Model Reduction using a Frequency-Limited H 2 -Cost

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1 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 petersson@isy.liu.se, johanl@isy.liu.se 7th March 212 Report no.: LiTH-ISY-R-345 Submitted to the 51st IEEE Conference on Decision and Control, Maui, Hawaii Address: Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden WWW: AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET Technical reports from the Automatic Control group in Linköping are available from

2 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 H 2 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. Keywords: Optimization, Model Reduction

3 Model Reduction using a Frequency-Limited H 2 -Cost Daniel Petersson and Johan Löfberg 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 H 2 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. 1 Introduction Given a linear time-invariant (lti dynamical model, ẋ(t = Ax(t + Bu(t, y(t = Cx(t + Du(t where A R n n, B R n m, C R p n and D R p m, the model reduction problem in this paper is to find a reduced order model ẋ r (t = A r x r (t + B r u(t, y r (t = C r x r (t + D r u(t, with A r R nr nr, B r R nr m, C r R p nr and D r R p m with n r < 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. [212] 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 frequencyweighted, see e.g. Gugercin and Antoulas [24] and Ghafoor and Sreeram [28]. Some of the most commonly used frequency-weighted methods, according to Gugercin and Antoulas [24], 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 Sweden {petersson@isy.liu.se,johanl@isy.liu.se} 1

4 Lin and Chiu [1992], which all use different balanced truncation approaches. In many of the frequency-weighted methods one need to specify input and output filter weights. In Gawronski and Juang [199] 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 highpass 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 [24], where they introduce a modification of the method in Gawronski and Juang [199], and derive H bound for the error. Sahlan et al. [212] presents a modification of the method from Gawronski and Juang [199], 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 [199], and does not have the drawback of sometimes finding unstable models. We use these Gramians to construct a frequency-limited H 2 -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, Ĝ, that is a good approximation on a given frequency interval, e.g. [, ω]. The objective is to minimize the error between the given model and the sought reduced model in a frequency-limited H 2 -norm, using the frequency-limited Gramians introduced in Gawronski and Juang [199]. We formulate the optimization problem ( where minimize Ĝ G Ĝ 2 = minimize E 2 H, (1 H 2,ω 2,ω Ĝ E 2 H = 1 ω 2,ω E(iν E(iνdν. (2 2π ω Assume that the system E is stable and described by E : { ẋ(t = AE x(t + B E u(t y(t = C E x(t + D E u(t (3 which will be denoted as [ AE B E : E C E D E ]. (4 Assuming G and Ĝ are represented as [ ] [ A B Â G :, C D Ĝ : ˆB Ĉ ˆD ], (5 2

5 then the error system can be realized in state space form as ( ( [ ] A AE B E : E = Â BˆB C E D E ( C Ĉ. (6 D ˆD 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 =, or assume that D = ˆD. We will also assume that the given model is stable. To be able to calculate the frequency-limited H 2 -norm we need expressions for the frequency-limited Gramians. We start by defining the standard Gramians, see Zhou et al. [1996], in time and frequency domain, as P E = Q E = e A Eτ B E B T Ee AT E τ dτ = 1 2π e AT E τ C T EC E e A Eτ dτ = 1 2π H(νB E B T EH (νdν, H (νc T EC E H(νdν, (7a (7b where H(ω = (iωi A E 1 and H (ω denotes the conjugate transpose of H(ω. The controllability and observability Gramians satisfies, respectively, the Lyapunov equations A E P E + P E A T E + B E B T E =, A T EQ E + Q E A E + C T EC E =. (8a (8b The H 2 -norm of E can be expressed as E 2 H 2 = tr = 1 2π = tr = 1 2π C E e A Eτ B E B T Ee AT E τ C T Edτ C E H(νB E B T EH (νc T Edν = tr C E P E C T E B T Ee AT E τ C T EC E e A Eτ B E dτ B T EH (νc T EC E H(νB E dν = tr B T EQ E B E. (9a (9b (9c (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 [199], as P E,ω = 1 ω H(νB E B T 2π EH (νdν, (1a ω Q E,ω = 1 ω H (νc T 2π EC E H(νdν, (1b ω where these Gramians satisfy the Lyapunov equations, see Gawronski and Juang [199], A E P E,ω + P E,ω A T E + S E,ω B E B T E + B E B T ES E,ω =, A T EQ E,ω + Q E,ω A E + S E,ωC T EC E + C T EC E S E,ω =, (11a (11b 3

6 with S E,ω = i 2π ln ( (A E + iωi(a E iωi 1 = i 2π ln f(a E. (12 Using the frequency-limited Gramians we can express the cost function of the optimization problem (1, i.e., the frequency-limited H 2 -norm for E, as E 2 H 2,ω = tr C EP E,ω C T E (13a = tr B T EQ E,ω B E. (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 partitioning the Gramians P E,ω and Q E,ω as ( ( Pω X ω Qω Y ω P E,ω =, Q ˆP E,ω =, (14 ω ˆQ ω X T ω Y T ω and S E,ω as ( Sω S E,ω =. (15 Ŝ ω P ω, Q ω, ˆP ω, ˆQ ω, X ω and Y ω satisfy, due to (11, the Sylvester and Lyapunov equations AP ω + P ω A T + S ω BB T + BB T S ω =, AX ω + X ω Â T + S ω B ˆB T + B ˆB T Ŝ ω =, Â ˆP ω + ˆP ω Â T + Ŝω ˆB ˆB T + ˆB ˆB T Ŝ ω =, A T Q ω + Q ω A + S ωc T C + C T CS ω =, A T Y ω + Y ω Â S ωc T Ĉ C T ĈŜω =, Â T ˆQω + ˆQ ω Â + Ŝ ωĉt Ĉ + ĈT ĈŜω =. (16a (16b (16c (16d (16e (16f Note that P ω and Q ω satisfy the Lyapunov equations for the frequency-limited controllability and observability Gramians for the given model, and ˆP ω and ˆQ ω satisfy the Lyapunov equations for the frequency-limited controllability and observability Gramians for the sought model. With the partitioning of P E,ω and Q E,ω it is possible to rewrite (13 in two alternative forms E 2 H = tr 2,ω (B T Q ω B + 2B T Y ω ˆB + ˆBT ˆQω ˆB, (17a E 2 H 2,ω = tr (CP ω C T 2CX ω Ĉ T + Ĉ ˆP ω Ĉ T. (17b Remark 1. We note that when solving the optimization problem (1 with the cost function (17 we do not need the term B T Q ω B or CP ω C T in the cost function as they do not depend on the optimization variables, Â, ˆB and Ĉ. 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, 4

7 Â, ˆB and Ĉ. In addition, the closed form expression obtained when differentiating the cost function is expressed in the given data (A, B and C, the optimization variables (Â, ˆB and Ĉ and solutions to equations in (16. To show this we start by differentiating with respect to ˆB and Ĉ. First we note that neither Q ω, Y ω nor ˆQ ω in 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 Ĉ. Hence, the derivative of the cost function with respect ˆB and Ĉ becomes E 2 H 2,ω ˆB E 2 H 2,ω Ĉ ( = 2 ˆQω ˆB + Y T ω B, (18a = 2 (Ĉ ˆPω CX ω. (18b For the more complicated case of differentiating with respect to Â we observe that ˆQ ω and Y ω do depend on Â, 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 where W = i π E 2 H 2,ω Â E 2 H 2,ω ˆB E 2 H 2,ω Ĉ ( =2 YωX T + ˆQ ω ˆP + W, (19a ( =2 ˆQω ˆB + Y T ω B, (19b =2 (Ĉ ˆPω CX ω, (19c T [ ( (I f(â L f(â, ĈT Ĉ ˆP T CX] ĈT ( Â iωi T, (2 with the function L(, being the Frechét derivative of the matrix logarithm, see Higham [28]. 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 Â-matrix. Remark 3. By using addition/subtraction of two or more different frequencylimited Gramians it is possible to focus on one or more arbitrary frequency ranges, e.g., you can construct the frequency-limited controllability Gramian, P Ω, for the interval ω [ω 1, ω 2 ] [ω 3, ω 4 ] as P Ω = P ω2 P ω1 + P ω4 P ω3. Remark 4. The proposed method can, analogous to what is done in Petersson [21], 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. 5

8 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 [199] (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 [24], 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 [, 1.7] to try to only capture the first model. 1 9 G = G 1 G 2 = s 2 +.2s + 1 s 2 +.3s + 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 indiscernible. Table 1: Numerical results for Example 1 G Ĝ G Ĝ H 2,ω Re λ max H2,ω G H2,ω Hankel 1.765e+ 1.6e e-3 Gawronski 9.14e e e-2 Prop. method 8.512e e e-2 Example 2 (Example 1 in Gawronski and Juang [199]. In this example we reuse Example 1 from Gawronski and Juang [199]. 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 [, 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 6

9 Magnitude plot for the given and the reduced models 4 2 Magnitude (db Hankel Gawronski Prop. method True Frequency (rad/s Figure 1: Magnitude plot of the given and the reduced models in example 1 for ω [, 1.7]. The dashed black vertical line denotes ω = 1.7. 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 indiscernible Magnitude plot for the error systems Magnitude (db Hankel Gawronski Prop. method Frequency (rad/s Figure 2: Magnitude plot of the error systems in example 1 for ω [, 1.7]. The dashed black vertical line denotes ω =

10 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. Magnitude plot for the given and the reduced models 4 2 Hankel Gawronski Prop. method True Magnitude (db Frequency (rad/s Figure 3: Magnitude plot of the given and the reduced models in example 2 for ω [, 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 Ĝ G Ĝ H 2,ω Re λ max H2,ω G H2,ω Hankel 1.62e e e-3 Gawronski 9.464e e e-3 Prop. method 3.926e e-3-3.8e-3 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. [212]. We will reduce this model to five states and limit the frequency range to ω [, 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 8

11 Magnitude plot for the error systems 1 1 Hankel Gawronski Prop. method 2 Magnitude (db Frequency (rad/s Figure 4: Magnitude plot of the error systems in example 2 for ω [, 1.3]. The dashed black vertical line denotes ω = 1.3. Table 3: Numerical results for Example 3 G Ĝ G Ĝ H 2,ω Re λ max H2,ω G H2,ω Hankel 3.7e e e+1 Gawronski 5.443e e-3 4.1e+3 Prop. method 5.899e e e+ Juang [199], 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 [199], and does not have the drawback of finding unstable models. We use these Gramians to construct a frequency-limited H 2 -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 [21] 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. 9

12 Magnitude plot for the given and the reduced models Hankel Gawronski Prop. method True Magnitude (db Frequency (rad/s Figure 5: Magnitude plot of the given and the reduced models in example 3 for ω [, 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 , Las Vegas, USA, Wodek Gawronski and Jer-Nan Juang. Model reduction in limited time and frequency intervals. International Journal of Systems Science, 21(2: , 199. Abdul Ghafoor and Victor Sreeram. A survey/review of frequency-weighted balanced model reduction techniques. Journal of Dynamic Systems, Measurement and Control, 13:614, 28. Serkan Gugercin and Athanasios C. Antoulas. A survey of model reduction by balanced truncation and some new results. International Journal of Control, 77(8: , 24. Nicholas J. Higham. Functions of Matrices: Theory and Computation. SIAM, 28. Ching-An Lin and Tai-Yih Chiu. Model reduction via frequency weighted balanced realization. Control Theory and Advanced Technology, 8: ,

13 Magnitude plot for the error systems 5 4 Hankel Gawronski Prop. method 3 Magnitude (db Frequency (rad/s Figure 6: Magnitude plot of the error systems in example 3 for ω [, 15]. The dashed black vertical line denotes ω = 15. Daniel Petersson. Nonlinear optimization approaches to H 2 -norm based LPV modelling and control. Licentiate thesis no. 1453, Department of Electrical Engineering, Linköping University, 21. Shafishuhaza Sahlan, Abdul Ghafoor, and Victor Sreeram. A new method for the model reduction technique via a limited frequency interval impulse response gramian. Mathematical and Computer Modelling, 55(3-4:134 14, 212. Andreas Varga, Anders Hansson, and Guilhem Puyou, editors. Optimization Based Clearance of Flight Control Laws. Lecture Notes in Control and Information Science. Springer, 212. 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: , Kemin Zhou, John C. Doyle, and Keith Glover. Robust and optimal control. Prentice-Hall, Inc., ISBN A Appendix In this appendix we present the differentiation of the cost function with respect to Â. The cost function is E 2 H = tr 2,ω (B T Q ω B + 2B T Y ω ˆB + ˆBT ˆQω ˆB (22 11

14 and by looking at the equations A T Y ω + Y ω Â S ωc T Ĉ C T ĈŜω =, Â T ˆQω + ˆQ ω Â + Ŝ ωĉt Ĉ + ĈT ĈŜω =, (23a (23b we observe that ˆQ ω and Y ω depend on Â which [ we need] to keep in mind when differentiating (22 with respect to Â. Hence, becomes where Yω in (23, Â T ˆQ ω E 2 H 2,ω Â [ ] ( 2 E H 2,ω = tr 2 ˆBB T Y ω + Â â ˆB ˆB T ˆQ ω, (24 ij ij and ˆQ ω Â T YT ω depend on Â via the differentiated versions of the equations + YT ω A + ÂT Y T ω Ŝ ω a ij Ĉ T C =, ij (25a + ˆQ ω Â + ÂT ˆQω + ˆQ ω Â + Ŝ ω a ij Ĉ T Ĉ + ĈT Ĉ Ŝω a ij =. (25b To be able to substitute ˆQ ω and Yω tractable we use the following lemma. to something more computationally Lemma 1. If M and N satisfy the Sylvester equations then tr CN = tr DM. AM + MB + C =, NA + BN + D =, Studying Lemma 1, the two factors in front of Yω and ˆQ ω in (24 and the structure of the Lyapunov/Sylvester equations in (25, ÂT + Â + = and Â T + A + =, brings us to the conclusion that, to do the substitution, we need to solve two additional Lyapunov/Sylvester equations, namely AX + XÂT + B ˆB T =, Â ˆP + ˆPÂT + ˆB ˆB T =. (26a (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 [ ] [ 2 E ( H 2,ω = 2 tr ÂT Y Â â ωx T + ˆQ ω ˆP + Ŝ ω (ĈT Ĉ ij â ˆP ] ĈT CX. ij ij (27 What remains is to rewrite the second term in (27, which includes Ŝ ω. Recall the definition of Ŝω, Ŝ ω = i ( 2π ln (Â + iωi(â iωi 1 (28 12

15 and differentiate with respect to an element in Â, i.e., a ij. This yields ( ( Ŝω = i a ij 2π L f(â f(â, = i a ij 2π L Â f(â, (I f(â (A iωi 1 a ij (29 where L(A, E is the Frechét derivative of the matrix logarithm, see Higham [28], with L(A, E = 1 (t(a I + I 1 E (t(a I + I 1 dt, (3 f(â =(Â + iωi(â iωi 1. (31 The function L(A, E can be efficiently evaluated using the algorithm by Higham [28]. By substituting (29 into (27 and using (3 with the fact that we can interchange the tr-operator and the integral we obtain [ ] 2 E H 2,ω Â where W = i π = 2 tr ij = 2 tr [ ( ÂT Y â ωx T + ˆQ ω ˆP + Ŝ ω (ĈT Ĉ ij â ˆP ] ĈT CX ij ( ( ÂT Y â ωx T + ˆQ ω ˆP ij ( [ L + i ( π tr Â T ( T I â f(â ij f(â, ĈT Ĉ ˆP ] T ĈT CX ( Â iωi T = tr ( [ ( ÂT 2 Y â ωx T + ˆQ ω ˆP + W], (32 ij T [ ( (I f(â L f(â, ĈT Ĉ ˆP T CX] ĈT ( Â iωi T. (33 13

17 Avdelning, Institution Division, Department Datum Date Division of Automatic Control Department of Electrical Engineering Språk Language Rapporttyp Report category ISBN Svenska/Swedish Licentiatavhandling ISRN Engelska/English Examensarbete C-uppsats D-uppsats Övrig rapport Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version LiTH-ISY-R-345 Titel Title Model Reduction using a Frequency-Limited H 2 -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 H 2 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. Nyckelord Keywords Optimization, Model Reduction

An efficient implementation of gradient and Hessian calculations of the coefficients of the characteristic polynomial of I XY

Technical report from Automatic Control at Linköpings universitet An efficient implementation of gradient and Hessian calculations of the coefficients of the characteristic polynomial of I XY Daniel Ankelhed