On Controllability of Linear Systems with Jumps in Parameters

  • Adam Czornik
  • Aleksander Nawrat
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 118)


In this paper we discuss different types of controllability of jump linear systems. We start with the weakest of the concepts of controllability for jump linear system namely the problem of controllability of the expectation of the final state. The main result of this section gives a necessary and sufficient condition for this type of controllability. Next we consider the possibility of reaching any deterministic target value from given deterministic initial condition in given time with prescribed probability. We have also investigated several variations of this problem such as: the case when the target or initial condition are zero and the case when we want to achieve only certain neighborhood of the target. Part of our results are devoted to the special cases of stochastic controllability. We also consider the concepts of controllability when the time of achieving the target value can be a random variable. Finally we discuss the relationships between the introduced types of controllability.


Initial Distribution Directly Controllable Stochastic Controllability Deterministic System Markovian Jump 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akella, R., Kumar, P.R.: Optimal control of production rate in failure prone manufacturing systems. IEEE Transactions on Automatic Control 31, 116–126 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Athans, M.: Command and control (C2) theory: a challange to control science. IEEE Transactions on Automatic Control 32(4), 286–293 (1987)CrossRefGoogle Scholar
  3. 3.
    Bar Shalom, Y., Fortman, T.E.: Tracking and Data Association. Academic Press, New York (1988)zbMATHGoogle Scholar
  4. 4.
    Bar Shalom, Y.: Tracking methods in a multitarget environment. IEEE Transactions on Automatic Control 23, 618–628 (1978)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bar Shalom, Y., Birmiwal, K.: Variable dimension for maneuvering target tracking. IEEE Transactions on Aero. Electr. Systems, AES 18, 621–628 (1982)CrossRefGoogle Scholar
  6. 6.
    Bielecki, T., Kumar, P.R.: Necessary and sufficient conditions for a zero inventory policy to be optimal in an unreliable manufacturing systems. In: Proccidings of 25 th IEEE Conference on Decision And Control, Athens, pp. 248–250 (1986)Google Scholar
  7. 7.
    Birdwel, J.G.: On reliable control systems design. Ph. D. Disertation, Electrical Systems Lab., Mass. Inst. Technology, report no. ESL-TH-821 (1978)Google Scholar
  8. 8.
    Birdwel, J.G., Castanon, D.A., Athans, M.: On reliable control system designs. IEEE Transaction on Systems, Man and Cybernetics 16(5), 703–711 (1986)CrossRefGoogle Scholar
  9. 9.
    Boukas, E.K., Liu, Z.K.: Production and maintenance control for manufacturing systems. IEEE Transactions on Automatic Control 46(9), 1455–1460 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Czornik, A., Swierniak, A.: On controllability with respect to the expectation of discrete time jump linear systems. Journal of the Franklin Institute 338, 443–453 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Czornik, A., Swierniak, A.: Controllability of discrete time jump linear systems, Dynamics of Continuous. Discrete and Impulsive Systems Series B: Applications and Algorithms 12(2), 165–189Google Scholar
  12. 12.
    Czornik, A., Swierniak, A.: On controllability of discrete time jump linear systems. In: Procedings of 4th World CSCC, Athens, Greece, pp. 3041–3044 (2000)Google Scholar
  13. 13.
    Czornik, A., Swierniak, A.: Controllability of continuous time jump linear systems. In: Proc. of 10th Mediterranean Conference on Control and Automation, CD-ROM, Portugal, Lisbon (2002)Google Scholar
  14. 14.
    Ehrhardt, M., Kliemann, W.: Controllability of linear stochastic system. Systems and Control Letters 2, 145–153 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Florentin, J.J.: Optimal control of continuous-time Markov stochastic systems. Journal of Electronics Control 10, 473–481 (1961)MathSciNetGoogle Scholar
  16. 16.
    Ji, Y., Chizeck, H.: Controllability, observability and discrete-time markovian jump linear quadratic control. International Journal of Control 48(2), 481–498 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ji, Y., Chizeck, H.J.: Controllability, stability, and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control 35, 777–788 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kazangey, T., Sworder, D.D.: Effective federal policies for regulating residential housing. In: Proc. Summer Comp. Simulation Conf., Los Angeles, pp. 1120–1128 (1971)Google Scholar
  19. 19.
    Kalman, R.E.: On the general theory of control systems. In: Proc. First IFAC Congress Automatic Control, Moscow, Butterworths, London, vol. 1, pp. 481–492 (1960)Google Scholar
  20. 20.
    Klamka, J., Socha, L.: Some remarks about stochastic controllability. IEEE Transactions on Automatic Control 22, 880–881 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Klamka, J., Dong, L.S.: Stochastic controllability of discrete time systems with jump Markov disturbances. Archiwum Automatyki i Telemechaniki 35, 67–74 (1990)zbMATHGoogle Scholar
  22. 22.
    Krasovski, N.N., Lidski, E.A.: Analytical design of controllers in systems with random attribute. Parts I-III, Automation and Remote Control (Part I) 22, 1021–1027, (Part II) pp.1141–1146, (Part III) pp.1289–1297 (1961) Google Scholar
  23. 23.
    Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker, New York and Besel (1990)Google Scholar
  24. 24.
    Montgomery, R.C.: Reliability considerations in placement of control systems components. In: Proc. AIAA Guidance and Control Conference, Gatlinburg (1983)Google Scholar
  25. 25.
    Rosenbrock, H.H., McMorran, P.D.: Good, bad, or optimal? IEEE Transactions on Automatic Control 16(6), 552–554 (1971)CrossRefGoogle Scholar
  26. 26.
    Siljak, D.D.: Reliable control using multiple control systems. International Journal on Control 31(2), 303–329 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Siljak, D.D.: Dynamic reliability using multiple control systems. In: Proc. 2nd Lawrece Syphosium on Systems Decision Sciences, Berkley (1978)Google Scholar
  28. 28.
    Swierniak, A., Simek, K., Czornik, A.: Fault tolerant control design for linear systems. In: 5th National Conference on Diagnostics of Industrial Processes, pp. 45–50. Tech. Univ. Press of Zielona Gora, Poland (2001)Google Scholar
  29. 29.
    Swierniak, A., Simek, K., Boukas, E.K.: Intelligent robust control of fault tolerant linear systems. In: Proceedings IFAC Symposium on Artificial Intelligence in Real-Time Control, pp. 245–248. Pergamon Press, Malaysia (1997)Google Scholar
  30. 30.
    Sworder, D.D., Rogers, R.O.: An LQ- solution to a control problem associated with solar thermal central receiver. IEEE Transactions on Automatic Control 28, 971–978 (1983)CrossRefGoogle Scholar
  31. 31.
    Willsky, A.S., Levy, B.C.: Stochastic stability research for complex power systems. Lab. Inf. Decision Systems, Mass. Inst. Technology, report no. ET-76-C-01-2295 (1979)Google Scholar
  32. 32.
    Zabczyk, J.: Controllability of stochastic linear systems. Systems and Control Letters 1, 25–31 (1981)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Silesian University of TechnologyGliwicePoland
  2. 2.WASKO S. A.GliwicePoland

Personalised recommendations