FREQUENCY-WEIGHTED MODEL REDUCTION METHOD WITH ERROR BOUNDS FOR 2-D SEPARABLE DENOMINATOR DISCRETE SYSTEMS

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1 INERNAIONAL JOURNAL OF INFORMAION AND SYSEMS SCIENCES Volume 1, Number 2, Pages c 2005 Institute for Scientific Computing and Information FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS FOR 2-D SEPARABLE DENOMINAOR DISCREE SYSEMS ABDUL GHAFOOR, JING WANG, AND VICOR SREERAM Abstract. Frequency weighted model reduction scheme for two-dimensional 2-D separable denominator discrete time systems is presented. he method yields the stable reduced order models for stable separable denominator original systems. he method is based on one-dimensional 1-D balanced truncation. It is easily extendable to singular perturbation and optimal Hankel norm based approximations. he bound on the approximation error is also derived. Key Words. frequency weighted, 2-D approximation, separable denominator. 1. INRODUCION he process of deriving low order model from high order model is known as model reduction with the objective that lower order model retains or closely approximates the input - output behavior of the original system. he balanced realization 13 has been a significant contribution to 1-D system theory, especially its application to model order reduction, since it can preserve stability 14 and give an explicit bound on frequency response error 3, 1. he balanced model reduction has been subject of intensive research during last two decades, a survey of balanced model reduction schemes can be found in 6. In model reduction, the error between the original system and the reduced order model needs to be small ideally for all frequencies. However, sometimes, the accuracy is more important over a certain frequency band, rather than for all frequencies. his is the motivation for introduction of frequency weightings to the model reduction procedures. he frequency weighted balanced truncation scheme was originally introduced by Enns 3. he main weaknesses of the Enns method are: i the stability of the reduced order models is not guaranteed in the case of two sided weightings, ii and there is no a prior error bound on frequency response error. Wang et al 19 modified Enns method to overcome these short comings. he state space modeling of 2-D filters have been studied by many researchers, and different models have been proposed 2, 4, 17. It was shown in 7 that the Roesser model is the most general 2-D model, and that other models can be embedded in this model. Given a 2-D separable denominator system, it is always possible to find a minimal separable realization 7. he separable denominator systems cover a broad range of 2-D systems D balanced model reduction concept has been extended to 2-D. he 2-D balanced model reduction problem has been investigated by many researchers 16, 21, 22, 9, 8, and the results have many applications including design and approximation of digital filters. Since balanced realization is determined by the controllability and observability Gramians of the system, and since there are several types of Gramians he work was supported by Australian Research Council under the Discovery Grants Scheme. 105

2 106 A. GHAFOOR, J. WANG, AND V. SREERAM that can be defined for a given 2-D system, there are different types of balanced realizations for a given 2-D discrete system, leading to different balanced approximations. For example, in 16 pseudo balanced approximation is used, in 21 quasi balancing is proposed, and in 12 structurally balanced approach is developed. In 22, 2-D approximation was considered based on 1-D approximation. his method is very useful, because it extends the important properties and approximation methods of 1-D systems to the 2-D case. Other related interesting results can be found in 9, 8. he 2-D frequency weighted balanced truncation problem has been studied in 12, 10, 11, 18. In 12, a frequency weighted structurally balanced approximation of 2-D discrete systems is considered which is extended to quasi balancing in 11. Although, the methods in 12, 11 are good starting points and motivate further research on this issue, but have many shortcomings: i he method in 12 uses Linear Matrix Inequalities due to which it is numerically exhaustive and computationally inefficient. However, 11 does not have this problem. ii Stability of the reduced order models in the case of two sided weightings is not guaranteed except for the single input single output case 10. iii here is no bound on approximation error available. Another scheme based on pseudo balancing is presented in 18. An obvious drawback of this scheme is that the weighting functions for this scheme need to be separable, and furthermore, there are no bounds on approximation error. he preliminary results of this work are presented in 5. he main contribution of this paper is the extension of the frequency weighted model reduction technique of 19 to the 2-D case. he advantages of the proposed technique include: i guaranteed stability in case of double-sided weighting, ii easily computable frequency response error bounds, iii and the weighting function need not necessarily be separable denominator system. 2. PRELIMINARIES he system configuration to be considered in this paper is shown in Figure 1, where Hz 1, z 2 R p q is stable, minimal and separable denominator transfer function matrix of the system of order m, n; W i z 1, z 2 R q t and W o z 1, z 2 R s p are stable and minimal input and output weights of orders m i, n i and m o, n o, respectively. Figure 1. A weighted 2-D discrete system he Roesser state-space model 17 to describe Hz 1, z 2, W i z 1, z 2 and W o z 1, z 2 can be written as 12: Hz 1, z 2 = Cz 1 I m z 2 I n A 1 B + D W i z 1, z 2 = C i z 1 I mi z 2 I ni A i 1 B i + D i W o z 1, z 2 = C o z 1 I mo z 2 I no A o 1 B o + D o

3 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS 107 where 1 A = A i = A o = A1 A, B = 0 A 4 Ai1 A i2, B A i3 A i = i4 Ao1 A o2, B A o3 A o = o4 B B 2, C = Bi1 C 1 B i2, C i = Bo1 C B o2, C o = C i1 C i2 C o1 C o2 the symbol denotes the direct sum, I is the identity matrix, A R m+n m+n, B R m+n q, C R p m+n, D R p q, A i R mi+ni mi+ni, B i R mi+ni t, C i R q mi+ni, D i R q t, A o R mo+no mo+no, B o R mo+no p, C o R s mo+no and D o R s p. An alternate form of Roesser state-space realization of equation 1 can also be given as follows: A1 0 B1 C 2 A =, B =, C = A A 4 B C2 Let the minimal rank decomposition see 22 for more details for the Roesser state-space realization in equation 1 be as follows: A B B1 = C2 D C D D 2 1 then we can write Hz 1, z 2 = H 1 z 1 H 2 z 2. Similarly, the minimal rank decomposition for equation 2 A B B2 = C1 D C D D 1 2 allows us to write Hz 1, z 2 = H 2 z 2 H 1 z 1 where H 1 z 1 = C 1 z 1 I A 1 1 B 1 + D 1 H 2 z 2 = C 2 z 2 I A 2 1 B 2 + D 2 Lemma 1 22: Let H r z 1, z 2 = H r1 z 1 H r2 z 2 be the 2-D reduced order model obtained from 1-D balanced truncation, then H r z 1, z 2 is 2-D stable. Furthermore, the frequency response error is bounded by Hz 1, z 2 H r z 1, z 2 n m 2 D µ i i=m r+1 m r σ i + 2 D σ i n i=n r+1 Alternatively for H r z 1, z 2 = H r2 z 2 H r1 z 1, the frequency response error is bounded by Hz 1, z 2 H r z 1, z 2 n r m 2 D µ i i=m r+1 σ i + 2 D m σ i n i=n r+1 where σ i and µ i are the Hankel singular values of the system H 1 z 1 and H 2 z 2, respectively. µ i µ i

