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Stabilization

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Analysis and Design of Singular Markovian Jump Systems

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Abstract

As one of the most important control problems, the stabilization problem is to design a controller such that the corresponding closed-loop system is stable and has desired performances. Due to singular Markovian jump systems containing singular derivative matrix and Markov property simultaneously, the control synthesis of stabilization of the SMJSs becomes complex especially when the underlying SMJSs have some general conditions. In this chapter, stabilization problem of SMJSs is focused and controllers are to be designed such that the closed-loop system is regular, impulse-free and stable. A robust stabilizing controller guaranteeing the closed-loop systems robustly stochastically admissible is designed based on LMI techniques. When a TRM can be designed, the stabilization for SMJSs is also discussed. Various controllers based on noise control, proportional-derivative (PD) control and partially mode-dependent (PMD) control, are proposed. These controllers are formulated in terms of LMIs or LMIs with equation constraints, which can be solved easily.

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References

  1. Uezato E, Ikeda M (1999) Strict LMI conditions for stability, robust stabilization and \(H_\infty \) control of descriptor systems. In Proceedings of the 38th IEEE conference on decision and control, Phoenix, AZ, pp 4092–4097

    Google Scholar 

  2. Xu SY, Lam J (2006) Control and filtering of singular systems. Springer, Berlin

    MATH  Google Scholar 

  3. Boukas EK (2008) Control of singular systems with random abrupt changes. Springer, Berlin

    MATH  Google Scholar 

  4. Boukas EK (2008) On stability and stabilization of continuous-time singular Markovian switching systems. IET Control Theor Appl 2:884–894

    Article  MathSciNet  Google Scholar 

  5. Xia YQ, Boukas EK, Shi P, Zhang JH (2009) Stability and stabilization of continuous-time singular hybrid systems. Automatica 45:1504–1509

    Article  MATH  MathSciNet  Google Scholar 

  6. Mao XR, Yuan CG (2006) Stability of stochastic differential equations with Markovian switching. Imperial College Press, London

    Book  Google Scholar 

  7. Appleby JAD, Mao XR, Rodkina A (2008) Stabilization and desabilization of nonlinear differential equations by noise. IEEE Trans Autom Control 53:683–691

    Article  MathSciNet  Google Scholar 

  8. Deng FQ, Luo Q, Mao XR (2012) Stochastic stabilization of hybrid differential equations. Automatica 48:2321–2328

    Article  MATH  MathSciNet  Google Scholar 

  9. Hu LJ, Mao XR (2008) Almost sure exponential stabilization of stochastic systems by state-feedback control. Automatica 44:465–471

    Article  MATH  MathSciNet  Google Scholar 

  10. Mao XR (1994) Stochastic stabilisation and destabilization. Syst Control Lett 23:279–290

    Article  MATH  Google Scholar 

  11. Mao XR, Yin GG, Yuan CG (2007) Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica 43:264–273

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu HP, Sun FC, Sun ZQ (2004) \(H_\infty \) control for Markovian jump linear singularly perturbed systems. IEE Proc Control Theory Appl 151:637–644

    Article  Google Scholar 

  13. Liu HP, Boukas EK, Sun FH (2006) \(H_\infty \) stabilization of Markovian jump singularly perturbed delayed systems. In Proceedings of 2006 American control conference, Minneapolis, Minnesota, pp 14–16

    Google Scholar 

  14. Nguang SK, Assawinchaichote W, Shi P (2007) Robust \(H_\infty \) control design for fuzzy singularly perturbed systems with Markovian jumps: an LMI approach. IET Control Theory Appl 1: 893–908

    Google Scholar 

  15. Mao XR (1999) Stability of stochastic differential equations with Markovian switching. Stoch Processes Appl 79:45–67

    Article  MATH  Google Scholar 

  16. Lin C, Wang QG, Lee TH (2005) Robust normalization and stabilization of uncertain descriptor systems with norm-bounded perturbations. IEEE Trans Autom Control 50:515–520

    Article  MathSciNet  Google Scholar 

  17. Lin C, Wang JL, Yang GH, Lam J (2000) Robust stabilization via state feedback for descriptor systems with uncertainties in the derivative matrix. Int J Control 73:407–415

    Article  MATH  MathSciNet  Google Scholar 

  18. Takaba KN, Katayama T (1998) \(H_2\) output feedback control for descriptor systems. Automatica 34:841–850

    Article  MATH  MathSciNet  Google Scholar 

  19. Ren JC, Zhang QL (2010) Robust \(H_\infty \) control for uncertaind escriptor systems by proportional-derivative state feedback. Int J Control 83:89–96

    Article  MATH  MathSciNet  Google Scholar 

  20. Huang LR, Mao XR (2010) Stability of singular stochastic systems with Markovian switching. IEEE Trans Autom Control 56:1213–1219

    Google Scholar 

  21. Wang GL (2011) Partially mode-dependent design of \(H_\infty \) filter for stochastic Markovian jump systems with mode-dependent time delays. J Math Anal Appl 383:573–584

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang GL, Zhang QL, Sreeram V (2010) Robust delay-range-dependent stabilization for Markovian jump systems with mode-dependent time delays and nonlinearities. Optimal Control Appl Methods 31:249–264

    MATH  MathSciNet  Google Scholar 

  23. Xu SY, Chen TW, Lam J (2003) Robust \(H_\infty \) filtering for uncertain Markovian jump systems with mode-dependent time delays. IEEE Trans Autom Control 48:900–907

    Article  MathSciNet  Google Scholar 

  24. Zhang LX, Lam J (2010) Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions. IEEE Trans Autom Control 55:1695–1701

    Article  MathSciNet  Google Scholar 

  25. Xiong JL, Lam J (2006) On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica 41:897–903

    Article  MathSciNet  Google Scholar 

  26. Xie LH, Fu MY, de Souza CE (1992) \(H_\infty \) control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans Autom Control 37:1253–1256

    Article  MATH  Google Scholar 

  27. Davis MHA (1993) Markov models and optimization. Chapman and Hall, London

    Book  MATH  Google Scholar 

  28. Wang GL(2012) Stabilization of singular Markovian jump systems via switching probability rate design. In Proceedings of the 31st Chinese control conference, Hefei, pp 2732–2736

    Google Scholar 

  29. Wang GL (2013) Robust stabilization of singular Markovian jump systems with uncertain switching. Int J Control Autom Syst 11:188–193

    Article  Google Scholar 

  30. Wang GL, Huang C, Zhang QL, Yang CY (2012) Stabilization bound of stochastic singularly perturbed systems with Markovian switching by noise control. IET Control Theory Appl 8(5):367–374. doi:10.1049/iet-cta.2013.0493

    Article  MathSciNet  Google Scholar 

  31. Wang GL, Zhang QL (2012) Robust control of uncertain singular stochastic systems with Markovian switching via proportional-derivative state feedback. IET Control Theory Appl 8:1089–1096

    Article  Google Scholar 

  32. Wang GL, Zhang QL, Yang CY (2012) Dissipative control for singular Markovian jump systems with time delay. Optimal Control Appl Methods 33:415–432

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Liaoning Shihua University, Fushun, China

    Guoliang Wang

  2. Institute of Systems Science, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China

    Qingling Zhang

  3. School of Engineering and Digital Arts, The University of Kent, Kent, CT, UK

    Xinngang Yan

Authors
  1. Guoliang Wang
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  2. Qingling Zhang
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  3. Xinngang Yan
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Corresponding author

Correspondence to Guoliang Wang .

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© 2015 Springer International Publishing Switzerland

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Wang, G., Zhang, Q., Yan, X. (2015). Stabilization. In: Analysis and Design of Singular Markovian Jump Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-08723-8_3

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  • DOI: https://doi.org/10.1007/978-3-319-08723-8_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08722-1

  • Online ISBN: 978-3-319-08723-8

  • eBook Packages: EngineeringEngineering (R0)

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