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.


  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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Information & Authors


Published In

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


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


  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



Anastasia Thöns-Zueva

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