4 108 A. GHAFOOR, J. WANG, AND V. SREERAM Lemma 2 22: Let H hr z 1, z 2 = H hr1 z 1 H hr2 z 2 be the 2-D reduced order model obtained from 1-D optimal Hankel norm approximation, then H hr z 1, z 2 is 2-D stable. Furthermore, the frequency response error is bounded by Hz 1, z 2 H hr z 1, z 2 n m m r D µ i σ i + D σ i + 3 m n σ i µ i i=m r+1 i=m r i=n r+1 Alternatively for H hr z 1, z 2 = H hr2 z 2 H hr1 z 1, the frequency response error is bounded by Hz 1, z 2 H hr z 1, z 2 n r n m m D µ i + 3 µ i σ i + D σ i n µ i i=n r+1 i=m r+1 i=n r+1 3. MAIN RESULS In this section we present a frequency weighted balanced technique for 2-D separable denominator discrete systems. his is based on an extension of the well-known frequency weighted balanced model reduction technique 19. As illustrated in Figure 2, the weighted-input-to-state and the state-to-weightedoutput auxiliary transfer function matrices H i z 1, z 2 and H o z 1, z 2 are defined as 3 4 H i z 1, z 2 = z 1 I m z 2 I n A 1 BW i z 1, z 2 = Ĉiz 1 I m+mi z 2 I n+ni Âi 1 ˆBi H o z 1, z 2 = W o z 1, z 2 Cz 1 I m z 2 I n A 1 = Ĉoz 1 I m+mo z 2 I n+no Âo 1 ˆBo where Figure 2. Auxiliary transfer function matrices

5 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS 109  i = ˆB i = Âi1  i2  i3 ˆBi1  i4 = ˆB i2 = B D i B i1 B 2 D i B i2 A 1 B C i1 A B C i2 0 A i1 0 A i2 0 B 2 C i1 A 4 B 2 C i2 0 A i3 0 A i4 Ĉ i = I Ĉ i1 Ĉ i2 = 0 0 I 0 A 1 0 A 0 Âo1   o = o2 = B o1 C 1 A o1 B o1 C A o2  o3  o4 0 0 A 4 0 B o2 C 1 A o3 B o2 C A o4 I 0 ˆBo1 ˆB o = = 0 0 ˆB o2 0 I 0 0 Ĉ o = Ĉ o1 Ĉ o2 = Do C 1 C o1 D o C C o2 It is obvious that auxiliary systems H i z 1, z 2 and H o z 1, z 2 are stable since systems Hz 1, z 2, W i z 1, z 2 and W o z 1, z 2 are stable. he 2-D frequency weighted Gramians 5 ˆP i = ˆPi1 ˆPi2 = ˆP i3 ˆPi4 ˆP i11 ˆPi12 ˆPi21 ˆPi22 ˆP i12 ˆP i14 ˆPi23 ˆPi24 ˆP i21 ˆP i23 ˆP i41 ˆPi42 6 ˆQ o = ˆQo1 ˆQo2 = ˆQ o3 ˆQo4 ˆP i22 ˆP i24 ˆP i42 ˆP i44 ˆQ o11 ˆQo12 ˆQo21 ˆQo22 ˆQ o12 ˆQ o14 ˆQo23 ˆQo24 ˆQ o21 ˆQ o23 ˆQ o41 ˆQo42 ˆQ o22 ˆQ o24 ˆQ o42 ˆQ o44 satisfy following Lyapunov equations 7 8  i ˆPi  i ˆP i + ˆB i ˆB i = 0  o ˆQ o  o ˆQ o + Ĉ o Ĉo = 0 he 1,1 block of equation 8 and 3,3 block of equation 7, respectively yield A 1 ˆQ o11 A 1 ˆQ o11 + Y 1 = 0 A 4 ˆPi41 A 4 ˆP i41 + X 4 = 0

6 110 A. GHAFOOR, J. WANG, AND V. SREERAM where 9 10 Y 1 = A 1 ˆQ o12 B o1 C 1 + A 1 ˆQ o22 B o2 C 1 + B o1 C 1 ˆQ o12 A 1 + B o1 C 1 ˆQo14 B o1 C 1 + B o1 C 1 ˆQo24 B o2 C 1 + B o2 C 1 ˆQ o22 A 1 + B o2 C 1 ˆQ o24 B o1 C 1 + B o2 C 1 ˆQo44 B o2 C 1 + D o C 1 D o C 1 X 4 = B 2 C i1 ˆPi14 B 2 C i1 + B 2 C i1 ˆPi23 A 4 + B 2 C i1 ˆPi24 B 2 C i2 + A 4 ˆP i23 B 2 C i1 + A 4 ˆPi42 B 2 C i2 + B 2 C i2 ˆP i24 B 2 C i1 + B 2 C i2 ˆP i42 A 4 + B 2 C i2 ˆPi44 B 2 C i2 + B 2 D i B 2 D i Since X 4 and Y 1 are symmetric, there are orthogonal matrices U 4, V 1, and diagonal matrices S 4, H 1, such that 11 X 4 = U 4 S 4 U4 12 Y 1 = V 1 H 1 V1 where S 4 = diags 41, s 42,, s 4n, H 1 = diagh 11, h 12,, h 1m, s 41 s 42 s 4n 0, h 11 h 12 h 1m 0, rank X 4 = i 4 and rank Y 1 = j 1. Let us define new matrices B 4 and C 1 as follows: B 4 := U 4 diag s , s ,, s 4i4 1 2, 0,, 0 C 1 := diag h , h ,, h 1j1 1 2, 0,, 0V 1 Lemma 3: Assume that then following relationships hold where rank B4 B 2 = rank B4 C1 rank = rank C C 1 1 B 2 = B 4 K 4 C 1 = L 1 C1 K 4 = diag s , s ,, s 4i4 1 2, 0,, 0U 4 B 2 L 1 = C 1 V 1 diag h , h ,, h 1j1 1 2, 0,, 0 Proof: Similar to that in 19. Remark 1: It is shown in 19 that the assumption 13 and 14 are almost always true. Note that in the expression 10, every term can be expressed as B 2 or B2 or B 2 B2, which is exactly same as in 19, here is some matrix which does not affect our analysis. So the assumption 13 is almost always true. Similar remark applies for 14. B Assume rank = rankb B 2 and rank C 1 C = rankc1, then there exist 2 matrices K 1 and L 2, such that 19 B = K 1 B 2 20 C = C 1 L 2 B Remark 2: he assumption rank = rankb B 2 will automatically be satisfied 2 when B 2 is full column rank. Similar remark also applies for rank C 1 C = rank C 1.

7 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS 111 Using the equations 15, 16, 19 and 20, we can define new matrices B 1new and C 4new as follows 21 B 1new := K 1 B4, C4new := C 1 L 2 then we have B B 2 C1 C = B1new B 4 K 4 := B new K 4 = L 1 C1 C4new := L1 Cnew heorem 1: he following conditions almost always hold: 1 rank B B new = rank Bnew C 2 rank = rank C C new new Proof: his result is an immediate consequence of Lemma 3 and the equations 22 and 23. Let us now consider the following minimal rank decomposition 22 A B1new B1n 24 = C4n D C 4new Dnew D 4n 1n then 25 where 26 H n z 1, z 2 = H 1n z 1 H 4n z 2 H n z 1, z 2 = C new z 1 I m z 2 I n A 1 Bnew + D new H 1n z 1 = C 1 z 1 I A 1 1 B 1n + D 1n H 4n z 2 = C 4n z 2 I A 4 1 B4 + D 4n D = L 1 Dnew K 4 Remark 3: he equation 26 is solvable for D new if and only if one of the following equivalent conditions holds 15: K4 1 rank L 1 = rank L 1 D and rank K 4 = rank. D 2 here exist matrices Y and Z such that D = L 1 Y and D = ZK 4. Remark 4: he conditions for the existence of solution for equation 26 will automatically be satisfied for strictly proper original systems. For proper systems, this condition will be satisfied when L 1 is full row rank, and K 4 is full column rank. We note that we can even get rid of this assumption see Remark 6. heorem 2: he realization { A, B new, C new, D new is stable, minimal and separable denominator. Proof: he stability and separability of the realization { A, B new, C new, D new follows from the stability and separability of the realization {A, B, C, D. Where as, the minimality of the realization { A, B new, C new, D new follows from the minimality of the realization {A, B, C, D and heorem 1. heorem 3: he realizations { A 1, B 1n, C 1, D 1n and { A4, B 4, C 4n, D 4n are stable and minimal. Proof: his result follows from equation 25 and the stability, minimality and separability of the realization { A, B new, C new, D new.

8 112 A. GHAFOOR, J. WANG, AND V. SREERAM Now let A 1 ˆP1 A 1 ˆP 1 + B 1n B 1n = 0 A 1 ˆQ 1 A 1 ˆQ 1 + C 1 C 1 = 0 A 4 ˆP4 A 4 ˆP 4 + B 4 B 4 = 0 A 4 ˆQ 4 A 4 ˆQ 4 + C 4nC 4n = 0 where ˆP 1, ˆP 4, ˆQ 1, and ˆQ 4 are positive definite. here exist two transformation matrices 1 and 4, such that 1 1 ˆP 1 1 = 1 ˆQ Σ = Σ = 0 Σ ˆP 4 4 = 4 ˆQ Λ = Λ = 0 Λ 2 where Now let A1b B 33 1nb C 1b D 1n 34 A4b B4b C 4nb D 4n Σ 1 = diagσ 1, σ 2,, σ mr Σ 2 = diagσ mr+1, σ mr+2,, σ m Λ 1 = diagλ 1, λ 2,, λ nr Λ 2 = diagλ nr+1, λ nr+2,, λ n σ 1 σ 2 σ mr > σ mr+1 σ m > 0 λ 1 λ 2 λ nr > λ nr+1 λ n > 0 = = 1 1 A B 1n = C 1 1 D 1n 1 4 A B 4 C 4n 4 D 4n = A 1b1 A 1b2 B 1nb1 A 1b3 A 1b4 B 1nb2 C 1b1 C1b2 D 1n A 4b1 A 4b2 B4b1 A 4b3 A 4b4 B4b2 C 4nb1 C 4nb2 D 4n Lemma 4: he realizations { A 1b1, B 1nb1, C 1b1, D 1n and { A4b1, B 4n1, C 4nb1, D 4n are stable. Proof: he stability of the the realizations { A 1b1, B 1nb1, C 1b1, D 1n and { A4b1, B 4n1, C 4nb1, D 4n follows from the stability 14 of the reduced order models obtained via 1-D unweighted balanced truncation 13. hen we can take the truncated system H r z 1, z 2 = A r, B r, C r, D r as the weighted reduced approximation of the original system Hz 1, z 2, where 35 A r = A1b1 B 1nb1 C 4nb1 0 A 4b1 36 B r = B1nb1 D 4n B 4b1 K 4 := B r K C r = L 1 C1b1 D 1n C 4nb1 := L1 Cr D r = L 1 D 1n D 4n K 4 = D Algorithm: Given the original system Hz 1, z 2 and the weights W i z 1, z 2 and W o z 1, z 2, the frequency weighted reduced-order model is obtained using the following steps: 1 Use formulas 7-8 to compute ˆP i and ˆQ o. 2 Use formulas 9-10 to compute Y 1 and X 4.

9 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS Use formulas 12 and 11 to decompose Y 1 and X 4, respectively, to obtain C 1 = H V1, B4 = U 4 S Use formulas to compute K 1, K 4, L 1, and L 2. 5 Use formulas 21 to compute B 1new and C 4new. 6 Solve Lyapunov equations to compute ˆP 1, ˆP4, ˆQ1 and ˆQ 4. 7 Find the transformation 1 and 4 to satisfy the equation 31 and 32, respectively. 8 Compute the 1-D frequency weighted balanced realization as in equation he reduced order model is obtained using equations { Remark 5: For input weighting only, the frequency weighted realization becomes A, Bnew, C, D and consequently C replaces C 4new in equation 24. Similar remark applies when only output weighting is present. Remark 6: Note here, we can even get rid of the assumption in Remark 4 by setting D new = 0 with compatible dimension in equation 24, and later setting D r = D in equation 38. Remark 7: Although, the above algorithm is explicitly given for balanced truncation, but it is straight forward to extend/define the algorithms for almost all 1-D based reduction schemes, such as, Hankel norm approximation and singular perturbation approximation etc. heorem 4: he reduced order models obtained using the above algorithm/procedure are stable. Proof: he result follows immediately from Lemma 4 and the equation 35. heorem 5: Let the reduced order models be obtained by balanced truncation, then frequency response error is bounded by alternatively W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 n m m r 2k D 4n + 2 λ i σ i + 2k D 1n + 2 σ i i=m r+1 W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 n r m m 2k D 4n + 2 λ i σ i + 2k D 1n + 2 σ i i=m r+1 n i=n r+1 n i=n r+1 where σ i and λ i are the Hankel singular values of the system H 1n z 1 and H 4n z 2, respectively, and k = W o z 1, z 2 L 1 K 4 W i z 1, z 2. Proof: W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 = W o z 1, z 2 Cz 1 I m z 2 I n A 1 B C r z 1 I mr z 2 I nr A r 1 B r W i z 1, z 2 = Wo z 1, z 2 L 1 Cnew z 1 I m z 2 I n A 1 Bnew K 4 L 1 Cr z 1 I mr z 2 I nr A r 1 Br K 4 W i z 1, z 2 = Wo z 1, z 2 L 1 C new z 1 I m z 2 I n A 1 Bnew λ i λ i C r z 1 I mr z 2 I nr A r 1 Br K 4 W i z 1, z 2 W o z 1, z 2 L 1 Cnew z 1 I m z 2 I n A 1 Bnew C r z 1 I mr z 2 I nr A r 1 Br K 4 W i z 1, z 2

10 114 A. GHAFOOR, J. WANG, AND V. SREERAM is its re- A Bnew Ar Br Since is a balanced realization and C new Dnew C r duced order model, we have the following from Lemma 1 Dnew Cnew z 1 I m z 2 I n A 1 Bnew C r z 1 I mr z 2 I nr A r 1 Br n m m r n 2 D 4n + 2 λ i σ i + 2 D 1n + 2 σ i i=m r+1 i=n r+1 λ i he result follows. heorem 6: If the reduced order models are obtained by optimal Hankel norm approximation, then frequency response error is bounded by W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 n m m r m k D 4n + 2 λ i σ i + k D 1n + 2 σ i + 3 σ i n λ i i=m r+1 i=m r+1 i=n r+1 alternatively W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 n r n m m k D 4n + 2 λ i + 3 σ i + k D 1n + 2 σ i λ i +n r i=m r+1 n λ i i=n r+1 where σ i and λ i are the Hankel Singular values of the system H 1n z 1 and H 4n z 2, respectively, and k = W o z 1, z 2 L 1 K 4 W i z 1, z 2. Proof: he proof is similar to the proof of heorem 5 and is therefore omitted. Corollary 1: When only input weighting is present, k becomes K 4 W i z 1, z 2, similarly when only output weighting is present, k becomes W o z 1, z 2 L 1. Moreover, when no weighting is present, k = 1.

11 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS Numerical Results Consider the following system matrices corresponding to Roesser model A 1 = A 2 = A 3 = A 4 = B 1 = B 2 = C 1 = C 2 = D = Let the weighting system matrices be as following A i1 = A o1 = A i4 = A o4 = A i2 = A o = A i3 = A o3 = B i1 = B o1 = B i2 = B o2 = C i1 = C o1 = C i2 = C o2 = D i = D o = he able 1 shows the frequency weighted approximation error and error bounds for different reduced order models. he approximation error criterion used for this

12 116 A. GHAFOOR, J. WANG, AND V. SREERAM able 1. Frequency weighted errors and error bounds. Order Approximation Error Bound m r, n r Error 2, , , , , , , example is: W o z 1, z 2 Hz 1, z 2 H r z 1, z 2 W i z 1, z 2 = max W o e j2πx/x, e j2πy/y 1 x X 1 y Y He j2πx/x, e j2πy/y H r e j2πx/x, e j2πy/y W i e j2πx/x, e j2πy/y he Figure 3, Figure 4 and Figure 5 show the frequency response of the original system, the input and output weights, and the reduced order model of order 3, 3, respectively. he frequency response of the input and output weights has low pass characteristics as shown in the Figure 4. Comparing the original system and the reduced order model frequency responses, it is clear that the approximation is better at low frequencies than at high frequencies. Figure 3. Original System 5. Conclusions A new frequency weighted model reduction scheme for 2-D separable denominator discrete time systems based on frequency weighted balanced truncation method of 19 is presented. he weighting function may not necessarily be separable denominator. he reduced order models are guaranteed to be stable. he method

13 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS 117 Figure 4. Input/Output Weighting System Figure 5. Reduced Order System can be easily extendable to frequency weighted optimal Hankel norm and singular perturbation approximation. he bound on the approximation error is also given. References 1 U.M. Al-Saggaf, and G.F. Franklin, An Error Bound for Discrete Reduced Order Model of a Linear Multivariable System, IEEE ransaction on Automatic Control, AC-32, 1987, pp S. Attasi, Modeling and Recursive Estimation for Double Indexed Sequences, System Identification: Advances and Case Studies, Newyork: Academic D. F. Enns, Model Reduction with Balanced Realizations: An Error Bound and a Frequency Weighted Generalization, Proceedings of 23rd IEEE Conference on Decision and Control, 1984, pp F. Fornasini, and G. Marchesini, State Space Realization heory of wo Dimensional Filters, IEEE rans. Automat. Contr, vol 21, 1976, pp A. Ghafoor, J. Wang, and V. Sreeram Frequency-Weighted Model Reduction Method with Error Bounds for 2-D Separable Denominator Discrete Systems, Proceedings of 20 th IEEE International Symposium on Intelligent Control, 2005, pages to appear.

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15 FREQUENCY-WEIGHED MODEL REDUCION MEHOD WIH ERROR BOUNDS 119 Abdul Ghafoor obtained his Bachelor in Electrical Engineering in 1994 from University of Engineering and echnology, Pakistan, and Master in Electrical Engineering in 2003 from National University of Sciences and echnology, Pakistan. From 1999 to 2002, he performed teaching assignments in National University of Sciences and echnology, Pakistan. Since 2003, he is a PhD candidate in the University of Western Australia. His research topic is frequency-weighted model reduction. Dr. Jing Wang, received the B.Sc and M.Sc from Information Engineering University, China in 1998 and 2001, respectively. She received her Ph.D degree from Northeastern University, China in Her PhD research focused on model reduction for descriptor systems. Since July 2004, she is a post-doctoral fellow for one year in Control Systems Research Group, School of Electrical, Electronic & Computer Engineering,University of Western Australia. Her research interests include descriptor system, model reduction, robust control and system identification. Victor Sreeram obtained Bachelor s degree in 1981 from Bangalore University, India, Master s degree in 1983 from Madras University, India, and Ph.D degree from University of Victoria, Canada in 1989, all in Electrical Engineering. He worked as a Project Engineer in the Indian Space Research Organisation from 1983 to He joined the Department of Electrical & Electronic Engineering, University of Western Australia in 1990 and he is now an Associate Professor. He has held Visiting Appointments at the Department of Systems Engineering, Australian National University during 1994, 1995 and 1996 and at the Australian elecommunication Research Institute in Curtin University of echnology during 1997 and His research interests are control and signal processing.