Structural Properties of Linear Stochastic Systems

  • Vasile Dragan
  • Toader Morozan
  • Adrian-Mihail Stoica
Chapter

Abstract

In this chapter we present the stochastic version of some basic concepts in control theory, namely the stabilizability, detectability, observability and controllability. All these concepts are defined both in Lyapunov operators terms and in stochastic systems terms. The definitions given in this chapter extend the corresponding definitions from the deterministic time-varying systems. Some examples will show that the stochastic observability does not always imply stochastic detectability and stochastic controllability does not necessarily imply stochastic stabilizability. As in the deterministic case the concepts of stochastic detectability and observability are used in some criteria of exponential stability in mean square

Bibliography

  1. 13.
    R.W. Brockett, Parametrically stochastic linear differential equations. Math. Program. Study 5, 8–21 (1979)MathSciNetCrossRefGoogle Scholar
  2. 15.
    H.F. Chen, On stochastic observability. Sci. Sin. 20, 305–323 (1977)MATHGoogle Scholar
  3. 16.
    H.F. Chen, On the stochastic observability and controllability, in Proceedings of 7th IFAC World Congress, Helsinki (1978), pp. 1115–1162Google Scholar
  4. 41.
    V. Dragan, T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equations with Markov jumping and multiplicative white noise. Stoch. Anal. Appl. 20(1), 33–92 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 45.
    V. Dragan, T. Morozan, The linear quadratic optimization problem and tracking problem for a class of linear stochastic systems with multiplicative white noise and Markovian jumping. IEEE-T-AC 49, 665–676 (2004)MathSciNetCrossRefGoogle Scholar
  6. 46.
    V. Dragan, T. Morozan, Stochastic observability and applications. IMA J. Math. Contr. Inf. 21, 323–344 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 56.
    M. Ehrhardt, W. Kliemann, Controllability of linear stochastic systems. Report no. 50, Institut of Dynamische Systems, Bremen, August 1981Google Scholar
  8. 63.
    M.D. Fragoso, J. Baczynski, Optimal control for continuous time LQ-problems with infinite Markov jump parameters. SIAM J. Contr. Optim. 40(1), 270–297 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 74.
    A. Halanay, Differential Equations, Stability, Oscillations, Time Lag (Academic, New York, 1966)Google Scholar
  10. 86.
    Y. Ji, H.J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Contr. 35(7), 777–788 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 89.
    R.E. Kalman, Contributions to the theory of optimal control. Bull. Soc. Math. Mex. 5, 102–119 (1960)MathSciNetGoogle Scholar
  12. 93.
    J. Klamka, L. Socha, Some remarks about stochastic controllability. IEEE Trans. Automat. Contr. 27, 880–881 (1977)MathSciNetCrossRefGoogle Scholar
  13. 94.
    J. Klamka, L. Socha, Stochastic controllability of dynamical systems (in Polish). Podstawy Sterowonia 8, 191–200 (1978)MathSciNetMATHGoogle Scholar
  14. 109.
    T. Morozan, Optimal stationary control for dynamic systems with Markov perturbations. Stoch. Anal. Appl. 1(3), 299–325 (1983)MathSciNetCrossRefMATHGoogle Scholar
  15. 110.
    T. Morozan, Stabilization of some control differential systems with Markov perturbations (in Romanian). Stud. Cerc. Mat. 37(3), 282–284 (1985)MathSciNetMATHGoogle Scholar
  16. 111.
    T. Morozan, Stochastic uniform observability and Riccati equations of stochastic control. Rev. Roum. Math. Pures Appl. 38(9), 771–781 (1993)MathSciNetMATHGoogle Scholar
  17. 112.
    T. Morozan, Stability and control for linear systems with jump Markov perturbations. Stoch. Anal. Appl. 13(1), 91–110 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 131.
    Y. Shunahara, S. Aihara, K. Kishino, On the stochastic observability and controllability of nonlinear systems. IEEE Trans. Automat. Contr. 19, 49–54 (1974)CrossRefGoogle Scholar
  19. 132.
    Y. Shunahara, S. Aihara, K. Kishino, On the stochastic observability and controllability of nonlinear systems. Int. J. Contr. 22, 65–82 (1975)CrossRefGoogle Scholar
  20. 142.
    V.M. Ungureanu, Stochastic uniform observability of linear differential equations with multiplicative noise. J. Math. Anal. Appl. 343(1), 446–463 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 143.
    V.M. Ungureanu, Stochastic uniform observability of general linear differential equations with multiplicative noise. Dyn. Syst. 23(3), 333–350 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 144.
    V.M. Ungureanu, V. Dragan, Nonlinear differential equations of Riccati type on ordered Banach spaces. Electron. J. Qual. Theor. Differ. Equat. Proc. 9th Coll. QTDE 17, 1–22 (2012)Google Scholar
  23. 155.
    J. Zabczyk, Controllability of stochastic linear systems. Syst. Contr. Lett. 1(1), 25–31 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vasile Dragan
    • 1
  • Toader Morozan
    • 1
  • Adrian-Mihail Stoica
    • 2
  1. 1.Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.University Politechnica of BucharestBucharestRomania

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