF 2 Rr with ron. The obvious meaning in filtering is to designan efficient filter to estimate the output z from the noise corrupted output y [12]. It is well-known that Kalmanfilter is a full-order optimal filter.

In order to consider reduced-order filters, we formulate the filtering problem above in terms of LMIs.Rewrite (1) and (2) in the following compact form:

_xcl ¼ Aclxcl þ Bclw,

ARTICLE IN PRESSZ. Duan et al. / Signal Processing 86 (2006) 1061–1075 1063

z� zF ¼ Cclxcl, (3)

where

xcl ¼x

xF

!; Acl ¼

A 0

BFC AF

!,

Bcl ¼B 0

0 BFD

!; w ¼

w1

w2

!,

Ccl ¼ ðL � LFÞ.

It is a standard result, e.g. [12,14], that for a given g240,

E½z2cl�og22, (4)

if and only if, the following matrix inequalities are feasible in the variables P;Z;AF;BF and CF:

PAcl þ ATclP PBcl

BTclP �I

!o0, (5)

P CTcl

Ccl Z

!40; trðZÞog22. (6)

By the canonical LMI method, the above feasibility problem of matrix inequalities can be converted to theproblem shown below, which is proved in [14].

Theorem 1. For a given g240, if there are S ¼ ST40; Q ¼ QTX0 and Z ¼ ZT40 such that

SAþ ATS SB

BTS �I

!o0, (7)

NT½C D0�

ðS þQÞAþ ATðS þQÞ ðS þQÞB0

BT0 ðS þQÞ �I

0@

1A

�N½C D0�o0, ð8Þ

and

S þQ LT

L Z

!40; trðZÞog22, (9)

where B0 ¼ ½B 0�; D0 ¼ ½0 D� and N½C D0� is any basis of the nullspace of ½C D0�, then there exists a filter in the

form (2) such that (4) holds.

From Theorem 1, we can get the following simplified result when D ¼ I .

Corollary 1. If D ¼ I in (1), then (8) holds if, and only if,

ðS þQÞAþ ATðS þQÞ � CTC ðS þQÞB

BðS þQÞ �I

!o0. (10)

Proof. By the canonical LMI method [20], (8) holds if, and only if, there exists a scalar l40 such that

ðS þQÞAþ ATðS þQÞ ðS þQÞB0

BT0 ðS þQÞ �I

0@

1A

� l½C D0�T½C D0�o0. ð11Þ


Using Schur complement and the supposition D ¼ I and taking l sufficiently large, the above inequality holdsif and only if (10) holds. &

Remark 1. From Corollary 1, we can see that by the condition D ¼ I and the use of (11), the matricesNT½C D0�

and N½C D0� in (8) are reduced to �CTC in (10). Hence the formula (10) is much simpler and we can see theeffect of the matrices NT

½C D0�and N½C D0� clearly in (10).

The matrix Q in (8) and (9) is crucial to the determination of the order of designed filter. In fact, the order ofdesigned filter is equal to the rank of Q. In order to design reduced-order filters, we have to get a singularmatrix Q that satisfies the matrix inequalities in Theorem 1. This is the well-known rank constraint in reduced-order filter/controller design, which is non-convex in nature and greatly increases design difficulty.

Intuitively, we can get reduced-order filters by tailoring Q. For example, when we get Q by solving the LMIsin Theorem 1, we decompose Q as Q ¼ USUT, where U satisfies UUT ¼ I and S is a diagonal matrix.Replacing small eigenvalues of Q in S by zeros, we can get a singular matrix S1. With this S1, we can get apositive semi-definite matrix Q1 ¼ US1UT, which will yield a reduced-order filter. Based on this simple idea,we can derive a design algorithm for reduced-order filters. But generally, this method is rather conservative ifthere is no other constraint on Q. Therefore, we introduce an upper bound constraint on Q to reduce theconservativeness. This leads to the following algorithm.

Design algorithm for reduced-order filters:Step 1: Compute the optimal value g2fo for full-order filters by solving the LMIs in Theorem 1.Step 2: Take a scalar g2 a little larger than g2fo. Then solve the inequalities (7), (8) and (9) again with the

chosen g2 and minimize the upper bound a such that

0oQpaI . (12)

In this way, we can get a positive definite matrix Q.Step 3: Decompose Q as Q ¼ USUT, where S ¼ diagðl1; . . . ; lr; lrþ1; . . . ; lnÞ. Suppose

l1X � � �XlrXlrþ1X � � �Xln without loss of generality and let U1 be the matrix composed of the first r

columns of U.Step 4: Minimize trðZÞ subject to (7) and

NT½C D0�

ðS þU1RUT1 ÞAþ ATðS þU1RUT

1 Þ ðS þU1RUT1 ÞB0

BT0 ðS þU1RUT

1 Þ �I

0@

1A

�N½C D0�o0 ð13Þ

S þU1RUT1 LT

L Z

!40 (14)

with matrix variables S40;R40 and Z40. Then we can get a positive semi-definite matrix U1RUT1 and the

corresponding optimal value g2ro (¼ trðZÞ) for reduced-order filters. Using this U1RUT1 , we can get a filter with

order r and estimation performance g2ro by the traditional LMI method given in [14,16].Step 5: If this g2ro is not satisfactory, then go back to Step 2 to increase g2 a little and repeat the other steps

above, stop when we get a comparatively optimal g2ro.

Remark 2. Compared with the reduced-order filter design method in [12], the method given above is moredirect and can achieve better result. The method in [12] is an indirect method. It designs the reduced order filterby a closed-loop order reduction of the full order Kalman filter. Because the order reduction imposes somestructural constraints inherited from full order Kalman filter, this method does not always give the best results,see [14] and Section 5 for demonstrative examples.

Remark 3. Compared with reduced-order filter design method in [14], the method given above is simpler andmore reliable. In [14], to find the Q that satisfies the rank constraint, a slack variable R satisfying Q ¼ RRT isintroduced and three optimization algorithms are suggested to minimize TrðQ� RRTÞ. All these algorithmsrequire searching (all the elements of) a positive semi-definite matrix Q with dimension n� n and a rectangular


matrix R with dimension n� r. These searches must be carried out at each computation of LMIs (7)–(9). Thusthey greatly increases computational requirements. Because these algorithms are coupled with LMIs (7)–(9),and because TrðQ� RRTÞ is nonlinear in R and non-convex in terms of Q and R, the entire optimizationproblem becomes nonlinear and non-convex. It is not only computationally very intensive and very hard tosolve, but also may not reliably generate the required suboptimal solution. In contrast, the algorithm proposedabove is based on minimizing the upper bound of Q during the normal LMI computation and on eliminatingthe small eigenvalues of Q after the minimization. Before each computation of LMIs (7)–(9), only a singleparameter g2 needs to be adjusted and the additional nonlinear computation of TrðQ� RRTÞ is completelyavoided. Since the initial value of the parameter g2 is g2fo, which is obtained from the full order filter design inStep 1 and is the global minimum value of filtering error, the adjustment of g2 is almost certain (normally to alarger value) and can be easily and reliably confined to the vicinity of the global minima by controlling the sizeof adjustment. Also, once the parameter g2 is specified, only the conventional LMI optimization is involved inthe subsequent computations, which is convex and guarantees a suboptimal solution if it exists. The constraintQoaI in Step 2 is crucial. This constraint guarantees that some eigenvalues of Q are near zero, so we caneliminate these eigenvalues to obtain reduced-order filters. If Q ¼ 0, it corresponds a zero filter. As will beshown in Section 5, the proposed method has an acceptable conservativeness.

Obviously the method above can also be applied to the H1 optimal filtering problem which is to find astable filter of the form:

_xF ¼ AFxF þ BFy,

zF ¼ LFxF þDFy (15)

to minimize the worst case estimation error energy kz� zFk2 over all bounded energy disturbances w1 and w2.Again, we formulate the H1 filtering problem above in terms of LMIs. Rewrite (1) and (15) in the following

compact form:

_xcl ¼ Aclxcl þ Bclw,

z� zF ¼ Cclxcl þDclw, (16)

where

xcl ¼x

xF

!; Acl ¼

A 0

BFC AF

!,

Bcl ¼B 0

0 BFD

!; w ¼

w1

w2

!,

Ccl ¼ ðL�DFC � LFÞ; Dcl ¼ �DFD0,

D0 ¼ ½0 D�.

It is a standard result, e.g. [10], that for given a g140, the following inequality holds for all non-zerow 2 L2½0; 1Þ:

kzclk2og1kwk2, (17)

if and only if, the following matrix inequality is feasible in the variables P;AF;BF;CF and DF:

PAcl þ ATclP PBcl CT

cl

BTclP �g1I DT

cl

Ccl Dcl �g1I

0B@

1CAo0. (18)

By the canonical LMI method, the above feasibility problem of matrix inequalities can be converted to theproblem shown below.


Theorem 2. For a given g140, if there are S ¼ ST40 and Q ¼ QTX0 such that

SAþ ATS SB

BTS �g1I

!o0, (19)

NT½C D0�

ðS þQÞAþ ATðS þQÞ þ g�11 LTL ðS þQÞB0

BT0 ðS þQÞ �g1I

0@

1A

�N½C D0�o0, ð20Þ

where B0; D0 and N½C D0� are given as in Theorem 1, then there exists a filter as in (15) such that (17) holds.

Proof. Write P and P�1 as

P ¼X M

MT �

� �; P�1 ¼

Y N

NT �

� �, (21)

where � denotes the submatrix which needs not to be written out. Then (18) can be written as

PAd þ ATd P PBd CT

d

BTd P �g1I DT

0

Cd D0 �g1I

0BB@

1CCA

þ BKC þ ðBKCÞTo0, ð22Þ

where

Ad ¼A 0

0 0

!; Bd ¼

B 0

0 0

!,

B ¼

0 P0

I

!

0 0

�I 0

0BBBBB@

1CCCCCA; C ¼

C 0 D0 0

0 I 0 0

!,

K ¼ ðDF

BF

LF

AFÞ and D0 is as given above. By the well-known projection lemma, (22) holds if and only if,

YAT þ AY B

BT �g1I

!o0 (23)

and

NT½C D0�

XAþ ATX þ g�11 LTL XB0

BT0 X �g1I

0@

1A

�N½C D0�o0, ð24Þ

where B0 is as given in Theorem 1. By (21), we know that X ¼ Y�1 þQ, where QX0. Let S ¼ Y�1, obviously(19) is equivalent to (23) and (20) is equivalent to (24). &

Similar to Corollary 1, we can get the following simplified result when D ¼ I .

Corollary 2. If D ¼ I in (1), then (20) holds if and only if,

ðS þQÞAþ ATðS þQÞ þ g�11 LTL� g1CTC ðS þQÞB

BðS þQÞ �g1I

!o0. (25)


Apparently the design algorithm for reduced-order filters carries over easily to the optimal H1 filteringproblem stated in Theorem 2. We just need to replace, in the algorithm, the LMIs (7), (8) and (9) with (19) and(20), and replace g2 with g1.

3. H2 and H1 reduced-order filtering for discrete-time systems

This section generalizes the result above to H2 and H1 filtering of discrete-time systems. Consider thefollowing discrete-time stable system:

xkþ1 ¼ Axk þ Bw1k,

yk ¼ Cxk þDw2k,

zk ¼ Lxk, (26)

where xk 2 Rn is the system state vector, yk 2 Rq is the measurement, zk 2 Rp is the signal to be estimated,

w1k 2 Rm1 is the input noise and w2k 2 Rm2 is the measurement noise. The filtering problem is to find a stablefilter described as

xkþ1 ¼ AFxk þ BFyk,

zk ¼ LFxk þDFyk, (27)

where the matrices ðAF;BF;LF;DFÞ are to be designed. Assume that w1k and w2k are mutually independentGaussian white noises with zero-mean and unit covariance. The aim of design is to minimize

E½ðz� zÞTðz� zÞ�,

where E½z� denotes mathematical expectation of the random variable z. The filter (27) is called a full-orderfilter when xF 2 Rn, or a reduced-order filter when xF 2 Rr with ron. In order to consider reduced-orderfilters, we formulate the filtering problem above in terms of LMIs. Rewrite (26) and (27) in the followingcompact form:

xkþ1 ¼ Aclxk þ Bclwk,

zk � zk ¼ Cclxk þDclwk, (28)

where xk ¼ ðxk

xkÞ; wk ¼ ð

w1k

w2kÞ, and Acl; Bcl; Ccl; Dcl are as given in (16). It has been established in [3,21] that for

given g240,

E½ðz� zÞTðz� zÞ�og22, (29)

if and only if, the following matrix inequalities are feasible in the variables P;Z;AF;BF and CF:

P�1 Acl Bcl

ATcl P 0

BTcl 0 I

0B@

1CA40 (30)

and

Z Ccl Dcl

CTcl P 0

DTcl 0 I

0B@

1CA40; trðZÞog22. (31)

Following the same line as the proof of Theorem 2 and Corollary 1, it is easy to establish the followingtheorem.


Theorem 3. For a given g240, if there are S ¼ ST40, Q ¼ QTX0 and a scalar l40 such that

S 0

0 I

� ��

AT

BT

!SðA BÞ40, (32)

S þQ 0 0

0 I 0

0 0 I

0BB@

1CCA�

AT

BT

0

0BB@

1CCAðS þQÞðA B 0Þ

þ l

CT

0

DT

0BB@

1CCAðC 0 DÞ40, ð33Þ

and

Z L 0

LT S þQ 0

0 0 I

0BB@

1CCAþ l

0

CT

DT

0BB@

1CCAð0 C DÞ40,

trðZÞog22, ð34Þ

then there exists a filter in the form (27) such that (29) holds.

Similar to Corollary 1, we get

Corollary 3. If D ¼ I , then (33) is equivalent to

S þQþ CTC 0

0 I

!�

AT

BT

!ðS þQÞðA BÞ40, (35)

and (34) is equivalent to

Z L

LT S þQþ CTC

!40; trðZÞog22, (36)

Obviously, the method above is also applicable to the following H1 filtering problems: Assume thatwk 2 l2½0; 1Þ in (28). Given a prescribed scalar g1, design a filter of the form (27) such that for all non-zerowk 2 l2½0; 1Þ,

kz� zk2og1kwk2. (37)

It is well known that (37) holds if and only if there exists P ¼ PT40 such that

P�1 0 Acl Bcl

0 g1I Ccl Dcl

ATcl CT

cl P 0

BTcl DT

cl 0 g1I

0BBBB@

1CCCCA40. (38)

The above condition can be described equivalently in the following form which is the H1 counterpart ofTheorem 3.

Theorem 4. Given g140, if there are S ¼ ST40, Q ¼ QTX0 and a scalar l40 such that

S 0

0 g1I

!�

AT

BT

!SðA BÞ40, (39)


and

S þQ� g�11 LTL 0 0

0 g1I 0

0 0 g1I

0BB@

1CCA�

AT

BT

0

0BB@

1CCA

�ðS þQÞðA B 0Þ þ l

CT

0

DT

0BB@

1CCAðC 0 DÞ40, ð40Þ

then there exists a filter of the form (27) such that (37) holds.

Corollary 4. If D ¼ I , then (40) is equivalent to

S þQ� g�11 LTLþ g1CTC 0

0 g1I

!�

AT

BT

!

�ðS þQÞðA BÞ40. ð41Þ

Design algorithm for H2 and H1 reduced-order filter: Obviously, the algorithm given in Section 2 is alsosuitable for the H2 and/or H1 reduced-order filter design of discrete-time linear systems based on the LMIsgiven in Theorems 3 and 4. Under an upper bound constraint of Q, if Q ¼ 0, it corresponds a constant filterDF in discrete-time systems.

4. Application to multirate filter bank design

As shown in [1], the design of synthesis filters in FBs can be cast into the following model matchingproblem: given transfer function matrices W ðzÞ and EðzÞ with compatible dimensions, find a transfer functionmatrix RðzÞ that minimizes

kW ðzÞ � RðzÞEðzÞk1.

In the above formula, EðzÞ, RðzÞ and W ðzÞ are, respectively, the polyphase representations of analysisfilters HiðzÞ, synthesis filters F iðzÞ; i ¼ 0; 1; . . . ;m� 1, and the required time delay between input andoutput signals, which are proper rational transfer functions, see [1] for details. To simplify presentation,in the sequel we will consider only two channel FBs. The results of this section carry over easily to multi-channel FBs.

In two-channel FBs, EðzÞ, RðzÞ and W ðzÞ are 2� 2 transfer function matrices. Generally, W ðzÞ is a transferfunction matrix composed of time-delay operators only and takes on the following form [1]:

W ðzÞ ¼z�d 0

0 z�d

!if m ¼ 2d þ 1,

W ðzÞ ¼0 z�dþ1

z�d 0

!if m ¼ 2d,

where m is the desired value of time delay. For full-order design, orderðRðzÞÞ ¼ orderðEðzÞÞ þ orderðW ðzÞÞ;see [1]. In this section, we design reduced-order RðzÞ for the model-matching problem minRðzÞkW ðzÞ�

RðzÞEðzÞk1. The reduced-order synthesis filters can be obtained directly from the designed RðzÞ using theformulae given in [1].

ARTICLE IN PRESS

Suppose that the state-space realizations of W ðzÞ, EðzÞ and RðzÞ are W (z ) :=

(A W B W

CW 0

);( ) ( )

Z. Duan et al. / Signal Processing 86 (2006) 1061–10751070

E (z ) :=A E B E

CE DE

; R (z ) :=A R B R

CR DR

;respectively. Then the state space realization of

W ðzÞ � RðzÞEðzÞ is

(A cl B cl

Ccl D cl

):=

⎛⎜⎜⎜⎜⎝

A W 0 0 B W

0 A E 0 B E

0 B R CE A R B R DE

CW − DR CE − CR − D R D E

:=

A 0

;

B

B R CE 0 A R B R D E

CW 0 − DR CE 0 − CR − DR DE

where

A ¼ ðAW

00

AEÞ; B ¼ ðBW

BEÞ; CE0 ¼ ð0 CEÞ; CW0 ¼ ðCW 0Þ.

Using Theorem 4, we can establish the following result.

Theorem 5. Suppose that EðzÞ and W ðzÞ are given as above. Then for a given g140, there exists RðzÞ

such that

kW ðzÞ � RðzÞEðzÞk1og1,

if and only if there are S ¼ ST40, Q ¼ QTX0 and a scalar l40 such that

S 0

0 g1I

!�

AT

BT

!SðA BÞ40, (42)

and

S þQ� g�11 CTW0CW0 þ lCT

E0CE0 lCTE0DE

lDTECE0 g1I þ lDT

EDE

0@

1A

�AT

BT

!ðS þQÞðA BÞ40. ð43Þ

Remark 4. Obviously, the key matrix Q which determines the order of RðzÞ is linearly separatedin the formulae of Theorem 5. So the algorithm given in Section 2 can be used to design reduced-ordersynthesis filter banks. This method can also be generalized to H2 reduced-order design of multi-ratefilter banks.

Remark 5. The FBs designed with H1 norm optimization [1] do not necessarily remove all aliasing. However,the aliasing distortion, measured by the H2 norm of aliasing component, is bounded by kW ðzÞ � RðzÞEðzÞk1,and it will be small if kW ðzÞ � RðzÞEðzÞk1 is small [22]. This is also true for the reduced order FBs designedwith Theorem 5 and the algorithm in Section 2, since they are based on the same H1 norm criterion. Thus, theprescribed g1 in the algorithm controls both the maximum filtering error and aliasing distortion of thedesigned FBs, and the H2 norm of the FBs’ aliasing component is always less than or equal to the achievedkW ðzÞ � RðzÞEðzÞk1 :¼ g1ro. For two channel FBs, the aliasing component is given by [18]H0ð�zÞF0ðzÞ þH1ð�zÞF 1ðzÞ, and its H2 norm satisfies kH0ð�zÞF0ðzÞ þH1ð�zÞF 1ðzÞk2pg1ro, see Section 5for demonstrative examples.


5. Examples

Example 1. Consider the following example system borrowed from [12,14]:

_x ¼

0 1 0:5

�5 �0:020

1:5 0 �0:1

0B@

1CAxþ

0

1

1

0B@

1CAw1,

y ¼ ð1 1 � 2Þxþ w2,

z ¼ ð1 1 � 2Þx. (44)

For this example, the optimal full-order H2 filtering performance is g2fo ¼ 1:87. In the reduced second-ordercase, the satisfying H2 filtering performance is g2ro ¼ 2:4183 obtained by the algorithm in Section 2 withg2 ¼ 2:02 and the constraint 0oQo5:2I . The corresponding singular matrix is

U1RUT1 ¼

2:5658 0:4796 �1:7438

0:4796 0:2989 �0:5272

�1:7438 �0:5272 1:3788

0B@

1CA.

