frequency weighted model reduction technique retaining hankel singular values

50 Asian Journal of Control, Vol. 9, No. 1, pp. 50-56, March 2007

Manuscript received March 7, 2005; revised November 2, 2005; accepted January 22, 2006.

The authors are with the School of Electrical, Electronics and Computer Engineering University of Western Australia, Crawley, WA 6009, Australia (e-mail: {aghafoor, sreeram, rtreasure}@ee.uwa.edu.au).

The work was supported by Australian Research Council under the Discovery Grants Scheme.

－Brief Paper－

FREQUENCY WEIGHTED MODEL REDUCTION TECHNIQUE RETAINING HANKEL SINGULAR VALUES

Abdul Ghafoor, Victor Sreeram, and Richard Treasure

ABSTRACT

In this paper, a modification to Lin and Chiu’s technique [1,2] of frequency weighted balanced truncation is presented. As in [1,2] the method is based on simulataneously diagonalizing the frequency weighted controllability and ob-servability Gramians. Instead of obtaining the reduced order models from the truncated frequency weighted balanced realization as in conventional techniques [1,2], the models are obtained from the truncated augmented system realizations which ensures matching of the dominant frequency weighted Hankel singular val-ues. The method is compared with the method of [2] using numerical examples.

KeyWords: Frequency weighted model reduction, Hankel singular values.

I. INTRODUCTION

Enns [3] has presented a scheme for reducing a stable high order model with frequency weighting, based on a modification of balanced truncation [4]. The method, known as frequency weighted balanced truncation, may use input weighting, output weighting, or both. With only one weighting present, stability of the reduced order model is guaranteed. With both weightings present, the method may yield unstable models. To overcome the potential drawback of instability, Lin and Chiu proposed a new frequency weighted balanced reduction technique [1]. They showed that the reduced-order models obtained by their technique are necessarily stable when both input and output weight-ings are included. However, their technique is only appli-cable when the weights are strictly proper. The requirement of no pole zero cancellations in the augmented system lim-its its application to solving controller reduction problems with certain types of weighting functions [5]. However, Lin and Chiu technique has numerous other applications e.g., low-order filter design [6]. Several extensions of these [1,3,4] techniques have been proposed which include gen-eralization to proper weights [2] and enhancement of accu-

racy [5]. Other interesting results on frequency weighted balanced truncation methods are also available [7-11].

Although, the frequency weighted model reduction techniques [1,2] are based on diagonalizing simultaneously the frequency weighted controllability and observabilty Gramians, they do not match the dominant frequency weighted Hankel singular values. The reduced order mod-els only approximate these singular values instead. We are aware from [9] that retaining dominant frequency weighted singular values may not guarantee low approximation error. Nevertheless, matching these values gives low approxima-tion error in most cases.

In this paper, we propose a modification to Lin and Chiu’s technique to match the most significant frequency weighted Hankel singular values. Preliminary results were presented in [12]. The method is based on simultaneously diagonalizing the frequency weighted Gramians. However, the reduced order models are not obtained by direct trunca-tion of frequency weighted balanced realization as in [1,2] but from the truncated augmented system realizations which guarantees retaining the most significant frequency weighted Hankel singular values. The method is illustrated by numerical examples.

II. PRELIMINARIES

In this section, we present generalized Lin and Chiu’s technique [2] of frequency weighted balanced truncation. Consider the transfer function of a linear time invariant system:

A. Ghafoor et al.: Frequency Weighted Model Reduction Technique Retaining Hankel Singular Values 51

1( ) ( )G s C sI A B D−= − + (1)

where {A, B, C, D} is its minimal realizations. Let the transfer functions of the stable input and output weights be given by Eqs. (2) and (3) respectively:

1( ) ( )V V V VV s C sI A B D−= − + (2) 1( ) ( )W W W WW s C sI A B D−= − + (3)

where {AV, BV, CV, DV} and {AW, BW, CW, DW} are their minimal realizations respectively. Assuming that there are no pole-zero cancellations in G(s)V(s) and W(s)G(s), the augmented systems given by

1( ) ( ) ( )i i i iG s V s C sI A B D−= − + (4) 1( ) ( ) ( )o o o oW s G s C sI A B D−= − + (5)

have the following minimal realizations:

0VV

iiV V

BDA BCBA BA

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

= , = ,

i V i VC C DC D DD⎡ ⎤⎢ ⎥⎣ ⎦= , =

and 0

ooW W W

W W Wo o

A BBA B C A B D

D C C D DDC

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤= , = ,⎢ ⎥

⎣ ⎦

= , =

The frequency weighted Gramians

11 1112 12

12 12

T

i oTV W

P P Q QQP P P Q Q

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= , =

satisfy the following Lyapunov equations:

0T Ti i i ii iP P B BA A+ + =

0TTo o oo oQ Q C CA A+ + =

The Gramians iP and oQ are positive definite if iA and oA are stable. Lin and Chiu’s method [1,2] is based on block diagonalizing the Gramians, iP and oQ :

111 12 121 0

0

TVT

i i i iV

P P P PD T P T

P

⎡ − ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−= =

and 1

11 12 12 00

TWT

o o ooW

Q Q Q QD QT T

Q

⎡ − ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−= =

using the transformations 1

121

12

0and

0V

i oW

II P PT T Q Q II

−

−

⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎣ ⎦⎣ ⎦

The state-space realizations corresponding to the diagonal-ized Gramians are:

121 1

112

1 11

21 12

,0

0,

i i i i i i iV V

V Vi i i

o o o o o o oW WW

o o o W

A X XA T A T B T B

A B

C C T C CP P DC

A BA T A T B T B

Y A Q Q B B D

C C T Y C

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ − ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥− −⎢ ⎥ −⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

= = = =

= = +

⎡ ⎤= = = = ⎢ ⎥+⎢ ⎥⎣ ⎦

= =

where 1 1

12 12 12V V V VX AP P BC P P A− −= + − 1

12V V VX BD P P B−= − 1 1

21 12 12W W W WY Q Q A B C A Q Q− −= + −

112W W WY D C C Q Q−= −

Note that the diagonal blocks of the system matrices iA and oA are not changed by the similarity transformations. The new realizations now satisfy the following Lyapunov equations:

0T Ti i i i i iA D D A B B+ + = (6)

0T To o o o o oA D D A C C+ + = (7)

Since Di and Do are positive definite and Ai and Ao are sta-ble, {Ai, Bi} is controllable and {Co, Ao} is observable. Ex-panding the (1,1) blocks of the Lyapunov equations (6) and (7), we get

1 111 12 12 11 12 12( ) ( )T T T T

V VA P P P P P P P P A XX− −− + − = −

1 111 12 12 11 12 12( ) ( )T T T T

W WA Q Q Q Q Q Q Q Q A Y Y− −− + − = −

For the stable A, since 111 12 12( )T

VP P P P−− and 11(Q − 1

12 12 )TWQ Q Q− are positive definite, we can say that the re-

alization {A, X, Y} is minimal. Lin and Chiu’s technique is based on diagonalizing the Gramians 1

11 12 12T

WQ Q Q Q−− and 1

11 12 12T

VP P P P−− :

1 1 111 12 12 11 12 12( ) ( )T T T T

W V LCT Q Q Q Q T T P P P P T− − − −− = − = Σ (8) where ΣLC = diag{σ1, σ2, …, σr, σr+1, …, σn}, σi ≥ σi + 1, σr > σr+1, and i = 1, 2, …, n−1.

Lin and Chiu’s [2], frequency weighted balanced re-alization { }A B C, , is given by

121

21 22

rA AA T ATA A

− ⎡ ⎤= = ⎢ ⎥

⎣ ⎦ (9)

52 Asian Journal of Control, Vol. 9, No. 1, March 2007

1

2

rBB T B

B

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

= = (10)

2rC CT C C⎡ ⎤⎢ ⎥⎣ ⎦= = (11)

where r rrA R ×∈ , Br ∈ R

r ×

p, Cr ∈ R

m ×r, and r < n. Trun-

cating, we get the reduced order model, { }r r rA B C, , .

Remark 1. Note that although, the realization { }A B C, , is known as the frequency weighted balanced realization, it is strictly speaking, not balanced.

III. MAIN RESULTS

Note that the transformation T in Lin and Chiu’s tech-nique balances {A, X, Y} and not {A, B, C}. Since, the reduced-order models { }r r rA B C, , are obtained directly from {T

−1AT, T −1B, CT} (see Eqs. (9)-(11)), they do not

match the frequency weighted Hankel singular values. In order to obtain the reduced order models which

match the frequency weighted Hankel singular values, we first block balance the augmented state-space realization using the transformation

00balT

TI

⎡ ⎤= ⎢ ⎥⎣ ⎦

(12)

to obtain the following:

121 1,0

xi bal i bal xi bal iVV

XA XA T AT B T BBA

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

= = = =

and

1

21

0,yo bal o bal yo o bal W

W

AA T A T C C T Y C

AY

⎡ ⎤⎢ ⎥− ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

= = = =

Note that the expressions for Cxi and Byo are not given here, since they are not required in the reduction procedure. The corresponding Gramians are given by

1 00

00

LCTxi bal i bal

V

LCTyo bal o bal

W

P T D TP

Q T D TQ

⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Σ= =

Σ= =

Assuming the model order as r we partition the above matrices as follows:

1212

122 221 22 ,0 0

r rr

xi xi

V V

X XA AA BX XA A

A B

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥

= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

12

, 221 22

21 212

00

r

Wryo yo

Wr

A AA C CY YA A

AY Y

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= =

2 2

0 0 0 00 0 , 0 00 0 0 0

LCr LCr

xi LC yo LC

V W

P QP Q

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤Σ Σ⎢ ⎥

= Σ = Σ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Truncating the second block rows and columns, we obtain

12

0r rr

xir xirV V

X XAA BA B

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= , = (13)

21

0ryor yor Wr

Wr

AA C CYAY

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

= , = (14)

0 00 0LCr LCr

xir yorV W

P QP Q

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Σ Σ= , =

Note that the Gramians Pxir and Qyor satisfy:

0T Txir xir xir xir xir xirA P P A B B+ + =

0T Tyor yor yor yor yor yorA Q Q A C C+ + =

The (1, 1) block corresponding to the above Lyapunov equations are given by

0T TLCr LCr r rr r X XA AΣ + Σ + = (15)

0T TLCr LCr r rr r Y YA AΣ + Σ + = (16)

It is obvious from the above that if we construct the re-duced order model such that their augmented state space realizations satisfy Eqs. (13) and (14), then the models will match frequency weighted Hankel singular values (due to Eqs. (15) and (16)). Note that the reduced order model sys-tem matrix, rA is known (see Eqs. (13) and (14)). Only the input and output matrices of the reduced order model have to be determined.

Let us suppose that 1( ) ( )r r rrG s C sI B DA −= − + be the reduced order model with Br and Cr yet to be deter-mined. Assuming there are no pole-zero cancellations, we can write the augmented systems, Gr (s)V(s) and W(s)Gr(s) by the following minimal realizations

0r Vr Vr

ririr VirV V

B DB CA C DCB CA BA

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

= , = , =

0 rrW r Woror or

W r W W

BA D C CB CA B C A B D

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤= , = , =⎢ ⎥

⎢ ⎥⎣ ⎦


Let

11 1112 12

12 12

Tr rr r

ir orTr V r W

P P Q QP Q

P P Q Q

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= , =

be the solutions of the following Lyapunov equations

0T Tir ir ir irir irP P B BA A+ + =

0TTor oror or or orQ Q C CA A+ + =

Block diagonalizing the Gramians using the transformation

112

112

0

0r V

ir orW r

II P PT T Q Q II

−

−

⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎣ ⎦⎣ ⎦

we get

121

0rr

ir ir irirV

XAA T TA A

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

= = (17)

1 rir ir ir

V

XB T B B

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

= = (18)

112ir rir r r V VirC C C P P DCC T ⎡ ⎤−

⎢ ⎥⎣ ⎦= = + (19)

1

21

0ror or oror

r W

AA T TA Y A

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

= = (20)

11

12

ror or or

W r r W

BB T B Q Q B B D

−−

⎡ ⎤= = ⎢ ⎥+⎣ ⎦

(21)

or ror WorC Y CC T ⎡ ⎤⎢ ⎥⎣ ⎦= = (22)

where 1 1

12 12 12r r V r V r V VrX P P B C P P AA − −= + − (23)

112r r V r V VX B D P P B−= − (24)

1 121 12 12r W r W r W W rrY Q Q B C A Q QA

− −= + − (25)

112r W r W W rY D C C Q Q−= − (26)

Comparing, {Air, Bir} and {Aor, Cor} (Eqs. (17), (18), (20), (22)) with {Axir, Bxir} and {Ayor, Cyor} (Eqs. (13)-(14)), they have the same structure. Note that all the entries in Eqs. (13) and (14) are known, where as the entries X12r, Xr, Y21r, and Yr in Eqs. (17), (18), (20), (22) are unknown since they depend on Br and Cr which are unknown. It is obvious that if the reduced order model has to match the frequency weighted Hankel singular values then the augmented state-space realization matrices {Air, Bir}, and {Aor, Cor} should be same as {Axir, Bxir} and {Ayor, Cyor} respectively. This implies

12 12r rX X= (27)

r rX X= (28)

21 21r rY Y= (29)

r rY Y= (30)

Substituting the above in Eqs. (23)-(26), we have

1 112 12 12r r r V r V r V VX A P P B C P P A− −= + −

112r r V r V VX B D P P B−= −

1 121 12 12r W r r W r W W rY Q Q A B C A Q Q− −= + −

112r W r W W rY D C C Q Q−= −

Rewriting the above equation by letting 112r VP P P−= and

112W rQ Q Q−= , we have

12r r r V VX A P B C PA= + − (31)

r r V VX B D PB= − (32)

21r r W r WY QA B C A Q= + − (33)

r W r WY D C C Q= − (34)

The Eqs. (33) and (34) can be written as

Vec( )Vec( )

TW Wr

rW W

QI A I I BAI C I D C

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤− ⊗ + ⊗ ⊗⎢ ⎥

− ⊗ ⊗ ⎣ ⎦

12Vec( )Vec( )

r

r

YY

⎡ ⎤= ⎢ ⎥⎣ ⎦

(35)

where Vec(X) denotes the vector formed by stacking the columns of X into one long vector. The coefficient matrix on the left of the above equation has full rank, guaranteeing solvability of the equation when

W W

W W

A I BC D

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− + λ−

has full rank for all ( )i rAλ = λ [13], where λi(X) denotes the eigenvalues of X. However, there is a unique solution if and only if the matrix on the left of (35) is square, i.e., if and only if W(s) is square. Similarly P and rB , provided they exist, are uniquely determined if and only if V(s) is square.

Remark 2. The condition that

W W

W W

A I BC D

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− + λ−

have full rank at some λi is effectively a condition that W(λi) have full rank there. This observation follows immediately from the identity:


1

0( ) 0 ( )

W W W W

W W i iW W

A I B I A I BC D C A I I W

⎡ ⎤⎢ ⎥⎢ ⎥ −⎢ ⎥⎣ ⎦

− + λ − + λ⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥− − λ λ⎣ ⎦ ⎣ ⎦

We say effectively, since there remains open the possibility that W(s) could have a pole at λi. A similar remark applies to the input weight V(λi).

Remark 3. Note that if the weights W(s) and V(s) have full row and column rank respectively, the requirement for them to have this property for the particular values of

( )i rAλ = λ will be generally satisfied.

Remark 4. It was noted during simulations, that the infin-ity norm of weighted approximation error can be signifi-cantly reduced by incorporating a direct component term

rD , obtained using the optimization scheme based on minimizing the following objective function:

2|| ( )( ( ) ( )) ( ) ||W s G s Gr s V s− =

[ ]{ }2

0 ( ) ( ) ( ) ( )rtr dtt g t g t v t∞ ω ∗ − ∗∫

where v(t) = L −1[V(s)], g(t) = L −1[G(s)], ω(t) = L −1[W(s)] and ( ) 11 1( ) [ ( )] [ ]r r r r rrg t G s C B DsI A

−− −= = +−L L .

IV. NUMERICAL EXAMPLES

Example 1. Consider a 6th order stable system having the following state-space realization:

13 1 0 0 0 0 0 175 0 1 0 0 0 1 0 6

210 0 0 1 0 0 2 0 9320 0 0 0 1 0 5 0 3600 0 0 0 0 1 1 0 680 0 0 0 0 0 0 5 1

1 0 1 0 1 0 0 01 0 2 0 3 0 4 0 5 0 6 0 0

A B

C D

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− .⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− .⎢ ⎥ ⎢ ⎥= , =− .⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− .⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− .⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥. . . . .⎣ ⎦ ⎣ ⎦

The state space realization of the input and output weights are respectively:

5 25 0 1 0 5 0 3 0 50 0 05 0 2 0 3 0 1 0 1

5 2 0 0 1 0 5 0 3 0 50 0 25 0 2 0 3 0 4 0 7

V V V

W W W

A B C

A B C

− . . . .⎡ ⎤ ⎡ ⎤ ⎡ ⎤= , = , =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− . . . . .⎣ ⎦ ⎣ ⎦ ⎣ ⎦

− . . . . .⎡ ⎤ ⎡ ⎤ ⎡ ⎤= , = , =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− . . . . .⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Figure 1 shows the maximum singular value of the input and output weight, σmax[V(s)] and σmax[W(s)], respec-tively.

