Advertisement

Stabilizability of Linear Discrete Time-Varying Systems

Conference paper
  • 373 Downloads
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)

Abstract

For linear discrete time-varying systems we discuss the relation between stabilizability, controllability and finiteness of quadratic cost functional. The role of the existence of global and bounded solutions of the discrete time-varying Riccati equation for stabilizability is also explained.

Keywords

Discrete time-varying system Controllability Stabilizability Riccati equation 

References

  1. 1.
    Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications. CRC Press, Boca Roton (2000)CrossRefGoogle Scholar
  2. 2.
    Alexandridis, A., Galanos, G.: Optimal pole-placement for linear multi-input controllable systems. IEEE Trans. Circ. Syst. 34(12), 1602–1604 (1987)CrossRefGoogle Scholar
  3. 3.
    Babiarz, A., Banshchikova, I., Czornik, A., Makarov, E., Niezabitowski, M., Popova, S.: Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems. SIAM J. Control Optim. 57(2), 1355–1377 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Babiarz, A., Banshchikova, I., Czornik, A., Makarov, E.K., Niezabitowski, M., Popova, S.: Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems. IEEE Trans. Autom. Control 63(11), 3825–3837 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Babiarz, A., Czornik, A., Makarov, E., Niezabitowski, M., Popova, S.: Pole placement theorem for discrete time-varying linear systems. SIAM J. Control Optim. 55(2), 671–692 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bhattacharyya, S., de Souza, E.: Pole assignment via Sylvester’s equation. Syst. Control Lett. 1(4), 261–263 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bittanti, S., Bolzern, P.: On the structure theory of discrete-time linear systems. Int. J. Syst. Sci. 17(1), 33–47 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dickinson, B.: On the fundamental theorem of linear state variable feedback. IEEE Trans. Autom. Control 19(5), 577–579 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Franklin, G.F., Powell, J.D., Emami-Naeini, A.: Feedback Control of Dynamic Systems. Prentice Hall Press, Upper Saddle River (2014)zbMATHGoogle Scholar
  10. 10.
    Furuta, K., Kim, S.: Pole assignment in a specified disk. IEEE Trans. Autom. Control 32(5), 423–427 (1987)CrossRefGoogle Scholar
  11. 11.
    Gaishun, I.: Discrete-time systems. In: Natsionalnaya Akademiya Nauk Belarusi. Institut Matematiki Minsk (2001)Google Scholar
  12. 12.
    Halanay, A., Ionescu, V.: Time-Varying Discrete Linear Systems: Input-Output Operators. Riccati Equations. Disturbance Attenuation, vol. 68. Birkhäuser, Basel (2012)Google Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  14. 14.
    Ichikawa, A., Katayama, H., et al.: Linear Time Varying Systems and Sampled-Data Systems, vol. 265. Springer, Heidelberg (2001)Google Scholar
  15. 15.
    Kalman, R.E.: On the general theory of control systems. In: Proceedings of the First International Congress on Automatic Control (1960)Google Scholar
  16. 16.
    Kalman, R.E.: Mathematical description of linear dynamical systems. J. Soc. Ind. Appl. Math. Ser. A Control 1(2), 152–192 (1963)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems, vol. 1. Wiley, Hoboken (1972)zbMATHGoogle Scholar
  18. 18.
    Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford Science (1995)Google Scholar
  19. 19.
    Ludyk, G.: Stability of Time-Variant Discrete-Time Systems, vol. 5. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  20. 20.
    Reza Moheimani, S.O., Petersen, I.R.: Quadratic guaranteed cost control with robust pole placement in a disk. IEE Proc. Control Theory Appl. 143(1), 37–43 (1996)CrossRefGoogle Scholar
  21. 21.
    Rugh, W.J.: Linear System Theory, vol. 2. Prentice Hall, Upper Saddle River (1996)zbMATHGoogle Scholar
  22. 22.
    Steele, J.: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. MAA problem books series. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  23. 23.
    Sugimoto, K.: Partial pole placement by LQ regulators: an inverse problem approach. IEEE Trans. Autom. Control 43(5), 706–708 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sugimoto, K., Yamamoto, Y.: On successive pole assignment by linear-quadratic optimal feedbacks. Linear Algebra Appl. 122–124, 697–723 (1989)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Weiss, L.: Controllability, realization and stability of discrete-time systems. SIAM J. Control 10(2), 230–251 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Automatic Control and RoboticsSilesian University of TechnologyGliwicePoland

Personalised recommendations