Example 2. Consider the following example system borrowed from [12,14]:

_x ¼

0 �0:1 0 0 0

1 �0:3 0 0 0

0 �0:2 0 0 0:016

0 �0:3 1 0 0:06

0 �0:1 0:1 �1:5 �0:9

0BBBBBB@

1CCCCCCA

xþ

0 1

0 0

1 0

0 0

0 1

0BBBBBB@

1CCCCCCA

w1,

y ¼0:1 0 0 �0:5 1:6

0:1 0:2 0 �0:3 0:12

� �xþ w2,

z ¼0:1 0 0 �0:5 1:6

0:1 0:2 0 �0:3 0:12

� �x. (45)

For this example, the optimal full-order H2 filtering performance is g2fo ¼ 1:78: In the reduced third-ordercase, the satisfying H2 filtering performance is g2ro ¼ 1:93 obtained by the algorithm in Section 2 withg2 ¼ 1:81 and the constraint 0oQo1:93I . Its corresponding singular matrix is

U1RUT1

¼

0:3152 �0:2997 �0:1880 �0:0754 �0:0999

�0:2997 0:2853 0:1820 0:0857 0:1166

�0:1880 0:1820 0:1997 0:0059 0:4261

�0:0754 0:0857 0:0059 0:6878 0:2040

�0:0999 0:1166 0:4261 0:2040 1:7465

0BBBBBBBB@

1CCCCCCCCA.

Observations: Table 1 compares the performance of the reduced-order filters designed for Examples 1 and 2using different methods. As seen from the table, for Example 1, the proposed method gives a betterperformance than the existing methods [14,12]. Note that for this example, the performance of our reducedorder filter is also better than ‘‘the true global optimal value 2.4253’’ calculated in [14] with the BB method.For Example 2, the performance is better than that of [12] but slightly worse than that of [14]. However, forthis example, the optimal full order filtering performance we calculated is g2fo ¼ 1:78, whereas that calculatedin [14] is g2fo ¼ 1:77. This discrepancy indicates some slight numerical errors in either calculations. Thus the

ARTICLE IN PRESS

Table 1

Comparison of H2 reduced-order filtering performances obtained by different methods

Method This paper [14] [12]

Example 1, 2nd-order 2.4183 2.4503 4.74

2, 3rd-order 1.93 1.912 2.06

Z. Duan et al. / Signal Processing 86 (2006) 1061–10751072

relative ratio g2fo=g2ro might be a fairer comparison. For the proposed method g2fo=g2ro ¼ 0:9222, while for themethod of [14] g2fo=g2ro ¼ 0:9257. The performances of the proposed method and that of [14] are almost thesame. In addition, we can see that for both examples, the reduced order filtering cost given by [12] is generallyworse, particularly for Example 1, it is far from its real global optimal value. This demonstrates that themethod of [12], which is based on a closed-loop order reduction of the full order Kalman filter, may notminimize the filtering error of the reduced order filter due to the structural constraints imposed in the orderreduction.

Example 3. Consider the following discrete-time system in the form of (26) with

A ¼

0:6 0:1 0:2 �0:3 �0:2 0

0 0:4 �0:3 0:2 0:1 0:1

0:3 �0:2 0:1 �0:1 0 �0:2

�0:1 0:3 0:1 �0:3 0:1 0:05

0:1 0:2 �0:1 0:1 0:3 0:1

0:3 0:1 �0:2 0:3 0:2 �0:3

0BBBBBBBBBBB@

1CCCCCCCCCCCA,

B ¼

1

0

0

0

1

0

0BBBBBBBBBBB@

1CCCCCCCCCCCA; CT ¼

2

0

2

0

0

0

0BBBBBBBBBBB@

1CCCCCCCCCCCA; LT ¼

1

0

0

0

0

1

0BBBBBBBBBBB@

1CCCCCCCCCCCA; D ¼ 1,

which is adapted from [21]. Note that we have taken D ¼ 1 here for simplicity of LMI formulae. The optimalfull-order H2 filtering cost is g2fo ¼ 0:4803. In the reduced third-order case, the satisfying H2 filteringperformance is g2ro ¼ 0:4809 obtained by the algorithm in Section 2 with g2 ¼ 0:4809 and the constraint0oQo25I . For this example, the optimal full-order H1 filtering cost is g1fo ¼ 0:56162. In the reduced third-order case, the satisfying H1 filtering value is still g1ro ¼ 0:56162 obtained by the algorithm in Section 2 withg1 ¼ 0:56163 and the constraint 0oQo5I .

From this example, we can see that our reduced-order H2 filtering cost is very close to the full-order filteringcost, and our reduced order H1 filter has achieved the global optimal value of filtering performance. So theconservativeness of the proposed method is acceptable.

Example 4 (FB design). This is Example 2 of [17]. The analysis filters are third-order IIR filters given by

H0ðzÞ ¼0:1412z3 þ 0:3805z2 þ 0:3805zþ 0:1412

z3 � 0:3011z2 þ 0:3694z� 0:0250,

H1ðzÞ ¼ H0ð�zÞ.


Using the proposed method (Theorem 5 plus Design Algorithm), we can obtain synthesis filters F0ðzÞ andF1ðzÞ with the order 19, the maximum reconstruction error g1ro ¼ 0:015 and the aliasing distortionkH0ð�zÞF0ðzÞ þH1ð�zÞF 1ðzÞk2 ¼ 0:0037, by taking the values of time delay m ¼ 10, g1 ¼ 0:015 and theconstraint 0oQo68I in the algorithm given in Section 2. For this example, the maximum reconstruction errorresulting from full-order H1 design is g1fo ¼ 0:002, and the order of this full-order H1 design is 31, see [1] forcalculation of the order. So the order of our synthesis filters is reduced by 12 at the sacrifice of maximumreconstruction error. Figs. 1 and 2 show the frequency responses of the designed synthesis filters which aresimilar to those of [17] obtained by mixed H2=H1 optimization design, but our H1 reconstruction error isless than 0.0186 given in [17]. Compared with the filters given in [17], the order of our synthesis filters is lowerand the frequency responses are better than those of [17] by H1 norm optimization.

Example 5 (FB design). In this Example H0ðzÞ is a low-pass elliptic filter obtained by MATLAB functionellip, and H1ðzÞ ¼ H0ð�zÞ,

H0ðzÞ ¼0:1396z3 þ 0:3686z2 þ 0:3686zþ 0:1396

z3 � 0:4276z2 þ 0:6066z� 0:1626.

Fig. 1. jH0ðzÞj and jH1ðzÞj in decibels versus w=2p.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-50

-40

-30

-20

-10

0

10

20

Fig. 2. jF0ðzÞj and jF 1ðzÞj in decibels versus w=2p.


Using the proposed method (Theorem 5 plus Design Algorithm), we can obtain synthesis filters F0ðzÞ andF1ðzÞ with the order 23, the maximum reconstruction error g1ro ¼ 0:005 and the aliasing distortionkH0ð�zÞF0ðzÞ þH1ð�zÞF 1ðzÞk2 ¼ 0:0036, by taking the values of time delay m ¼ 10, g1 ¼ 0:01 and theconstraint 0oQo110I in the algorithm given in Section 2. For this example, the maximum reconstructionerror resulting from full-order H1 design is g1fo ¼ 0:002, and the order of this full-order H1 design is 31, see[1] for calculation of the order. So the order of our synthesis filters is reduced by 8 with some minor sacrifice ofmaximum reconstruction error. Figs. 3 and 4 show the frequency responses of the analysis and designedsynthesis filters.

It is worth pointing out that for this example, the order reduction method suggested in [1] does not work.Firstly, there are no zero-pole cancellation in the full-order filters designed with the H1 optimization method[1]. That is, the order of the designed filters cannot be reduced by zero-pole cancellation. Secondly, using thebalanced truncation tool of model reduction (Matlab function balmr), we can only get synthesis filters with theorder 29 and the maximum reconstruction error g1ro ¼ 0:05. The benefit of the proposed method is obvious.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Fig. 3. jH0ðzÞj and jH1ðzÞj in decibels versus w=2p.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-50

-40

-30

-20

-10

0

10

20

Fig. 4. jF0ðzÞj and jF1ðzÞj in decibels versus w=2p.


6. Conclusion

Compared with H2 and H1 control problems, H2 and H1 filtering problems are easier. The key matrix Q

which determines the order of the designed filters can be separated explicitly. Using this property, a newmethod is proposed for the design of reduced-order filters. Different from the positive semi-definite matrixsearch method given in [14], the proposed method first minimizes the upper bound of Q and then eliminates itsnear-zero eigenvalues. Therefore, it greatly simplifies the design procedure. The proposed method is applicableto both continuous and discrete systems, to H2 and H1 filtering problems, and to multirate filter bank design.The effectiveness of the method has been demonstrated by examples.

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Multi-rate Filter 1