Learning Optimal Controllers for Linear Systems with Multiplicative Noise via Policy Gradient

Benjamin Gravell, Peyman Mohajerin Esfahani, Tyler H. Summers

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)
16 Downloads (Pure)


The linear quadratic regulator (LQR) problem has reemerged as an important theoretical benchmark for reinforcement learning-based control of complex dynamical systems with continuous state and action spaces. In contrast with nearly all recent work in this area, we consider multiplicative noise models, which are increasingly relevant because they explicitly incorporate inherent uncertainty and variation in the system dynamics and thereby improve robustness properties of the controller. Robustness is a critical and poorly understood issue in reinforcement learning; existing methods which do not account for uncertainty can converge to fragile policies or fail to converge at all. Additionally, intentional injection of multiplicative noise into learning algorithms can enhance robustness of policies, as observed in ad hoc work on domain randomization. Although policy gradient algorithms require optimization of a non-convex cost function, we show that the multiplicative noise LQR cost has a special property called gradient domination, which is exploited to prove global convergence of policy gradient algorithms to the globally optimum control policy with polynomial dependence on problem parameters. Results are provided both in the model-known and model-unknown settings where samples of system trajectories are used to estimate policy gradients.

Original languageEnglish
Pages (from-to)5283-5298
JournalIEEE Transactions on Automatic Control
Issue number11
Publication statusPublished - 2021

Bibliographical note

Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Additive noise
  • Convergence
  • Covariance matrices
  • gradient methods
  • noise
  • optimal control
  • Reinforcement learning
  • Robustness
  • Stability analysis
  • Stochastic processes
  • stochastic systems
  • uncertain systems
  • Uncertainty


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