System Theory pp 131-143 | Cite as

Risk-Averse Designs: From Exponential Cost to Stochastic Games

  • Tamer Başar
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 518)

Abstract

We discuss the relationship between risk-averse designs based on exponential cost functions and a class of stochastic games, which yields a robustness interpretation for risk-averse decision rules through a stochastic dissipation inequality. In particular, we prove the equivalence between risk-averse linear filter designs and saddle-point solutions of a particular stochastic differential game with asymmetric information for the players. A byproduct of this study is that risk-averse filters for linear signal-measurement models are robust (through a stochastic dissipation inequality) to unmodeled perturbations in both the signal and the measurement processes.

Keywords

Attenuation Covariance Acoustics Estima 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Başar and P. Bernhard, H -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Birkhäuser, Boston, MA, 2nd edition, 1995.Google Scholar
  2. [2]
    W. H. Fleming and W. M. McEneaney, “Risk Sensitive Control and Differential Games,” Lecture Notes in Control and Information Science, Springer, Berlin, Germany, Vol. 184, pp. 185–197, 1992.MathSciNetCrossRefGoogle Scholar
  3. [3]
    W. H. Fleming and W. M. McEneaney, “Risk Sensitive Control on an Infinite Time Horizon,” SIAM J. Control and Optimization, vol. 33, pp. 1881–1915, 1995.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. H. Fleming and W. M. McEneaney, Risk Sensitive and Robust Nonlinear Filtering, Proceedings of the 36th IEEE CDC, San Diego, CA, 1997.Google Scholar
  5. [5]
    W. H. Fleming and S. K. Mitter, “Optimal control and nonlinear filtering for nondegenerate diffusion processes,” Stochastics, vol. 8, pp. 63–77, 1982.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    D. H. Jacobson, “Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games,” IEEE Transactions on Automatic Control, Vol. 18, pp. 124–131, 1973.MATHCrossRefGoogle Scholar
  7. [7]
    M. R. James and J. S. Baras, “Partially observed differential games, infinite dimensional HJI equations, and nonlinear H control,”, SIAM J. Control and Optim., vol. 34, pp. 1342–1364, 1996.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    M. R. James, J. Baras, and R. J. Elliott, “Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems,” IEEE Transactions on Automatic Control, vol. AC-39, no. 4, pp. 780–792, April 1994.MathSciNetCrossRefGoogle Scholar
  9. [9]
    S. K. Mitter, “Filtering and stochastic control: A historical perspective,” IEEE Control Systems Magazine, pp. 67–76, June 1996.Google Scholar
  10. [10]
    Z. Pan and T. Bagar, “Model simplification and optimal control of stochastic singularly perturbed systems under exponentiated quadratic cost,” SIAM Journal on Control and Optimization, Vol. 34, pp. 1734–1766, 1996.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    T. Runolfsson, “The equivalence between infinite-horizon optimal control of stochastic systems with exponential-of-integral performance index and stochastic differential games,” IEEE Transactions on Automatic Control, Vol. 39, pp. 1551–1563, 1994.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    P. Whittle. Risk-Sensitive Optimal Control, John Wiley and Sons, Chichester, England, 1990.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Tamer Başar
    • 1
  1. 1.Coordinated Science Laboratory and Department of Electrical and Computer EngineeringUniversity of IIIinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations