Skip to main content
Springer Nature Link
Log in

Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method

  • Research Article
  • Published:
International Journal of Automation and Computing Aims and scope Submit manuscript

Abstract

Methods to stabilize discrete-time linear control systems subject to variable sampling rates, i.e., using state feedback controllers, are well known in the literature. Several recent works address the use of the Tikhonov regularization method, originally designed to attenuate the noise effects on ill-posed problems, with the aim of improving performance and stabilizing approximately controllable dynamical systems. Inspired by these works, we propose the use of a feedback controller designed using the Tikhonov method to regularize discrete-time linear systems subject to varying sampling rates. The goal is to minimize an error function, thus improving the performance of the closed loop system and reducing the possibility of instability. Illustrative examples show the effectiveness of the proposed method.

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

Access this article

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

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. B. Wittenmark, J. Nilsson, M. Törngren. Timing problems in real-time control systems. In Proceedings of American Control Conference, IEEE, Seattle, USA, pp. 2000–2004, 1995. DOI: https://doi.org/10.1109/ACC.1995.531240.

    Google Scholar 

  2. A. Rabello, A. Bhaya. Stability of asynchronous dynamical systems with rate constraints and applications. IEE Proceedings-Control Theory and Applications, vol. 150, no. 5, pp. 546–550, 2003. DOI: https://doi.org/10.1049/ip-cta:20030704.

    Article  Google Scholar 

  3. A. Bhaya, P. R. Medeiros. On the stability of asynchronous multirate linear systems. In Proceedings of the 36th IEEE Conference on Decision and Control, IEEE, San Diego, USA, pp. 2041–2042, 1997. DOI: https://doi.org/10.1109/CDC.1997.657066.

    Chapter  Google Scholar 

  4. V. S. Ritchey, G. F. Franklin. A stability criterion for asynchronous multirate linear systems. IEEE Transactions on Automatic Control, vol.34, no.5, pp.529–535, 1989. DOI: https://doi.org/10.1109/9.24205.

    Article  MathSciNet  Google Scholar 

  5. L. Hetel, J. Daafouz, C. Iung. Analysis and control of LTI and switched systems in digital loops via an event-based modelling. International Journal of Control, vol.81, no.7, pp. 1125–1138, 2008. DOI: https://doi.org/10.1080/00207170701670442.

    Article  MathSciNet  Google Scholar 

  6. R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King. Stability criteria for switched and hybrid systems. SIAM Review, vol.49, no.4, pp.545–592, 2007. DOI: https://doi.org/10.1137/05063516X.

    Article  MathSciNet  Google Scholar 

  7. V. D. Blondel, Y. Nesterov, J. Theys, Approximations of the rate of growth of switched linear systems. In Proceedings of the 7th International Workshop on Hybrid Systems: Computation and Control, Springer, Philadelphia, USA, vol.2993, pp. 173–186, 2004. DOI: https://doi.org/10.1007/978-3-540-24743-212.

    MATH  Google Scholar 

  8. M. Schinkel, W. H. Chen, A. Rantzer. Optimal control for systems with varying sampling rate. In Proceedings of American Control Conference, IEEE, Anchorage, USA, pp. 2979–2984, 2002. DOI: https://doi.org/10.1109/ACC.2002.1025245.

    Google Scholar 

  9. F. E. Felicioni, S. J. Junco. A Lie algebraic approach to design of stable feedback control systems with varying sampling rate. In Proceedings of the 17th World Congress of the International Federation of Automatic Control, IFAC, Seoul, South Korea, pp. 4881–4886, 2008.

    Google Scholar 

  10. M. Schinkel, W. H. Chen. Control of sampled-data systems with variable sampling rate. International Journal of Systems Science, vol.37, no.9, pp.609–618, 2006. DOI: https://doi.org/10.1080/00207720600681161.

    Article  MathSciNet  Google Scholar 

  11. F. Felicioni. Stabilitéet Performance des Systèmes Distribués de Contrôle-Commande, Ph.D. dissertation, Nancy University, Lorraine, France, 2011. (in Spanish)

    Google Scholar 

  12. A. Sala. Computer control under time-varying sampling period: An LMI gridding approach. Automatica, vol.41, no. 12, pp.2077–2082, 2005. DOI: https://doi.org/10.1016/j.automatica.2005.05.017.

    Article  MathSciNet  Google Scholar 

  13. F. Pazos, A. Zanini, A. Bhaya. Stabilization of discrete-time linear systems with varying sampling rates via a state feedback controller. In Proceedings of the 14th Brazilian Conference on Dynamics, Control and Applications, Säo Carlos, Brazil, 2019. (to be published)

    Google Scholar 

  14. D. Liberzon, R. Tempo. Common Lyapunov functions and gradient algorithms. IEEE Transactions on Automatic Control, vol.49, no.6, pp.990–994, 2004. DOI: https://doi.org/10.1109/TAC.2004.829632.

    Article  MathSciNet  Google Scholar 

  15. B. Tang, Q. J. Zeng, D. F. He, Y. Zhang. Random stabilization of sampled-data control systems with nonuniform sampling. International Journal of Automation and Computing, vol.9, no.5, pp.492–500, 2012. DOI: https://doi.org/10.1007/s11633-012-0672-y.

    Article  Google Scholar 

  16. C. C. Hua, S. C. Yu, X. P. Guan. Finite-time control for a class of networked control systems with short time-varying delays and sampling jitter. International Journal of Automation and Computing, vol.12, no. 4, pp. 448–454, 2015. DOI: https://doi.org/10.1007/s11633-014-0849-7.

    Article  Google Scholar 

  17. C. C. Sun. Stabilization for a class of discrete-time switched large-scale systems with parameter uncertainties. International Journal of Automation and Computing, vol.16, no.4, pp.543–552, 2019. DOI: https://doi.org/10.1007/s11633-016-0966-6.

    Article  MathSciNet  Google Scholar 

  18. L. G. Wu, Y. B. Gao, J. X. Liu, H. Y. Li. Event-triggered sliding mode control of stochastic systems via output feedback. Automatica, vol.82, pp.79–92, 2017. DOI: https://doi.org/10.1016/j.automatica.2017.04.032.

    Article  MathSciNet  Google Scholar 

  19. J. X. Liu, L. G. Wu, C. W. Wu, W. S. Luo, L. G. Fran-quelo. Event-triggering dissipative control of switched stochastic systems via sliding mode. Automatica, vol. 103, pp. 261–273, 2019. DOI: https://doi.org/10.1016/j.automatica.2019.01.029.

    Article  MathSciNet  Google Scholar 

  20. M. T. Nair, N. Sukavanam, R. Katta. Computation of control for linear approximately controllable system using Tikhonov regularization. Numerical Functional Analysis and Optimization, vol.39, no.3, pp.308–321, 2018. DOI: https://doi.org/10.1080/01630563.2017.1361440.

    Article  MathSciNet  Google Scholar 

  21. R. Katta, G. D. Reddy, N. Sukavanam. Computation of control for linear approximately controllable system using weighted Tikhonov regularization. Applied Mathematics and Computation, vol.317, pp. 252–263, 2018. DOI: https://doi.org/10.1016/j.amc.2017.08.012.

    Article  MathSciNet  Google Scholar 

  22. R. Katta, N. Sukavanam. Computation of control for nonlinear approximately controllable system using Tikhonov regularization. Cogent Mathematics, vol.3, no. 1, Article number 1216241, 2016 DOI: https://doi.org/10.1080/23311835.2016.1216241.

    Google Scholar 

  23. M. Loaiza. A short introduction to Hilbert space theory. Journal of Physics: Conference Series, vol.839, no. 1, Article number 012002, 2017. DOI: https://doi.org/10.1088/1742-6596/839/1/012002.

    Google Scholar 

  24. M. T. Nair. On controllability of linear systems, Department of Mathematics, HT Madras, Madras, India, Technical Report, [Online], Available: https://math.iitm.ac.in/public_html/mtnair/IIST-control-mtn.pdf, November 28–29, 2012.

    Google Scholar 

  25. R. F. Curtain, H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, New York, USA: Springer-Verlag, 1995. DOI: https://doi.org/10.1007/978-1-4612-4224-6.

    Book  Google Scholar 

  26. C. T. Chen. Linear System Theory and Design, 3rd ed., New York, USA: Oxford University Press, 1999.

    Google Scholar 

  27. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, London, UK: Kluwer Academic Publishers, 1996.

    Book  Google Scholar 

  28. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, Washington, USA: Winston and Sons, 1977.

    MATH  Google Scholar 

  29. V. A. Morozov, Regularization Methods for Ill-Posed Problems, Boca Raton, USA: CRC Press, 1993.

    MATH  Google Scholar 

  30. G. H. Golub, M. Heath, G. Wahba. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, vol.21, no.2, pp.215–223, 1979. DOI: https://doi.org/10.1080/00401706.1979.10489751.

    Article  MathSciNet  Google Scholar 

  31. P. C. Hansen. The L-curve and its use in the numerical treatment of inverse problems. Computational Inverse Problems in Electrocardiology, P. Johnston, Ed., Southampton, UK: IMM, 2001.

    Google Scholar 

  32. P. C. Hansen, D. P. O’Leary. The use of the L-curve in the regularization of discrete ill-posed problems.. SIAM Journal on Scientific Computing, vol.14, no. 6, pp. 1487–1503, 1993. DOI: https://doi.org/10.1137/0914086.

    Article  MathSciNet  Google Scholar 

  33. F. Pazos, A. Bhaya. Adaptive choice of the Tikhonov regularization parameter to solve ill-posed linear algebraic equations via Liapunov optimizing control. Journal of Computational and Applied Mathematics, vol.279, pp. 123–132, 2015. DOI: https://doi.org/10.1016/j.cam.2014.10.022.

    Article  MathSciNet  Google Scholar 

  34. F. Bauer, M. A. Lukas. Comparingparameter choice methods for regularization of ill-posed problems. Mathematics and Computers in Simulation, vol.81, no. 9, pp. 1795–1841, 2011. DOI: https://doi.org/10.1016/j.matcom.2011.01.016.

    Article  MathSciNet  Google Scholar 

  35. K. Kunisch, J. Zou. Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, vol.14, no. 5, pp. 1247–1264, 1998. DOI: https://doi.org/10.1088/0266-5611/14/5/010.

    Article  MathSciNet  Google Scholar 

  36. H. Gfrerer. An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Mathematics of Computation, vol.49, no. 180, pp.507–522, 1987. DOI: https://doi.org/10.2307/2008325.

    Article  MathSciNet  Google Scholar 

  37. L. Reichel, G. Rodriguez. Old and new parameter choice rules for discrete ill-posed problems. Numerical Algorithms, vol.63, no. 1, pp.65–87, 2013. DOI: https://doi.org/10.1007/s11075-012-9612-8.

    Article  MathSciNet  Google Scholar 

  38. T. Reginska. A regularization parameter in discrete ill-posed problems. SIAM Journal on Scientific Computing, vol.17, no. 3, pp. 740–749, 1996. DOI: https://doi.org/10.1137/S1064827593252672.

    Article  MathSciNet  Google Scholar 

  39. D. Zhang, T. Z. Huang. Generalized Tikhonov regularization method for large-scale linear inverse problems. Journal of Computational Analysis and Applications, vol. 15, no. 7, pp. 1317–1331, 2013.

    Article  MathSciNet  Google Scholar 

  40. K. J. Åström, B. Wittenmark, Computer Controlled Systems: Theory and Design, Englewood Cliffs, USA: Prentice-Hall, 1984.

    Google Scholar 

  41. G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 3rd ed., Menlo Park, USA: Ad-dison-Wesley, 1997.

    MATH  Google Scholar 

  42. P. C. Hansen. Regularization tools version 4.0 for Matlab 7.3. Numerical Algorithms, vol.46, no.2, pp. 189–194, 2007. DOI: https://doi.org/10.1007/s11075-007-9136-9.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and helpful suggestions that have contributed to the improvement of the paper.

Author information

Authors and Affiliations

  1. Department of Electronics and Telecommunication Engineering, State University of Rio de Janeiro, Rua São Fransciso Xavier 524, Rio de Janeiro, 20550-900, Brazil

    Fernando Agustín Pazos

  2. Universidad de Buenos Aires - ITHES-UBA- CONICET, Departamento de Ingeniería Química, Pabellón de Industrias, Ciudad Universitaria, Buenos Aires, C1428EGA, Argentina

    Anibal Zanini

  3. Department of Electrical Engineering, Federal University of Rio de Janeiro, PEE/COPPE/UFRJ, Rio de Janeiro, 21945-970, Brazil

    Amit Bhaya

Authors
  1. Fernando Agustín Pazos
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Anibal Zanini
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Amit Bhaya
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Fernando Agustín Pazos.

Additional information

Fernando Agustín Pazos received the B. Sc. degree in electronic engineering from the University of Buenos Aires, Argentina in 1990, the M.Sc. and D.Sc. degrees in electrical engineering from the Federal University of Rio de Janeiro, Brazil in 2000 and 2007, respectively. He is a professor in the Electronic and Telecommunications Engineering Department, the State University of Rio de Janeiro, Brazil. He is an author of “Automação de Sistemas & Robótica” (Axcel Books, Rio de Janeiro, Brazil 2002).

His research interest is development of algorithms to solve several optimization problems interpreting them as closed loop control systems.

Anibal Zanini received the B.Eng. degree in electrical engineering from Universidad Nacional de Rosario, Argentina in 1977, and recived the Ph. D. degree in electrical engineering at the Escuela Técnica Superior de Ingenieros Industriales of the Universidad Politécnica de Madri, Spain in 1983. He has worked for 20 years as project leader and senior researcher in the automation area at metallurgical industry. Since 1998, his main activity is at Universidad de Buenos Aires. He is a professor at the Engineering School of “Universidad de Buenos Aires”.

His research interests include industrial control, identification and adaptive control.

Amit Bhaya received the B.Eng. degree in eletrical engineering from the Indian Institute of Technology, India in 1981, and received the Ph.D. degree in eletrical engineering from the University of California at Berkeley, USA in 1986. He is a professor in the Electrical Engineering Department, Graduate School of Engineering, the Federal University of Rio de Janeiro (COPPE/UFRJ), Brazil. He coauthored the books “Matrix Diagonal Stability in Systems and Computation” (Birkhäuser, 2000), and “Control Perspectives on Numerical Algorithms and Matrix Problems” (SIAM, 2006). He was an associate editor of the IEEE Transactions on Neural Networks and Learning Systems from 2010 to 2012.

His research interests include systems and control theory, parallel computation, neural networks, matrix stability theory, mathematical ecology and optimization of economic dynamic systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pazos, F.A., Zanini, A. & Bhaya, A. Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method. Int. J. Autom. Comput. 17, 453–463 (2020). https://doi.org/10.1007/s11633-019-1205-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11633-019-1205-8

Keywords