Skip to main content
SpringerLink
Log in

Decentralized Control of Markovian Jump Systems

  • Chapter
Decentralized Systems with Design Constraints

  • 705 Accesses

Abstract

This chapter deals with systems having Markovian jump parameters. There are basically two types of models. The first type describes interconnected systems with Markovian jump parameters for which the problems of stochastic stability and stabilization are examined and a set of feedback controls is conveniently developed. In the second type, we deal with systems with Markov chains exhibiting slow-fast separation. An appropriate averaging and aggregation technique is developed for this purpose.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akella, R. and P. R. Kumar, “Optimal Control of Production Rate in a Failure Prone Manufacturing System”, IEEE Trans. Autom. Control, vol. 31, 1986, pp. 116–126.

    Article  MathSciNet  MATH  Google Scholar 

  2. Basar, T. and P. Bernhard, \({\mathcal{H}}_{\infty}\) -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach (2nd ed.), Birkhäuser, Boston, 1995.

    Google Scholar 

  3. Basar, T. and A. Haurie, “Feedback Equilibria in Differential Games with Structural and Modal Uncertainties”, Advances in Large Scale Systems, Cruz, J. B., Jr. (Ed.), JAI Press, London, 1984, pp. 163–201.

    Google Scholar 

  4. Basar, T., “Minimax Control of Switching Systems Under Sampling”, Proc. the 33rd IEEE Conference on Decision and Control, Orlando, 1994, pp. 716–721.

    Google Scholar 

  5. Boyd S., L. ElGhaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.

    Book  MATH  Google Scholar 

  6. Davis, M., Markov Models and Optimization, Chapman and Hall, London, 1992.

    Google Scholar 

  7. Delebecque, F. and J. P. Quadrat, “Optimal Control of Markov Chains Admitting Strong and Weak Interactions”, Automatica, vol. 17, no. 2, 1981, pp. 281–296.

    Article  MathSciNet  MATH  Google Scholar 

  8. de Souza, C. E. and M. Fragoso, “\({\mathcal{H}}_{\infty}\) Control of Linear Systems with Markovian Jumping Parameters”, Control Theory Adv. Technol., vol. 9, no. 2, 1993, pp. 457–466.

    Google Scholar 

  9. Doyle, J., K. Glover, P. Khargonekar and B. Francis, “State-Space Solutions to Standard \({\mathcal{H}}_{2}\) and \({\mathcal{H}}_{\infty}\) Control Problems”, IEEE Trans. Autom. Control, vol. 34, no. 8, 1989, pp. 831–847.

    Article  MathSciNet  MATH  Google Scholar 

  10. Elliott, R. J. and D. D. Sworder, “Control of Hybrid Conditionally Linear Gaussian Processes”, J. Optim. Theory Appl., vol. 74, 1992, pp. 75–85.

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng, X., K. A. Loparo, Y. Ji and H. J. Chizeck, “Stochastic Stability Properties of Jump Linear Systems”, IEEE Trans. Autom. Control, vol. 37, 1992, pp. 38–53.

    Article  MathSciNet  MATH  Google Scholar 

  12. Fleming, W., S. Sethi and M. Soner, “An Optimal Stochastic Production Planning Problem with Randomly Fluctuating Demand”, SIAM J. Control Optim., vol. 25, 1987, pp. 1494–1502.

    Article  MathSciNet  MATH  Google Scholar 

  13. Geromel, J. C., J. Bernussou and P. L. D. Peres, “Decentralized Control Through Parameter Space Optimization”, Automatica, vol. 30, 1994, pp. 1565–1578.

    Article  MathSciNet  MATH  Google Scholar 

  14. Gershwin, S. B., “Hierarchical Flow Control: A Framework for Scheduling and Planning Discrete Events in Manufacturing Systems”, Proc. the IEEE 28th Conference on Decision and Control, Tampa, FL, 1999, pp. 195–209.

    Google Scholar 

  15. Ismail, A., M. S. Mahmoud and P. Shi, “Robust \({\mathcal{H}}_{\infty}\) Analysis and Synthesis for Jumping Time-Delay Systems Using Transformation Methods”, Nonlinear Dyn. Syst. Theory, vol. 4, no. 3, 2004, pp. 333–356.

    MathSciNet  MATH  Google Scholar 

  16. Ismail, A. and M. S. Mahmoud, “A Descriptor Approach to Simultaneous \({\mathcal{H}}_{2}/{\mathcal{H}}_{\infty}\) Control of Jumping Time-Delay Systems”, IMA J. Math. Control Inf., vol. 22, no. 2, 2004, pp. 95–114.

    Article  MathSciNet  Google Scholar 

  17. Ismail, A., M. S. Mahmoud and P. Shi, “Output Feedback Stabilization and Disturbance Attenuation of Time-Delay Systems with Markovian Jump Parameters”, Control Intell. Syst., vol. 32, no. 3, 2004, pp. 193–206.

    MathSciNet  MATH  Google Scholar 

  18. Ji, Y. and H. J. Chizeck, “Controllability, Stabilizability, and Continuous-Time Markovian Jump linear Quadratic Control”, IEEE Trans. Autom. Control, vol. 35, 1990, pp. 777–788.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kokotovic, P. V. and R. A. Yackel, “Singular Perturbation of Linear Regulators: Basic Theorems”, IEEE Trans. Autom. Control, vol. 17, 1972, pp. 29–37.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kozin, F., “A Survey of Stability of Stochastic Systems”, Automatica, vol. 5, 1969, pp. 95–112.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kushner, H. J., Stochastic Stability and Control, Academic Press, New York, 1967.

    MATH  Google Scholar 

  22. Mahmoud, M. S. and M. G. Singh, Large-Scale Systems Modelling, Pergamon Press, London, 1981.

    MATH  Google Scholar 

  23. Mahmoud, M. S., “Stabilizing Control for a Class of Uncertain Interconnected Systems”, IEEE Trans. Autom. Control, vol. AC-39, 1994, pp. 2484–2488.

    Article  Google Scholar 

  24. Mahmoud, M. S., “Guaranteed Stabilization of Interconnected Discrete-Time Uncertain Systems”, Int. J. Syst. Sci., vol. 26, 1995, pp. 337–358.

    Article  MATH  Google Scholar 

  25. Mahmoud, M. S., “Adaptive Stabilization of a Class of Interconnected Systems”, Comput. Electr. Eng., vol. 23, 1997, pp. 225–238.

    Article  Google Scholar 

  26. Mahmoud, M. S. and S. Bingulac, “Robust Design of Stabilizing Controllers for Interconnected Time-Delay Systems”, Automatica, vol. 34, 1998, pp. 795–800.

    Article  MathSciNet  MATH  Google Scholar 

  27. Mahmoud, M. S. and M. Zribi, “Robust and \({\mathcal{H}}_{\infty}\) Stabilization of Interconnected Systems with Delays”, IEE Proc., Control Theory Appl., Vol. 145, pp. 559–567, 1998.

    Article  Google Scholar 

  28. Mahmoud, M. S., Robust Control and Filtering for Time-Delay Systems, Marcel Dekker Inc., New York, 2000.

    MATH  Google Scholar 

  29. Mahmoud, M. S. and P. Shi, “Robust Control of Markovian Jumping Linear Discrete-Time Systems with Unknown Nonlinearities”, IEEE Trans. Circuits Systems I, vol. 49, 2002, pp. 538–542.

    Article  MathSciNet  Google Scholar 

  30. Mahmoud, M. S. and P. Shi, “Robust Control of Markovian Jumping Linear Discrete-Time with Unknown Nonlinearities”, IEEE Trans. Circuits Systems I, vol. 49, no. 4, 2002, pp. 538–542.

    Article  MathSciNet  Google Scholar 

  31. Mahmoud, M. S. and P. Shi, Methodologies for Control of Jumping Time-Delay Systems, Kluwer Academic, Amsterdam, 2003.

    Google Scholar 

  32. Mahmoud, M. S. and P. Shi, “Robust Stability, Stabilization and \({\mathcal{H}}_{\infty}\) Control of Time-Delay Systems with Markovian Jump Parameters”, Int. J. Robust Nonlinear Control, vol. 13, 2003, pp. 755–784.

    Article  MathSciNet  MATH  Google Scholar 

  33. Mahmoud, M. S. and P. Shi, “Robust Kalman Filtering for Continuous Time-Lag Systems with Markovian Jump Parameters”, IEEE Trans. Circuits Systems I, vol. 50, 2003, pp. 98–105.

    Article  MathSciNet  Google Scholar 

  34. Mahmoud, M. S., P. Shi and Y. Shi, “Output Feedback Stabilization and Disturbance Attenuation of Time-Delay Jumping Systems”, IMA J. Math. Control Inf., vol. 20, no. 2, 2003, pp. 179–199.

    Article  MathSciNet  MATH  Google Scholar 

  35. Mahmoud, M. S. and P. Shi, “Control of Markovian Jump Uncertain Discrete Time-Delay Systems by Guaranteed Cost Approach”, Int. J. Hybrid Intell. Syst., vol. 3, no. 2–3, 2003, pp. 217–236.

    Google Scholar 

  36. Mahmoud, M. S., P. Shi and A. Ismail, “Robust \({\mathcal{H}}_{\infty}\) Filtering for a Class of Linear Jumping Discrete-Time Delay Systems”, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, vol. 10, no. 5, 2003, pp. 647–662.

    MathSciNet  MATH  Google Scholar 

  37. Mahmoud, M. S. and A. Ismail, “Interconnected Jumping Time-Delay Systems: Robust and \({\mathcal{H}}_{\infty}\) Control Schemes”, IMA J. Math. Control Inf., vol. 20, no. 4, 2003, pp. 411–440.

    Article  MathSciNet  MATH  Google Scholar 

  38. Mahmoud, M. S., “Uncertain Jumping Systems with Strong and Weak Functional Delays”, Automatica, vol. 40, no. 3, 2004, pp. 501–510.

    Article  MathSciNet  MATH  Google Scholar 

  39. Mahmoud, M. S., P. Shi and A. Ismail, “Robust Kalman Filtering for Discrete-Time Markovian Jump Systems with Parameter Uncertainty”, J. Comput. Appl. Math., vol. 169, no. 1, 2004, pp. 53–69.

    Article  MathSciNet  MATH  Google Scholar 

  40. Mahmoud, M. S., A. Ismail, P. Shi and R. K. Agarwal, “State Transformation for \({\mathcal{H}}_{\infty}\) Control of Uncertain Jumping Delayed Systems”, IMA J. Math. Control Inf., vol. 21, no. 4, 2004, pp. 419–431.

    Article  MathSciNet  MATH  Google Scholar 

  41. Mahmoud, M. S., P. Shi, R. K. Agarwal, P. Shi and Y. Shi, “\({\mathcal{H}}_{\infty}\) and Robust Control of Interconnected Systems with Markovian Jump Parameters”, Discrete Contin. Dyn. Syst., Ser. B, vol. 5, no. 2, 2005, pp. 365–384.

    Article  MathSciNet  MATH  Google Scholar 

  42. Mahmoud, M. S. and A. Ismail, “Robust Performance Analysis and Synthesis for Multi-State-Delay Jumping Systems”, IMA J. Math. Control Inf., vol. 22, no. 2, 2005, pp. 200–225.

    Article  MathSciNet  MATH  Google Scholar 

  43. Mahmoud, M. S., J. Yi and A. Ismail, “Worst Case Control of Uncertain Jumping Systems with Multi-State and Input Delay Information”, Inf. Sci., vol. 176, no. 2, 2006, pp. 186–200.

    Article  MathSciNet  MATH  Google Scholar 

  44. Mahmoud, M. S., K. Nguang and A. Ismail, “Robust Filtering for Jumping Systems with Mode-Dependent Delays”, J. Signal Process., vol. 86, no. 1, 2006, pp. 140–152.

    Article  MATH  Google Scholar 

  45. Mahmoud, M. S., P. Shi and Y. Shi, “Robust Observers and Stabilization for Uncertain Neutral Jumping Systems”, Inf. Sci., vol. 176, 2006, pp. 2355–2385.

    Article  MathSciNet  MATH  Google Scholar 

  46. Mahmoud, M. S., P. Shi and A. Ismail, “Control of Interconnected Systems Jumping Systems: a \({\mathcal{H}}_{\infty}\) Approach”, Asian J. Control, vol. 6, no. 1, 2004, pp. 1–8.

