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Multi-Cumulant and Pareto Solutions for Tactics Change Prediction and Performance Analysis in Stochastic Multi-Team Noncooperative Games

  • Khanh D. Piam
  • Stanley R. Liberty
  • Gang Jin
Chapter
Part of the Systems & Control: Foundations & Applications book series (SCFA)

Summary

New solution concepts, called the multi-cumulant Pareto Nash and minimax strategies, are proposed for quadratic decision problems where multiple teams of decision makers are interested in strategies that not only ensure cooperation within each team and competition among different teams but also provide noncooperative teams the capability of assessing team performance and predicting tactics via complete statistical descriptions. Analytical expressions for higher-order statistics associated with strategy selection and performance assessment as well as closed-form feedback Nash equilibrium and minimax solutions to the special linear-quadratic class of stochastic multi-team games are also presented. The capability to shape the probability density function of the team performance measure is possible because not only is the first performance statistic considered as in the special case of the linear-quadratic-gaussian problem, but also some finite linear combinations of other performance statistics are included. In all decision strategies developed here, the decision feedback gains are explicitly dependent upon the “information” statistics which are then used to directly target the uncertainty of team performance and decision laws. It is concluded that the need to account for the reduction of performance uncertainty gives rise to the interaction between dual decision control functions: reducing uncertainty and exercising control. As the result, the certainty equivalence property is no longer available for the class of statistical control problems considered here.

Keywords

Nash Equilibrium Team Performance Noncooperative Game Feedback Nash Equilibrium Nash Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [Bas82]
    T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, London, 1982.MATHGoogle Scholar
  2. [Cru01]
    J. B. Cruz, Jr., M. A. Simaan, A. Gacic, H. Jiang, B. Letellier, M. Li, and Y. Liu, Game-Theoretic Modeling and Control of Military Operations, IEEE Transactions on Aerospace and Electronic Systems, Vol. 37, No. 4, pp. 1393–1405, October 2001.CrossRefGoogle Scholar
  3. [Dav77]
    M. H. A. Davis, Linear Estimation and Stochastic Control, A Halsted Press, John Wiley & Sons, New York, 1977.MATHGoogle Scholar
  4. [Die60]
    J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York and London, 1960.MATHGoogle Scholar
  5. [DiS05]
    R. W. Diersing and M. K. Sain, The Third Generation Wind Structural Benchmark: A Nash Cumulant Robust Approach, Proceedings of the American Control Conference, pp. 3078–3083, Portland, Oregon, June 8–10, 2005.Google Scholar
  6. [DiS06]
    R. W. Diersing and M. K. Sain, Nash and Minimax Bi-Cumulant Games, The 45th IEEE Conference on Decision and Control, pp. 2571–2576, San Diego, California, December 13–15, 2006.Google Scholar
  7. [DSPW07]
    R. Diersing, M. K. Sain, K. D. Pham, and C.-H. Won, Output Feedback Multiobjective Cumulant Control with Structural Applications, Proceedings of the American Control Conference, pp. 2666–2671, New York City, New York, 2007.Google Scholar
  8. [FlR75]
    W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.MATHCrossRefGoogle Scholar
  9. [Jac73]
    D. H. Jacobson, Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Games, IEEE Transactions on Automatic Control, Vol. AC-18, pp. 124–131, 1973.CrossRefGoogle Scholar
  10. [Kli64]
    A. Klinger, Vector-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-9, No. 1, pp. 117–118, 1964.CrossRefGoogle Scholar
  11. [LiH76]
    S. R. Liberty and R. C. Hartwig, On the Essential Quadratic Nature of LQG Control-Performance Measure Cumulants, Information and Control, Vol. 32, No. 3, pp. 276–305, 1976.MathSciNetMATHCrossRefGoogle Scholar
  12. [LSC03]
    Y. Liu, M. A. Simaan, and J. B. Cruz, Jr., Game Theoretic Approach to Cooperative Teaming and Tasking in the Presence of an Adversary, Proceedings of the American Control Conference, pp. 5375–5380, Denver, Colorado, June 4–6, 2003.Google Scholar
  13. [PLSS99]
    K. D. Pham, S. R. Liberty, M. K. Sain, and B. F. Spencer, Jr., Generalized Risk Sensitive Building Control: Protecting Civil Structures with Multiple Cost Cumulants, Proceedings of the American Control Conference, pp. 500–504, San Diego, California, June 1999.Google Scholar
  14. [PLS99]
    K. D. Pham, S. R. Liberty, and M. K. Sain, Evaluating Cumulant Controllers on a Benchmark Structure Protection Problem in the Presence of Classic Earthquakes, Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, pp. 617–626, Monticello, Illinois, September 22–24, 1999.Google Scholar
  15. [PLSS00]
    K. D. Pham, S. R. Liberty, M. K. Sain, and B. F. Spencer, Jr., First Generation Seismic-AMD Benchmark: Robust Structural Protection by the Cost Cumulant Control Paradigm, Proceedings of the American Control Conference, pp. 1–5, Chicago, Illinois, June 28–30, 2000.Google Scholar
  16. [PSL02a]
    K. D. Pham, M. K. Sain, and S. R. Liberty, Robust Cost-Cumulants Based Algorithm for Second and Third Generation Structural Control Benchmarks, Proceedings of the American Control Conference, pp. 3070–3075, Anchorage, Alaska, May 08–10, 2002.Google Scholar
  17. [PSL02b]
    K. D. Pham, M. K. Sain, and S. R. Liberty, Finite Horizon Full-State Feedback kCC Control in Civil Structures Protection, Stochastic Theory and Adaptive Control, Lecture Notes in Control and Information Sciences, Proceedings of the Workshop Held in Lawrence, Kansas, Edited by B. Pasik-Duncan, Springer-Verlag, Berlin Heidelberg, Germany, Vol. 280, pp. 369–383, September 2002.Google Scholar
  18. [PSL02c]
    K. D. Pham, M. K. Sain, and S. R. Liberty, Cost Cumulant Control: State-Feedback, Finite-Horizon Paradigm with Application to Seismic Protection, Special Issue of Journal of Optimization Theory and Applications, Edited by A. Miele, Kluwer Academic/Plenum Publishers, New York, Vol. 115, No. 3, pp. 685–710, December 2002.Google Scholar
  19. [PJSSL04]
    K. D. Pham, G. Jin, M. K. Sain, B. F. Spencer, Jr., and S. R. Liberty, Generalized LQG Techniques for the Wind Benchmark Problem, Special Issue of ASCE Journal of Engineering Mechanics on the Structural Control Benchmark Problem, Vol. 130, No. 4, pp. 466–470, April 2004.Google Scholar
  20. [Pha04]
    K. D. Pham, Statistical Control Paradigms for Structural Vibration Suppression, Ph.D. Dissertation, Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana, May 2004.Google Scholar
  21. [PSL04]
    K. D. Pham, M. K. Sain, and S. R. Liberty, Infinite Horizon Robustly Stable Seismic Protection of Cable-Stayed Bridges Using Cost Cumulants, Proceedings of the American Control Conference, pp. 691–696, Boston, Massachusetts, June 30, 2004.Google Scholar
  22. [PSL05]
    K. D. Pham, M. K. Sain, and S. R. Liberty, Statistical Control for Smart Base-Isolated Buildings via Cost Cumulants and Output Feedback Paradigm, Proceedings of the American Control Conference, pp. 3090–3095, Portland, Oregon, June 8–10, 2005.CrossRefGoogle Scholar
  23. [Pha05]
    K. D. Pham, Minimax Design of Statistics-Based Control with Noise Uncertainty for Highway Bridges, Proceedings of DETC 2005/2005 ASME 20th Biennial Conference on Mechanical Vibration and Noise: Active Control of Vibration and Acoustics I, DETC2005-84593, Long Beach, California, September 24–28, 2005.Google Scholar
  24. [PhRo7]
    K. D. Pham and L. Robertson, Statistical Control Paradigm for Aerospace Structures Under Impulsive Disturbances, The 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA 2007-1755, Honolulu, Hawaii, April 23–26, 2007. Google Scholar
  25. [Pha07b]
    K. D. Pham, Cost Cumulant-Based Control for a Class of Linear-Quadratic Tracking Problems, Proceedings of the American Control Conference, pp. 335–340, New York, New York, 2007.CrossRefGoogle Scholar
  26. [Pha08b]
    K. D. Pham, On Statistical Control of Stochastic Servo-Systems: Performance-Measure Statistics and State-Feedback Paradigm, The 17th International Federation of Automatic Control, Accepted, Seoul, Korea, 2008.Google Scholar
  27. [Pha07c]
    K. D. Pham, Multi-Cumulant Control for Zero-Sum Differential Games: Performance-Measure Statistics and State-Feedback Paradigm, The 7th International Conference on Cooperative Control and Optimization, In Press, Gainesville, Florida, January 31–February 02, 2007.Google Scholar
  28. [Pha07d]
    K. D. Pham, Cooperative Solutions in Multi-Person Quadratic Decision Problems: Finite-Horizon and State-Feedback Cost-Cumulant Control Paradigm, The 46th IEEE Conference on Decision and Control, Accepted, pp. 2484–2490, New Orleans, Loussiana, December 12–14, 2007.Google Scholar
  29. [PLR08c]
    K. D. Pham, S. Lacy, and L. Robertson, Multi-Cumulant and Non-Inferior Stategies for Multi-Player Pursuit-Evasion, Proceedings of American Control Conference, Accepted, Seattles, Washington, 2008.Google Scholar
  30. [Pha08d]
    K. D. Pham, Non-Cooperative Outcomes for Stochastic Multi-Player Nash Games: Novel Decision Strategies for Multi-Resolution Performance Robustness, The 17th International Federation of Automatic Control, Accepted, Seoul, Korea, 2008.Google Scholar
  31. [Zad63]
    L. A. Zadeh, Optimality and Non-Scalar-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-8, No. 1, pp. 59–60, 1963.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Khanh D. Piam
    • 1
  • Stanley R. Liberty
    • 2
  • Gang Jin
    • 3
  1. 1.Space Vehicles DirectorateAir Force Research LaboratoryKirtland AFBUSA
  2. 2.Office of PresidentKettering UniversityFlintUSA
  3. 3.Electronics & Electrical EngineeringFord Motor CompanyDearbornUSA

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