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Partial Fraction Expansion Based Frequency Weighted Balanced Singular Perturbation Approximation Model Reduction Technique with Error Bounds
6th IFAC Symposium on System Structure and Control 6th IFAC Symposium on System Structure and Control June 22-24, 2016. Istanbul, TurkeyStructure 6th IFAC Symposium on System System and Control Control 6th IFAC Symposium on June 22-24, 2016. Istanbul, TurkeyStructure and Available online at www.sciencedirect.com June June 22-24, 22-24, 2016. 2016. Istanbul, Istanbul, Turkey Turkey
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Partial Fraction Expansion Based Partial Fraction Expansion Based Partial Fraction Expansion Based Frequency Weighted Balanced Singular Frequency Weighted Balanced Singular Frequency Weighted Balanced Singular Perturbation Approximation Perturbation Approximation Model Model Perturbation Approximation Model Reduction Technique with Error Bounds Reduction Technique with Error Reduction Technique with Error Bounds Bounds a b,c Deepak , Victor Sreeramb b, Deepak Kumar Kumaraaa ,, Ahmad Ahmad Jazlan Jazlanb,c b,cb, Victor Sreeramb b, b,c Deepak Kumar Kumar ,, Roberto Ahmad Jazlan Victor Sreeram Togneri b,,, Victor Deepak Ahmad Jazlan Sreeram ,, Roberto Tognerib , b Roberto Roberto Togneri Togneri ,, a a Electrical Engineering Department, M. N. National Institute of Electrical Engineering Department, M. N. National Institute of a a Electrical Allahabad, Department, M. National Technology India (e-mail: [email protected]) Electrical Engineering Engineering Department, M. N. N. [email protected]) National Institute Institute of of Technology India (e-mail: deepak deepak b Technology Allahabad, Allahabad, India (e-mail: deepak [email protected]) of Electrical and Electronics Engineering, University of b Department Technology Allahabad, India (e-mail: deepak [email protected]) Department of Electrical and Electronics Engineering, University of b b Department of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: Department of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: [email protected], [email protected], Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) c [email protected]) Engineering, Faculty c Department of Mechatronics [email protected]) Department of Mechatronics Engineering, Faculty of of Engineering, Engineering, c c Department of Mechatronics Engineering, Faculty of Engineering, Engineering, International Islamic University Malaysia, Jalan Gombak, 53100 Department of Mechatronics Engineering, Faculty of International Islamic University Malaysia, Jalan Gombak, 53100 International University Malaysia, Jalan Kuala Lumpur, Malaysia International Islamic Islamic University Malaysia, Jalan Gombak, Gombak, 53100 53100 Kuala Lumpur, Malaysia Kuala Kuala Lumpur, Lumpur, Malaysia Malaysia Abstract: In In this this paper, paper, aa new new frequency frequency weighted weighted partial partial fraction fraction expansion expansion based based model model Abstract: Abstract: In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to Abstract: In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation further reduce frequency weighted singular perturbation is incorporated incorporated into the algorithm. algorithm. This technique techniqueerror, results in stable stable reduced approximation order models models further reduce the theinto frequency weighted approximation approximation error, singular perturbation approximation is the This results in reduced order is incorporated into the algorithm. This technique results in stable reduced order models regardless if single sided or double sided weights are used. Error bounds are also derived for is incorporated intosided the or algorithm. Thisweights technique in stable reduced order models regardless if single double sided are results used. Error bounds are also derived for regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free the proposedare method. For minimization minimization of the the A frequency weighted approximation error, free parameters aremethod. introduced into the the algorithm. algorithm. A numerical exampleapproximation is provided provided in inerror, orderfree to the proposed For of frequency weighted parameters introduced into numerical example is order to parameters introduced into validate the the are proposed algorithm. parameters are introduced into the the algorithm. algorithm. A A numerical numerical example example is is provided provided in in order order to to validate proposed algorithm. validate validate the the proposed proposed algorithm. algorithm. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Model Reduction, Frequency Weighting, Weighting, Singular Singular Perturbation Perturbation Approximation, Approximation, Keywords: Model Reduction, Frequency Keywords: ModelExpansion, Reduction,Square Frequency Weighting, Singular Perturbation Perturbation Approximation, Approximation, Partial Fraction Expansion, Square RootWeighting, Balancing.Singular Keywords: Model Reduction, Frequency Partial Fraction Root Balancing. Partial Fraction Expansion, Square Root Balancing. Partial Fraction Expansion, Square Root Balancing. 1. INTRODUCTION INTRODUCTION Singular perturbation perturbation approximation approximation by by Liu Liu and and AnderAnder1. Singular 1. INTRODUCTION INTRODUCTION Singular perturbation approximation by Liu Liu and method Anderson (1989) (1989) is applied applied approximation to the the framework framework of the the method 1. Singular perturbation by and Anderson is to of son (1989) and is applied applied to (2009) the framework framework of improve the method method by Sahlan Sahlan and Sreeramto (2009) in order order to to improve the Model reduction reduction methods methods which which emphasize emphasize particular particular frefre- son (1989) is the of the by Sreeram in the Model by Sahlan and Sreeram (2009) in order to improve the frequency weighted approximation error. Two free paramModel reduction methods which emphasize particular frequency intervals continue to receive interest from the by Sahlan and Sreeram (2009) in order to improve the Model emphasize particular quencyreduction intervals methods continuewhich to receive interest from frethe frequency weighted approximation error. Two free paramfrequency weighted approximation error. Two free free parameters α α and andweighted β are are also also included in in the the algorithm to paramfurther quency intervals continue due to receive receive interest from from the control theory theory community due to the the importance importance of the ob- frequency approximation error. Two quency intervals continue to interest eters β included algorithm to further control community to of obeters α and and β frequency are also also included included inapproximation the algorithm algorithm to to further minimize theβ frequency weighted approximation error. Ercontrol theory community due to the importance of obtaining such reduced order models in a variety of apeters α are in the further control theory community due to the importance of obminimize the weighted error. Ertaining such reduced order models in a variety of apminimize the frequency weighted approximation error. Error bounds are developed for the proposed method. Simutaining such reduced order models in a variety of applications (Du and Yang (2010); Li et al. (2014); Ding the weighted error. Ertaining such order models ap- minimize ror bounds arefrequency developed for the approximation proposed method. Simuplications (Dureduced and Yang (2010); Li in et aal.variety (2014);ofDing ror bounds aredemonstrate developed for for thereduced proposed method. SimuSimulation results demonstrate that reduced approximation erplications (Du Shaker and Yang Yang (2010); Li (2014, et al. al. (2014); (2014); Ding et al. al. (2015); (2015); Shaker and(2010); Tahavori (2014, 2013)). Ding Fre- ror bounds are developed the proposed method. plications (Du and Li et lation results that approximation eret and Tahavori 2013)). Frelation results demonstrate that reduced approximation errors relative to the original method by Sahlan and Sreeram et al. (2015); Shaker and Tahavori (2014, 2013)). Frequency weighted model reduction methods based on the lation results demonstrate that reduced approximation eret al. (2015); Shaker and Tahavori (2014, 2013)). Frequency weighted model reduction methods based on the rors relative to the original method by Sahlan and Sreeram rors relative toachieved the original original method byproposed Sahlan and and Sreeram (2009) can be be achieved by method using the theby proposed approach. quency weightedexpansion model reduction reduction methods based on the the rors partial fraction fraction expansion approachmethods originally introduced relative to the Sahlan Sreeram quency weighted model based on (2009) can by using approach. partial approach originally introduced (2009) can be achieved by using the proposed approach. partial fraction expansion approach originally introduced by Saggaf and Franklin (1986, 1988) are an important (2009) can be achieved by using the proposed approach. partial fraction approach originally by Saggaf and expansion Franklin (1986, 1988) are an introduced important by Saggafof and Franklin (1986, which 1988) have are an an important category ofand reduction methods which have been further 2. PRELIMINARIES PRELIMINARIES by Saggaf Franklin (1986, 1988) are important category reduction methods been further 2. category of reduction methods which have been further developed to have desirable properties such as guaranteed 2. PRELIMINARIES category of reduction methods which have been further 2. PRELIMINARIES developed to have desirable properties such as guaranteed developed tothe have desirable properties such as as guaranteed guaranteed stability for forto the case of double double sided weightings weightings and the the exex- In this section, existing frequency weighted balanced trundeveloped have desirable properties such stability case of sided and this section, existing frequency weighted balanced trunstability for the case of double sided weightings and the theand ex- In istence of error bounds (Sreeram et al. (2013); Sahlan and stability the bounds case of double sided and exIn this section, section, existing frequencynamely weighted balanced truncation techniques are reviewed reviewed namely the methods by In this existing frequency weighted balanced trunistence offor error (Sreeram et weightings al. (2013); Sahlan cation techniques are the methods by istence of(2009); error bounds bounds (Sreeram et al. al. (2013); (2013); Sahlan and cation Sreeramof (2009); Sreeram and Anderson Anderson (1995); Ghafoor istence error (Sreeram et Sahlan and cation techniques are reviewed namely the methods by Enns (1984), Sreeram and Anderson (1995), Sreeram et al. techniques are reviewed namely the methods by Sreeram Sreeram and (1995); Ghafoor (1984), Sreeram and Anderson (1995), Sreeram et al. Sreeram (2009); Sreeram and and Anderson Anderson (1995); (1995); Ghafoor Ghafoor Enns and Sreeram Sreeram (2007)). Sreeram (2009); Sreeram Enns (1984), Sreeram and Anderson (1995), Sreeram et et al. al. (2013)(1984), and Sahlan Sahlan andand Sreeram (2009). An understanding understanding Enns Sreeram Anderson (1995), Sreeram and (2007)). (2013) and and Sreeram (2009). An and Sreeram (2007)). and Sreeram (2007)). (2013) and Sahlan and Sreeram (2009). An understanding of these techniques is required as a prerequisite for the (2013) and Sahlan and Sreeram (2009). An understanding Sahlan and and Sreeram Sreeram presented presented aa new new approach approach by by decomdecom- of these techniques is required as a prerequisite for the Sahlan of these presented techniquesin isthis required as aa prerequisite prerequisite for for the the material presented inis this paper.as these techniques required Sahlan and Sreeram presented a new approach by decomposition of the input-output augmented system using par- of material paper. Sahlan and Sreeram presented a new approach by decomposition of the input-output augmented system using parmaterial presented presented in in this this paper. paper. material position of the input-output augmented system using partial fraction expansion to form a new augmented system position of the input-output augmented system using par- Let Let G(s) G(s) be be the the transfer transfer function function of of aa stable stable original original tial fraction expansion to form a new augmented system tial fraction expansion tois form new augmented augmented system where the system system matrixto isform blockaa diagonal diagonal (Sreeramsystem et al. al. Let tial fraction expansion new Let G(s) be the transfer function of a stable original system with the following minimal realization G(s) be the transfer function of a stable original where the matrix block (Sreeram et with the following minimal realization where the system matrix is block block diagonal (Sreeram et al. al. system (2013);the Sahlan and Sreeram (2009)). Standard balanced where system matrix is diagonal (Sreeram et system with with the the following following minimal minimal realization system realization (2013); Sahlan and Sreeram (2009)). Standard balanced A B A B (2013); Sahlan and Sreeram Sreeram (2009)). Standard balanced truncation by Moore Moore (1981) is is then applied applied to balanced the augaugG(s) = = A B (2013); Sahlan and (2009)). Standard truncation by (1981) then to the G(s) C D A D B truncation by Moore Moore (1981) (1981) is is then then applied applied to to the the augaugmented system. system. G(s) = = C truncation by G(s) mented C D C D mented system. mented system. In this this paper paper frequency weighted weighted model model reduction reduction methods methods Similaly Similaly let let V V (s) (s) and and W W (s) (s) be be the the transfer transfer functions functions In frequency Similaly let V (s) (s) and W (s) (s) weights be the the with transfer functions In this paper frequency weighted model reduction methods Similaly based on partial fraction expansion are firstly reviewed. reviewed. of stable stable let input and output weights with the functions following In this on paper frequency weighted model reduction methods V and W be transfer based partial fraction expansion are firstly of input and output the following of stable input and output weights with the following based on partial fraction expansion are firstly reviewed. based on partial fraction expansion are firstly reviewed. of stable input and output weights with the following
Copyright 2016 IFAC 45 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 45 Peer review© of International Federation of Automatic Copyright 2016 IFAC 45 Copyright ©under 2016 responsibility IFAC 45 Control. 10.1016/j.ifacol.2016.07.488
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minimal realizations V (s) = W (s) =
Deepak Kumar et al. / IFAC-PapersOnLine 49-9 (2016) 045–050
AV BV CV DV AW BW CW DW
2.2 Sreeram and Anderson’s Partial Fraction Expansion Based Technique
The state space realization of the augmented W (s)G(s)V (s) is then given by AW BW C BW DCV BW DDV ˆ A BCV BDV Aˆ B 0 = 0 ˆ ˆ 0 A BV V C D CW DW C DW DCV DW DDV
The partial fraction expansion frequency weighted model reduction method which was first proposed by Saggaf and Franklin (1986, 1988) was only applicable to be used with single sided frequency weighting. Sreeram and Anderson (1995) presented a generalization of the partial fraction expansion frequency weighted model reduction method which is applicable for the case of double sided weighting. A frequency response error bound for this generalized technique was also presented by Sreeram and Anderson (1995). The main idea of this method is presented as follows.
system
(1)
The controllability and observability of the augmented ˆ B, ˆ C, ˆ D} ˆ is given as realization {A, QW Q12 Q13 PW P12 P13 T ˆ = QT12 QE Q23 PE P23 , Q (2) Pˆ = P12 T T P13 P23 PV QT13 QT23 QV
The augmented system matrix in (1) is block diagonalized I −Y R by the similarity transformation matrix T˜ = 0 I X 0 0 I which yields
(6)
AW BW C BW DCV BW DDV A BCV BDV 0 = 0 0 AV BV CW DW C DW DCV DW DDV −1 ˆ T˜ AˆT˜ T˜−1 B = ˆ Cˆ T˜ D X1 AW X12 X13 X2 0 A X23 = 0 0 AV BV CW Y1 Y2 DW DDV AW 0 0 X1 X2 0 A 0 = 0 0 A BV V CW Y1 Y2 DW DDV AW X1 AV BV A X2 = + + CW 0 Y1 DW DDV Y2 0 = Wdiag (s) + Gdiag (s) + Vdiag (s)
(7)
where X, Y and R are obtained by solving the following matrix equations
ˆ satisfy the following Lyapunov such that both Pˆ and Q equations ˆB ˆT = 0 AˆPˆ + Pˆ AˆT + B (3) T T ˆ+Q ˆ Aˆ + Cˆ Cˆ = 0 Aˆ Q (4)
Assuming that there are no pole-zero cancellations in ˆ are W (s)G(s)V (s), the augmented gramians Pˆ and Q positive definite. 2.1 Enns Technique The frequency weighted balanced truncation technique by Enns (1984) is presented in this section. Expanding the (2,2) blocks respectively of equations (3) and (4) yields the following: APE + PE AT + XE = 0 T
A QE + QE A + Y E = 0
(5)
where T + P23 CVT B T + BDV DVT B T XE = BCV P23 T T T T YE = C BW Q12 + QT12 BW C + C T DW DW C
(11)
(12)
(13)
(14)
(8) X12 = Y A − AW Y + BW C = 0 X23 = AX − XAV + BCV = 0 X13 = AW R − RAV + BW CX + Y AX + ... BW DCV + Y BCV − Y XAV = 0 X1 = BW DDV + Y BDV − Y XBV X2 = BDV − XBV Y1 = D W C − C W Y Y2 = DW CX + DW DCV
Simultaneously diagonalizing the weighted gramians PE and QE , we get T T QE T = T −1 PE T −T = diag{σ1 , σ2 , ...σn }
(10)
(9)
where σ1 ≥ σ2 ≥ ... ≥ σr > σr+1 ≥ ... ≥ σn > 0. The order of the reduced order model is denoted by r whereas the order of the original model is denoted by n. Transforming and partitioning the original system we get A11 A12 ˆ B1 Aˆ = T −1 AT = , B = T −1 B = A21 A22 B2 ˆ =D Cˆ = [C1 C2 ] , D
(15) (16) (17) (18) (19) (20) (21)
In this method the following gramians T + XPV X T PSA = PE − P23 XPT F − XP23
The reduced order model is given by Gr (s) = C1 (sI − A11 )−1 B1 + D. The stability of the reduced order model obtained by using Enns technique is not guaranteed to be stable for the case of double sided input weighting since the matrices XE and YE are not guaranteed to be positive definite.
QSA = QE − Q12 Y − Y
T
QT12
T
+ Y QW Y
which satisfy the following Lyapunov equations APSA + PSA AT + X2 X2T = 0 AT QSA + QSA A + Y1T Y1 = 0 46
(22) (23)
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C ¯W C¯ X12 = [BW AW I] −Y = B YA CV ¯ C¯V X23 = [B −X AX] AV = B I DV ¯D ¯V X2 = [B −X AX] BV = B 0 C ¯ W C¯ Y1 = [DW CW 0] −Y = D YA D 0 CX CV 0 R AV X13 = [BW AW I] 0 Y B −R − Y X Y AX I ¯ C¯V = 0 ¯W D =B D 0 CX DV 0 R BV X1 = [BW AW I] 0 Y B −R − Y X Y AX 0 ¯ ¯ ¯ = B W D DV D 0 CX CV 0 R AV Y2 = [DW CW 0] 0 Y B −R − Y X Y AX I ¯ C¯V ¯W D =D
are simultaneously diagonalized Since the realization {A, X2 , Y1 } is minimal and the diagonalized gramians satisfy the Lyapunov equations, this partial fraction expansion based technique yields stable reduced order models for the case of double sided weightings. 2.3 Ghafoor and Sreeram’s Partial Fraction Expansion Based Technique Ghafoor and Sreeram (2007) had further developed the original method by Sreeram and Anderson as follows. Instead of simultaneously diagonalizing the gramians PSA and QSA as with the method by Sreeram and Anderson (1995), Ghafoor and Sreeram had simultaneously diagonalized PX = P + α2 PSA QX = Q + α2 QSA where α and β are real constants and both P and Q are the unweighted gramians satisfying AP + P AT + BB T = 0 AT Q + QA + C T C = 0 The gramians PX and QX satisfy the following Lyapunov equations T =0 APX + PX AT + BX BX
where BX
47
Standard balanced truncation can then be applied to the ¯ ¯ C, ¯ D} ¯ to obtain and new original system G(s) = {A, B, th ¯ ¯r , C¯r , D ¯ r }. r order reduced order model Gr (s) = {Ar , B The final reduced order model Gr (s) = {A, Br , Cr , Dr } is then obtained by deleting the extra rows in C¯r and extra ¯r and both extra rows and columns in D ¯ r. columns in B
AT QY + QY A + CYT CT = 0 C . = [B αX2 ] and CY = βY1
2.4 Sahlan and Sreeram’s Partial Fraction Expansion Based Technique
3. MAIN WORK
Sahlan and Sreeram (2009) had also proposed a method based on decomposing the augmented system W (s)G(s)V (s) to be equal to Wdiag (s) + Gdiag (s) + Vdiag (s) by using partial fraction expansion where the system matrix is block diagonalized. The decomposed system (Wdiag (s) + Gdiag (s) + Vdiag (s)) is then expressed as a new augmented ¯ (s)G(s) ¯ V¯ (s) such that the system matrix of system W Wdiag (s) + Gdiag (s) + Vdiag (s) is the same as the system ¯ (s)G(s) ¯ V¯ (s) and is block diagonal such that matrix of W W (s)G(s)V (s) = Wdiag (s) + Gdiag (s) + Vdiag (s) ¯ (s)G(s) ¯ V¯ (s) =W ¯ ¯ C, ¯ D} ¯ is the new original system, where G(s) = {A, B, ¯ V } and W ¯ (s) = {AW , B ¯ W , CW , D ¯W } V¯ (s) = {AV , BV , C¯V , D are the input and output weights respectively where ¯ W = [DW CW 0] ¯W = [BW AW I] , D B C DV ¯ = [B −X AX] , C¯ = −Y , D ¯ V = BV B YA 0 CV D 0 CX ¯= 0 0 R D , C¯V = AV I Y B −R − Y X Y AX The equations in (15) to (21) can then be factorized as follows: 47
In the original partial fraction expansion method by Sahlan and Sreeram (2009) described in the previous section, standard truncation by Moore (1981) was applied to the augmented system. In this section we propose the following algorithm which involves applying singular perturbation approximation by Liu and Anderson (1989) to the augmented system instead of standard truncation in order to further improve the frequency weighted approximation error relative to the original method by Sahlan and Sreeram (2009). Error bounds for the proposed method are also derived. 3.1 Proposed Algorithm Step 1: Compute the matrices X and Y from (16) and (15) respectively. ¯P F and Step 2: Calculate the fictitious input matrix B output matrix C¯P F by using suitable values of the free parameters α and β as follows. ¯P F = [B −αX AX] B C C¯P F = −βY YA Step 3: Weighted controllability and observability grami¯ QF are obtained by solving the following ans P¯P F and Q Lyapunov equations.
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¯P F B ¯PT F = 0 AP¯P F + P¯P F AT + B (24) T ¯ T ¯ ¯ ¯ A Q P F + Q P F A + C P F CP F = 0 (25) Step 4: Calculate the transformation matrix T which simultaneously diagonalizes the matrices PP F and QP F such that ¯ P F T = T −1 P¯P F T −T TTQ = diag{σ1 , σ2 , ...σr , σr+1 , ...σn } where σi ≥ σi+1 , i = 1, 2, ..., n − 1 and σr > σr+1 Step 5: Compute the frequency weighted balanced realization. ˆ = T −1 B, Cˆ = CT Aˆ = T −1 AT, B ˆ B, ˆ C} ˆ as follows Step 6: Partition {A, B1 ˆ A11 A12 ˆ ,B = , C = [C1 C2 ] Aˆ = B2 A21 A22
||W (s)[G(s) − Grspa (s)]V (s)||∞ = ||W (s)[C(sI − A)−1 B + D − ... (Cspa (sI − Aspa )−1 Bspa − Dspa )]V (s)||∞
= ||W (s)[C(sI − A)−1 B + D − (C1 − C2 A−1 22 A21 ) × ...
−1 (sI − Aspa )−1 (B1 − A12 A−1 22 B2 ) − D + C2 A22 B2 ]V (s)||∞ ¯P F KP F + ... = ||W (s)[LP F C¯P F (sI − A)−1 B ¯2P F KP F − LP F (C¯1P F − C¯2P F A−1 A21 )× LP F C¯2P F A−1 B 22 −1
22
−1 ¯ ¯ ¯1P F − A12 A−1 B ¯ (B 22 2P F ) + C2P F A22 B2P F }KP F V (s)||∞ ¯P F − ... ≤ ||W (s)LP F ||∞ ||C¯P F (sI − A)−1 B −1 (C¯1P F − C¯2P F A A21 )(sI − Aspa )−1 × ...
where A11 ∈ Rr×r , B1 ∈ Rr×p , C1 ∈ Rq×r and r < n
22
−1 ¯ ¯ ¯1P F − A12 A−1 B ¯ (B 22 2P F ) + C2P F A22 B2P F ||∞ × ... 1 ¯P F − ... ||KP F V (s)||∞ ≤ ||W (s)||∞ ||C¯P F (sI − A)−1 B β ¯spa P F + ... (C¯spa P F (sI − Aspa )−1 B 1 ¯ C¯2P F A−1 22 B2P F ||∞ ||V (s)||∞ β
Step 7: The reduced order model is given by Grspa (s) = Cspa (sI − Aspa )−1 Bspa + Dspa , where Aspa = A11 − A12 A−1 22 A21
Bspa = B1 − A12 A−1 22 B2 Cspa = C1 − C2 A−1 22 A21 Dspa = D − C2 A−1 22 B2
¯P F , C¯P F } is a balanced realized model and Since {A, B ¯spa P F , C¯spa P F , −C¯2P F A˜−1 B ¯ {Aspa , B 22 2P F } is its corresponding reduced order model, then the from Liu and Anderson (1989) the following holds true ¯P F − (C¯spa P F (sI − Aspa )−1 B ¯spa P F ... ||C¯P F (sI − A)−1 B
3.2 Error Bounds In this section the error bounds for the reduced order models obtained using the proposed algorithm is derived. To establish the relationship between the input and output matrices (B and C), and the new fictitious input and ¯P F and C¯P F ), we define two constant output matrices (B I/α matrices, KP F = and LP F = [I/β 0] respectively 0 where I is an identity matrix with the appropriate dimension. It follows that the following relationships hold true between the input and output matrices B and C and ¯P F and C¯P F as the fictitious input and output matrices B follows ¯ P F KP F (26) B=B (27) C = LP F C¯P F
¯ + C¯2P F A˜−1 22 B2P F ||∞ n ≤2 σk k=r+1
it follows that
||W (s)(G(s) − Grspa (s)V (s)||∞ ≤ δ with δ =
σk
k=r+1
(s)||∞
ki =r+1
and
k=r+1
2 αβ ||W (s)||∞ ||V
2 αβ ||W (s)||∞ ||V
n
Corollary 1 : The error bounds when only input or output weights are used are as follows: n 2 ||(G(s) − Grspa (s)V (s)||∞ ≤ ||V (s)||∞ σki α
Theorem 1: Let G(s) be a stable and strictly proper transfer function whereas V (s) and W (s) are the input and output weights respectively. If Gspa (s) is a reduced order model obtained by using the algorithm presented in Section 3.1, it follows that the following error bound holds true: n ||W (s)(G(s) − Grspa (s)V (s)||∞ ≤ δ σk (28) where δ =
22
¯1P F − A12 A−1 B ¯ (sI − Aspa ) (B 22 2P F )KP F ]V (s)||∞ −1 ¯P F − ... = ||W (s)LP F {C¯P F (sI − A) B −1 (C¯1P F − C¯2P F A˜ A21 )(sI − Aspa )−1 × ...
||W (s)(G(s) − Grspa (s)||∞ ≤
(s)||∞
n 2 ||W (s)||∞ σko β ko =r+1
where σki and σko are the Hankel singular values of the ¯P F , C} and {A, B, C¯P F } respectively. systems {A, B
Proof: ¯ ¯ Upon partitioning the fictitious matrices BP F and CP F as ¯1P F B ¯ ¯ ¯ ¯P F = B ¯2P F and CP F = [C1P F C2P F ] respectively B ¯1P F KP F , B2 = B ¯2P F KP F and and substituting B1 = B C1 = LP F C¯1P F , C2 = LP F C¯2P F , we have
4. NUMERICAL EXAMPLE Considering the 4th order system originally from Enns (1984) 48
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Table 3. Weighted Approximation Error Obtained by Sahlan and Sreeram (2009); Sreeram et al. (2013)
Table 1. Weighted Approximation Error Obtained by Various Techniques Order
Enns W.E. 2.1291 0.2660 0.1131
1 2 3
Lin and Chiu’s W.E. 2.5744 0.5607 0.1645
Wang et al
Imran et al
W.E. 2.1213 0.2720 0.1151
W.E 2.1234 0.2424 0.1075
E.B. 7.2898 1.4895 0.3228
Order 1
Table 2. Weighted Approximation Error Obtained by Ghafoor and Sreeram (2007)
2
Order
3
1
2
3
Ghafoor and Sreeram’s Method α
β
W.E. 1
W.E 2
Error Bound
0.4 0.4 0.5 0.8 0.9 0.7 0.4 0.8 1
0.4 0.5 0.4 2.9 2.7 0.7 0.4 0.5 0.6
2.1126 2.1165 2.1131 0.2709 0.2710 0.2784 0.1009 0.1084 0.1117
1.4564 1.4877 1.4833 0.3116 0.3238 0.2543 0.0668 0.0714 0.0750
3.4848 3.6212 3.6229 2.3312 2.3242 0.8949 0.1589 0.1786 0.1942
−1 0 0 0 0 0 −2 0 0 1/2 A= ,B = 0 0 −3 0 1 0 0 0 −4 −1/2 1 0 10 00 C= ,D = 4/15 1 0 1 00
49
Sreeram and Sahlan’s Method α β Weighted Error 8.1 6.7 2.1158 8.2 6.9 2.1161 8.0 7.0 2.1201 180 110 0.3149 90 110 0.3219 147.5 32.5 0.3166 103.9 35.8 0.1081 105.7 35.7 0.1082 105.3 35.3 0.1082
Error Bound 13.922 13.488 13.580 0.5290 0.5686 0.6996 0.3104 0.3092 0.3120
Table 4. Weighted Approximation Error Obtained Using the Proposed Method Order 1
−5/2 −3/2 −5 1/6
2
3
Proposed technique α β Weighted Error 0.1 0.1 1.2361 0.2 0.1 1.2363 0.2 0.2 1.2375 4.9 4.9 0.2430 4.5 4.5 0.2432 5.5 5.5 0.2433 6.0 2.0 0.0614 7.0 3.0 0.0648 5.0 1.0 0.0719
Error Bound 6.9011 6.9040 6.9112 2.4105 2.1981 2.7572 0.5388 0.6221 0.4599
and the 2nd order input and output weighting functions as −4.5 0 30 , BV = BW = , AV = AW = 0 −4.5 03 1.5 0 10 C V = CW = , DV = DW = 0 1.5 01 Table 1 shows the results of the weighted approximation errors - ||W (s)[G(s)−Gr (s)]V (s)||∞ for the existing methods by Enns (1984); Lin and Chiu (1992); Wang et al. (1999); Imran et al. (2014) for benchmarking. Weighted approximation error and error bounds are denoted as W.E and E.B. respectively. Table 2 shows the weighted approximation error obtained using both methods described in Ghafoor and Sreeram (2007). W.E.1 and W.E. 2 denote the weighted approximation errors obtained using method 1 and method 2 in Ghafoor and Sreeram (2007). Table 3 shows the results obtained by using the method by Sahlan and Sreeram (2009). Applying the proposed method with various values of the parameters α and β yields the results in Table 4. It can be observed that the proposed approach gives reduced frequency weighted approximation error relative to the method by Sahlan and Sreeram (2009). Figure 1 shows the variation of the frequency weighted approximation error relative to the variation of the free parameters α and β for a first order controller using the proposed method.
Fig. 1. Variation of Weighted Approximation Error due to Variation of α and β model reduction method is developed which is a further development to the method proposed by Sahlan and Sreeram (2009). By applying singular perturbation approximation instead of standard truncation, it has been demonstrated that the frequency weighted approximation error can be reduced. An error bound has also been developed for the proposed technique. REFERENCES
5. CONCLUSION
Ding, D.W., Du, X., and Li, X. (2015). Finite frequency model reduction of two-dimensional digital filters. IEEE Transactions on Automatic Control, 60(6), 1624–1629. Du, X. and Yang, G. (2010). H-infinity model reduction of linear continuous-time systems over finite frequency
In this paper, a frequency weighted balanced singular perturbation approximation model reduction method which belongs to the class of partial fraction expansion based 49
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interval. IET Control Theory and Applications, 4(3), 499–508. Enns, D. (1984). Model reduction with balanced realizations: An error bound and a frequency weighted generalization. In The 23rd IEEE Conference on Decision and Control, 1984, 127–132. Ghafoor, A. and Sreeram, V. (2007). Partial fraction expansion based frequency weighted model reduction techniquee with error bounds. IEEE Transactions on Automatic Control, 52(10), 1942–1948. Imran, M., Ghafoor, A., and Sreeram, V. (2014). A frequency weighted model order reduction technique and error bounds. Automatica, 50(12), 3304 – 3309. Li, X., Yin, S., and Gao, H. (2014). Passivity-preserving model reduction with finite frequency approximation performance. Automatica, 50(9), 2294–2303. Lin, C. and Chiu, T. (1992). Model reduction via frequency weighted balanced realization. Control Theory and Advanced Technology, 8, 341–451. Liu, Y. and Anderson, B. (1989). Singular perturbation approximation of balanced systems. International Journal of Control, 50(4), 1339–1405. Moore, B. (1981). Principal component analysis in linear system: Controllability, observability, and model reduction. IEEE Trans. Automat. Contr, vol.AC-26, 17–32. Saggaf, U. and Franklin, G. (1986). On model reduction. In Proc. of the 23rd IEEE Conf. on Decision and Control, 1064–1069. Saggaf, U. and Franklin, G. (1988). Model reduction via balanced realization. IEEE Trans. on Autom. Control, AC-33, 687–692. Sahlan, S. and Sreeram, V. (2009). New results on partial fraction expansion based frequency weighted balanced truncation. In American Control Conference, 2009. ACC ’09., 5695–5700. Shaker, H.R. and Tahavori, M. (2013). Frequency-interval control configuration selection for multivariable bilinear systems. Journal of Process Control, 23 (6), 894–904. Shaker, H. and Tahavori, M. (2014). Frequency interval model reduction of bilinear systems. IEEE Transactions on Automatic Control, 59 (7), 1948–1953. Sreeram, V. and Anderson, B. (1995). Frequency weighted balanced reduction technique: A generalization and an error bound. In Proceedings of the 34th IEEE Conference on Decision and Control, 3576–3581. Sreeram, V., Sahlan, S., Muda, W., Fernando, T., and Iu, H. (2013). A generalised partial-fraction-expansion based frequency weighted balanced truncation technique. International Journal of Control, 86(5), 833–843. Wang, G., Sreeram, V., and Liu, W. (1999). A new frequency weighted balanced truncation method and error bound. IEEE Transactions on Automatic Control, 44(9), 1734–1737.
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ScienceDirect IFAC-PapersOnLine 49-9 (2016) 045–050
Partial Fraction Expansion Based Partial Fraction Expansion Based Partial Fraction Expansion Based Frequency Weighted Balanced Singular Frequency Weighted Balanced Singular Frequency Weighted Balanced Singular Perturbation Approximation Perturbation Approximation Model Model Perturbation Approximation Model Reduction Technique with Error Bounds Reduction Technique with Error Reduction Technique with Error Bounds Bounds a b,c Deepak , Victor Sreeramb b, Deepak Kumar Kumaraaa ,, Ahmad Ahmad Jazlan Jazlanb,c b,cb, Victor Sreeramb b, b,c Deepak Kumar Kumar ,, Roberto Ahmad Jazlan Victor Sreeram Togneri b,,, Victor Deepak Ahmad Jazlan Sreeram ,, Roberto Tognerib , b Roberto Roberto Togneri Togneri ,, a a Electrical Engineering Department, M. N. National Institute of Electrical Engineering Department, M. N. National Institute of a a Electrical Allahabad, Department, M. National Technology India (e-mail: [email protected]) Electrical Engineering Engineering Department, M. N. N. [email protected]) National Institute Institute of of Technology India (e-mail: deepak deepak b Technology Allahabad, Allahabad, India (e-mail: deepak [email protected]) of Electrical and Electronics Engineering, University of b Department Technology Allahabad, India (e-mail: deepak [email protected]) Department of Electrical and Electronics Engineering, University of b b Department of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: Department of Electrical and Electronics Engineering, University of Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: [email protected], [email protected], Western Australia, 35 Stirling Highway, WA 6009, Australia (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) c [email protected]) Engineering, Faculty c Department of Mechatronics [email protected]) Department of Mechatronics Engineering, Faculty of of Engineering, Engineering, c c Department of Mechatronics Engineering, Faculty of Engineering, Engineering, International Islamic University Malaysia, Jalan Gombak, 53100 Department of Mechatronics Engineering, Faculty of International Islamic University Malaysia, Jalan Gombak, 53100 International University Malaysia, Jalan Kuala Lumpur, Malaysia International Islamic Islamic University Malaysia, Jalan Gombak, Gombak, 53100 53100 Kuala Lumpur, Malaysia Kuala Kuala Lumpur, Lumpur, Malaysia Malaysia Abstract: In In this this paper, paper, aa new new frequency frequency weighted weighted partial partial fraction fraction expansion expansion based based model model Abstract: Abstract: In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to Abstract: In this paper, a new frequency weighted partial fraction expansion based model reduction technique is developed based on the partial fraction expansion approach. In order to reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation reduction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation further reduce frequency weighted singular perturbation is incorporated incorporated into the algorithm. algorithm. This technique techniqueerror, results in stable stable reduced approximation order models models further reduce the theinto frequency weighted approximation approximation error, singular perturbation approximation is the This results in reduced order is incorporated into the algorithm. This technique results in stable reduced order models regardless if single sided or double sided weights are used. Error bounds are also derived for is incorporated intosided the or algorithm. Thisweights technique in stable reduced order models regardless if single double sided are results used. Error bounds are also derived for regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free regardless if single sided or double sided weights are used. Error bounds are also derived for the proposed method. For minimization of the frequency weighted approximation error, free the proposedare method. For minimization minimization of the the A frequency weighted approximation error, free parameters aremethod. introduced into the the algorithm. algorithm. A numerical exampleapproximation is provided provided in inerror, orderfree to the proposed For of frequency weighted parameters introduced into numerical example is order to parameters introduced into validate the the are proposed algorithm. parameters are introduced into the the algorithm. algorithm. A A numerical numerical example example is is provided provided in in order order to to validate proposed algorithm. validate validate the the proposed proposed algorithm. algorithm. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Model Reduction, Frequency Weighting, Weighting, Singular Singular Perturbation Perturbation Approximation, Approximation, Keywords: Model Reduction, Frequency Keywords: ModelExpansion, Reduction,Square Frequency Weighting, Singular Perturbation Perturbation Approximation, Approximation, Partial Fraction Expansion, Square RootWeighting, Balancing.Singular Keywords: Model Reduction, Frequency Partial Fraction Root Balancing. Partial Fraction Expansion, Square Root Balancing. Partial Fraction Expansion, Square Root Balancing. 1. INTRODUCTION INTRODUCTION Singular perturbation perturbation approximation approximation by by Liu Liu and and AnderAnder1. Singular 1. INTRODUCTION INTRODUCTION Singular perturbation approximation by Liu Liu and method Anderson (1989) (1989) is applied applied approximation to the the framework framework of the the method 1. Singular perturbation by and Anderson is to of son (1989) and is applied applied to (2009) the framework framework of improve the method method by Sahlan Sahlan and Sreeramto (2009) in order order to to improve the Model reduction reduction methods methods which which emphasize emphasize particular particular frefre- son (1989) is the of the by Sreeram in the Model by Sahlan and Sreeram (2009) in order to improve the frequency weighted approximation error. Two free paramModel reduction methods which emphasize particular frequency intervals continue to receive interest from the by Sahlan and Sreeram (2009) in order to improve the Model emphasize particular quencyreduction intervals methods continuewhich to receive interest from frethe frequency weighted approximation error. Two free paramfrequency weighted approximation error. Two free free parameters α α and andweighted β are are also also included in in the the algorithm to paramfurther quency intervals continue due to receive receive interest from from the control theory theory community due to the the importance importance of the ob- frequency approximation error. Two quency intervals continue to interest eters β included algorithm to further control community to of obeters α and and β frequency are also also included included inapproximation the algorithm algorithm to to further minimize theβ frequency weighted approximation error. Ercontrol theory community due to the importance of obtaining such reduced order models in a variety of apeters α are in the further control theory community due to the importance of obminimize the weighted error. Ertaining such reduced order models in a variety of apminimize the frequency weighted approximation error. Error bounds are developed for the proposed method. Simutaining such reduced order models in a variety of applications (Du and Yang (2010); Li et al. (2014); Ding the weighted error. Ertaining such order models ap- minimize ror bounds arefrequency developed for the approximation proposed method. Simuplications (Dureduced and Yang (2010); Li in et aal.variety (2014);ofDing ror bounds aredemonstrate developed for for thereduced proposed method. SimuSimulation results demonstrate that reduced approximation erplications (Du Shaker and Yang Yang (2010); Li (2014, et al. al. (2014); (2014); Ding et al. al. (2015); (2015); Shaker and(2010); Tahavori (2014, 2013)). Ding Fre- ror bounds are developed the proposed method. plications (Du and Li et lation results that approximation eret and Tahavori 2013)). Frelation results demonstrate that reduced approximation errors relative to the original method by Sahlan and Sreeram et al. (2015); Shaker and Tahavori (2014, 2013)). Frequency weighted model reduction methods based on the lation results demonstrate that reduced approximation eret al. (2015); Shaker and Tahavori (2014, 2013)). Frequency weighted model reduction methods based on the rors relative to the original method by Sahlan and Sreeram rors relative toachieved the original original method byproposed Sahlan and and Sreeram (2009) can be be achieved by method using the theby proposed approach. quency weightedexpansion model reduction reduction methods based on the the rors partial fraction fraction expansion approachmethods originally introduced relative to the Sahlan Sreeram quency weighted model based on (2009) can by using approach. partial approach originally introduced (2009) can be achieved by using the proposed approach. partial fraction expansion approach originally introduced by Saggaf and Franklin (1986, 1988) are an important (2009) can be achieved by using the proposed approach. partial fraction approach originally by Saggaf and expansion Franklin (1986, 1988) are an introduced important by Saggafof and Franklin (1986, which 1988) have are an an important category ofand reduction methods which have been further 2. PRELIMINARIES PRELIMINARIES by Saggaf Franklin (1986, 1988) are important category reduction methods been further 2. category of reduction methods which have been further developed to have desirable properties such as guaranteed 2. PRELIMINARIES category of reduction methods which have been further 2. PRELIMINARIES developed to have desirable properties such as guaranteed developed tothe have desirable properties such as as guaranteed guaranteed stability for forto the case of double double sided weightings weightings and the the exex- In this section, existing frequency weighted balanced trundeveloped have desirable properties such stability case of sided and this section, existing frequency weighted balanced trunstability for the case of double sided weightings and the theand ex- In istence of error bounds (Sreeram et al. (2013); Sahlan and stability the bounds case of double sided and exIn this section, section, existing frequencynamely weighted balanced truncation techniques are reviewed reviewed namely the methods by In this existing frequency weighted balanced trunistence offor error (Sreeram et weightings al. (2013); Sahlan cation techniques are the methods by istence of(2009); error bounds bounds (Sreeram et al. al. (2013); (2013); Sahlan and cation Sreeramof (2009); Sreeram and Anderson Anderson (1995); Ghafoor istence error (Sreeram et Sahlan and cation techniques are reviewed namely the methods by Enns (1984), Sreeram and Anderson (1995), Sreeram et al. techniques are reviewed namely the methods by Sreeram Sreeram and (1995); Ghafoor (1984), Sreeram and Anderson (1995), Sreeram et al. Sreeram (2009); Sreeram and and Anderson Anderson (1995); (1995); Ghafoor Ghafoor Enns and Sreeram Sreeram (2007)). Sreeram (2009); Sreeram Enns (1984), Sreeram and Anderson (1995), Sreeram et et al. al. (2013)(1984), and Sahlan Sahlan andand Sreeram (2009). An understanding understanding Enns Sreeram Anderson (1995), Sreeram and (2007)). (2013) and and Sreeram (2009). An and Sreeram (2007)). and Sreeram (2007)). (2013) and Sahlan and Sreeram (2009). An understanding of these techniques is required as a prerequisite for the (2013) and Sahlan and Sreeram (2009). An understanding Sahlan and and Sreeram Sreeram presented presented aa new new approach approach by by decomdecom- of these techniques is required as a prerequisite for the Sahlan of these presented techniquesin isthis required as aa prerequisite prerequisite for for the the material presented inis this paper.as these techniques required Sahlan and Sreeram presented a new approach by decomposition of the input-output augmented system using par- of material paper. Sahlan and Sreeram presented a new approach by decomposition of the input-output augmented system using parmaterial presented presented in in this this paper. paper. material position of the input-output augmented system using partial fraction expansion to form a new augmented system position of the input-output augmented system using par- Let Let G(s) G(s) be be the the transfer transfer function function of of aa stable stable original original tial fraction expansion to form a new augmented system tial fraction expansion tois form new augmented augmented system where the system system matrixto isform blockaa diagonal diagonal (Sreeramsystem et al. al. Let tial fraction expansion new Let G(s) be the transfer function of a stable original system with the following minimal realization G(s) be the transfer function of a stable original where the matrix block (Sreeram et with the following minimal realization where the system matrix is block block diagonal (Sreeram et al. al. system (2013);the Sahlan and Sreeram (2009)). Standard balanced where system matrix is diagonal (Sreeram et system with with the the following following minimal minimal realization system realization (2013); Sahlan and Sreeram (2009)). Standard balanced A B A B (2013); Sahlan and Sreeram Sreeram (2009)). Standard balanced truncation by Moore Moore (1981) is is then applied applied to balanced the augaugG(s) = = A B (2013); Sahlan and (2009)). Standard truncation by (1981) then to the G(s) C D A D B truncation by Moore Moore (1981) (1981) is is then then applied applied to to the the augaugmented system. system. G(s) = = C truncation by G(s) mented C D C D mented system. mented system. In this this paper paper frequency weighted weighted model model reduction reduction methods methods Similaly Similaly let let V V (s) (s) and and W W (s) (s) be be the the transfer transfer functions functions In frequency Similaly let V (s) (s) and W (s) (s) weights be the the with transfer functions In this paper frequency weighted model reduction methods Similaly based on partial fraction expansion are firstly reviewed. reviewed. of stable stable let input and output weights with the functions following In this on paper frequency weighted model reduction methods V and W be transfer based partial fraction expansion are firstly of input and output the following of stable input and output weights with the following based on partial fraction expansion are firstly reviewed. based on partial fraction expansion are firstly reviewed. of stable input and output weights with the following
Copyright 2016 IFAC 45 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 45 Peer review© of International Federation of Automatic Copyright 2016 IFAC 45 Copyright ©under 2016 responsibility IFAC 45 Control. 10.1016/j.ifacol.2016.07.488
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minimal realizations V (s) = W (s) =
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AV BV CV DV AW BW CW DW
2.2 Sreeram and Anderson’s Partial Fraction Expansion Based Technique
The state space realization of the augmented W (s)G(s)V (s) is then given by AW BW C BW DCV BW DDV ˆ A BCV BDV Aˆ B 0 = 0 ˆ ˆ 0 A BV V C D CW DW C DW DCV DW DDV
The partial fraction expansion frequency weighted model reduction method which was first proposed by Saggaf and Franklin (1986, 1988) was only applicable to be used with single sided frequency weighting. Sreeram and Anderson (1995) presented a generalization of the partial fraction expansion frequency weighted model reduction method which is applicable for the case of double sided weighting. A frequency response error bound for this generalized technique was also presented by Sreeram and Anderson (1995). The main idea of this method is presented as follows.
system
(1)
The controllability and observability of the augmented ˆ B, ˆ C, ˆ D} ˆ is given as realization {A, QW Q12 Q13 PW P12 P13 T ˆ = QT12 QE Q23 PE P23 , Q (2) Pˆ = P12 T T P13 P23 PV QT13 QT23 QV
The augmented system matrix in (1) is block diagonalized I −Y R by the similarity transformation matrix T˜ = 0 I X 0 0 I which yields
(6)
AW BW C BW DCV BW DDV A BCV BDV 0 = 0 0 AV BV CW DW C DW DCV DW DDV −1 ˆ T˜ AˆT˜ T˜−1 B = ˆ Cˆ T˜ D X1 AW X12 X13 X2 0 A X23 = 0 0 AV BV CW Y1 Y2 DW DDV AW 0 0 X1 X2 0 A 0 = 0 0 A BV V CW Y1 Y2 DW DDV AW X1 AV BV A X2 = + + CW 0 Y1 DW DDV Y2 0 = Wdiag (s) + Gdiag (s) + Vdiag (s)
(7)
where X, Y and R are obtained by solving the following matrix equations
ˆ satisfy the following Lyapunov such that both Pˆ and Q equations ˆB ˆT = 0 AˆPˆ + Pˆ AˆT + B (3) T T ˆ+Q ˆ Aˆ + Cˆ Cˆ = 0 Aˆ Q (4)
Assuming that there are no pole-zero cancellations in ˆ are W (s)G(s)V (s), the augmented gramians Pˆ and Q positive definite. 2.1 Enns Technique The frequency weighted balanced truncation technique by Enns (1984) is presented in this section. Expanding the (2,2) blocks respectively of equations (3) and (4) yields the following: APE + PE AT + XE = 0 T
A QE + QE A + Y E = 0
(5)
where T + P23 CVT B T + BDV DVT B T XE = BCV P23 T T T T YE = C BW Q12 + QT12 BW C + C T DW DW C
(11)
(12)
(13)
(14)
(8) X12 = Y A − AW Y + BW C = 0 X23 = AX − XAV + BCV = 0 X13 = AW R − RAV + BW CX + Y AX + ... BW DCV + Y BCV − Y XAV = 0 X1 = BW DDV + Y BDV − Y XBV X2 = BDV − XBV Y1 = D W C − C W Y Y2 = DW CX + DW DCV
Simultaneously diagonalizing the weighted gramians PE and QE , we get T T QE T = T −1 PE T −T = diag{σ1 , σ2 , ...σn }
(10)
(9)
where σ1 ≥ σ2 ≥ ... ≥ σr > σr+1 ≥ ... ≥ σn > 0. The order of the reduced order model is denoted by r whereas the order of the original model is denoted by n. Transforming and partitioning the original system we get A11 A12 ˆ B1 Aˆ = T −1 AT = , B = T −1 B = A21 A22 B2 ˆ =D Cˆ = [C1 C2 ] , D
(15) (16) (17) (18) (19) (20) (21)
In this method the following gramians T + XPV X T PSA = PE − P23 XPT F − XP23
The reduced order model is given by Gr (s) = C1 (sI − A11 )−1 B1 + D. The stability of the reduced order model obtained by using Enns technique is not guaranteed to be stable for the case of double sided input weighting since the matrices XE and YE are not guaranteed to be positive definite.
QSA = QE − Q12 Y − Y
T
QT12
T
+ Y QW Y
which satisfy the following Lyapunov equations APSA + PSA AT + X2 X2T = 0 AT QSA + QSA A + Y1T Y1 = 0 46
(22) (23)
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C ¯W C¯ X12 = [BW AW I] −Y = B YA CV ¯ C¯V X23 = [B −X AX] AV = B I DV ¯D ¯V X2 = [B −X AX] BV = B 0 C ¯ W C¯ Y1 = [DW CW 0] −Y = D YA D 0 CX CV 0 R AV X13 = [BW AW I] 0 Y B −R − Y X Y AX I ¯ C¯V = 0 ¯W D =B D 0 CX DV 0 R BV X1 = [BW AW I] 0 Y B −R − Y X Y AX 0 ¯ ¯ ¯ = B W D DV D 0 CX CV 0 R AV Y2 = [DW CW 0] 0 Y B −R − Y X Y AX I ¯ C¯V ¯W D =D
are simultaneously diagonalized Since the realization {A, X2 , Y1 } is minimal and the diagonalized gramians satisfy the Lyapunov equations, this partial fraction expansion based technique yields stable reduced order models for the case of double sided weightings. 2.3 Ghafoor and Sreeram’s Partial Fraction Expansion Based Technique Ghafoor and Sreeram (2007) had further developed the original method by Sreeram and Anderson as follows. Instead of simultaneously diagonalizing the gramians PSA and QSA as with the method by Sreeram and Anderson (1995), Ghafoor and Sreeram had simultaneously diagonalized PX = P + α2 PSA QX = Q + α2 QSA where α and β are real constants and both P and Q are the unweighted gramians satisfying AP + P AT + BB T = 0 AT Q + QA + C T C = 0 The gramians PX and QX satisfy the following Lyapunov equations T =0 APX + PX AT + BX BX
where BX
47
Standard balanced truncation can then be applied to the ¯ ¯ C, ¯ D} ¯ to obtain and new original system G(s) = {A, B, th ¯ ¯r , C¯r , D ¯ r }. r order reduced order model Gr (s) = {Ar , B The final reduced order model Gr (s) = {A, Br , Cr , Dr } is then obtained by deleting the extra rows in C¯r and extra ¯r and both extra rows and columns in D ¯ r. columns in B
AT QY + QY A + CYT CT = 0 C . = [B αX2 ] and CY = βY1
2.4 Sahlan and Sreeram’s Partial Fraction Expansion Based Technique
3. MAIN WORK
Sahlan and Sreeram (2009) had also proposed a method based on decomposing the augmented system W (s)G(s)V (s) to be equal to Wdiag (s) + Gdiag (s) + Vdiag (s) by using partial fraction expansion where the system matrix is block diagonalized. The decomposed system (Wdiag (s) + Gdiag (s) + Vdiag (s)) is then expressed as a new augmented ¯ (s)G(s) ¯ V¯ (s) such that the system matrix of system W Wdiag (s) + Gdiag (s) + Vdiag (s) is the same as the system ¯ (s)G(s) ¯ V¯ (s) and is block diagonal such that matrix of W W (s)G(s)V (s) = Wdiag (s) + Gdiag (s) + Vdiag (s) ¯ (s)G(s) ¯ V¯ (s) =W ¯ ¯ C, ¯ D} ¯ is the new original system, where G(s) = {A, B, ¯ V } and W ¯ (s) = {AW , B ¯ W , CW , D ¯W } V¯ (s) = {AV , BV , C¯V , D are the input and output weights respectively where ¯ W = [DW CW 0] ¯W = [BW AW I] , D B C DV ¯ = [B −X AX] , C¯ = −Y , D ¯ V = BV B YA 0 CV D 0 CX ¯= 0 0 R D , C¯V = AV I Y B −R − Y X Y AX The equations in (15) to (21) can then be factorized as follows: 47
In the original partial fraction expansion method by Sahlan and Sreeram (2009) described in the previous section, standard truncation by Moore (1981) was applied to the augmented system. In this section we propose the following algorithm which involves applying singular perturbation approximation by Liu and Anderson (1989) to the augmented system instead of standard truncation in order to further improve the frequency weighted approximation error relative to the original method by Sahlan and Sreeram (2009). Error bounds for the proposed method are also derived. 3.1 Proposed Algorithm Step 1: Compute the matrices X and Y from (16) and (15) respectively. ¯P F and Step 2: Calculate the fictitious input matrix B output matrix C¯P F by using suitable values of the free parameters α and β as follows. ¯P F = [B −αX AX] B C C¯P F = −βY YA Step 3: Weighted controllability and observability grami¯ QF are obtained by solving the following ans P¯P F and Q Lyapunov equations.
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¯P F B ¯PT F = 0 AP¯P F + P¯P F AT + B (24) T ¯ T ¯ ¯ ¯ A Q P F + Q P F A + C P F CP F = 0 (25) Step 4: Calculate the transformation matrix T which simultaneously diagonalizes the matrices PP F and QP F such that ¯ P F T = T −1 P¯P F T −T TTQ = diag{σ1 , σ2 , ...σr , σr+1 , ...σn } where σi ≥ σi+1 , i = 1, 2, ..., n − 1 and σr > σr+1 Step 5: Compute the frequency weighted balanced realization. ˆ = T −1 B, Cˆ = CT Aˆ = T −1 AT, B ˆ B, ˆ C} ˆ as follows Step 6: Partition {A, B1 ˆ A11 A12 ˆ ,B = , C = [C1 C2 ] Aˆ = B2 A21 A22
||W (s)[G(s) − Grspa (s)]V (s)||∞ = ||W (s)[C(sI − A)−1 B + D − ... (Cspa (sI − Aspa )−1 Bspa − Dspa )]V (s)||∞
= ||W (s)[C(sI − A)−1 B + D − (C1 − C2 A−1 22 A21 ) × ...
−1 (sI − Aspa )−1 (B1 − A12 A−1 22 B2 ) − D + C2 A22 B2 ]V (s)||∞ ¯P F KP F + ... = ||W (s)[LP F C¯P F (sI − A)−1 B ¯2P F KP F − LP F (C¯1P F − C¯2P F A−1 A21 )× LP F C¯2P F A−1 B 22 −1
22
−1 ¯ ¯ ¯1P F − A12 A−1 B ¯ (B 22 2P F ) + C2P F A22 B2P F }KP F V (s)||∞ ¯P F − ... ≤ ||W (s)LP F ||∞ ||C¯P F (sI − A)−1 B −1 (C¯1P F − C¯2P F A A21 )(sI − Aspa )−1 × ...
where A11 ∈ Rr×r , B1 ∈ Rr×p , C1 ∈ Rq×r and r < n
22
−1 ¯ ¯ ¯1P F − A12 A−1 B ¯ (B 22 2P F ) + C2P F A22 B2P F ||∞ × ... 1 ¯P F − ... ||KP F V (s)||∞ ≤ ||W (s)||∞ ||C¯P F (sI − A)−1 B β ¯spa P F + ... (C¯spa P F (sI − Aspa )−1 B 1 ¯ C¯2P F A−1 22 B2P F ||∞ ||V (s)||∞ β
Step 7: The reduced order model is given by Grspa (s) = Cspa (sI − Aspa )−1 Bspa + Dspa , where Aspa = A11 − A12 A−1 22 A21
Bspa = B1 − A12 A−1 22 B2 Cspa = C1 − C2 A−1 22 A21 Dspa = D − C2 A−1 22 B2
¯P F , C¯P F } is a balanced realized model and Since {A, B ¯spa P F , C¯spa P F , −C¯2P F A˜−1 B ¯ {Aspa , B 22 2P F } is its corresponding reduced order model, then the from Liu and Anderson (1989) the following holds true ¯P F − (C¯spa P F (sI − Aspa )−1 B ¯spa P F ... ||C¯P F (sI − A)−1 B
3.2 Error Bounds In this section the error bounds for the reduced order models obtained using the proposed algorithm is derived. To establish the relationship between the input and output matrices (B and C), and the new fictitious input and ¯P F and C¯P F ), we define two constant output matrices (B I/α matrices, KP F = and LP F = [I/β 0] respectively 0 where I is an identity matrix with the appropriate dimension. It follows that the following relationships hold true between the input and output matrices B and C and ¯P F and C¯P F as the fictitious input and output matrices B follows ¯ P F KP F (26) B=B (27) C = LP F C¯P F
¯ + C¯2P F A˜−1 22 B2P F ||∞ n ≤2 σk k=r+1
it follows that
||W (s)(G(s) − Grspa (s)V (s)||∞ ≤ δ with δ =
σk
k=r+1
(s)||∞
ki =r+1
and
k=r+1
2 αβ ||W (s)||∞ ||V
2 αβ ||W (s)||∞ ||V
n
Corollary 1 : The error bounds when only input or output weights are used are as follows: n 2 ||(G(s) − Grspa (s)V (s)||∞ ≤ ||V (s)||∞ σki α
Theorem 1: Let G(s) be a stable and strictly proper transfer function whereas V (s) and W (s) are the input and output weights respectively. If Gspa (s) is a reduced order model obtained by using the algorithm presented in Section 3.1, it follows that the following error bound holds true: n ||W (s)(G(s) − Grspa (s)V (s)||∞ ≤ δ σk (28) where δ =
22
¯1P F − A12 A−1 B ¯ (sI − Aspa ) (B 22 2P F )KP F ]V (s)||∞ −1 ¯P F − ... = ||W (s)LP F {C¯P F (sI − A) B −1 (C¯1P F − C¯2P F A˜ A21 )(sI − Aspa )−1 × ...
||W (s)(G(s) − Grspa (s)||∞ ≤
(s)||∞
n 2 ||W (s)||∞ σko β ko =r+1
where σki and σko are the Hankel singular values of the ¯P F , C} and {A, B, C¯P F } respectively. systems {A, B
Proof: ¯ ¯ Upon partitioning the fictitious matrices BP F and CP F as ¯1P F B ¯ ¯ ¯ ¯P F = B ¯2P F and CP F = [C1P F C2P F ] respectively B ¯1P F KP F , B2 = B ¯2P F KP F and and substituting B1 = B C1 = LP F C¯1P F , C2 = LP F C¯2P F , we have
4. NUMERICAL EXAMPLE Considering the 4th order system originally from Enns (1984) 48
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Table 3. Weighted Approximation Error Obtained by Sahlan and Sreeram (2009); Sreeram et al. (2013)
Table 1. Weighted Approximation Error Obtained by Various Techniques Order
Enns W.E. 2.1291 0.2660 0.1131
1 2 3
Lin and Chiu’s W.E. 2.5744 0.5607 0.1645
Wang et al
Imran et al
W.E. 2.1213 0.2720 0.1151
W.E 2.1234 0.2424 0.1075
E.B. 7.2898 1.4895 0.3228
Order 1
Table 2. Weighted Approximation Error Obtained by Ghafoor and Sreeram (2007)
2
Order
3
1
2
3
Ghafoor and Sreeram’s Method α
β
W.E. 1
W.E 2
Error Bound
0.4 0.4 0.5 0.8 0.9 0.7 0.4 0.8 1
0.4 0.5 0.4 2.9 2.7 0.7 0.4 0.5 0.6
2.1126 2.1165 2.1131 0.2709 0.2710 0.2784 0.1009 0.1084 0.1117
1.4564 1.4877 1.4833 0.3116 0.3238 0.2543 0.0668 0.0714 0.0750
3.4848 3.6212 3.6229 2.3312 2.3242 0.8949 0.1589 0.1786 0.1942
−1 0 0 0 0 0 −2 0 0 1/2 A= ,B = 0 0 −3 0 1 0 0 0 −4 −1/2 1 0 10 00 C= ,D = 4/15 1 0 1 00
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Sreeram and Sahlan’s Method α β Weighted Error 8.1 6.7 2.1158 8.2 6.9 2.1161 8.0 7.0 2.1201 180 110 0.3149 90 110 0.3219 147.5 32.5 0.3166 103.9 35.8 0.1081 105.7 35.7 0.1082 105.3 35.3 0.1082
Error Bound 13.922 13.488 13.580 0.5290 0.5686 0.6996 0.3104 0.3092 0.3120
Table 4. Weighted Approximation Error Obtained Using the Proposed Method Order 1
−5/2 −3/2 −5 1/6
2
3
Proposed technique α β Weighted Error 0.1 0.1 1.2361 0.2 0.1 1.2363 0.2 0.2 1.2375 4.9 4.9 0.2430 4.5 4.5 0.2432 5.5 5.5 0.2433 6.0 2.0 0.0614 7.0 3.0 0.0648 5.0 1.0 0.0719
Error Bound 6.9011 6.9040 6.9112 2.4105 2.1981 2.7572 0.5388 0.6221 0.4599
and the 2nd order input and output weighting functions as −4.5 0 30 , BV = BW = , AV = AW = 0 −4.5 03 1.5 0 10 C V = CW = , DV = DW = 0 1.5 01 Table 1 shows the results of the weighted approximation errors - ||W (s)[G(s)−Gr (s)]V (s)||∞ for the existing methods by Enns (1984); Lin and Chiu (1992); Wang et al. (1999); Imran et al. (2014) for benchmarking. Weighted approximation error and error bounds are denoted as W.E and E.B. respectively. Table 2 shows the weighted approximation error obtained using both methods described in Ghafoor and Sreeram (2007). W.E.1 and W.E. 2 denote the weighted approximation errors obtained using method 1 and method 2 in Ghafoor and Sreeram (2007). Table 3 shows the results obtained by using the method by Sahlan and Sreeram (2009). Applying the proposed method with various values of the parameters α and β yields the results in Table 4. It can be observed that the proposed approach gives reduced frequency weighted approximation error relative to the method by Sahlan and Sreeram (2009). Figure 1 shows the variation of the frequency weighted approximation error relative to the variation of the free parameters α and β for a first order controller using the proposed method.
Fig. 1. Variation of Weighted Approximation Error due to Variation of α and β model reduction method is developed which is a further development to the method proposed by Sahlan and Sreeram (2009). By applying singular perturbation approximation instead of standard truncation, it has been demonstrated that the frequency weighted approximation error can be reduced. An error bound has also been developed for the proposed technique. REFERENCES
5. CONCLUSION
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