Nonlinear Second Cumulant/H-infinity Control with Multiple Decision Makers

Chukwuemeka Aduba

Abstract

This paper studies a second cumulant/h-infinity control problem with multiple players for a nonlinear stochastic system on a finite-horizon. The second cumulant/h-infinity control problem, which is a generalization of the higher-order multi-objective control problem, involves a control method with multiple performance indices. The necessary condition for the existence of Nash equilibrium strategies for the second cumulant/h-infinity control problem is given by the coupled Hamilton-Jacobi-Bellman (HJB) equations. In addition, a threeplayer Nash strategy is derived for the second cumulant/h-infinity control problem. A simulation example is given to illustrate the application of the proposed theoretical formulations.

References

  1. Aduba, C. and Won, C.-H. (2015). Two-Player Ad Hoc Output-Feedback Cumulant Game Control. In Proceedings of the 12th International Conference On Informatics in Control, Automation and Robotics, pages 53-59, INSTICC, IFAC, Colmar, Alsace, France.
  2. Al'brekht, E. G. (1961). On the Optimal Stabilization of Nonlinear Systems. Journal of Applied Mathematics and Mechanics, 25(5):836-844.
  3. Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications. John Wiley & Sons Inc., New York, NY.
  4. Basar, T. and Olsder, G. J. (1999). Dynamic Noncooperative Game Theory. SIAM, Philadelphia, PA.
  5. Bauso, D., Giarré, L., and Pesenti, R. (2008). Consensus in non-cooperative dynamic games: A multiretailer inventory application. IEEE Transactions on Automatic Control, 53(4):998-1003.
  6. Beard, R. W., Saridis, G. N., and Wen, J. T. (1998). Approximate Solutions to the Time-Invariant HamiltonJacobi-Bellman Equation. PMM - Journal of Optimization Theory and Applications, 96(3):589-626.
  7. Bernstein, D. S. and Hassas, W. M. (1989). LQG Control with an H8 Performance Bound: A Riccati Equation Approach. IEEE Transactions on Automatic Control, 34(3):293-305.
  8. Charilas, D. E. and Panagopoulos, A. D. (2010). A survey on game theory applications in wireless networks. Computer Networks, 54(18):3421-3430.
  9. Chen, T., Lewis, F. L., and Abu-Khalaf, M. (2007). A Neural Network Solution for Fixed-Final Time Optimal Control of Nonlinear Systems. Automatica, 43(3):482-490.
  10. Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles. Academic Press, New York, NY.
  11. Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer-Verlag, New York, NY.
  12. Kappen, H. J. (2005). A Linear Theory for Control of Nonlinear Stochastic Systems. Physical Review Letters, 95(20).
  13. Lee, J., Won, C., and Diersing, R. (2010). Two Player Statistical Game with Higher Order Cumulants. In Proc. of the American Control Conference, pages 4857- 4862, Baltimore, MD.
  14. Limebeer, D. J. N., Anderson, B. D. O., and Hendel, D. (1994). A Nash Game Approach to Mixed H2/H8 control. IEEE Transactions on Automatic Control, 39(1):69-82.
  15. Sain, M. K. (1966). Control of Linear Systems According to the Minimal Variance Criterion-A New Approach to the Disturbance Problem. IEEE Transactions on Automatic Control, AC-11(1):118-122.
  16. Sain, M. K. and Liberty, S. R. (1971). Performance Measure Densities for a Class of LQG Control Systems. IEEE Transactions on Automatic Control, AC-16(5):431- 439.
  17. Sain, M. K., Won, C.-H., Spencer, Jr., B. F., and Liberty, S. R. (2000). Cumulants and risk-sensitive control: A cost mean and variance theory with application to seismic protection of structures. In Filar, J., Gaitsgory, V., and Mizukami, K., editors, Advances in Dynamic Games and Applications, volume 5 of Annals of the International Society of Dynamic Games, pages 427- 459. Birkhuser Boston.
  18. Smith, P. J. (1995). A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa. The American Statistician, (49):217-219.
  19. Song, W. and Dyke, S. J. (2011). Application of Pseudospectral Method in Stochastic Optimal Control of Nonlinear Structural Systems. In Proc. of the American Control Conference, pages 4857-4862, San Francisco, CA.
  20. Won, C.-H., Diersing, R. W., and Kang, B. (2010). Statistical Control of Control-Affine Nonlinear Systems with Nonquadratic Cost Function: HJB and Verification Theorems. Automatica, 46(10):1636-1645.
Download


Paper Citation


in Harvard Style

Aduba C. (2016). Nonlinear Second Cumulant/H-infinity Control with Multiple Decision Makers . In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-758-198-4, pages 31-37. DOI: 10.5220/0005955400310037


in Bibtex Style

@conference{icinco16,
author={Chukwuemeka Aduba},
title={Nonlinear Second Cumulant/H-infinity Control with Multiple Decision Makers},
booktitle={Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2016},
pages={31-37},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005955400310037},
isbn={978-989-758-198-4},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Nonlinear Second Cumulant/H-infinity Control with Multiple Decision Makers
SN - 978-989-758-198-4
AU - Aduba C.
PY - 2016
SP - 31
EP - 37
DO - 10.5220/0005955400310037