Double-stepped adaptive control for hybrid systems with unknown Markov jumps and stochastic noises

Shuping Tan; Ji-Feng Zhang

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 4, page 969-993
  • ISSN: 1292-8119

Abstract

top
This paper is concerned with the sampled-data based adaptive linear quadratic (LQ) control of hybrid systems with both unmeasurable Markov jump processes and stochastic noises. By the least matching error estimation algorithm, parameter estimates are presented. By a double-step (DS) sampling approach and the certainty equivalence principle, a sampled-data based adaptive LQ control is designed. The DS-approach is characterized by a comparatively large estimation step for parameter estimation and a sufficient small control step for control updating. Under mild conditions, the closed-loop system is shown to be stable. It is found that the key factor determining the performance index is the estimation step rather than the control step. When the estimation step becomes too small, the system performance will become worse. When the estimation step is fixed, the system performance can indeed be improved by reducing the control step, but cannot reach the optimal value. The index difference between the sampled-data based adaptive LQ control and the conventional LQ optimal control is asymptotically bounded by a constant depending on the estimation step and the priori information of the parameter set.

How to cite

top

Tan, Shuping, and Zhang, Ji-Feng. "Double-stepped adaptive control for hybrid systems with unknown Markov jumps and stochastic noises." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2008): 969-993. <http://eudml.org/doc/90946>.

@article{Tan2008,
abstract = { This paper is concerned with the sampled-data based adaptive linear quadratic (LQ) control of hybrid systems with both unmeasurable Markov jump processes and stochastic noises. By the least matching error estimation algorithm, parameter estimates are presented. By a double-step (DS) sampling approach and the certainty equivalence principle, a sampled-data based adaptive LQ control is designed. The DS-approach is characterized by a comparatively large estimation step for parameter estimation and a sufficient small control step for control updating. Under mild conditions, the closed-loop system is shown to be stable. It is found that the key factor determining the performance index is the estimation step rather than the control step. When the estimation step becomes too small, the system performance will become worse. When the estimation step is fixed, the system performance can indeed be improved by reducing the control step, but cannot reach the optimal value. The index difference between the sampled-data based adaptive LQ control and the conventional LQ optimal control is asymptotically bounded by a constant depending on the estimation step and the priori information of the parameter set. },
author = {Tan, Shuping, Zhang, Ji-Feng},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Sampling control system; Markov jump parameter; adaptive control; double-step approach; stochastic noise; least matching error estimation; sampling control system; double-step approach; least matching error estimation},
language = {eng},
month = {8},
number = {4},
pages = {969-993},
publisher = {EDP Sciences},
title = {Double-stepped adaptive control for hybrid systems with unknown Markov jumps and stochastic noises},
url = {http://eudml.org/doc/90946},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Tan, Shuping
AU - Zhang, Ji-Feng
TI - Double-stepped adaptive control for hybrid systems with unknown Markov jumps and stochastic noises
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/8//
PB - EDP Sciences
VL - 15
IS - 4
SP - 969
EP - 993
AB - This paper is concerned with the sampled-data based adaptive linear quadratic (LQ) control of hybrid systems with both unmeasurable Markov jump processes and stochastic noises. By the least matching error estimation algorithm, parameter estimates are presented. By a double-step (DS) sampling approach and the certainty equivalence principle, a sampled-data based adaptive LQ control is designed. The DS-approach is characterized by a comparatively large estimation step for parameter estimation and a sufficient small control step for control updating. Under mild conditions, the closed-loop system is shown to be stable. It is found that the key factor determining the performance index is the estimation step rather than the control step. When the estimation step becomes too small, the system performance will become worse. When the estimation step is fixed, the system performance can indeed be improved by reducing the control step, but cannot reach the optimal value. The index difference between the sampled-data based adaptive LQ control and the conventional LQ optimal control is asymptotically bounded by a constant depending on the estimation step and the priori information of the parameter set.
LA - eng
KW - Sampling control system; Markov jump parameter; adaptive control; double-step approach; stochastic noise; least matching error estimation; sampling control system; double-step approach; least matching error estimation
UR - http://eudml.org/doc/90946
ER -

References

top
  1. K.J. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design. Third edition, Tsinghua University Press (2002).  
  2. R. Bhatia, Matrix Analysis. Springer Verlag (1996).  Zbl0863.15001
  3. P.E. Caines and J.F. Zhang, On the adaptive control of jump parameter systems via nonlinear filtering. SIAM J. Contr. Opt.33 (1995) 1758–1777.  Zbl0843.93076
  4. Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales. Springer Verlag (1978).  
  5. E.A. Coddington and R. Carlson, Linear Ordinary Differential Equations. SIAM (1997).  Zbl0871.34001
  6. F. Dufour and P. Betrand, Stabilizing control law for hybrid models. IEEE Trans. Automat. Contr.39 (1994) 2354–2357.  Zbl0825.93698
  7. R.J. Elliott and V. Krishnamurthy, Exact finite dimensional filters for maximum likelihood parameter estimation of continuous time linear-Gaussian systems. SIAM. J. Contr. Opt.35 (1997) 1908–1923.  Zbl0935.93061
  8. R.J. Elliott and V. Krishnamurthy, New finite dimensional filters for parameter estimation of discrete time linear Gaussian models. IEEE. Trans. Automat. Contr.44 (1999) 938–951.  Zbl0959.93055
  9. L.S. Hu, Y.-Y. Cao and H.-H. Shao, Constrained robust sampled-data control for nonlinear uncertain systems. Int. J. Robust Nonlinear Contr.12 (2002) 447–464.  Zbl1026.93035
  10. M.Y. Huang and L. Guo, Stabilization of stochastic systems with hidden Markov jumps. Science in China (Series F)44 (2001) 104–118.  Zbl1125.93488
  11. A. Ilchmann and S. Townley, Adaptive sampling control of high-gain stabilizable systems. IEEE Trans. Automat. Contr.44 (1999) 1961–1966.  Zbl0956.93056
  12. Y.D. Ji and H.J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Contr.35 (1990) 777–788.  Zbl0714.93060
  13. Y.D. Ji and H.J. Chizeck, Jump linear quadratic Guassian control in continuous time. IEEE Trans. Automat. Contr.37 (1992) 1884–1892.  Zbl0773.93052
  14. A. Kanchanaharuthai, Optimal sampled-data controller design with time-multiplied performance index for load-frequency control, in Proceedings of the 2004 IEEE international conference on control applications (2004) 655–660.  
  15. G. Kreisselmeier and R. Lozano, Adaptive control of continuous-time overmodeled plants. IEEE Trans. Automat. Contr.41 (1996) 1779–1794.  Zbl0876.93051
  16. H.J. Kushner, Stochastic Stability and Control. Academic Press (1967).  Zbl0244.93065
  17. R.S. Lipster and A.N. Shiryaev, Statistics of Random Processes I. General Theory. Second edition, Springer Verlag (2001).  
  18. M. Mariton, Jump Linear Systems in Automatic Control. Marcel Dekker Inc. (1990).  
  19. O. Ocah and M.E. Sezer, Robust adaptive sampled-data control of a class of systems under structured nonlinear perturbations. IEEE Trans. Automat. Contr.42 (1997) 553–558.  Zbl0944.93513
  20. Y. Oishi, A bound of conservativeness in sampled-data robust stabilization and its dependence on sampling periods. Systems Control Lett. 32 (1997) 11–19.  Zbl0901.93039
  21. R. Ortega and G. Kreisselmeier, Discrete-time model reference adaptive control for continuous-time systems using generalized sampled-Data hold functions. IEEE Trans. Automat. Contr.35 (1990) 334–338.  Zbl0707.93034
  22. P. Protter, Stochastic Integration and Differential Equations: A New Approach. Springer-Verlag (1990).  Zbl0694.60047
  23. D.D. Sworder, Hybrid adaptive control. Appl. Math. Comput.45 (1991) 173–192.  Zbl0738.93047
  24. S.P. Tan, J.-F. Zhang and L.L. Yao, Optimality analysis of adaptive sampled control of hybrid systems with quadratic index. IEEE Trans. Automat. Contr.50 (2005) 1044–1051.  
  25. W.M. Wonham, Random Differential Equations in Control Theory, in Probabilistic Methods in Applied Mathematics 2, A.T. Bharucha-Reid Ed., Academic Press (1970).  Zbl0235.93025
  26. F. Xue and L. Guo, Necessary and sufficient conditions for adaptive stability of jump linear systems. Communications in Information and Systems1 (2001) 205–224.  Zbl1059.93131
  27. L.L. Yao and J.F. Zhang, Sampled-data-based LQ control of stochastic linear continuous-time systems. Science in China (Series F)45 (2002) 383–396.  Zbl1185.93078
  28. R. Yu, O. Ocali and M.E. Sezer, Adaptive robust sampled-data control of a class of systems under structured perturbations. IEEE Trans. Automat. Contr.38 (1993) 1707–1713.  Zbl0790.93097
  29. C. Zhang, R.H. Middleton and R.J. Evans, An algorithm for multirate sampling adaptive control. IEEE Trans. Automat. Contr.34 (1989) 792–795.  Zbl0687.93053

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.