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Robust Control and Dynamic Games

  • Pierre Bernhard
Reference work entry

Abstract

We describe several problems of “robust control” that have a solution using game theoretical tools. This is by no means a general overview of robust control theory beyond that specific purpose nor a general account of system theory with set description of uncertainties.

Keywords

Robust control \(\mathcal {H}^\infty \)-optimal control Nonlinear \(\mathcal {H}^\infty \) control Robust control approach to option pricing 

References

  1. Akian M, Quadrat J-P, Viot M (1998) Duality between probability and optimization. In: Gunawardena J (ed) Indempotency. Cambridge University Press, CambridgeGoogle Scholar
  2. Aubin J-P, Bayen AM, Saint-Pierre P (2011) Viability theory. New directions. Springer, Heidelberg/New YorkGoogle Scholar
  3. Baccelli F, Cohen G, Olsder G-J, Quadrat J-P (1992) Synchronization and linearity. Wiley, Chichester/New YorkGoogle Scholar
  4. Ball JA, Helton W, Walker ML (1993) H control for nonlinear systems with output feedback. IEEE Trans Autom Control 38:546–559Google Scholar
  5. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhaüser, BostonGoogle Scholar
  6. Barles G (1994) Solutions de viscosité des équations de Hamilton-Jacobi. Springer, Paris/BerlinGoogle Scholar
  7. Başar T, Bernhard P (1995) H-optimal control and related minimax design problems: a differential games approach, 2nd edn. Birkhäuser, BostonGoogle Scholar
  8. Başar T, Olsder G-J (1982) Dynamic noncooperative game theory. Academic Press, London/ New YorkGoogle Scholar
  9. Bensoussan A, Bernhard P (1993) On the standard problem of H-optimal control for infinite dimensional systems. In: Identification and control of systems governed by patial differential equations (South Hadley, 1992). SIAM, Philadelphia, pp 117–140Google Scholar
  10. Berkowitz L (1971) Lectures on differential games. In: Kuhn HW, Szegö GP (eds) Differential games and related topics. North Holland, Amsterdam, pp 3–45Google Scholar
  11. Bernhard P (1994) A min-max certainty equivalence principle for nonlinear discrete time control problems. Syst Control Lett 24:229–234Google Scholar
  12. Bernhard P, Rapaport A (1996) Min-max cetainty equivalence principle and differential games. Int J Robust Nonlinear Control 6:825–842Google Scholar
  13. Bernhard P, Engwerda J, Roorda B, Schumacher H, Kolokoltsov V, Aubin J-P, Saint-Pierre P (2013) The interval market model in mathematical finance: a game theoretic approach. Birkhaüser, New YorkGoogle Scholar
  14. Corless MJ, Leitman G (1981) Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans Autom Control 26:1139–1144Google Scholar
  15. Coutinho DF, Trofino A, Fu M (2002) Nonlinear H-infinity control: an LMI approach. In: 15th trienal IFAC world congress. IFAC, Barcelona, SpainGoogle Scholar
  16. Daum FE (2015) Nonlinear filters. In: Samad T, Ballieul J (eds) Encyclopedia of systems and control. Springer, London, pp 870–876Google Scholar
  17. Didinsky G, Başar T, Bernhard P (1993) Structural properties of minimax policies for a class of differential games arising in nonlinear H-optimal control. Syst Control Lett 21:433–441Google Scholar
  18. Doyle JC (1982) Analysis of feedback systems with structured uncertainties. IEEE Proc 129: 242–250Google Scholar
  19. Doyle JC, Glover K, Kargonekar PP, Francis BA (1989) State-space solutions to the standard \(\mathcal {H}_2\) and \(\mathcal {H}_\infty \) control problems. IEEE Trans Autom Control 34:831–847Google Scholar
  20. El Ghaoui L, Scorletti G (1996) Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 32:1273–1284Google Scholar
  21. Georgiou TT, MC Smith (1997) Robustness analysis of nonlinear feedback systems: an input-output approach. IEEE Trans Autom Control 42:1200–1221Google Scholar
  22. Glover K (2015) H-infinity control. In: Bailleul J, Samad T (eds) Encyclopaedia of systems and control. Springer, London, pp 520–526Google Scholar
  23. Gutman S (1979) Uncertain dynamical systems: a Lyapunov min-max approach. IEEE Trans Autom Control 24:437–443Google Scholar
  24. Gutman S, Leitmann G (1976) Stabilizing feedback control for dynamical systems with bounded uncertainty. In: Proceedings of the conference on decision and control. IEEE, Clearwater, FloridaGoogle Scholar
  25. Kang-Zhi L (2015) Lmi approach to robust control. In: Bailleul J, Samad T (eds) Encyclopaedia of systems and control. Springer, London, pp 675–681Google Scholar
  26. Kurzhanski AB, Varaiya P (2014) Dynamics and control of trajectory tubes. Birkhäuser, ChamGoogle Scholar
  27. Qiu L (2015) Robust control in gap metric. In: Bailleul J, Samad T (eds) Encyclopaedia of systems and control. Springer, London, pp 1202–1207Google Scholar
  28. Rapaport A, Gouzé J-L (2003) Parallelotopic and practical observers for nonlinear uncertain systems. Int J Control 76:237–251Google Scholar
  29. Samad T, Ballieul J (2015) Encyclopedia of systems and control. Springer, LondonGoogle Scholar
  30. Stoorvogel AA (1992) The H-infinity control problem. Prentice Hall, NeW YorkGoogle Scholar
  31. van der Schaft A (1996) L2-gain and passivity techniques in nonlinear control. Lecture notes in control and information sciences, vol 18. Springer, LondonGoogle Scholar
  32. Vidyasagar M (1993) Nonlinear systems analysis. Prentice Hall, Englewood CliffGoogle Scholar
  33. Zames G (1981) Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans Autom Control 26:301–320Google Scholar
  34. Zames G, El-Sakkary AK (1980) Unstable ststems and feedback: the gap metric. In: Proceedings of the 16th Allerton conference, Monticello, pp 380–385Google Scholar
  35. Zhou K, JC Doyle, Glover K (1996) Robust and optimal control. Prentice Hall, Upper Saddle RiverGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Emeritus Senior Scientist, Biocore TeamUniversit de la Cte d’Azur-INRIASophia Antipolis CedexFrance

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