Abstract
In this paper, we discuss truncated Gramians (TGrams) for bilinear control systems and their relations to Lyapunov equations. We show how TGrams relate to input and output energy functionals, and we also present interpretations of controllability and observability of the bilinear systems in terms of these TGrams. These studies allow us to determine those states that are less important for the system dynamics via an appropriate transformation based on the TGrams. Furthermore, we discuss advantages of the TGrams over the Gramians for bilinear systems as proposed in Al-baiyat and Bettayeb (Proceedings of 32nd IEEE CDC, pp. 22–27, 1993). We illustrate the efficiency of the TGrams in the framework of model order reduction via a couple of examples, and compare to the approach based on the full Gramians for bilinear systems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Al-baiyat, S. A, Bettayeb, M.: A new model reduction scheme for k-power bilinear systems. In: Proceedings of 32nd IEEE CDC, pp. 22–27. IEEE, Piscataway (1993)
Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)
Bai, Z., Skoogh, D.: A projection method for model reduction of bilinear dynamical systems. Linear Algebra Appl. 415(2–3), 406–425 (2006)
Benner, P., Breiten, T.: On \(\mathcal{H}_{2}\)-model reduction of linear parameter-varying systems. In: Proceedings of Applied Mathematics and Mechanics, vol. 11, pp. 805–806 (2011)
Benner, P., Breiten, T.: Interpolation-based \(\mathcal{H}_{2}\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33(3), 859–885 (2012)
Benner, P., Breiten, T.: Low rank methods for a class of generalized Lyapunov equations and related issues. Numer. Math. 124(3), 441–470 (2013)
Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Cont. Optim. 49(2), 686–711 (2011)
Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin/Heidelberg (2005)
Benner, P., Redmann, M.: Model reduction for stochastic systems. Stoch. PDE: Anal. Comp. 3(3), 291–338 (2015)
Breiten, T., Damm, T.: Krylov subspace methods for model order reduction of bilinear control systems. Syst. Control Lett. 59(10), 443–450 (2010)
Bruni, C., DiPillo, G., Koch, G.: On the mathematical models of bilinear systems. Automatica 2(1), 11–26 (1971)
Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations. Numer. Linear Algebra Appl. 15(9), 853–871 (2008)
Flagg, G., Gugercin, S.: Multipoint Volterra series interpolation and \(\mathcal{H}_{2}\) optimal model reduction of bilinear systems. SIAM. J. Matrix Anal. Appl. 36(2), 549–579 (2015)
Fujimoto, K., Scherpen, J.M.A.: Balanced realization and model order reduction for nonlinear systems based on singular value analysis. SIAM J. Cont. Optim. 48(7), 4591–4623 (2010)
Gray, W.S., Mesko, J.: Energy functions and algebraic Gramians for bilinear systems. In: Preprints of the 4th IFAC Nonlinear Control Systems Design Symposium, Enschede, The Netherlands, pp. 103–108 (1998)
Gray, W.S., Scherpen, J.M.A.: On the nonuniqueness of singular value functions and balanced nonlinear realizations. Syst. Control Lett. 44(3), 219–232 (2001)
Hartmann, C., Schäfer-Bung, B., Thons-Zueva, A.: Balanced averaging of bilinear systems with applications to stochastic control. SIAM J. Cont. Optim. 51(3), 2356–2378 (2013)
Mohler, R.R.: Bilinear Control Processes. Academic Press, New York (1973)
Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control AC-26(1), 17–32 (1981)
Redmann, M., Benner, P.: Approximation and model order reduction for second order systems with Levy-noise. In: Dynamical Systems, Differential Equations and Applications AIMS Proceedings, pp. 945–953, 2015
Rugh, W.J.: Nonlinear System Theory. The Johns Hopkins University Press, Baltimore (1981)
Scherpen, J.M.A.: Balancing for nonlinear systems. Syst. Control Lett. 21, 143–153 (1993)
Schilders, W.H.A., van der Vorst, H. A., Rommes, J.: Model Order Reduction: Theory, Research Aspects and Applications. Springer, Berlin/Heidelberg (2008)
Shank, S.D., Simoncini, V., Szyld, D.B.: Efficient low-rank solution of generalized Lyapunov equations. Numer. Math. 134(2), 327–342 (2016)
Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)
Acknowledgements
The authors thank Dr. Stephen D. Shank for providing the MATLAB implementation to compute the low-rank solutions of the generalized Lyapunov equations.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Benner, P., Goyal, P., Redmann, M. (2017). Truncated Gramians for Bilinear Systems and Their Advantages in Model Order Reduction. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-58786-8_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58785-1
Online ISBN: 978-3-319-58786-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)