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H ∞-Control for Markovian Jumping Linear Systems with Parametric Uncertainty

Author

Listed:
  • P. Shi

    (University of South Australia)

  • E. K. Boukas

    (Ecole Polytechnique de Montréal, and GERAD)

Abstract

This paper studies the problem of H ∞-control for linear systems with Markovian jumping parameters. The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable. Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H ∞-performance are achieved. We also deal with the robust H ∞-control problem for linear systems with both Markovian jumping parameters and parameter uncertainties. The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix. Both the finite-horizon and infinite-horizon cases are analyzed. We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case. Particularly, robust H ∞-controllers are also designed when the jumping rates have parameter uncertainties.

Suggested Citation

  • P. Shi & E. K. Boukas, 1997. "H ∞-Control for Markovian Jumping Linear Systems with Parametric Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 75-99, October.
    • Handle: RePEc:spr:joptap:v:95:y:1997:i:1:d:10.1023_a:1022683311870
    DOI: 10.1023/A:1022683311870
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    Citations

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    Cited by:

    1. Yan, Huaicheng & Huang, Xinhan & Zhang, Hao & Wang, Min, 2009. "Delay-dependent robust stability criteria of uncertain stochastic systems with time-varying delay," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1668-1679.
    2. E. K. Boukas, 2004. "Nonfragile Controller Design for Linear Markovian Jumping Parameters Systems," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 241-255, August.
    3. Quanxin Zhu & Fubao Xi & Xiaodi Li, 2012. "Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 154-174, July.
    4. WonIl Lee & JaeWook Shin & BumYong Park, 2024. "The Strictly Dissipative Condition of Continuous-Time Markovian Jump Systems with Uncertain Transition Rates," Mathematics, MDPI, vol. 12(5), pages 1-14, February.
    5. E. K. Boukas, 2008. "On State Feedback Stabilization of Singular Systems with Random Abrupt Changes," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 335-345, May.
    6. T. Senthilkumar & P. Balasubramaniam, 2011. "Delay-Dependent Robust Stabilization and H ∞ Control for Nonlinear Stochastic Systems with Markovian Jump Parameters and Interval Time-Varying Delays," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 100-120, October.
    7. E. K. Boukas, 2004. "Nonfragile Robust Controller for Linear Markovian Jumping Parameter Systems with Multiplicative Brownian Disturbance," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 455-469, September.
    8. O. L. V. Costa & E. K. Boukas, 1998. "Necessary and Sufficient Condition for Robust Stability and Stabilizability of Continuous-Time Linear Systems with Markovian Jumps," Journal of Optimization Theory and Applications, Springer, vol. 99(2), pages 359-379, November.

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