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Robust Designs Through Risk Sensitivity: An Overview

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Abstract

This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown (adversarial) term and some classes of stochastic games. In particular, the paper discusses the equivalences between risk-averse controller and filter designs and saddle-point solutions of some corresponding risk-neutral stochastic differential games with different information structures for the players. One of the by-products of these analyses is that risk-averse controllers and filters (or estimators) for control and signal-measurement models are robust, through stochastic dissipation inequalities, to unmodeled perturbations in controlled system dynamics as well as signal and the measurement processes. The paper also discusses equivalences between risk-sensitive stochastic zero-sum differential games and some corresponding risk-neutral three-player stochastic zero-sum differential games, as well as robustness issues in stochastic nonzero-sum differential games with finite and infinite populations of players, with the latter belonging to the domain of mean-field games.

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References

  1. Başar T and Bernhard P, H-infinity Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Birkhäuser, Boston, Massachusetts, 1995.

    MATH  Google Scholar 

  2. Hansen L P and Sargent T J Robustness, Princeton University Press, Princeton, New Jersey, 2008.

    MATH  Google Scholar 

  3. Başar T and Olsder G J, Dynamic Noncooperative Game Theory, SIAM Series in Classics in Applied Mathematics, Philadelphia, Pennsylvania, 1999.

    MATH  Google Scholar 

  4. Başar T, Risk-averse designs: From exponential cost to stochastic games, System Theory: Modeling, Analysis and Control (Ed. by Djaferis T E and Schick I C), Kluwer, 2000, 131–144.

  5. Pan Z and Başar T, Model simplification and optimal control of stochastic singularly perturbed systems under exponentiated quadratic cost, SIAM J. Control and Optimization, 1996, 34(5): 1734–1766.

    Article  MathSciNet  Google Scholar 

  6. Başar T, Nash equilibria of risk-sensitive nonlinear stochastic differential games, J. Optimization Theory and Applications, 1999, 100(3): 479–498.

    Article  MathSciNet  Google Scholar 

  7. Başar T and Tang C, Locally optimal risk-sensitive controllers for nonlinear strict-feedback systems, J. Optimization Theory and Applications, 2000, 105(3): 521–541.

    Article  MathSciNet  Google Scholar 

  8. Arslan G and Başar T, Risk-sensitive adaptive trackers for strict-feedback systems with output measurements, IEEE Trans. Automatic Control, 2002, 47(10): 1754–1758.

    Article  MathSciNet  Google Scholar 

  9. Arslan G and Başar T, Decentralized risk-sensitive controller design for strict-feedback systems, Systems and Control Letters, 2003, 50(5): 383–393.

    Article  MathSciNet  Google Scholar 

  10. Runolfsson T, The equivalence between infinite-horizon optimal control of stochastic systems with exponential-of-integral performance index and stochastic differential games, IEEE Trans. Autom. Control, 1994, 39: 1551–1563.

    Article  MathSciNet  Google Scholar 

  11. Moon J and Başar T, Risk-sensitive control of Markov jump linear systems: Caveats and difficulties, Internat J Control, Automation, and Systems, 2017, 15(1): 462–467.

    Article  Google Scholar 

  12. Jacobson D H, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Autom. Control, 1973, 18(2): 124–131.

    Article  MathSciNet  Google Scholar 

  13. Fleming W H and McEneaney W M, Risk-sensitive control and differential games, Lecture Notes in Control and Information Science, Springer, Berlin, Germany, 1992, 184: 185–197.

    MATH  Google Scholar 

  14. Fleming W H and McEneaney W M, Risk-sensitive control on an infinite time horizon, SIAM J. Control and Optimization, 1996, 33: 1881–1915.

    Article  MathSciNet  Google Scholar 

  15. Whittle P, Risk-sensitive linear quadratic Gaussian control, Adv. Appl. Prob., 1981, 13: 764–777.

    Article  MathSciNet  Google Scholar 

  16. Whittle P, Risk-Sensitive Optimal Control, John Wiley & Sons, Chichester, England, 1990.

    MATH  Google Scholar 

  17. James M R, Baras J, and Elliott R J, Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems, IEEE Trans. Autom. Control, 1994, 39(4): 780–792.

    Article  MathSciNet  Google Scholar 

  18. Moon J, Duncan T E, and Başar T, Risk-sensitive zero-sum differential games, IEEE Trans. Automatic Control, 2019, 64(4): 1503–1518.

    Article  MathSciNet  Google Scholar 

  19. Lasry J M and Lions P L, Mean field games, Jpn. J. Math., 2007, 2: 229–260.

    Article  MathSciNet  Google Scholar 

  20. Malhame R P, Huang M Y, and Caines P E, Large population stochastic dynamic games: Closed-loop McKeanVlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 2006, 6(3): 221–252.

    MathSciNet  MATH  Google Scholar 

  21. Huang M, Caines P E, and Malhamé R P, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized Nash equilibria, IEEE Trans. Autom. Control, 2007, 52(9): 1560–1571.

    Article  MathSciNet  Google Scholar 

  22. Tembine H, Zhu Q, and Başar T, Risk-sensitive mean-field games, IEEE Trans. Automatic Control, 2014, 59(4): 835–850.

    Article  MathSciNet  Google Scholar 

  23. Moon J and Başar T, Linear quadratic risk-sensitive and robust mean field games, IEEE Trans. Automatic Control, 2017, 62(3): 1062–1077.

    Article  MathSciNet  Google Scholar 

  24. Moon J and Başar T, Risk-sensitive mean field games via the stochastic maximum principle, J Dynamic Games and Applications, 2019, 9: 1100–1125.

    Article  MathSciNet  Google Scholar 

  25. Yong J and Zhou X Y, Stochastic Controls, Springer, New York, NY, 1999.

    Book  Google Scholar 

  26. Saldi N, Başar T, and Raginsky M, Approximate Markov-Nash equilibria for discrete-time risk-sensitive mean-field games, Mathematics of Operations Research, 2020, 45(4): 1596–1620.

    Article  MathSciNet  Google Scholar 

  27. Krainak J C, Speyer J L, Marcus S I, Static team problems, part I: Sufficient conditions and the exponential cost criterion, IEEE Trans. Autom. Control, 1982, 27: 839–848.

    Article  Google Scholar 

  28. Krainak J C, Speyer J L, Marcus S.I, Static team problems part II: Affine control laws, projections, algorithms, and the LEGT problem, IEEE Trans. Autom. Control, 1982, 27: 848–859.

    Article  Google Scholar 

  29. Başar T, Variations on the theme of the Witsenhausen counterexample, Proc. 47th IEEE Conf Decision and Control CDC’08, Dec 9–11, 2008, Cancun, Mexico, 2008, 1614–1619.

  30. Yüksel S and Başar T, Stochastic Networked Control Systems: Stabilization and Optimization under Information Constraints, Systems & Control: Foundations and Applications Series, Birkhäuser, Boston, MA, 2013.

    Book  Google Scholar 

  31. Başar T, Decentralized multicriteria optimization of linear stochastic systems, IEEE Trans. Autom. Control, 1978, 23(2): 233–243.

    Article  MathSciNet  Google Scholar 

  32. Akyol E, Langbort C, and Başar T, Information-theoretic approach to strategic communication as a hierarchical game, Proceedings of the IEEE, 2017, 105(2): 205–218.

    Article  Google Scholar 

  33. Sayin M O and Başar T, On the optimality of linear signaling to deceive Kalman filters over finite/infinite horizons, Proc. GameSec 2019 (10th International Conference on Decision and Game Theory for Security), Eds. by Alpcan T, Vorobeychik Y, Baras J, et al., Springer LNCS 11836, 2019, 459–478.

  34. Sayin M O and Başar T, Persuasion-based robust sensor design against attackers with unknown control objectives, IEEE Transactions on Automatic Control, 2021, DOI: https://doi.org/10.1109/TAC.2020.3030861.

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Acknowledgements

This research was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant FA9550-19-1-0353, and in part by the Army Research Office MURI Grant AG285.

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Authors and Affiliations

  1. Coordinated Science Laboratory, University of Illinois, 1308 West Main, Urbana, IL, 61801, USA

    Tamer Başar

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  1. Tamer Başar
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Correspondence to Tamer Başar.

Additional information

This research was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-19-1-0353, and in part by the Army Research Office MURI under Grant No. AG285.

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Başar, T. Robust Designs Through Risk Sensitivity: An Overview. J Syst Sci Complex 34, 1634–1665 (2021). https://doi.org/10.1007/s11424-021-1242-6

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  • DOI: https://doi.org/10.1007/s11424-021-1242-6

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