Skip to main content
SpringerLink
Log in

Information-Theoretic Lower Bounds of the Quadratic Cost in Stochastic Control with Partial Observation

  • Conference paper
  • First Online:
Advanced, Contemporary Control (PCC 2023)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 709))

Included in the following conference series:

  • 172 Accesses

Abstract

Stochastic control with quadratic cost and partial, noisy observation has been considered. It has been proven, under rather natural assumptions, that the cost function is lower bounded by two types of bounds. The first one is convex increasing function of the mutual information between observations and state of the system. The second bound is decreasing function of the mutual information between control sequence and the state. Considerations are illustrated with simple example.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alpcan, T., Shames, I., Cantoni, M., Nair, G.: An information-based learning approach to dual control. IEEE Trans. on Neural Netw. Learn. Syst. 26(11), 2736–2748 (2015)

    Google Scholar 

  2. Bania, P.: An information based approach to stochastic control problems int. J. Appl. Math. Comput. Sci. 30(1), 23–34 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Bania, P.: Bayesian input design for linear dynamical model discrimination. Entropy 21(4), 1–13 (2019). https://doi.org/10.3390/e21040351

  4. Bania, P.: Example for equivalence of dual and information based optimal control. Int. J. Control 38(5), 787–803 (2018)

    MathSciNet  Google Scholar 

  5. Bania, P.: Simple example of dual control problem with almost analytical solution. In: Proceedings of the 19th Polish Control Conference, Krakow, Poland, 18-21 June, pp. 55–64 (2017(

    Google Scholar 

  6. BarShalom, Y., Tse, E.: Caution, probing, and the value of information in the control of uncertain systems. Annals Econ. Soc. Measurem.5(3), 323–337 (1976)

    Google Scholar 

  7. Cao, F.J., Feito, M.: Open problems on information and feedback controlled systems. Entropy 14(12), 834–847 (2012)

    Article  MATH  Google Scholar 

  8. Cover, T.M., Thomas, J.A.: Elements of Information Theory, \(2^{nd}\) edition. John Wiley & Sons Inc., Hoboken, New Jersey, USA (2006)

    Google Scholar 

  9. Delvenne, J.C., Sandberg, H.: Towards a thermodynamics of control: entropy, energy and Kalman filtering. In: Proceedings of the 52nd IEEE Conference on Decision and Control, 10-13 December, Florence, Italy, pp. 3109–3114 (2013)

    Google Scholar 

  10. Filatov, N.M., Unbehauen, H.: Adaptive Dual Control: Theory and Applications. Springer-Verlaag, Berlin-Heidelberg (2004). https://doi.org/10.1007/b96083

  11. Kozlowski, E., Banek, T.: Active learning in discrete time stochastic systems. In: J.J., O. D. (eds.). Knowledge-Based Intelligent System Advancements: Systemic and Cybernetic Approaches., Information Science References, New York, USA, pp. 350–371 (2011)

    Google Scholar 

  12. Lozano-Dur’an, A., Arranz, G.: Information-theoretic formulation of dynamical systems: causality, modelling, and control. Phys. Rev. Res. 4(2), 023195 (2022)

    Article  Google Scholar 

  13. Mitter, S.K., Newton, N.J.: Information and entropy flow in the Kalman-Bucy filter. J. Stat. Phys. 118(1), 145–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sagawa, T., Ueda, M.: Role of mutual information in entropy production under information exchanges.. New J. Phys.15(125012), 2–23 (2013)

    Google Scholar 

  15. Tatjewski, P., Lawrynczuk, M.: Algorithms with state estimation in linear and nonlinear model predictive control. Comput. Chem. Eng. 143, 107065 (2020), ISSN 0098-1354

    Google Scholar 

  16. Touchette, H., Lloyd, S.: Information-theoretic limits of control. Phys. Rev. Lett. 84(6), 1156–1159 (2000)

    Google Scholar 

  17. Touchette, H., Lloyd, S.: Information-theoretic approach to the study of control systems. Physica A.331(1), 140–172 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Automatic Control and Robotics, AGH University of Science and Technology in Krakow, Mickiewicza 30 Avenue, 30-059, Krakow, Poland

    Piotr Bania

Authors
  1. Piotr Bania
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Piotr Bania .

Editor information

Editors and Affiliations

  1. Silesian University of Technology, Gliwice, Poland

    Marek Pawelczyk

  2. Silesian University of Technology, Gliwice, Poland

    Dariusz Bismor

  3. Silesian University of Technology, Gliwice, Poland

    Szymon Ogonowski

  4. Systems Research Institute, Polish Academy of Science, Warsaw, Poland

    Janusz Kacprzyk

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bania, P. (2023). Information-Theoretic Lower Bounds of the Quadratic Cost in Stochastic Control with Partial Observation. In: Pawelczyk, M., Bismor, D., Ogonowski, S., Kacprzyk, J. (eds) Advanced, Contemporary Control. PCC 2023. Lecture Notes in Networks and Systems, vol 709. Springer, Cham. https://doi.org/10.1007/978-3-031-35173-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation