Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games

  • T-Y. Li
  • Z. Gajic
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 3)

Abstract

In this paper we study the symmetric coupled algebraic Riccati equations corresponding to the steady state Nash strategies. Under control-oriented assumptions, imposed on the problem matrices, the Lyapunov iterations are constructed such that the proposed algorithm converges to the nonnegative (positive) definite stabilizing solution of the coupled algebraic Riccati equations. In addition, the problem order reduction is achieved since the obtained Lyapunov equations are of the reduced-order and can be solved independently. As a matter of fact a parallel synchronous algorithm is obtained. A high-order numerical example is included in order to demonstrate the efficiency of the proposed algorithm. In the second part of this paper we have proposed an algorithm, in terms of the Lyapunov iterations, for finding the positive semidefinite stabilizing solution of the algebraic Riccati equation of the zero-sum differential games. The similar algebraic Riccati type equations appear in the H optimal control and related problems.

Keywords

Attenuation Assure Nash Nite 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Abou-Kandil, G. Freiling, and G. Jank, Necessary conditions for constant solutions of coupled Riccati equations in Nash games, Systems & Control Letters, 21 (1993), 295–306.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    T. Başar, Generalized Riccati equations in dynamic games, in The Riccati Equation, S. Bittanti, A. Laub, and J. Willems, eds., Springer- Verlag, 1991.Google Scholar
  3. [3]
    T. Başar and R Bernhard, H∞ Optimal Control and Related Minimax Design problems: A Dynamic Game Approach, Birkhauser, Boston, 1991.Google Scholar
  4. [4]
    T. Başar, A counterexample in linear-quadratic games: existence of non-linear Nash strategies, J. of Optimization Theory and Applications, 14 (1974), 425–430.MATHCrossRefGoogle Scholar
  5. [5]
    R. Bellman, Monotone approximation in dynamic programming and calculus of variations, Proc. The National Academy of Science USA, 44 (1954), 1073–1075.CrossRefGoogle Scholar
  6. [6]
    R. Bellman, Dynamic Programming, Princeton University Press, 1957.Google Scholar
  7. [7]
    R. Bellman, Adaptive Control Processes: A Guided Tour, Princeton University Press, 1961.Google Scholar
  8. [8]
    D. Bernstein and W. Haddad, LQG control with an H∞ performance bound: A Riccati equation approach, IEEE Trans. Automatic Control, 34 (1989), 293–305.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    D. Bertsekas, Dynamic Programming: Deterministic and Stochastic Models, Prentice Hall, Englewood Cliffs, 1987.MATHGoogle Scholar
  10. [10]
    D. Bertsekas and J. Tsitsiklis, Some aspects of parallel and distributed iterative algorithms—A survey, Automatica, 27 (1991), 3–21.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    S. Bingulac and H. Vanlandingham, Algorithms for Computer- Aided Design of Multivariate Control Systems, Marcel Dekker, New York, 1993.Google Scholar
  12. [12]
    Z. Gajic and T-Y. Li, Simulation results for two new algorithms for solving coupled algebraic Riccati equations, Third Int. Symp. on Differential Games, Sophia Antipolis, Prance, June 1988.Google Scholar
  13. [13]
    Z. Gajic and X. Shen, Parallel Algorithms for Optimal Control of Large Scale Linear Systems, Springer Verlag, London, 1993.MATHGoogle Scholar
  14. [14]
    G. Hewer, Existence theorems for positive semidefinite and sign indefinite stabilizing solutions of H Riccati equations, SIAM J. Control and Optimization, 31 (1993), 16–29.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    L. Jodar and H. Abou-Kandil, Kronecker products and coupled matrix Riccati differential equations, Linear Algebra and Its Applications, 121 (1989), 39–51.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    L. Kantorovich and G. Akilov, Functional Analysis in Normed Spaces, Macmillan, New York, 1964.MATHGoogle Scholar
  17. [17]
    D. Kleinman, On an iterative techniques for Riccati equation computations, IEEE Trans. Automatic Control, 13 (1968), 114–115.CrossRefGoogle Scholar
  18. [18]