Fig. 1. Maximum singular value: Input/output weighting filter.

The diagonalized Gramians found from Eq. (8) are:

diag(1 9350 0 3041 0 3001 0 0060 0 0015 0 0001)LCΣ = . , . , . , . , . , .

The state space realization of 3rd order reduced order model is:

0 1433 0 0758 0 06430 0754 0 0853 2 06260 0318 1 9518 0 0554

0 0687 1 63860 9267 3 17955 1947 11 0309

0 3445 1 8194 0 20091 0097 0 9155 0 7268

4 7914 134 03134 09

r

r

r

r

A

B

C

D

− . − . − .⎡ ⎤⎢ ⎥= − . − . − .⎢ ⎥⎢ ⎥− . . − .⎣ ⎦

− . − .⎡ ⎤⎢ ⎥= . − .⎢ ⎥⎢ ⎥. − .⎣ ⎦

− . . .⎡ ⎤= ⎢ ⎥− . . − .⎣ ⎦

. − .=

− . 99 69 1340⎡ ⎤⎢ ⎥.⎣ ⎦

The weighted Hankel singular values of the reduced order model are:

diag(1 9350 0 3041 0 3001)LCrΣ = . , . , .

It can also be seen that the reduced order model retains the first 3 weighted Hankel singular values of the original sys-tem.

Figures 2 and 3 respectively show the comparison of the maximum singular value of the weighted approxima-tion error, σmax[W(s)(G(s) − Gr(s))V(s)] for 3rd and 5th order models obtained using the proposed method (with

0Dr = and Dr obtained using optimization), Lin & Chiu’s technique and the unweighted balanced truncation method.

Table 1 shows the frequency weighted errors, ||W(s)(G(s) − Gr(s))V(s)|| ∞ for models obtained using the proposed and Lin and Chiu’s techniques. It can be seen that proposed scheme produces lower approximation error than


Fig. 2. Weighted model reduction error for 3rd order

approximation.

Fig. 3. Weighted model reduction error for 5th order

approximation.

Table 1. Frequency weighted errors for the reduced-order models.

Proposed Method with Order 0rD = rD obtained using optimization

Lin and Chiu Technique

1 31.0415 0.6039 38.0495 2 34.7550 0.6089 213.7808 3 17.9146 0.0373 20.6704 4 13.2955 0.0146 16.5731 5 2.6799 0.0092 6.4662

Lin and Chiu’s method. Furthermore, by including a direct component term (obtained using optimization) in the re-duced order model, the approximation error is significantly reduced.

Example 2. For the system in Example 1, let us consider the output (single sided) weight only.

The diagonalized Gramians found from Eq. (8) are:

diag(12 1854 11 6316 3 6121 0 6527 0 1703 0 0128)LCΣ =

. , . , . , . , . , .

The state space realization of 3rd order reduced order model is:

0 0164 2 0199 0 08852 0192 0 1368 0 17250 0678 0 0959 0 2409

0 3662 0 51590 7570 1 61530 3912 1 2600

10 7324 3 8382 23 30345 2555 7 1188 11 9062

15 8731 65 01828 4477 27 140

r

r

r

r

A

B

C

D

− . − . .⎡ ⎤⎢ ⎥= . − . .⎢ ⎥− . . − .⎣ ⎦

− . .⎡ ⎤⎢ ⎥= . − .⎢ ⎥. .⎣ ⎦

. − . .⎡ ⎤= ⎢ ⎥. . − .⎣ ⎦

− . − .= . . 1⎡ ⎤⎢ ⎥⎣ ⎦

The weighted Hankel singular values of the reduced order model are:

diag(12 1854 11 6316 3 6121)LCrΣ = . , . , .

It can also be seen that the reduced order model retains the first 3 weighted Hankel singular values of the original sys-tem.

Figure 4 shows the comparison of the maximum sin-gular value of the weighted approximation error, σmax[W(s)(G(s) − Gr(s))] for 3rd order models obtained using the proposed method (with 0Dr = and Dr ob-tained using optimization), Lin & Chiu’s technique and the unweighted balanced truncation method.

Table 2 shows the frequency weighted errors, ||W(s)(G(s) − Gr(s))|| ∞ for models obtained using the pro-posed and Lin and Chiu’s techniques. It can be seen that proposed scheme mostly produces lower approximation error than Lin and Chiu’s method.

Table 2. Frequency weighted errors for the reduced-order models.

Proposed Method with Order 0rD = rD obtained using optimization

Lin and Chiu Technique

1 31.6993 28.3259 472.9125 2 8.5404 8.0139 8.5859 3 15.8263 0.9788 15.8476 4 8.5252 0.3619 20.2152 5 9.9002 0.0255 9.0761

Fig. 4. Weighted model reduction error for 3rd order approximation.


Remark 5. The above examples indicate that by merely matching or retaining the Hankel singular values in the reduced order models does not necessarily guarantee a bet-ter frequency weighted error in all model orders. However, it was clear from the examples that a further reduction in frequency weighted error can be achieved by introducing a direct component term (obtained based on optimization) in the reduced order model.

V. CONCLUSIONS

In this paper an extension to Lin and Chiu’s [1] method of frequency weighted balanced truncation is pre-sented. This method is based on retaining the most signifi-cant frequency weighted Hankel singular values in the re-duced order model. The numerical examples show that the frequency weighted error in the case of the proposed tech-nique is mostly better than the generalized Lin and Chiu’s technique. Furthermore, a significant reduction in the fre-quency weighted error is achieved by incorporating a direct component term.

REFERENCES

1. Lin, C.-A. and T.-Y. Chiu, “Model Reduction via Fre-quency Weighted Balanced Realization,” Contr.-Theory Adv. Technol., Vol. 8, pp. 341-451 (1992).

2. Sreeram, V., B.D. Anderson, and A.G. Madievski, “New Results on Frequency Weighted Balanced Reduction Technique,” Proc. Amer. Contr. Conf., Seattle, Wash-ington, pp. 4004-4009 (1995).

3. Enns, D.F., “Model Reduction with Balanced Realiza-tions: An Error Bound and a Frequency Weighted Gen-eralization,” Proc. 23rd IEEE Conf. Decis. Contr., Las Vegas, pp. 127-132 (1984).

4. Moore, B.C., “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Re-duction,” IEEE Trans. Automat. Contr., Vol. AC-26, pp. 17-32 (1981).

5. Varga, A. and B.D.O. Anderson, “Accuracy- En-hancing Methods for Balancing-Related Frequency- Weighted Model and Controller Reduction,” Automatica, Vol. 39, pp. 919-927 (2003).

6. Holford, S. and P. Agathoklis, “The Use of Model Re-duction Techniques for Designing IIR Filters with Lin-ear Phase in the Passband,” IEEE Trans. Signal Process., Vol. 44, pp. 2396−2404 (1996).

7. Al-Saggaf, U.M. and G.F. Franklin, “Model Reduction via Balanced Realization: An Extension and Frequency Weighted Techniques,” IEEE Trans. Automat. Contr., Vol. AC-33, pp. 687-692 (1988).

8. Sreeram, V., “On the Properties of Frequency Weighted Balanced Truncation Techniques,” Proc. Amer. Contr. Conf., Anchorage, Alaska, pp. 1153-1154 (2002).

9. Gestal, T.V., B. De Moor, B.D.O. Anderson, and P. Van Overschee, “On Frequency Weighted Balanced Trunca-tion: Hankel Singular Values and Error Bounds,” Europ. J. Contr., Vol. 7, pp. 584-593 (2001).

10. Wang, G., V. Sreeram, and W.Q. Liu, “A New Fre-quency Weighted Balanced Truncation Method and an Error Bound,” IEEE Trans. Automat. Contr., Vol. 44, pp. 1734-1737 (1999).

11. Gugercin, S. and A.C. Antoulas, “A Survey of Model Reduction by Balanced Truncation and Some New Re-sults,” Int. J. Contr., Vol. 77, No. 8, pp. 748-766 (2004).

12. Ghafoor, A., V. Sreeram and R. Treasure “On Fre-quency Weighted Balanced Truncation Technique Re-taining Hankel Singular Values,” Proc. IEEE Conf. Contr. Appl., Taipei, Taiwan, pp. 638-642 (2004).

13. Zhou, K., “Frequency Weighted L∞ Norm Optimal Hankel Norm Model Reduction,” IEEE Trans. Automat. Contr., Vol. 40, pp. 1687-1699 (1995).

frequency weighted model reduction technique retaining hankel singular values

Documents