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Characterizing regions of attraction for piecewise affine systems by continuity of discrete transition functions

Abstract

Based on an analysis of the continuity of discrete transition functions, this paper presents a new approach for characterizing piecewise affine (PWA) systems’ regions of attraction (RoAs). First, the RoA boundary is proven to consist of the points where the discrete transition function is discontinuous and of the specific parts of switching surfaces. Then, the formulas and numerical algorithm for computing states with discontinuous discrete transition functions are developed. The proposed approach enables the characterization of the entire RoA for both equilibrium points and limit cycles. Finally, the proposed method is used to analyze multiple examples. For clarity, the results of a third-order inductor-capacitor-inductor resonant inverter are provided in a video. Although we focus on PWA systems, the key concept behind continuity and stability also applies to other hybrid dynamical systems, enabling the broader application of the proposed RoA computing method.

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References

  1. 1.

    de Braga, D., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dyn. 73, 1283–1288 (2013)

  2. 2.

    Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn. 82, 1159–1175 (2015)

  3. 3.

    Wu, T., Wang, L., Yang, X.-S.: Chaos generator design with piecewise affine systems. Nonlinear Dyn. 84, 817–832 (2016)

  4. 4.

    Hetel, L., Bernuau, E.: Local stabilization of switched affine systems. IEEE Trans. Autom. Control 60, 1158–1163 (2015)

  5. 5.

    Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43, 555–559 (1998)

  6. 6.

    Hajiahmadi, M., De Schutter, B., Hellendoorn, H.: Design of stabilizing switching laws for mixed switched affine systems. IEEE Trans. Autom. Control 61(6), 1676–1681 (2016)

  7. 7.

    Rubagotti, M., Zaccarian, L., Bemporad, A.: A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains. Int. J. Control 89(5), 950–959 (2016)

  8. 8.

    Goncalves, J.M.: Regions of stability for limit cycle oscillations in piecewise linear systems. IEEE Trans. Autom. Control 50, 1877–1882 (2005)

  9. 9.

    Tang, J.Z., Manchester, I.R.: Transverse contraction criteria for stability of nonlinear hybrid limit cycles. In: 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), pp. 31–36. IEEE (2014)

  10. 10.

    Gering, S., Eciolaza, L., Adamy, J., Sugeno, M.: A piecewise approximation approach to nonlinear systems: stability and region of attraction. IEEE Trans. Fuzzy Syst. 23, 2231–2244 (2015)

  11. 11.

    Dezuo, T., Rodrigues, L., Trofino, A.: Stability analysis of piecewise affine systems with sliding modes. In: American Control Conference (ACC), vol. 2014, pp. 2005–2010 (2014)

  12. 12.

    Genesio, R., Tartaglia, M., Vicino, A.: On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE Trans. Autom. Control 30, 747–755 (1985)

  13. 13.

    Noldus, E., Loccufier, M.: A new trajectory reversing method for the estimation of asymptotic stability regions. Int. J. Control 61, 917–932 (1995)

  14. 14.

    Jerbi, H., Braiek, N.B., Bacha, A.B.B.: A method of estimating the domain of attraction for nonlinear discrete-time systems. Arab. J. Sci. Eng. 39, 3841–3849 (2014)

  15. 15.

    Iwatani, Y., Hara, S.: Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems. Automatica 42, 1685–1695 (2006)

  16. 16.

    Camlibel, M.K.: Well-posed bimodal piecewise linear systems do not exhibit Zeno behavior. In: Proceedings of 17th IFAC World Congress on Automatic Control. pp. 7973–7978 (2008)

  17. 17.

    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides: Control Systems. Springer Science & Business Media, Berlin (1988)

  18. 18.

    Thuan, L.Q.: Non-Zenoness of piecewise affine dynamical systems and affine complementarity systems with inputs. Control Theory Technol. 12, 35–47 (2014)

  19. 19.

    Imura, J., van der Schaft, A.: Characterization of well-posedness of piecewise-linear systems. IEEE Trans. Autom. Control 45, 1600–1619 (2000)

  20. 20.

    Şahan, G., Eldem, V.: Well posedness conditions for bimodal piecewise affine systems. Syst. Control Lett. 83, 9–18 (2015)

  21. 21.

    Piecewise Linear Control Systems: A Computational Approach. Springer, Berlin (2003)

  22. 22.

    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry. Springer, Berlin (2008)

  23. 23.

    De Berg, M.: Linear size binary space partitions for uncluttered scenes. Algorithmica 28, 353–366 (2000)

  24. 24.

    Tóth, C.D.: Binary space partitions: recent developments. Comb. Comput. Geom. MSRI Publ. 52, 529–556 (2005)

  25. 25.

    Agarwal, P.K., Suri, S.: Surface approximation and geometric partitions. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 24–33. Society for Industrial and Applied Mathematics, Philadelphia, PA (1994)

  26. 26.

    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)

  27. 27.

    Van Loan, C.: Computing Integrals Involving the Matrix Exponential. Cornell University, Ithaca, NY (1977)

  28. 28.

    Bell, P.C., Delvenne, J.-C., Jungers, R.M., Blondel, V.D.: The continuous Skolem–Pisot problem. Theor. Comput. Sci. 411, 3625–3634 (2010)

  29. 29.

    Johansson, M.: Piecewise quadratic estimates of domains of attraction for linear systems with saturation. Presented at the July 21 (2002)

  30. 30.

    Park, T.J., Kim, T.W., Han, M.H.: Load estimation and effective heating method of LCL-resonant inductive heater. In: 8th International Conference on Power Electronics—ECCE Asia. pp. 1576–1578 (2011)

  31. 31.

    Wang, C.-S., Covic, G.A., Stielau, O.H.: Investigating an LCL load resonant inverter for inductive power transfer applications. IEEE Trans. Power Electron. 19, 995–1002 (2004)

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61573074, 51277192, 51477020 and by the National High-Tech R & D Program of China under Grant 2015AA010402.

Author information

Correspondence to Yue Sun.

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Supplementary material 1 (mp4 101106 KB)

Supplementary material 1 (mp4 101106 KB)

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Chen, Y., Sun, Y., Tang, C. et al. Characterizing regions of attraction for piecewise affine systems by continuity of discrete transition functions. Nonlinear Dyn 90, 2093–2110 (2017). https://doi.org/10.1007/s11071-017-3786-5

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Keywords

  • Piecewise affine systems
  • Hybrid dynamical systems
  • Switched systems
  • Stability
  • Region of attraction