Skip to main content
SpringerLink
Log in

Practical Stability Preservation Under Sampling, Actuation Disturbance and Measurement Noise, for Globally Lipschitz Time-Delay Systems

  • Chapter
  • First Online:
Accounting for Constraints in Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 12))

  • 301 Accesses

Abstract

In this chapter we deal with the problem of global exponential practical stability preservation under sampling, for globally Lipschitz time-delay systems with state feedback controllers affected by measurement noises and actuation disturbances. It is shown that, if the continuous-time closed-loop system at hand is globally exponentially stable and the maps describing the dynamics and the continuous-time state feedback are globally Lipschitz, then, under suitably fast sampling, the global exponential practical stability of the sampled-data closed-loop system is preserved even in the case of bounded measurement noises and bounded actuation disturbances.

This work is supported in part by the Atheneum Project RIA 2018, and by the Italian Ministry of Education, University and Research MIUR, project FFABR 2017.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Ahmed-Ali, E. Fridman, F. Giri, L. Burlion, F. Lamnabhi-Lagarrigue, Using exponential time-varying gains for sampled-data stabilization and estimation. Automatica 67, 244–251 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. C.T.H. Baker, E. Buckwar, Exponential stability in pth mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math. 184, 404–427 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Briat, Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints. Automatica 49, 3449–3457 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Carnevale, A.R. Teel, D. Nesic, A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems. IEEE Trans. Automat. Control 52, 892–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. F.H. Clarke, Discontinuous feedback and nonlinear systems, in Plenary Lecture at IFAC Conference on Nonlinear Control Systems (2010)

    Google Scholar 

  6. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, A.I. Subbotin, Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42, 1394–1407 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. J.M. Gomes da Silva, I. Queinnec, A. Seuret, S. Tarbouriech, Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation. IEEE Trans. Automat. Control 61, 4176–4182 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Di Ferdinando, P. Pepe, Sampled-Data emulation of dynamic output feedback controllers for nonlinear time-delay systems. Automatica 99(1), 120–131 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Di Ferdinando, P. Pepe, Robustification of sample-and-hold stabilizers for control-affine time-delay systems. Automatica 83, 141–154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.D. Driver, Existence and stability of solutions of a delay-differential system. Arch. Ration. Mech. Anal. 10, 401–426 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Fridman, Introduction to time-delay systems: analysis and control. Birkhauser (2014)

    Google Scholar 

  12. E. Fridman, A refined input delay approach to sampled-data control. Automatica 46, 421–427 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Fridman, A. Seuret, J.P. Richard, Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40, 1441–1446 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Grune, D. Nesic, Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models. SIAM J. Control Optim. 42, 98–122 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Halanay, Differential Equations, Stability, Oscillations, Time Lags (Academic Press, New York, 1966)

    MATH  Google Scholar 

  16. J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer Verlag, New York, 1993)

    Google Scholar 

  17. W.P.M.H. Heemels, A.R. Teel, N. Van De Wouw, D. Nesic, Networked control systems with communication constraints: tradeoff between transmission intervals, delays and performance. IEEE Trans. Autom. Control 55, 1781–1796 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Herrmann, S.K. Spurgeon, C. Edwards, Discretization of sliding mode based control schemes, in 38th IEEE Conference on Decision and Control, vol. 5, pp. 4257–4262 (1999)

    Google Scholar 

  19. J. Hespanha, A model for stochastic hybrid systems with application to communication networks. Nonlinear Anal. Spec. Issue Hybrid Syst. 62, 1353–1383 (2005)

    MathSciNet  MATH  Google Scholar 

  20. L. Hetel, C. Fiter, H. Omran, A. Seuret, E. Fridman, J.P. Richard, S.I. Niculescu, Recent developments on the stability of systems with aperiodic sampling: an overview. Automatica 76, 309–335 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. L.V. Hien, V.N. Phat, H. Trinh, New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems. Nonlinear Dyn. 82, 563–575 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Hsu, S. Sastry, The effect of discretized feedback in a closed loop system, in 26th IEEE Conference on Decision and Control, vol. 26, pp. 1518–1523 (1987)

    Google Scholar 

  23. I. Karafyllis, K. Kravaris, Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization. Int. J. Robust Nonlinear Control 19, 1105–1128 (2009)

    Article  MATH  Google Scholar 

  24. I. Karafyllis, M. Krstic, Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order-hold. IEEE Trans. Automat. Control 57, 1141–1154 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. I. Karafyllis, P. Pepe, Z.P. Jiang, Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations. Eur. J. Control 6, 516–536 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. H.K. Khalil, Nonlinear Systems, 3rd edn. (Prentice Hall, Upper Saddle River, NJ, 2000)

    Google Scholar 

  27. V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations (Kluwer Academic, Dordrecht, 1999)

    Book  MATH  Google Scholar 

  28. N.N. Krasovskii, Stability of Motion (Stanford University Press, 1963)

    Google Scholar 

  29. D. S. Laila, D. Nesic, A. Astolfi, Sampled-data control of nonlinear systems, in Lecture Notes in Control and Information Sciences, ed. by A. Loria, F. Lamnabhi-Lagarrigue, E. Panteley (Eds.), vol. 328, pp. 91–137 (2006)

    Google Scholar 

  30. D.S. Laila, D. Nesic, A.R. Teel, Open and closed loop dissipation inequalities under sampling and controller emulation. Eur. J. Control 18, 109–125 (2002)

    Article  MATH  Google Scholar 

  31. Y.S. Ledyaev, E.D. Sontag, A Lyapunov characterization of robust stabilization. Nonlinear Anal. Ser. A: Theory Methods 37, 813–840 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Malisoff, E.D. Sontag, Asymptotic controllability and input-to-state stabilization: the effect of actuator errors, in Optimal Control, Stabilization and Nonsmooth Analysis, vol. 301. Lecture Notes in Control and Information Sciences, pp. 855–171 (2004)

    Google Scholar 

  33. F. Mazenc, M. Malisoff, T.H. Dinh, Robustness of nonlinear systems with respect to delay and sampling of the controls. Automatica 49, 1925–1931 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Naghshtabrizi, J. Hespanha, A.R. Teel, On the robust stability and stabilization of sampled-data systems: a hybrid system approach, in Proceedings of the 45th Conference on Decision and Control (2006)

    Google Scholar 

  35. D. Nesic, A.R. Teel, A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Trans. Automat. Control 49, 1103–1122 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Nesic, A.R. Teel, Input-to-state stability of networked control systems. Automatica 40, 2121–2128 (2004)

    MathSciNet  MATH  Google Scholar 

  37. D. Nesic, A.R. Teel, P.V. Kokotovic, Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. Syst. Control Lett. 45, 259–270 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Omran, L. Hetel, J.P. Richard, F. Lamnabhi-Lagarrigue, Stability analysis of bilinear systems under a periodic sampled-data control. Automatica 50, 1288–1295 (2014)

    Article  MATH  Google Scholar 

  39. P. Pepe, On Lyapunov-Krasovskii functionals under Caratheodory conditions. Automatica J. IFAC 43, 701–706 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. P. Pepe, Stabilization in the sample-and-hold sense of nonlinear retarded systems. SIAM J. Control Optim. 52, 3053–3077 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Pepe, Robustification of nonlinear stabilizers in the sample-and-hold sense. J. Franklin Inst. 42, 4107–4128 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Pepe, On stability preservation under sampling and approximation of feedbacks for retarded systems. SIAM J. Control Optim. 54, 1895–1918 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. P. Pepe, On control Lyapunov-Razumikhin functions, Nonconstant delays, Nonsmooth feedbacks, and nonlinear sampled-data stabilization. IEEE Trans. Autom. Control 62, 5604–5619 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Pepe, E. Fridman, On global exponential stability preservation under sampling for globally lipschitz delay-free and retarded systems, in \(13{\rm th}\) IFAC Workshop on Time-Delay Systems Istanbul, IFAC-PapersOnLine, vol. 49, pp. 41–46 (2016)

    Google Scholar 

  45. P. Pepe, E. Fridman, On global exponential stability preservation under sampling for globally Lipschitz time-delay systems. Automatica 82, 295–300 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Pepe, I. Karafyllis, Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale’s form. Internat. J. Control 86, 232–243 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Postoyan, T. Ahmed-Ali, F. Lamnabhi-Lagarrigue, Robust backstepping for the Euler approximate model of sampled-data strict-feedback systems. Automatica 45, 2164–2168 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Seuret, A novel stability analysis of linear systems under asynchronous samplings. Automatica 48, 177–182 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Seuret, C. Briat, Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals. Automatica 55, 274–278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. E.D. Sontag, Clocks and insensitivity to small measurement errors. ESAIM Control Optim. Calc. Var. 4, 537–557 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  51. E.D. Sontag, Stability and stabilization: discontinuities and the effect of disturbances. Math. Phys. Sci. 528, 551–598 (1999)

    MathSciNet  MATH  Google Scholar 

  52. E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34, 435–443 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  53. T. Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan (1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, Via Vetoio (Coppito 1), 67100, L’Aquila, Italy

    Mario Di Ferdinando & Pierdomenico Pepe

  2. School of Electrical Engineering, Tel Aviv University, 69978, Tel Aviv, Israel

    Emilia Fridman

Authors
  1. Mario Di Ferdinando
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierdomenico Pepe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilia Fridman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Di Ferdinando .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CentraleSupélec, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Wim Michiels

  3. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L'Aquila, Italy

    Pierdomenico Pepe

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Di Ferdinando, M., Pepe, P., Fridman, E. (2022). Practical Stability Preservation Under Sampling, Actuation Disturbance and Measurement Noise, for Globally Lipschitz Time-Delay Systems. In: Valmorbida, G., Michiels, W., Pepe, P. (eds) Accounting for Constraints in Delay Systems. Advances in Delays and Dynamics, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-89014-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation