Technical report from Automatic Control at Linköpings universitet
Model Reduction using a
Frequency-Limited
H
2
-Cost
Daniel Petersson, Johan Löfberg
Division of Automatic Control
E-mail: petersson@isy.liu.se, johanl@isy.liu.se
7th March 2012
Report no.: LiTH-ISY-R-3045
Submitted to the 51st IEEE Conference on Decision and Control, Maui,
Hawaii
Address:
Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.
Abstract
We propose a method for model reduction on a given frequency range, without having to specify input and output filter weights. The method uses a nonlinear optimization approach where we formulate a H2 like cost
function which only takes the given frequency range into account. We derive a gradient of the proposed cost function which enables us to use off-the-shelf optimization software.
Model Reduction using a
Frequency-Limited H
2
-Cost
Daniel Petersson and Johan Löfberg
∗2012-03-07
Abstract
We propose a method for model reduction on a given frequency range, without having to specify input and output filter weights. The method uses a nonlinear optimization approach where we formulate a H2like cost
function which only takes the given frequency range into account. We derive a gradient of the proposed cost function which enables us to use off-the-shelf optimization software.
1
Introduction
Given a linear time-invariant (lti) dynamical model, ˙
x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t)
where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m, the model reduction problem in this paper is to find a reduced order model
˙
xr(t) = Arxr(t) + Bru(t),
yr(t) = Crxr(t) + Dru(t),
with Ar ∈ Rnr×nr, Br ∈ Rnr×m, Cr ∈ Rp×nr and Dr ∈ Rp×m with nr <
n, where this reduced order model describes the original model well in some metric. In this paper we are interested in a reduced model that describes the model well on a given frequency range. This is motivated by situations where the given model is valid only for a certain frequency range, for example as in Varga et al. [2012] where models coming from aerodynamical and structural mechanics computations describing a flexible structure are only valid up to a certain frequency.
For a review of model reduction approaches, both ordinary and frequency-weighted, see e.g. Gugercin and Antoulas [2004] and Ghafoor and Sreeram [2008]. Some of the most commonly used frequency-weighted methods, accord-ing to Gugercin and Antoulas [2004], are Wang et al. [1999], Enns [1984] and
∗D. Petersson and J. Löfberg are with the Division of Automatic Control,
Department of Electrical Engineering, Linköpings universitet, SE-581 83 Sweden
Lin and Chiu [1992], which all use different balanced truncation approaches. In many of the frequency-weighted methods one need to specify input and out-put filter weights. In Gawronski and Juang [1990] they introduce a method which does not need these weighting functions, by introducing frequency-limited Gramians. This method can be interpreted as using ideal low-, band- or high-pass filters as weights. However, this method has the drawback of not always producing stable models. One approach to remedy this has been presented in Gugercin and Antoulas [2004], where they introduce a modification of the method in Gawronski and Juang [1990], and derive H∞ bound for the error.
Sahlan et al. [2012] presents a modification of the method from Gawronski and Juang [1990], however this method is only applicable to siso models.
In this paper we propose a method, based on nonlinear optimization, that uses the frequency-limited Gramians introduced in Gawronski and Juang [1990], and does not have the drawback of sometimes finding unstable models. We use these Gramians to construct a frequency-limited H2-norm which describes the
cost function of the optimization problem. We derive a gradient of the proposed cost function which enables us to use off-the-shelve optimization software to solve the problem efficiently.
2
Frequency-Limited Model Reduction
The method proposed in this paper is a model reduction method that, given a model G, finds a reduced order model, ˆG, that is a good approximation on a given frequency interval, e.g. [0, ω]. The objective is to minimize the error between the given model and the sought reduced model in a frequency-limited H2-norm, using the frequency-limited Gramians introduced in Gawronski and
Juang [1990]. We formulate the optimization problem () minimize ˆ G G − ˆG 2 H2,ω = minimize ˆ G ||E||2H 2,ω, (1) where ||E||2H 2,ω = 1 2π Z ω −ω E(iν)∗E(iν)dν. (2)
Assume that the system E is stable and described by E : ˙ x(t) = AEx(t) + BEu(t) y(t) = CEx(t) + DEu(t) (3) which will be denoted as
E : AE BE CE DE . (4)
Assuming G and ˆG are represented as
G : A B C D , ˆG : ˆ A Bˆ ˆ C Dˆ , (5)
then the error system can be realized in state space form as E : AE BE CE DE = A 0 0 Aˆ B ˆ B C − ˆC D − ˆD . (6)
This realization of the error system will later prove beneficial when rewriting the optimization problem. Throughout the paper we will assume that the given model is strictly proper, i.e., D = 0, or assume that D = ˆD. We will also assume that the given model is stable.
To be able to calculate the frequency-limited H2-norm we need expressions
for the frequency-limited Gramians. We start by defining the standard Grami-ans, see Zhou et al. [1996], in time and frequency domain, as
PE= Z ∞ 0 eAEτB EBTEe ATEτdτ = 1 2π Z ∞ −∞ H(ν)BEBTEH ∗(ν)dν, (7a) QE= Z ∞ 0 eATEτCT ECEeAEτdτ = 1 2π Z ∞ −∞ H∗(ν)CTECEH(ν)dν, (7b)
where H(ω) = (iωI − AE)−1 and H∗(ω) denotes the conjugate transpose of
H(ω). The controllability and observability Gramians satisfies, respectively, the Lyapunov equations
AEPE+ PEATE+ BEBTE = 0, (8a)
ATEQE+ QEAE+ CTECE = 0. (8b)
The H2-norm of E can be expressed as
||E||2H 2 = tr Z ∞ 0 CEeAEτBEBTEe AT EτCT Edτ (9a) =1 2π Z ∞ −∞ CEH(ν)BEBTEH∗(ν)C T Edν = tr CEPECTE (9b) = tr Z ∞ 0 BTEeATEτCT ECEeAEτBEdτ (9c) =1 2π Z ∞ −∞ BTEH∗(ν)CTECEH(ν)BEdν = tr BTEQEBE. (9d)
Now we want to narrow the frequency band, from (−∞, ∞) to (−ω, ω) where ω < ∞. To do this we define the frequency-limited Gramians, see Gawronski and Juang [1990], as PE,ω = 1 2π Z ω −ω H(ν)BEBTEH∗(ν)dν, (10a) QE,ω = 1 2π Z ω −ω H∗(ν)CTECEH(ν)dν, (10b)
where these Gramians satisfy the Lyapunov equations, see Gawronski and Juang [1990],
AEPE,ω+ PE,ωATE+ SE,ωBEBTE+ BEBTES ∗
E,ω = 0, (11a)
ATEQE,ω+ QE,ωAE+ S∗E,ωC T
with SE,ω = i 2πln (AE+ iωI)(AE− iωI) −1 = i 2πln f (AE). (12) Using the frequency-limited Gramians we can express the cost function of the optimization problem (1), i.e., the frequency-limited H2-norm for E, as
||E||2H
2,ω = tr CEPE,ωC
T
E (13a)
= tr BTEQE,ωBE. (13b)
Now we want to rewrite the cost function (13) to a more computationally tractable form. This is done by using the realization given in (6) and by parti-tioning the Gramians PE,ω and QE,ω as
PE,ω = Pω Xω XT ω Pˆω , QE,ω = Qω Yω YT ω Qˆω , (14) and SE,ω as SE,ω = Sω 0 0 Sˆω . (15)
Pω, Qω, ˆPω, ˆQω, Xω and Yω satisfy, due to (11), the Sylvester and Lyapunov
equations APω+ PωAT+ SωBBT+ BBTS∗ω= 0, (16a) AXω+ XωAˆT+ SωB ˆBT+ B ˆBTSˆ∗ω= 0, (16b) ˆ A ˆPω+ ˆPωAˆT+ ˆSωB ˆˆBT+ ˆB ˆBTSˆ∗ω= 0, (16c) ATQω+ QωA + S∗ωC TC + CTCS ω= 0, (16d) ATYω+ YωA − Sˆ ∗ωC TC − Cˆ TCˆˆS ω= 0, (16e) ˆ ATQˆω+ ˆQωA + ˆˆ S∗ωCˆ TC + ˆˆ CTCˆˆS ω= 0. (16f)
Note that Pωand Qω satisfy the Lyapunov equations for the frequency-limited
controllability and observability Gramians for the given model, and ˆPω and
ˆ
Qωsatisfy the Lyapunov equations for the frequency-limited controllability and
observability Gramians for the sought model.
With the partitioning of PE,ω and QE,ω it is possible to rewrite (13) in two
alternative forms ||E||2H 2,ω = tr BTQωB + 2BTYωB + ˆˆ BTQˆωBˆ , (17a) ||E||2H 2,ω = tr CPωCT− 2CXωCˆT+ ˆC ˆPωCˆT . (17b)
Remark 1. We note that when solving the optimization problem (1) with the cost function (17) we do not need the term BTQ
ωB or CPωCT in the cost
function as they do not depend on the optimization variables, ˆA, ˆB and ˆC.
2.1
Gradient of the Cost Function
An appealing feature of the proposed nonlinear optimization approach to solve the problem is that the equations (17) are differentiable in the system matrices,
ˆ
A, ˆB and ˆC. In addition, the closed form expression obtained when differ-entiating the cost function is expressed in the given data (A, B and C), the optimization variables ( ˆA, ˆB and ˆC) and solutions to equations in (16).
To show this we start by differentiating with respect to ˆB and ˆC. First we note that neither Qω, Yω nor ˆQωin equation (17a) depends on ˆB which means
that the equation is quadratic in ˆB. Analogous observations can be made with equation (17b) and the variable ˆC. Hence, the derivative of the cost function with respect ˆB and ˆC becomes
∂ ||E||2H 2,ω ∂ ˆB = 2 ˆQωB + Yˆ T ωB , (18a) ∂ ||E||2H 2,ω ∂ ˆC = 2 ˆC ˆPω− CXω . (18b)
For the more complicated case of differentiating with respect to ˆA we observe that ˆQω and Yω do depend on ˆA, see the equations in (16). The calculations
of this part of the gradient are lengthy and can be found in the appendix. The complete gradient becomes
∂ ||E||2H 2,ω ∂ ˆA =2 YωTX + ˆQωPˆ + W, (19a) ∂ ||E||2H 2,ω ∂ ˆB =2 ˆQωB + Yˆ T ωB , (19b) ∂ ||E||2H 2,ω ∂ ˆC =2 ˆC ˆPω− CXω , (19c) where W = i π I − f ( ˆA) Th Lf ( ˆA), ˆCTC ˆˆP − ˆCTCXi T ( ˆA − iωI)−T, (20) with the function L(·, ·) being the Frechét derivative of the matrix logarithm, see Higham [2008].
Remark 2. By using nonlinear optimization and the fact that the derivation of the gradient is done element-wise, it is possible to impose structure in the system matrices of the reduced model, e.g., to impose a block structure in the
ˆ
A-matrix.
Remark 3. By using addition/subtraction of two or more different frequency-limited Gramians it is possible to focus on one or more arbitrary frequency ranges, e.g., you can construct the frequency-limited controllability Gramian, PΩ, for the interval ω ∈ [ω1, ω2] ∪ [ω3, ω4] as PΩ= Pω2− Pω1+ Pω4− Pω3. Remark 4. The proposed method can, analogous to what is done in Petersson [2010], easily be extended to a method for identifying lpv-models over a limited frequency domain.
Remark 5. By supplying the cost function and its gradient in computationally efficient forms this method can be used in any off-the-shelf Quasi-Newton solver. Remark 6. By using a stable model, e.g., a model from a Hankel reduction, as an initial point in the optimization and using a line-search we can limit the search to stable models.
3
Numerical Examples
In this section three examples are used to illustrate the applicability of the method and to compare it with other methods. In the examples we will use three different methods; Truncation of Hankel singular values (will be called Hankel), the method proposed in Gawronski and Juang [1990] (called Gawronski) and the proposed method (called Prop. method). Both the proposed method and the Gawronski method will take the limited frequency range into account, but Hankel will not. The reason that we use these methods for comparison is to have one standard method (Hankel) and one method that takes the limited frequency range into account. The Gawronski method is a representative method among frequency-weighted methods, see Gugercin and Antoulas [2004], with the benefit of not having to design weighting functions, but with the drawback that it cannot guarantee that the resulting model is stable.
The proposed method uses a cost function which is non-convex, which makes it important to use a good initial point. For the examples presented here we have used the model obtained by the Hankel method as an initial point for the optimization in the proposed method.
Example 1 (Small illustrative example). This example addresses a small model with four states. The model is two second order models in series, one with a resonance frequency at ω = 1 and the other at ω = 3. We will try to limit the frequency rage to ω ∈ [0, 1.7] to try to only capture the first model.
G = G1G2=
1 s2+ 0.2s + 1
9
s2+ 0.003s + 9. (21)
The results from the different methods can be seen in Figures 1 and 2 and Table 1. As can be seen in the result we are successful in finding a good model for the first model with both the proposed method and the Gawronski method. All the reduced models are stable. The Hankel method captures the wrong resonance mode (from our perspective) and fails completely in the lower frequency regions. The proposed method and the Gawronski method returns models essentially in-discernible.
Table 1: Numerical results for Example 1 G − ˆG H 2,ω ||G− ˆG|| H2,ω ||G||H2,ω Re λmax
Hankel 1.765e+00 1.006e+00 -1.59e-03 Gawronski 9.140e-02 5.209e-02 -9.88e-02 Prop. method 8.512e-02 4.851e-02 -9.94e-02
Example 2 (Example 1 in Gawronski and Juang [1990]). In this example we reuse Example 1 from Gawronski and Juang [1990]. The model, which is a spring-damper model with three masses, has six states and we will reduce the model to three states and limit the frequency interval to ω ∈ [0, 1.3]. The results from the different methods can be seen in Figures 3 and 4 and Table 2. The proposed method and the Gawronski method are also in this example successful in finding low order models that approximates the given model on the given frequency range, and all the reduced models are stable. All the methods find the
100 101 −80 −60 −40 −20 0 20 40
Magnitude plot for the given and the reduced models
Frequency (rad/s) Magnitude (dB) Hankel Gawronski Prop. method True
Figure 1: Magnitude plot of the given and the reduced models in example 1 for ω ∈ [0, 1.7]. The dashed black vertical line denotes ω = 1.7. The Han-kel method captures the wrong resonance mode (from our perspective) and fails completely in the lower frequency regions. The proposed method and the Gawronski method returns models essentially indiscernible.
100 101 −60 −50 −40 −30 −20 −10 0 10 20 30 40
Magnitude plot for the error systems
Frequency (rad/s)
Magnitude (dB)
Hankel Gawronski Prop. method
Figure 2: Magnitude plot of the error systems in example 1 for ω ∈ [0, 1.7]. The dashed black vertical line denotes ω = 1.7.
resonance mode, but once again the Hankel method fails in the lower frequencies. The proposed method, the Gawronski method and the true model are indiscernible up to the limit frequency ω = 1.3.
10−1 100 101 102 −80 −60 −40 −20 0 20 40
Magnitude plot for the given and the reduced models
Frequency (rad/s) Magnitude (dB) Hankel Gawronski Prop. method True
Figure 3: Magnitude plot of the given and the reduced models in example 2 for ω ∈ [0, 1.3]. The black vertical line denotes ω = 1.3. All the methods find the resonance mode, but once again the Hankel method fails in the lower frequencies. The proposed method, the Gawronski method and the true model are indiscernible up to the limit frequency ω = 1.3.
Table 2: Numerical results for Example 2 G − ˆG H2,ω ||G− ˆG|| H2,ω ||G||H2,ω Re λmax
Hankel 1.062e+00 5.452e-01 -3.54e-03 Gawronski 9.464e-03 4.858e-03 -3.79e-03 Prop. method 3.926e-03 2.015e-03 -3.80e-03
Example 3 (Aircraft example). The model in this example is a model with 22 states that describes the longitudinal motion of an aircraft, see Varga et al. [2012]. We will reduce this model to five states and limit the frequency range to ω ∈ [0, 15]. The results from the different methods can be seen in Figures 5 and 6 and Table 3. The proposed method and Gawronski method yields models with good agreement up to the limit frequency. Note though that the Gawronski method gives a model with a peculiar resonance mode at ω ≈ 11 rad/s, and the model turns out to be unstable. The Hankel method returns a model structurally different from the two other methods.
In all three of the above examples we observe that the proposed method finds an at least as good model as the method proposed in Gawronski and
10−1 100 101 102 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10
Magnitude plot for the error systems
Frequency (rad/s)
Magnitude (dB)
Hankel Gawronski Prop. method
Figure 4: Magnitude plot of the error systems in example 2 for ω ∈ [0, 1.3]. The dashed black vertical line denotes ω = 1.3.
Table 3: Numerical results for Example 3 G − ˆG H 2,ω ||G− ˆG|| H2,ω ||G||H2,ω Re λmax
Hankel 3.070e+02 1.495e-02 -2.18e+01 Gawronski 5.443e+01 2.651e-03 4.10e+03 Prop. method 5.899e+00 2.873e-04 -1.45e+00
Juang [1990], but with the proposed model we can guarantee that the reduced model is stable and also impose structure in the system matrices. In the first two examples it takes less than one second and in the third example about seven seconds for the proposed method to find a reduced model.
4
Conclusions
In this paper we have proposed a method, based on nonlinear optimization, that uses the frequency-limited Gramians introduced in Gawronski and Juang [1990], and does not have the drawback of finding unstable models. We use these Gramians to construct a frequency-limited H2-norm which describes the cost
function of the optimization problem. We derive a gradient of the proposed cost function which enables us to use off-the-shelve optimization software to solve the problem efficiently. The derivation of the method also enables us to impose structural constraints, e.g., upper triangular A-matrix, in the system matrices. The derivation follows closely the technique in Petersson [2010] and it is easy to extend the method to identifying lpv-models. We also present three examples of different sizes and characteristics to show the applicability of the method.
10−1 100 101 79.35 79.4 79.45 79.5 79.55 79.6
Magnitude plot for the given and the reduced models
Frequency (rad/s) Magnitude (dB) Hankel Gawronski Prop. method True
Figure 5: Magnitude plot of the given and the reduced models in example 3 for ω ∈ [0, 15]. The dashed black vertical line denotes ω = 15. The proposed method and Gawronski method yields models with good agreement up to the limit frequency. Note though that the Gawronski method gives a model with a peculiar resonance mode at ω ≈ 11 rad/s, and the model turns out to be unstable. The Hankel method returns a model structurally different from the two other methods.
References
Dale F. Enns. Model reduction with balanced realizations: An error bound and a frequency weighted generalization. In Proceedings of the 23rd IEEE Conference on Decision and Control, pages 127 – 132, Las Vegas, USA, 1984. Wodek Gawronski and Jer-Nan Juang. Model reduction in limited time and frequency intervals. International Journal of Systems Science, 21(2):349–376, 1990.
Abdul Ghafoor and Victor Sreeram. A survey/review of frequency-weighted balanced model reduction techniques. Journal of Dynamic Systems, Mea-surement and Control, 130:061004, 2008.
Serkan Gugercin and Athanasios C. Antoulas. A survey of model reduction by balanced truncation and some new results. International Journal of Control, 77(8):748–766, 2004.
Nicholas J. Higham. Functions of Matrices: Theory and Computation. SIAM, 2008.
Ching-An Lin and Tai-Yih Chiu. Model reduction via frequency weighted bal-anced realization. Control Theory and Advbal-anced Technology, 8:341–351, 1992.
10−1 100 101 −20 −10 0 10 20 30 40 50
Magnitude plot for the error systems
Frequency (rad/s)
Magnitude (dB)
Hankel Gawronski Prop. method
Figure 6: Magnitude plot of the error systems in example 3 for ω ∈ [0, 15]. The dashed black vertical line denotes ω = 15.
Daniel Petersson. Nonlinear optimization approaches to H2-norm based LPV
modelling and control. Licentiate thesis no. 1453, Department of Electrical Engineering, Linköping University, 2010.
Shafishuhaza Sahlan, Abdul Ghafoor, and Victor Sreeram. A new method for the model reduction technique via a limited frequency interval impulse re-sponse gramian. Mathematical and Computer Modelling, 55(3-4):1034–1040, 2012.
Andreas Varga, Anders Hansson, and Guilhem Puyou, editors. Optimization Based Clearance of Flight Control Laws. Lecture Notes in Control and Infor-mation Science. Springer, 2012.
G. Wang, Victor Sreeram, and W. Q. Liu. A new frequency-weighted balanced truncation method and an error bound. IEEE Transactions on Automatic Control, 44(9):1734 – 1737, 1999.
Kemin Zhou, John C. Doyle, and Keith Glover. Robust and optimal control. Prentice-Hall, Inc., 1996. ISBN 0-13-456567-3.
A
Appendix
In this appendix we present the differentiation of the cost function with respect to ˆA.
The cost function is ||E||2H 2,ω= tr BTQωB + 2BTYωB + ˆˆ BTQˆωBˆ (22)
and by looking at the equations ATYω+ YωA − Sˆ ∗ωC TC − Cˆ TCˆˆS ω= 0, (23a) ˆ ATQˆω+ ˆQωA + ˆˆ S∗ωCˆ Tˆ C + ˆCTCˆˆSω= 0, (23b)
we observe that ˆQωand Yω depend on ˆA which we need to keep in mind when
differentiating (22) with respect to ˆA. Hence, ∂||E||2 H2,ω ∂ ˆA ij becomes " ∂ ||E||2H 2,ω ∂ ˆA # ij = tr 2 ˆBBT∂Yω ∂ˆaij + ˆB ˆBT∂ ˆQω ∂ˆaij ! , (24) where ∂Yω ∂ˆaij and ∂ ˆQω
∂ˆaij depend on ˆA via the differentiated versions of the equations in (23), ˆ AT∂Y T ω ∂ˆaij +∂Y T ω ∂ˆaij A +∂ ˆA T ∂ˆaij YTω−∂ ˆS ∗ ω ∂aij ˆ CTC = 0, (25a) ˆ AT∂ ˆQω ∂ˆaij +∂ ˆQω ∂ˆaij ˆ A +∂ ˆA T ∂ˆaij ˆ Qω+ ˆQω ∂ ˆA ∂ˆaij +∂ ˆS ∗ ω ∂aij ˆ CTC + ˆˆ CTCˆ∂ ˆSω ∂aij = 0. (25b) To be able to substitute ∂ ˆQω ∂ˆaij and ∂Yω
∂ˆaij to something more computationally tractable we use the following lemma.
Lemma 1. If M and N satisfy the Sylvester equations AM + MB + C = 0, NA + BN + D = 0, then tr CN = tr DM.
Studying Lemma 1, the two factors in front of∂Yω
∂ˆaij and
∂ ˆQω
∂ˆaij in (24) and the structure of the Lyapunov/Sylvester equations in (25), ˆAT· + · ˆA + ? = 0 and
ˆ
AT· + · A + ? = 0, brings us to the conclusion that, to do the substitution, we
need to solve two additional Lyapunov/Sylvester equations, namely
AX + X ˆAT + B ˆBT = 0, (26a) ˆ
A ˆP + ˆP ˆAT + ˆB ˆBT = 0. (26b) Note that ˆP in (26b) happens to be the controllability Gramian for the reduced model.
Rewriting (24) using Lemma 1, (25) and (26) leads to " ∂ ||E||2H 2,ω ∂ ˆA # ij = 2 tr " ∂ ˆAT ∂ˆaij YωTX + ˆQωPˆ +∂ ˆS ∗ ω ∂ˆaij ˆCTC ˆˆP − ˆCTCX # . (27) What remains is to rewrite the second term in (27), which includes ∂ ˆS∗ω
∂ˆaij. Recall the definition of ˆSω, ˆ Sω= i 2πln ( ˆA + iωI)( ˆA − iωI)−1 (28)
and differentiate with respect to an element in ˆA, i.e., aij. This yields ∂ ˆSω ∂aij = i 2πL f ( ˆA), ∂f ( ˆA) ∂aij ! = i 2πL f ( ˆA), (I − f ( ˆA)) ∂ ˆA ∂aij (A − iωI)−1 ! (29) where L(A, E) is the Frechét derivative of the matrix logarithm, see Higham [2008], with L(A, E) = Z 1 0 (t(A − I) + I)−1E (t(A − I) + I)−1dt, (30) f ( ˆA) =( ˆA + iωI)( ˆA − iωI)−1. (31) The function L(A, E) can be efficiently evaluated using the algorithm by Higham [2008].
By substituting (29) into (27) and using (30) with the fact that we can interchange the tr-operator and the integral we obtain
" ∂ ||E||2H 2,ω ∂ ˆA # ij = 2 tr " ∂ ˆAT ∂ˆaij YTωX + ˆQωPˆ +∂ ˆS ∗ ω ∂ˆaij ˆCTC ˆˆP − ˆCTCX # = 2 tr ∂ ˆA T ∂ˆaij YTωX + ˆQωPˆ ! + i πtr ∂ ˆAT ∂ˆaij I − f ( ˆA) T ×hLf ( ˆA), ˆCTC ˆˆP − ˆCTCXi T ( ˆA − iωI)−T = tr ∂ ˆA T ∂ˆaij h 2YTωX + ˆQωPˆ + Wi ! , (32) where W = i π I − f ( ˆA) Th Lf ( ˆA), ˆCTC ˆˆP − ˆCTCXi T ( ˆA − iωI)−T. (33)
Avdelning, Institution Division, Department
Division of Automatic Control Department of Electrical Engineering
Datum Date 2012-03-07 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport
URL för elektronisk version http://www.control.isy.liu.se
ISBN — ISRN
—
Serietitel och serienummer Title of series, numbering
ISSN 1400-3902
LiTH-ISY-R-3045
Titel Title
Model Reduction using a Frequency-Limited H2-Cost
Författare Author
Daniel Petersson, Johan Löfberg
Sammanfattning Abstract
We propose a method for model reduction on a given frequency range, without having to specify input and output filter weights. The method uses a nonlinear optimization approach
where we formulate a H2 like cost function which only takes the given frequency range
into account. We derive a gradient of the proposed cost function which enables us to use off-the-shelf optimization software.
