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Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

  • Paresh Date
  • Bujar Gashi
Article
  • 301 Downloads

Abstract

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.

Keywords

Linear time-varying systems Controllablility Kalman canonical structure Stabilisation  Observer 

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References

  1. 1.
    Athreya, K.B., Lahiri, S.N.: Measure Theory and Probability Theory. Springer (2006)Google Scholar
  2. 2.
    Chai, W., Loh, N.K.: Design of minimal-order state observers for time-varying multivariable systems. I. J. Syst. Sci. 23(4), 581–592 (1992)MATHMathSciNetGoogle Scholar
  3. 3.
    Chang, A.: An algebraic characterization of controllability. IEEE Trans. Automat. Contr. 10, 112–113 (1965)CrossRefGoogle Scholar
  4. 4.
    Chen, M.-S., Yen, J.-Y.: Application of the least squares algorithm to the observer design for linear time-varying systems. IEEE Trans. Automat. Contr. 44(9), 1742–1745 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    D’Azzo, J.J., Houpis, C.H.: Linear Control System Analysis and Design: Conventional and Modern, 3rd edn. McGraw-Hill book Company (1988)Google Scholar
  6. 6.
    Gilbert, E.G.: Controllablity and observability in multivariable control systems. SIAM J. Control Ser. A, 1(2), 128–151 (1963)MATHGoogle Scholar
  7. 7.
    Kalman, R.E.: Contributions to the theory of optimal control. Boletin-Sociedad Matematica Mexicana, pp. 102–119 (1960)Google Scholar
  8. 8.
    Kalman, R.E.: Canonical structure of linear dynamical systems. Proc. Natl. Acad. Sci. USA 48, 596–600 (1962)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kalman, R.E.: Mathmatical description of linear dynamical systems. SIAM J. Control Ser. A, 1(2), 152–192 (1963)MATHMathSciNetGoogle Scholar
  10. 10.
    Lee, H.C., Choi, J.W.: Ackermann-like eigenvalue assignment formulae for linear time-varying systems. IEE Proc.-Contr. Theor. Appl. 152(4), 427–434 (2005)CrossRefGoogle Scholar
  11. 11.
    Leiva, H., Zambrano, H.: Rank condition for the controllablity of linear time-varying system. Int. J. Control 72(10), 929–931 (1999)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Lovass-Nagy, V., Miller, R.J., Mukundan, R.: On the application of matrix generalized inverses to the design of observer for time-varying and time-invariant linear systems. IEEE Trans. Automat. Contr. 25(6), 1213–1218 (1980)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Malek-Zavarei, M.: The stability of linear time-varying systems. I. J. Control 27(5), 809–815 (1978)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Nguyen, C., Lee, T.N.: Design of a state estimator for a class of time-varying mutivariable systems. IEEE Trans. Automat. Contr. 30(2), 179–182 (1985)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Nguyen, C.C.: Canonical transformation for a class of time-varying multivariable systems. I. J. Control 43(4), 1061–1074 (1986)CrossRefMATHGoogle Scholar
  16. 16.
    Nguyen, C.C.: Arbitrary eigenvalue assignments for linear time-varying multivariable control systems. I. J. Control 45(3), 1051–1057 (1987)CrossRefMATHGoogle Scholar
  17. 17.
    Nguyen, C.C.: Design of reduced-order state estimators for linear time-varying multivariable systems. I. J. Control 46(6), 2113–2126 (1987)CrossRefMATHGoogle Scholar
  18. 18.
    Phat, V.N.: Global stabilization for linear continuous time-varying systems. Appl. Math. Comput. 175, 1730–1743 (2006)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Silverman, L.M., Meadows, H.E.: Controllability and observability in time-variable linear systems. SIAM J. Control 5, 64–73 (1967)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer (1998)Google Scholar
  21. 21.
    Stubberud, A.R.: A controllablity criterion for a class of linear systems. IEEE Trans. Appl. Industry 68, 411–413 (1964)CrossRefGoogle Scholar
  22. 22.
    Tsui, C.C.: Function-observer design for a class of linear time-varying system. I. J. Control 44(1), 277–282 (1986)CrossRefMATHGoogle Scholar
  23. 23.
    Valášek, M., Olgac, N.: Efficient eigenvalue assignment for general linear MIMO systems. Automatica 31(11), 1605–1617 (1995)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Weiss, L., Kalman, R.E.: Contributions to linear system theory. Int. J. Eng. Sci. 3, 141–171 (1965)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Weiss, L.: On the sturucture theory of linear differential systems. SIAM J. Control 6(4), 659–680 (1968)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Weiss, L., Falb, P.L.: Doležal’s theorem, linear algebra with continuously parametrized elemnets, and time-varying systems. Math. Syst. Theory 3, 67–75 (1969)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Wonham, W.M.: On pole assignment in multi-input controllable linear systems. IEEE Trans. Automat. Contr. 12(6), 660–665 (1967)CrossRefGoogle Scholar
  28. 28.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall (1996)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.The Centre for the Analysis of Risk and Optimisation Modelling Applications (CARISMA), Department of Mathematical SciencesBrunel UniversityUxbridgeUK
  2. 2.Institute of Financial and Actuarial Mathematics (IFAM), Department of Mathematical SciencesThe University of LiverpoolLiverpoolUK

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