Skip to main content
SpringerLink
Log in

Accurate Calculation of the Settling Time of a Linear System Using New Expressions and Iterative Algorithms

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The purpose of this paper is to achieve an exhaustive and generalized method for the accurate calculation of the linear system settling time. Thus, the classical settling time expressions for the second-order linear systems are reviewed. Then, new expressions are proposed and developed for the step response of the second-order and for higher-order under-damped and over-damped linear systems. The proposed expressions are different from all the existing ones, and they are more accurate. In addition, iterative algorithms that can be easily implemented are proposed in order to achieve accurate calculation of the settling time. The accuracy of the new expressions and the proposed iterative algorithms are illustrated by several numerical simulation examples.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. W.H.T.M. Aangenent, W.P.M.H. Heemels, M.J.G. Van de Molengraft, D. Henrion, M. Steinbuch, Linear control of time domain constrained system. Automatica 48(5), 736–746 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Aminzadeh, M. Banihashemi, Miller compensation: optimal design for operational amplifiers with a required settling time. Circuits Syst. Signal Process. 33, 2675–2694 (2014)

    Article  Google Scholar 

  3. H. Aminzadeh, M. Danaie, R. Lotfi, Design of high-speed two-stage cascode-compensated operational amplifiers based on settling time and open-loop parameters. Integr. VLSI 41, 183–192 (2008)

    Article  Google Scholar 

  4. H. Aminzadeh, R. Lotfi, K. Mafinezhad, Design of low-power single-stage operational amplifiers based on an optimized settling model. Analog Integr. Circuits Signal Process. 58, 153–160 (2009)

    Article  Google Scholar 

  5. H. Aminzadeh, Three-stage nested-Miller-compensated operational amplifiers: analysis, design, and optimization based on settling time. Int. J. Circuit Theory Appl. 39(6), 573–587 (2010)

    Article  Google Scholar 

  6. C. Bert, An improved approximation for settling time of second order linear systems. IEEE Trans. Autom. Control 31(7), 642–643 (1986)

    Article  MATH  Google Scholar 

  7. J.R. Carstens, Automatic control systems and components (Prentice Hall, New Jersey, 1990)

  8. S. Cheng, Z. Zhangming, Y. Yintang, A fast-settling three-stage amplifier using regular Miller plus reversed indirect compensation. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-016-0314-7

    Google Scholar 

  9. V.N. Chetnov, Synthesis of a \(\text{ H }_{\infty }\)-controllers with given stability margin radius and settling time. Autom. Remote Control 75(2), 1593–1607 (2014)

    Article  MathSciNet  Google Scholar 

  10. L. Deng, P. Yang, Analysis of a general multi-capacity process standard form, in Chinese Control Conference (2015), pp. 332–337

  11. R. Dorf, R. Bishop, Modern Control Systems (Prentice Hall, Englewood Cliffs, NJ, 2008)

    MATH  Google Scholar 

  12. J. Dorsey, Continuous and Discrete Control Systems: Modeling, Identification, Design and Implementation (McGraw-Hill, New York, 2002)

    Google Scholar 

  13. Y. Granjon, Automatique: systèmes linéaires, non linéaires, à temps continu, à temps discret, représentation d’état (Dunod, Paris, 2003)

    Google Scholar 

  14. R.C. Ionel, S. Ionel, A. Ignea, Calculation of the second order settling time in SISO linear systems. Circuits Syst. Signal Process. 32, 375–385 (2013)

    Article  MathSciNet  Google Scholar 

  15. M.V. Khlebnikov, Settling time in a linear dynamic system with bounded external disturbances. Autom. Remote Control 73(6), 937–948 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. B.C. Kuo, F. Golnaraghi, Automatic Control Systems (Prentice Hall, Englewood Cliffs, NJ, 2002)

    Google Scholar 

  17. S. LeBallois, P. Codron, Automatique: systèmes linéaires et continus (Dunod, Paris, 2006)

    Google Scholar 

  18. A. Mandal, Introduction to Control Engineering (New Age, New Delhi, 2006)

    Google Scholar 

  19. A. Marques, Y. Geerts, M. Steyaert, W. Sansen, Settling time analysis of third order systems, in IEEE International Symposium of Circuits Systems (1998), pp. 505–508

  20. N. Nise, Control Systems Engineering (Wiley, Hoboken, NJ, 2004)

    MATH  Google Scholar 

  21. J. Nocedal, S.J. Wright, Numerical Optimization (Springer, New York, 2006). ISBN 978-0-387-30303-1

    MATH  Google Scholar 

  22. K. Ogata, Modern Control Engineering (Prentice Hall, Englewood Cliffs, NJ, 2005)

    MATH  Google Scholar 

  23. R. Piché, Comments on “an improved approximation for settling time of second order linear systems”. IEEE Trans. Autom. Control 32(8), 747–748 (1987)

    Article  Google Scholar 

  24. A. Pugliese, F. Amoroso, G. Cappuccino, G. Cocorullo, Settling-time-oriented design procedure for two-stage amplifiers with current-buffer Miller compensation. Rev. Roum. Sci. Tech. Electrotechn. Et Eng. 54(4), 375–384 (2009)

  25. C.A. Ramos-Paja, D. González, A.J. Saavedra-Montes, Accurate calculation of settling time in second order systems: a photovoltaic application. Rev. Fac. Univ. Antioq. 66, 104–117 (2013)

    Google Scholar 

  26. Z. Zuo, Nonsingular fixed-time consensus tracking for second order multi-agent networks. Automatica 54, 305–309 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to address great thanks to Professor Mekki Ksouri for his encouragements and to all reviewers for their constructive comments.

Author information

Authors and Affiliations

  1. LR11ES20, Laboratoire Analyse, Conception et Commande des Systèmes, Ecole Nationale d’Ingénieurs de Tunis, Institut Préparatoire aux Etudes d’Ingénieurs d’El Manar, Université de Tunis El Manar, BP 37, le Belvédère, 1002, Tunis, Tunisia

    Wassila Chagra

Authors
  1. Wassila Chagra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wassila Chagra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chagra, W. Accurate Calculation of the Settling Time of a Linear System Using New Expressions and Iterative Algorithms. Circuits Syst Signal Process 37, 408–431 (2018). https://doi.org/10.1007/s00034-017-0560-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-017-0560-3

Keywords

Search

Navigation