    H. Khalil and P. Kokotovic, Feedback and well-posedness of singularly perturbed Nash games, IEEE Trans. Automatic Control, 24 (1979), 699–708.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    H. Khalil, Multimodel design of a Nash strategy, J. Optimization Theory and Application, (1980), 553–564.Google Scholar
  20. [20]
    D. Kirk, Optimal Control Theory, Prentice Hall, Englewood Cliffs, 1970.Google Scholar
  21. [21]
    N. Krikelis and A. Rekasius, On the solution of the optimal linear control problems under conflict of interest, IEEE Trans. Automatic Control, 16 (1971), 140–147.MathSciNetCrossRefGoogle Scholar
  22. [22]
    V. Kucera, A contribution to matrix quadratic equations, IEEE Trans. Automatic Control, 17 (1972), 344–347.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    R. Larson, A survey of dynamic programming computational procedures, IEEE Trans. Aut. Control, 12 (1967), 767–774.CrossRefGoogle Scholar
  24. [24]
    R. Leake and R. Liu, Construction of suboptimal control sequences, SIAM J. Control, 5 (1967), 54–63.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    M. Levine and T. Vilis, On-line learning optimal control using successive approximation techniques, IEEE Trans. Aut. Control, 19, (1973) 279–284.CrossRefGoogle Scholar
  26. [26]
    E. Mageriou, Values and strategies for infinite time linear quadratic games, IEEE Trans. Automatic Control, 21 (1976), 547–550.MathSciNetCrossRefGoogle Scholar
  27. [27]
    E. Mageriou, Iterative techniques for Riccati game equations, J. Optimization Theory and Applications, 22 (1977), 51–61.MathSciNetCrossRefGoogle Scholar
  28. [28]
    E. Mageriou and H. Ho, Decentralized stabilization via game theoretic methods, Automatica, 13 (1977), 393–399.CrossRefGoogle Scholar
  29. [29]
    M. Mariton, Jump Linear Systems in Automatic Control, Marcell Dekker, New York, Basel, 1990.Google Scholar
  30. [30]
    G. Mil’shtein, Successive approximation for solution of one optimum problem, Auto, and Rem. Control, 25 (1964), 298–306.Google Scholar
  31. [31]
    G. Olsder, Comment on a numerical procedure for the solution of differential games, IEEE Trans. Automatic Control, 20 (1975), 704–705.MATHCrossRefGoogle Scholar
  32. [32]
    U. Ozguner and W. Perkins, A series solution to the Nash strategy for large scale interconnected systems, Automatica, 13 (1977), 313–315.CrossRefGoogle Scholar
  33. [33]
    G. Papavassilopoulos, J. Medanic, and J. Cruz, On the existence of Nash strategies and solutions to coupled Riccati equations in linear-quadratic games, J. of Optimization Theory and Applications, 28 (1979), 49–75.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    I. Peterson, Disturbance attenuation and H∞ optimization: A design method based on the algebraic Riccati equation, IEEE Trans. Automatic Control, 32 (1987), 427–429.CrossRefGoogle Scholar
  35. [35]
    I. Peterson, Some new results on algebraic Riccati equations arising in linear quadratic differential games and stabilization of uncertain linear systems, Systems & Control Letters, 10 (1988), 341–348.MathSciNetCrossRefGoogle Scholar
  36. [36]
    Peterson and C. Hollot, A Riccati approach to the stabilization of uncertain linear systems, Automatica 22 (1986), 397–411.CrossRefGoogle Scholar
  37. [37]
    B. Petrovic and Z. Gajic, The recursive solution of linear quadratic Nash games for weakly interconnected systems, J. Optimization Theory and Application, 56 (1988), 463–477.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    A. Starr and Y. Ho, Nonzero-sum differential games, J. of Optimization Theory and Applications, 3 (1969), 184–206.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    D. Tabak, Numerical solution of differential game problems, Int. J. Systems Sci., 6 (1975), 591–599, 1975.MATHCrossRefGoogle Scholar
  40. [40]
    E. Vaisbord, An approximate method for the synthesis of optimal control, Auto, and Rem. Control, 24 (1963), 1626–1632.Google Scholar
  41. [41]
    W. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. on Control, 6 (1968), 681–697.MathSciNetCrossRefGoogle Scholar
  42. [42]
    K. Zhou and P. Khargonekar, An algebraic Riccati equation approach to H∞ optimization, Systems & Control Letters, 11 (1987), 85–91.MathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • T-Y. Li
    • 1
  • Z. Gajic
    • 2
  1. 1.Department of MathematicsMichigan State UniversityE. LansingUSA
  2. 2.Department of Electrical and Computer EngineeringRutgers UniversityPiscatawayUSA

Personalised recommendations