    Article  MathSciNet  Google Scholar 

  47. Mahmoud, M. S. and A. Ismail, “Interconnected Jumping Time-Delay Systems: Robust and \({\mathcal{H}}_{\infty}\) Control Schemes”, IMA J. Math. Control Inf., vol. 20, no. 4, 2003, pp. 411–440.

    Article  MathSciNet  MATH  Google Scholar 

  48. Mahmoud, M. S., P. Shi and Y. Shi, “\({\mathcal{H}}_{\infty}\) and Robust Control of Interconnected Systems with Markovian Jump Parameters”, Discrete Contin. Dyn. Syst., Ser. B, vol. 5, no. 2, 2005, pp. 365–384.

    Article  MathSciNet  MATH  Google Scholar 

  49. Mahmoud, M. S., “Decentralized Stabilization of Interconnected Systems with Time-Varying Delays”, IEEE Trans. Autom. Control, vol. 54, no. 11, 2009, pp. 2663–2668.

    Article  Google Scholar 

  50. Mahmoud, M. S., “Decentralized Reliable Control of Interconnected Systems with Time-Varying Delays”, J. Optim. Theory Appl., vol. 143, no. 11, 2009, pp. 497–518.

    Article  MathSciNet  MATH  Google Scholar 

  51. Mahmoud, M. S. and N. B. Almutairi, “Decentralized Stabilization of Interconnected Systems with Time-Varying Delays”, Eur. J. Control, vol. 15, no. 6, 2009, pp. 624–633.

    Article  MathSciNet  MATH  Google Scholar 

  52. Mao, C. and J. H. Yang, “Decentralized Output Tracking for Linear Uncertain Interconnected Systems”, Automatica, vol. 31, 1995, pp. 151–154.

    Article  MathSciNet  MATH  Google Scholar 

  53. Pan, Z. and T. Basar, “Model Simplification and Optimal Control of Stochastic Singularly Perturbed Systems under Exponentiated Quadratic Cost”, SIAM J. Control Optim., vol. 34, 1996, pp. 1734–1766.

    Article  MathSciNet  MATH  Google Scholar 

  54. Phillips, R. G. and P. V. Kokotovic, “A Singular Perturbation Approach to Modeling and Control of Markov Chains”, IEEE Trans. Autom. Control, vol. 26, 1981, pp. 1087–1094.

    Article  MATH  Google Scholar 

  55. Sethi, S. P. and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems. Birkhäuser, Boston, 1994.

    Book  MATH  Google Scholar 

  56. Sethi, S. P. and Q. Zhang, “Asymptotic Optimal Controls in Stochastic Manufacturing Systems with Machine Failures Dependent on Production Rates”, Stoch. Stoch. Rep., vol. 48, 1994, pp. 97–121.

    MathSciNet  MATH  Google Scholar 

  57. Soner, H. M., “Singular Perturbations in Manufacturing”, SIAM J. Control Optim., vol. 31, 1993, pp. 132–146.

    Article  MathSciNet  MATH  Google Scholar 

  58. Wonham, W. M., “Random Differential Equations in Control Theory”, Probabilistic Methods in Applied Mathematics, Bharucha-Reid, A. T. (Ed.), pp. 131–212, Academic Press, New York, 1969.

    Google Scholar 

  59. Xiong, J., V. A. Ugrinovskii and I. R. Petersen, “Local Mode Dependent Decentralized Stabilization of Uncertain Markovian Jump Large-Scale Systems”, IEEE Trans. Autom. Control, vol. 54, no. 11, 2009, pp. 2632–2637.

    Article  MathSciNet  Google Scholar 

  60. Zames, G., “Feedback and Optimal Sensitivity: Model Reference Transformation, Multiplicative Seminorms and Approximate Inverses”, IEEE Trans. Autom. Control, vol. 26, 1981, pp. 301–320.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Systems Engineering, King Fahd Univ. of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

    Magdi S. Mahmoud

Authors
  1. Magdi S. Mahmoud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magdi S. Mahmoud .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag London Limited

About this chapter

Cite this chapter

Mahmoud, M.S. (2011). Decentralized Control of Markovian Jump Systems. In: Decentralized Systems with Design Constraints. Springer, London. https://doi.org/10.1007/978-0-85729-290-2_7

Download citation

  • DOI: https://doi.org/10.1007/978-0-85729-290-2_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-289-6

  • Online ISBN: 978-0-85729-290-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation