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Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input

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Abstract

A class of homogeneous polynomial functions is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameter-dependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

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References

  1. C.K. Ahn, An \(H_{\infty }\) approach to stability analysis of switched Hopfield neural networks with time-delay. Nonlinear Dyn. 60, 703–711 (2010)

    Article  MATH  Google Scholar 

  2. C.K. Ahn, Passive learning and input-to-state stability of switched Hopfield neural networks with time-delay. Inf. Sci. 180, 4582–4594 (2010)

    Article  MATH  Google Scholar 

  3. C.K. Ahn, Exponentially convergent state estimation for delayed switched recurrent neural networks. Eur. Phys. J. E 34, 122 (2011)

    Article  Google Scholar 

  4. C.K. Ahn, Switched exponential state estimation of neural networks based on passivity theory. Nonlinear Dyn. 67, 573–586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. C.K. Ahn, Linear matrix inequality optimization approach to exponential robust filtering for switched Hopfield neural networks. J. Optim. Theory Appl. 154, 573–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. C.K. Ahn, Receding horizon disturbance attenuation for Takagi–Sugeno fuzzy switched dynamic neural networks. Inf. Sci. 280, 53–63 (2014)

    Article  MathSciNet  Google Scholar 

  7. C.K. Ahn, M.K. Song, \(L_2-L_{\infty }\) filtering for time-delayed switched Hopfield neural networks. Int. J. Innov. Comput. Inf. Control 7, 1831–1844 (2011)

    Google Scholar 

  8. F. Blanchini, S. Miani, F. Mesquine, A separation principle for linear switching systems and parametrization of all stabilizing controllers. IEEE Trans. Autom. Control 54, 279–292 (2009)

    Article  MathSciNet  Google Scholar 

  9. P.-A. Bliman, R.C.L.F. Oliveira, V.F. Montagner, P.L.D. Peres, Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex. In IEEE Conference on Decision and Control, San Diego (2006), pp. 1486–1491

  10. Y. Cao, Z. Lin, Y. Shamash, Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation. Syst. Control Lett. 46, 137–151 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Cao, T. Hu, Stability analysis of linear time-delay systems subject to input saturation. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49, 233–240 (2002)

    Article  MathSciNet  Google Scholar 

  12. Y. Cao, Z. Lin, stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function. Automatica 39, 1235–1241 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Y. Chen, S. Fei, K. Zhang, Stabilization of impulsive switched linear systems with saturated control input. Nonlinear Dyn. 69, 793–804 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Chen, S. Fei, K. Zhang, Z. Fu, Control synthesis of discrete-time switched linear systems with input saturation based on minimum dwell time approach. Circuits Syst. Signal Process. 31, 779–795 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Chesi, A. Garulli, A. Tesi, A. Vicino, Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE Trans. Autom. Control 50, 365–370 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Chesi, Establishing stability and instability of matrix hypercubes. Syst. Control Lett. 54, 381–388 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Chesi, Establishing tightness in robust H-infinity analysis via homogeneous parameter-dependent Lyapunov functions. Automatica 43, 1992–1995 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Chesi, On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems. Automatica 44, 2973–2976 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Chesi, Tightness conditions for semidefinite relaxations of forms minimizations. IEEE Trans. Circuits Syst. II 55, 1299–1303 (2008)

    Article  Google Scholar 

  20. G. Chesi, A. Tesi, A. Garulli, A. Vicino, Homogneous Polynomial Forms for Robustness Analysis of Uncertain Systems (Springer, Berlin, 2009), pp. 1–26

    MATH  Google Scholar 

  21. Z.H. Guan, D.J. Hill, X. Shen, On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans. Autom. Control 50, 1058–1062 (2005)

    Article  MathSciNet  Google Scholar 

  22. T. Hu, Z. Lin, Ben M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38, 351–359 (2002)

    Article  MATH  Google Scholar 

  23. T. Hu, Z. Lin, B.M. Chen, Analysis and design for discrete-time linear systems subject to actuator saturation. Syst. Control Lett. 45, 97–112 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Hu, Z. Lin, Exact characterization of invariant ellipsoids for single input linear systems subject to actuator saturation. IEEE Trans. Autom. Control 47, 164–169 (2002)

    Article  MathSciNet  Google Scholar 

  25. T. Hu, Z. Lin, Composite quadratic Lyapunov functions for constrained control systems. IEEE Trans. Autom. Control 48, 440–450 (2003)

    Article  MathSciNet  Google Scholar 

  26. S. Ma, E.-K. Boukas, Stability and H\(_{\infty }\) control for discrete-time singular systems subject to actuator saturation. In Proceedings of the 2009 American Control Conference, St. Louis (2009), pp. 1244–1249

  27. W. Ni, D. Cheng, Control of switched linear systems with input saturation. Int. J. Syst. Sci. 41, 1057–1065 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. R.C.L.F. Oliveira, M.C. de Oliveira, P.L.D. Peres, Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions. Syst. Control Lett. 57, 680–689 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. R.C.L.F. Oliveira, P.L.D. Peres, Parameter-dependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations. IEEE Trans. Autom. Control 52, 1334–1340 (2007)

    Article  MathSciNet  Google Scholar 

  30. T. Shi, H. Su, J. Chu, Stability analysis for continuous-time systems with actuator saturation. J. Control Theory Appl. 7, 352–358 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This project is supported by Major Program of National Natural Science Foundation of China under Grant 11190015, National Natural Science Foundation of China under Grant 61374006 and Graduate Student Innovation Foundation of Jiangsu province under Grant 3208004904.

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Authors and Affiliations

  1. Key Laboratory of Measurement and Control of CSE Ministry of Education School of Automation, Southeast University, Nanjing, 210096, China

    Guochen Pang, Kanjian Zhang & Haikun Wei

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  1. Guochen Pang
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  2. Kanjian Zhang
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  3. Haikun Wei
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Correspondence to Kanjian Zhang.

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Pang, G., Zhang, K. & Wei, H. Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input. Circuits Syst Signal Process 35, 1091–1121 (2016). https://doi.org/10.1007/s00034-015-0104-7

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