Advertisement

Decentralized Control of Markovian Jump Systems

  • Magdi S. Mahmoud

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.

Keywords

Jump Linear System Aggregate Problem Markov Chain Process Markovian Jump Parameter Singular Perturbation Parameter 
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.

References

  1. 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. MathSciNetMATHCrossRefGoogle Scholar
  2. 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. 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. 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. 5.
    Boyd S., L. ElGhaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. MATHCrossRefGoogle Scholar
  6. 6.
    Davis, M., Markov Models and Optimization, Chapman and Hall, London, 1992. Google Scholar
  7. 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. MathSciNetMATHCrossRefGoogle Scholar
  8. 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. 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. MathSciNetMATHCrossRefGoogle Scholar
  10. 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. MathSciNetMATHCrossRefGoogle Scholar
  11. 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. MathSciNetMATHCrossRefGoogle Scholar
  12. 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. MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Geromel, J. C., J. Bernussou and P. L. D. Peres, “Decentralized Control Through Parameter Space Optimization”, Automatica, vol. 30, 1994, pp. 1565–1578. MathSciNetMATHCrossRefGoogle Scholar
  14. 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. 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. MathSciNetMATHGoogle Scholar
  16. 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. MathSciNetCrossRefGoogle Scholar
  17. 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. MathSciNetMATHGoogle Scholar
  18. 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. MathSciNetMATHCrossRefGoogle Scholar
  19. 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. MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kozin, F., “A Survey of Stability of Stochastic Systems”, Automatica, vol. 5, 1969, pp. 95–112. MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kushner, H. J., Stochastic Stability and Control, Academic Press, New York, 1967. MATHGoogle Scholar
  22. 22.
    Mahmoud, M. S. and M. G. Singh, Large-Scale Systems Modelling, Pergamon Press, London, 1981. MATHGoogle Scholar
  23. 23.
    Mahmoud, M. S., “Stabilizing Control for a Class of Uncertain Interconnected Systems”, IEEE Trans. Autom. Control, vol. AC-39, 1994, pp. 2484–2488. CrossRefGoogle Scholar
  24. 24.
    Mahmoud, M. S., “Guaranteed Stabilization of Interconnected Discrete-Time Uncertain Systems”, Int. J. Syst. Sci., vol. 26, 1995, pp. 337–358. MATHCrossRefGoogle Scholar
  25. 25.
    Mahmoud, M. S., “Adaptive Stabilization of a Class of Interconnected Systems”, Comput. Electr. Eng., vol. 23, 1997, pp. 225–238. CrossRefGoogle Scholar
  26. 26.
    Mahmoud, M. S. and S. Bingulac, “Robust Design of Stabilizing Controllers for Interconnected Time-Delay Systems”, Automatica, vol. 34, 1998, pp. 795–800. MathSciNetMATHCrossRefGoogle Scholar
  27. 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. CrossRefGoogle Scholar
  28. 28.
    Mahmoud, M. S., Robust Control and Filtering for Time-Delay Systems, Marcel Dekker Inc., New York, 2000. MATHGoogle Scholar
  29. 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. MathSciNetCrossRefGoogle Scholar
  30. 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. MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mahmoud, M. S. and P. Shi, Methodologies for Control of Jumping Time-Delay Systems, Kluwer Academic, Amsterdam, 2003. Google Scholar
  32. 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. MathSciNetMATHCrossRefGoogle Scholar
  33. 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. MathSciNetCrossRefGoogle Scholar
  34. 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. MathSciNetMATHCrossRefGoogle Scholar
  35. 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. 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. MathSciNetMATHGoogle Scholar
  37. 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. MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Mahmoud, M. S., “Uncertain Jumping Systems with Strong and Weak Functional Delays”, Automatica, vol. 40, no. 3, 2004, pp. 501–510. MathSciNetMATHCrossRefGoogle Scholar
  39. 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. MathSciNetMATHCrossRefGoogle Scholar
  40. 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. MathSciNetMATHCrossRefGoogle Scholar
  41. 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. MathSciNetMATHCrossRefGoogle Scholar
  42. 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. MathSciNetMATHCrossRefGoogle Scholar
  43. 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. MathSciNetMATHCrossRefGoogle Scholar
  44. 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. MATHCrossRefGoogle Scholar
  45. 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. MathSciNetMATHCrossRefGoogle Scholar
  46. 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. MathSciNetCrossRefGoogle Scholar
  47. 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. MathSciNetMATHCrossRefGoogle Scholar
  48. 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. MathSciNetMATHCrossRefGoogle Scholar
  49. 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. CrossRefGoogle Scholar
  50. 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. MathSciNetMATHCrossRefGoogle Scholar
  51. 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. MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Mao, C. and J. H. Yang, “Decentralized Output Tracking for Linear Uncertain Interconnected Systems”, Automatica, vol. 31, 1995, pp. 151–154. MathSciNetMATHCrossRefGoogle Scholar
  53. 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. MathSciNetMATHCrossRefGoogle Scholar
  54. 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. MATHCrossRefGoogle Scholar
  55. 55.
    Sethi, S. P. and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems. Birkhäuser, Boston, 1994. MATHCrossRefGoogle Scholar
  56. 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. MathSciNetMATHGoogle Scholar
  57. 57.
    Soner, H. M., “Singular Perturbations in Manufacturing”, SIAM J. Control Optim., vol. 31, 1993, pp. 132–146. MathSciNetMATHCrossRefGoogle Scholar
  58. 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. 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. MathSciNetCrossRefGoogle Scholar
  60. 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. MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Systems EngineeringKing Fahd Univ. of Petroleum & MineralsDhahranSaudi Arabia

Personalised recommendations