Abstract

We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.

Keywords

  1. bilinear systems
  2. model order reduction
  3. balanced truncation
  4. averaging method
  5. Hankel singular values
  6. generalized Lyapunov equations
  7. stochastic control

MSC codes

  1. 35Q84
  2. 93B11
  3. 93B20
  4. 93C70
  5. 93E24

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References

1.
S.A. Al-Baiyat and M. Bettayeb, A new model reduction scheme for k-power bilinear systems, in Proceedings of the 32nd IEEE Conference on Decision and Control, IEEE, New York, 1993, pp. 22--27.
2.
A.C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control, SIAM, Philadelphia, 2005.
3.
Z. Artstein, Invariant measures of differential inclusions applied to singular perturbations, Weizman Institute Research Report, 1998.
4.
Z. Bai and D. Skoogh, A projection method for model reduction of bilinear dynamical systems, Linear Algebra Appl., 415 (2006), pp. 406--425.
5.
D. Bakry and M. Emery, Hypercontractivité des semi-groupes de diffusion, C.R. Acad. Sci. Paris Sér. I Math., 299 (1984), pp. 775--778.
6.
P. Benner, J.-R. Li, and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra Appl., 15 (2008), pp. 755--777.
7.
P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM J. Control Optim., 49 (2011), pp. 686--711.
8.
A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
9.
T. Breiten and P. Benner, Interpolation-based $\mathcal{H}_{2}$-model reduction of bilinear control system, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 859--885.
10.
T. Breiten and T. Damm, Krylov subspace methods for model order reduction of bilinear control systems, Systems Control Lett., 59 (2011), pp. 443--450.
11.
N. Bogolyubov and Y. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.
12.
T. Breiten and T. Damm, Krylov subspace methods for model order reduction of bilinear control systems, Systems Control Lett., 59 (2010), pp. 443--450.
13.
M. Condon and R. Ivanov, Empirical balanced truncation for nonlinear systems, J. Nonlinear Sci., 14 (2004), pp. 405--414.
14.
M. Condon and R. Ivanov, Nonlinear systems---algebraic Gramians and model reduction, COMPEL, 24 (2005), pp. 202--219.
15.
M. Condon and R. Ivanov, Krylov subspace from bilinear representations of nonlinear systems, COMPEL, 26 (2007), pp. 11--26.
16.
P. D'Alessandro, A. Isidori, and A. Ruberti, Realization and structure theory of bilinear dynamical systems, SIAM J. Control, 12 (1974), pp. 517--535.
17.
T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numer. Linear Algebra Appl., 15 (2008), pp. 853--871.
18.
T. Damm, Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Inform. Sci. 297, Springer, Berlin, 2004.
19.
S. Djennoune and M. Bettayeb, On the structure of energy functions of singularly perturbed bilinear systems, J. Robust Nonlinear Control, 15 (2005), pp. 601--618.
20.
A. Dontchev, T. Dontchev, and I. Slavov, A Tichonov-type theorem for singularly perturbed differential inclusion, Nonlinear Anal., 26 (1996), pp. 1547--1554.
21.
L. Feng and P. Benner, A note on projection techniques for model order reduction of bilinear systems, in International Conference on Numerical Analysis and Applied Mathematics, AIP Conf. Proc. 936, 2007, pp. 208--211.
22.
K. Fernando and H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Trans. Automat. Control, AC-27 (1982), pp. 466--468.
23.
G.M. Flagg, Interpolation Methods for the Model Reduction of Bilinear Systems, Ph.D. Thesis, Virginia Tech, Blacksburg, VA, 2012.
24.
E.L. Florin, V.T. Moy, and H.E. Gaub, Adhesion forces between individual ligand-receptor pairs, Science, 264 (1994), pp. 415--417.
25.
M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer, New York, 1998.
26.
K. Fujimoto and J.M.A. Scherpen, Singular value analysis and balanced realizations for nonlinear systems, in Model Order Reduction: Theory, Research Aspects and Applications, W.H.A. Schilders, H.A. Vorst, J. van der Rommes, eds., Springer, Berlin, 2008, pp. 251--272.
27.
K. Fujimoto and J.M.A. Scherpen, Balanced realization and model order reduction for nonlinear systems based on singular value analysis, SIAM J. Control Optim., 48 (2010), pp. 4591--4623.
28.
V. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J. Control Optim., 30 (1992), pp. 1228--1249.
29.
Z. Gajic and M. Lelic, Singular perturbation analysis of system order reduction via system balancing, Proc. Amer. Control Conf., 4 (2000), pp. 2420--2424.
30.
K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^\infty$-error bounds, Internat. J. Control, 39 (1984), pp. 1115--1193.
31.
G. Grammel, Averaging of singularly perturbed systems, Nonlinear Anal., 28 (1997), pp. 1851--1865.
32.
W.S. Gray and J. Mesko, Energy functions and algebraic Gramians for bilinear systems, in Preprints of the 4th IFAC Nonlinear Control Systems Design Symposium, Enschede, Netherlands, 1998, Pergamon, Oxford, 1998, pp. 103--108.
33.
W.S. Gray and J. Mesko, Observability functions for linear and nonlinear systems, Systems Control Lett., 38 (1999), pp. 99--113.
34.
W.S. Gray and J.M.A. Scherpen, On the nonuniqueness of singular value functions and balanced nonlinear realizations, Systems Control Lett., 44 (2001), pp. 219--232.
35.
C. Hartmann, V.-M. Vulcanov, and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach, Multiscale Model. Simul., 8 (2010), pp. 1348--1367.
36.
C. Hartmann, Balanced model reduction of partially-observed Langevin equations: An averaging principle, Math. Comput. Modelling Dynam., 17 (2011), pp. 463--490.
37.
P. Holmes, J.L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996.
38.
A. Isidori and A. Ruberti, Realization theory of bilinear systems, in Geometric Methods in System Theory, D.Q. Mayne and R.W. Brocken, eds., Reidel, Dordrecht, 1973, pp. 81--130.
39.
M.S.Z. Kellermayer, S.B. Smith, H.L. Granzier, and C. Bustamante, Folding-unfolding transitions in single Titin molecules characterized with laser tweezers, Science, 276 (1997), pp. 1112--1116.
40.
R.Z. Khas'minskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), pp. 390--406.
41.
E. Klipp, R. Herg, A. Kowald, C. Wierling, and H. Lehrach, Systems Biology in Practice: Concepts, Implementation and Application, Wiley-VCH, Weinheim, Germany, 2005.
42.
P.V. Kokotović, Applications of singular perturbation techniques to control problems, SIAM Rev., 26 (1984), pp. 501--550.
43.
H.J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York, 2001.
44.
S. Lall, J.E. Marsden, and S. Glavaški, A subspace approach to balanced truncation for model reduction of nonlinear control systems, Internat. J. Robust Nonlinear Control, 12 (2002), pp. 519--535.
45.
Y.Q. Lin, L. Bao, and Y. Wei, A model-order reduction method based on Krylov subspaces for MIMO bilinear dynamical systems, J. Appl. Math. Comput., 25 (2007), pp. 293--304.
46.
Y.Q. Lin, L. Bao, and Y. Wei, Order reduction of bilinear MIMO dynamical systems using new block Krylov subspaces, Comput. Math. Appl., 58 (2009), pp. 1093--1102.
47.
Y. Liu and B.D.O. Anderson, Singular perturbation approximation of balanced systems, Internat. J. Control, 50 (1989), pp. 1379--1405.
48.
H. Mabuchi and N. Khaneia, Principles and applications of control in quantum systems, Internat. J. Robust Nonlinear Control, 15 (2005), pp. 647--667.
49.
R. Merkel, P. Nassoy, A. Leung, K. Ritchie, and E. Evans, Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy, Nature, 397 (1999), pp. 50--53.
50.
B.C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, AC-26 (1981), pp. 17--32.
51.
B. Ø ksendal, Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, 2003.
52.
G.C. Papanicolaou, Some probabilistic problems and methods in singular perturbations, Rocky Mountain J. Math., 6 (1976), pp. 653--674.
53.
G.A. Pavliotis and A.M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, New York, 2008.
54.
J.R. Phillips, Projection-based approaches for model reduction of weakly nonlinear, time-varying systems, IEEE Trans. Comp. Aided Design, 22 (2003), pp. 171--187.
55.
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Clarendon Press, Oxford, 1999.
56.
M. Rief, M. Gautel, F. Oesterhelt, J.M. Fernandez, and H.E. Gaub, Reversible unfolding of individual Titin Immunoglobulin domains by AFM, Science, 276 (1997), pp. 1109--1112.
57.
H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer, Berlin, 1996.
58.
C.W. Rowley, Model reduction for fluids using balanced proper orthogonal decomposition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 997--1013.
59.
R. Samar, I. Postlethwaite, and D.-W. Gu, Applications of the singular perturbation approximation of balanced systems, in Proceedings of the Third IEEE Conference on Control Applications, Piscataway, NJ, 1994, pp. 1823--1828.
60.
B. Schäfer-Bung, C. Hartmann, B. Schmidt and C. Schütte, Dimension reduction by balanced truncation: Application to light-induced control of open quantum systems, J. Chem. Phys., 135 (2011), 014112.
61.
J.M.A. Scherpen, Balancing for nonlinear systems, Systems Control Lett., 21 (1993), pp. 143--153.
62.
H. Schneider, Positive operators and an inertia theorem, Numer. Math., 7 (1965), pp. 11--17.
63.
M. Sznaier, A. Doherty, M. Barahona, H. Mabuchi, and J.C. Doyle, A new bound of the $L^{2}[0,T]$-induced norm and applications to model reduction, in Proceedings of the 2002 American Control Conference, IEEE, Piscataway, NJ, 2002, pp. 1180--1185.
64.
A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions, SIAM J. Control Optim., 35 (1997), pp. 1--28.
65.
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, AMS, Providence, RI, 2003.
66.
E. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), pp. 87--90.
67.
X. Wang and Y. Jiang, Model reduction of bilinear systems based on Laguerre series expansion, J. Franklin Inst., 349 (2012), pp. 1231--1246.
68.
F. Watbled, On singular perturbations for differential inclusions on the infinite time interval, J. Math. Anal. Appl., 310 (2005), pp. 362--378.
69.
L. Zhang and J. Lam, On $H_{2}$ model reduction of bilinear systems, Automatica J. IFAC, 38 (2002), pp. 205--216.
70.
K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, NJ, 1998.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2356 - 2378
ISSN (online): 1095-7138

History

Submitted: 1 June 2010
Accepted: 8 April 2013
Published online: 4 June 2013

Keywords

  1. bilinear systems
  2. model order reduction
  3. balanced truncation
  4. averaging method
  5. Hankel singular values
  6. generalized Lyapunov equations
  7. stochastic control

MSC codes

  1. 35Q84
  2. 93B11
  3. 93B20
  4. 93C70
  5. 93E24

Authors

Affiliations

Anastasia Thöns-Zueva

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