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Design of finite-/fixed-time ISS-Lyapunov functions for mechanical systems

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Abstract

For a canonical form of mechanical systems defined through gradients of potential energy and dissipative terms, the conditions of finite-time and fixed-time (integral) input-to-state stability are derived by finding suitable Lyapunov functions. The proposed stability conditions are constructive, which is demonstrated in several applications.

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Notes

  1. In this work, the abbreviation as (n)FxTS means that both properties, nFxTS and FxTS, are referenced.

References

  1. Aleksandrov AY (2020) Stability analysis and synthesis of stabilizing controls for a class of nonlinear mechanical systems. Nonlinear Dyn 100(4):3109–3119

    Article  Google Scholar 

  2. Aleksandrov AY, Efimov D, Dashkovskiy S (2022) On strict Iss-Lyapunov functions for mechanical systems. Int J Robust Nonlinear Control

  3. Aleksandrov AY, Kosov AA (2010) The stability and stabilization of non-linear, non-stationary mechanical systems. J Appl Math Mech 74(5):553–562

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrieu V, Praly L, Astolfi A (2008) Homogeneous approximation, recursive observer design, and output feedback. SIAM J Control Optim 47(4):1814–1850

    Article  MathSciNet  MATH  Google Scholar 

  5. Angeli D, Sontag ED, Wang Y (2000) A characterization of integral input-to-state stability. IEEE Trans Automat Control 45(6):1082–1097

    Article  MathSciNet  MATH  Google Scholar 

  6. Bacciotti A, Rosier L (2001) Liapunov functions and stability in control theory, vol 267. Lecture notes in control and information science. Springer, Berlin

    MATH  Google Scholar 

  7. Beletsky VV (1966) Motion of an artificial satellite about its center of mass. Israel Program for Scientific Translation, Jerusalem

  8. Bhat SP, Bernstein DS (2000) Finite time stability of continuous autonomous systems. SIAM J Control Optim 38(3):751–766

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaillet A, Angeli D, Ito H (2012) Strong iISS: combination of iISS and ISS with respect to small inputs. In: 2012 IEEE 51st IEEE conference on decision and control (CDC), pp 2256–2261

  10. Cortes J, Martinez S, Karatas T, Bullo F (2004) Coverage control for mobile sensing networks. IEEE Trans Robot Autom 20(2):243–255

    Article  Google Scholar 

  11. Cruz-Zavala E, Sanchez T, Moreno JA, Nuno E (2018) Strict Lyapunov functions for homogeneous finite-time second-order systems. In: 2018 IEEE conference on decision and control (CDC), pp 1530–1535

  12. Dashkovskiy S (2019) Practical examples of ISS systems. IFAC Pap 52(16):1–6

    Article  Google Scholar 

  13. Dashkovskiy SN, Efimov DV, Sontag ED (2011) Input to state stability and allied system properties. Autom Remote Control 72(8):1579–1614

    Article  MathSciNet  MATH  Google Scholar 

  14. Du HB, Li SH (2012) Finite-time attitude stabilization for aspacecraft using homogeneous method. J Guid Control Dyn 35(3):740–748

    Article  Google Scholar 

  15. Efimov D, Aleksandrov A (2021) Analysis of robustness of homogeneous systems with time delays using Lyapunov–Krasovskii functionals. Int J Robust Nonlinear Control 31:3730–3746

    Article  MathSciNet  Google Scholar 

  16. Efimov D, Polyakov A (2021) Finite-time stability tools for control and estimation. Found Trends Syst Control 9(2–3):171–364

    Article  Google Scholar 

  17. Giri DK, Sinha M (2014) Magneto-coulombic attitude control of earth-pointing satellites. J Guid Control Dyn 37(6):1946–1960

    Article  Google Scholar 

  18. Holloway J, Krstic M (2019) Prescribed-time output feedback for linear systems in controllable canonical form. Automatica 107:77–85

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong Y, Jiang Z, Feng G (2010) Finite-time input-to-state stability and applications to finite-time control design. SIAM J Control Optim 48(7):4395–4418

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang X, Lin W, Yang B (2005) Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 41(5):881–888

    Article  MathSciNet  MATH  Google Scholar 

  21. Khalil HK (2002) Nonlinear systems. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  22. Lageman C, Helmke U, Anderson BDO (2015) Formation control on lines, circles and ellipses: genericity results and Morse theoretic ideas. In: 2015 54th IEEE conference on decision and control (CDC), pp 4278–4283

  23. Leonard NE, Olshevsky A (2013) Nonuniform coverage control on the line. IEEE Trans Automat Control 58(11):2743–2755

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin Y, Sontag ED, Wang Y (1996) A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim 34(1):124–160

    Article  MathSciNet  MATH  Google Scholar 

  25. Lopez-Ramirez F, Efimov D, Polyakov A, Perruquetti W (2019) Conditions for fixed-time stability and stabilization of continuous autonomous systems. Syst Control Lett 129:26–35

    Article  MathSciNet  MATH  Google Scholar 

  26. Lopez-Ramirez F, Efimov D, Polyakov A, Perruquetti W (2020) Finite-time and fixed-time input-to-state stability: explicit and implicit approaches. Syst Control Lett 130

  27. Malisoff M, Mazenc F (2009) Constructions of strict Lyapunov functions. Springer, London

    Book  MATH  Google Scholar 

  28. Moulay E, Perruquetti W (2006) Finite time stability and stabilization of a class of continuous systems. J Math Anal Appl 323(2):1430–1443

    Article  MathSciNet  MATH  Google Scholar 

  29. Moulay E, Perruquetti W (2008) Finite time stability conditions for non autonomous continuous systems. Int J Control 81(5):797–803

    Article  MathSciNet  MATH  Google Scholar 

  30. Nersesov SG, Haddad WM, Hui Q (2008) Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions. J Franklin Inst 345:819–837

    Article  MathSciNet  MATH  Google Scholar 

  31. Parsegov S, Polyakov A, Shcherbakov P (2012) Nonlinear fixed-time control protocol for uniform allocation of agents on a segment. In: Proceedings of 51st IEEE conference on decision and control (CDC), pp 7732–7737

  32. Polyakov A (2012) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control 57(8):2106–2110

    Article  MathSciNet  MATH  Google Scholar 

  33. Qian C, Lin W (2001) A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans Autom Control 46(7):1061–1079

    Article  MathSciNet  MATH  Google Scholar 

  34. Roxin E (1966) On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo 15:273–283

    Article  MathSciNet  MATH  Google Scholar 

  35. Shen Y, Xia X (2008) Semi-global finite-time observers for nonlinear systems. Automatica 44(12):3152–3156

    Article  MathSciNet  MATH  Google Scholar 

  36. Sontag ED (2007) Input to state stability: basic concepts and results. In: Nistri P, Stefani G (eds) Nonlinear and optimal control theory. Springer, Berlin, pp 163–220

    Google Scholar 

  37. Terushkin M, Fridman E (2021) Network-based deployment of nonlinear multi agents over open curves: a PDE approach. Automatica 129:109697

    Article  MathSciNet  MATH  Google Scholar 

  38. Wagner IA, Bruckstein AM (1997) Row straightening via local interactions. Circuits Syst Signal Process 16:287–305

    Article  MathSciNet  MATH  Google Scholar 

  39. Zavala-Río A, Sanchez T, Zamora-Gómez GI (2022) On the continuous finite-time stabilization of the double integrator. SIAM J Control Optim 60(2):699–719

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhao Z-L, Jiang Z-P, Liu T, Chai T (2021) Global finite-time output-feedback stabilization of nonlinear systems under relaxed conditions. IEEE Trans Autom Control 66(9):4259–4266

    Article  MathSciNet  MATH  Google Scholar 

  41. Zimenko K, Polyakov A, Efimov D, Kremlev A (2022) Homogeneity based finite/fixed-time observers for linear mimo systems. Int J Robust Nonlinear Control

  42. Zimenko K, Polyakov A, Efimov D, Perruquetti W (2020) Robust feedback stabilization of linear mimo systems using generalized homogenization. IEEE Trans Autom Control 65(12):5429–5436

    Article  MathSciNet  MATH  Google Scholar 

  43. Zubov VI (1975) Lectures on control theory. Nauka, Moscow (Russian)

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Authors and Affiliations

  1. Saint Petersburg State University, 7-9 Universitetskaya nab., Saint Petersburg, 199034, Russia

    Alexander Aleksandrov

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, 199178, Russia

    Alexander Aleksandrov

  3. Inria, Univ. Lille, CNRS, UMR 9189, CRIStAL, 59000, Lille, France

    Denis Efimov

  4. Institute of mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074, Würzburg, Germany

    Sergey Dashkovskiy

Authors
  1. Alexander Aleksandrov
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  2. Denis Efimov
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  3. Sergey Dashkovskiy
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Correspondence to Denis Efimov.

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The paper is partly supported by the Ministry of Science and Higher Education of Russian Federation, passport of Goszadanie No. 2019-0898. Section 5 was performed in IPME RAS and supported by the Ministry of Science and Higher Education of the Russian Federation under Project 075-15-2021-573.

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Aleksandrov, A., Efimov, D. & Dashkovskiy, S. Design of finite-/fixed-time ISS-Lyapunov functions for mechanical systems. Math. Control Signals Syst. 35, 215–235 (2023). https://doi.org/10.1007/s00498-022-00338-x

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  • DOI: https://doi.org/10.1007/s00498-022-00338-x

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