Skip to main content
SpringerLink
Log in

Stability of 2-D Discrete Systems in the Presence of Saturation Function and Delays

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

A variety of economically relevant processes are better modeled by using two-dimensional (2-D) models, which is the case, for instance, in metal rolling and the pharmaceutical industries. Typically Roesser and Fornasini Marchesini Local State Space (FMSLSS) models incorporate delayed states leading to a challenging stability analysis task. The main contribution of this work is to provide new convex stability analysis criteria for discrete-time FMSLSS models under both saturating function and time-varying delays. In particular, the saturating function tailors the digital control behavior, capturing the effect of finite wordlength. With Wirtinger and reciprocally convex sum lemmas, these new conditions can achieve less conservative results than other recent literature methods. Consequently, as illustrated by examples, this proposal enlarges the class of systems that can be stable-certified, still keep reasonable the associated computational effort.

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

Similar content being viewed by others

References

  1. F. Wang, Z. Wang, J. Liang, and X. Liu, “Event-triggered recursive filtering for shift-varying linear repetitive processes,” IEEE Transactions on Cybernetics, vol. 50, no. 4, 1761–1770, 2020.

    Article  Google Scholar 

  2. S. B. Mohamed, B. Boussaid, M. N. Abdelkrim, and C. Tahri, “1D and 2D modelling of a chemical process using the FM-II model: Rotary drum filter,” Transactions of the Institute of Measurement and Control, vol. 40, no. 4, 1309–1319, 2018.

    Article  Google Scholar 

  3. T. Zhou, “Regularity/controllability/observability of an NDS with descriptor form subsystems and generalized LFTs,” Automatica, vol. 121, 109153, 2020.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Chang, C. Hua, and C. Z. Yong, “F-M II model realization of 2D radar based on array theory method,” in 2020 IEEE 6th International Conference on Control Science and Systems Engineering (ICCSSE), pp. 10–16, 2020.

    Google Scholar 

  5. P. Kokil, S. Jogi, C. K. Ahn, and H. Kar, “Stability of digital filters with state-delay and external interference,” Circuits, Systems, and Signal Processing, vol. 40, pp. 3866–3883, 2021.

    Article  MATH  Google Scholar 

  6. V. K. R. Kandanvli and H. Kar, “Global asymptotic stability of 2-D digital filters with a saturation operator on the state-space,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 67, no. 11}, 2742–2746, 2020.

  7. P. Kokil, S. Jogi, C. K. Ahn, and H. Kar, “An improved local stability criterion for digital filters with interference and overflow nonlinearity,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 67, no. 3}, 595–599, 2020.

  8. S. Pandey and S. K. Tadepalli, “Improved criterion for stability of 2-D discrete systems involving saturation nonlinearities and variable delays,” ICIC Express Letters, vol. 15, no. 3, 273–283, 2021.

    Google Scholar 

  9. K. Singh, V. K. R. Kandanvli, and H. Kar, “Delay partitioning approach to the robust stability of discrete-time systems with finite wordlength nonlinearities and time-varying delays,” Transactions of the Institute of Measurement and Control, vol. 43, no. 4, 958–974, 2021.

    Article  Google Scholar 

  10. P. Rani, M. K. Kumar, and H. Kar, “Hankel norm performance of interfered fixed-point state-space digital filters with quantization/overflow nonlinearities,” Circuits, Systems, and Signal Processing, vol. 38, pp. 3762–3777, 2019.

    Article  Google Scholar 

  11. C. K. Ahn and H. Kar, “Passivity and finite-gain performance for two-dimensional digital filters: The FM LSS model case,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 62, no. 9}, 871–875, 2015.

  12. J. Liu and J. Zhang, “Note on stability of discrete-time time-varying delay systems,” IET Control Theory & Applications, vol. 6, no. 2, 335–339, 2012.

    Article  MathSciNet  Google Scholar 

  13. P. T. Nam, P. N. Pathirana, and H. Trinh, “Discrete wirtinger-based inequality and its application,” Journal of the Franklin Institute, vol. 352, pp. 1893–1905, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Seuret, F. Gouaisbaut, and E. Fridman, “Stability of discrete-time systems with time-varying delays via a novel summation inequality,” IEEE Transactions on Automatic Control, vol. 60, no. 10, 2740–2745, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. V. Hien and H. Trinh, “New finite-sum inequalities with applications to stability of discrete time-delay systems,” Automatica, vol. 71, pp. 197–201, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Li, Z. Xiang, and J. Zhang, “Dwell-time conditions for exponential stability and standard l1-gain performance of discrete-time singular switched positive systems with timevarying delays,” Nonlinear Analysis: Hybrid Systems, vol. 38, 100939, 2020.

    MathSciNet  MATH  Google Scholar 

  17. S. K. Tadepalli and V. J. S. Leite, “Robust stabilization of uncertain 2-D discrete delayed systems,” Journal of Control, Automation, and Electrical Systems, vol. 29, no. 5}, 280–291, 2018.

  18. P. Kokil, “An improved criterion for the global asymptotic stability of 2-D discrete state-delayed systems with saturation nonlinearities,” Circuits, Systems, and Signal Processing, vol. 36, no. 6, 2209–2222, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. K. Tadepalli, V. K. R. Kandanvli, and H. Kar, “Stability criterion for uncertain 2-D discrete systems with intervallike time-varying delay employing quantization/overflow nonlinearities,” Turkish Journal of Electrical Engineering & Computer Sciences, vol. 24, no. 5, 3543–3551, 2016.

    Article  Google Scholar 

  20. A. Dey and H. Kar, “LMI-based criterion for robust stability of 2-D discrete systems with interval time-varying delays employing quantisation/overflow nonlinearities,” Multidimensional Systems and Signal Processing, vol. 25, no. 3, 473–492, 2014.

    Article  MATH  Google Scholar 

  21. S. Huang and Z. Xiang, “Delay-dependent robust h∞ control for 2-D discrete nonlinear systems with state delays,” Multidimensional Systems and Signal Processing, vol. 25, pp. 775–794, 2014.

    Article  MATH  Google Scholar 

  22. P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, 235–238, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Shen, Z. Yuan, and X. Wang, “Stability analysis for digital filters with multiple saturation nonlinearities,” Automatica, vol. 48, no. 10, 2717–2720, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Lofberg, “Yalmip: A toolbox for modeling and optimization in MATLAB,” Proc. of Computer Aided Control Systems Design Conference, Taipei, Taiwan, pp. 284–289, 2004.

    Google Scholar 

  25. J. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11–12, pp. 625–653, 1999, version 1.05 available from http://fewcal.kub.nl/sturm.

    Article  MathSciNet  MATH  Google Scholar 

  26. S.-F. Chen, “Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities,” Signal Processing, vol. 90, no. 7, 2265–2275, 2010.

    Article  MATH  Google Scholar 

  27. V. K. R. Kandanvli and H. Kar, “Novel realizability criterion for saturation overflow oscillation-free 2-D digital filters based on the fornasini-marchesini second model,” Circuits, Systems, and Signal Processing, vol. 40, pp. 520–5233, 2021.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Telecommunication Engineering, Bhilai Institute of Technology, Durg, India

    Suchitra Pandey

  2. Department of Electronics Engineering, National Institute of Technology, Srinagar, Uttarakhand, India

    Siva Kumar Tadepalli & Rishi Nigam

  3. Department of Mechatronics Engineering, CEFET-MG, Campus Divinopolis, Rua Alvares Azevedo, 400, Divinopolis, MG, 35503-822, Brazil

    Valter J. S. Leite

  4. Department of Electrical and Electronics Engineering, Bhilai Institute of Technology, Durg, India

    Surekha Bhusnur

Authors
  1. Suchitra Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siva Kumar Tadepalli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valter J. S. Leite
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rishi Nigam
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Surekha Bhusnur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suchitra Pandey.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author(s) received no financial support for the research, authorship, and/or publication of this article from any funding agency in the public, commercial, or not-for-profit sectors. The authors thank the Editor and the anonymous Reviewers for their constructive comments and suggestions.

Suchitra Pandey was born in Chhattisgarh, India in 1978. She received her B.E. degree from Pt. Ravi Shankar Shukla University, India, in 2003 and an M.Tech. degree from Bhilai Institute of Technology, Durg (BIT, Durg) affiliated to Chhattisgarh Swami Vivekanand Technical University (CSVTU), Bhilai, India, in 2008. She is currently working towards her Ph.D. degree at CSVTU, Bhilai. Her research interests include delayed systems, finite wordlength effects, multidimensional systems, and nonlinear systems.

Siva Kumar Tadepalli was born in Warangal, India, in 1978. He received his B.E. degree from Pt. Ravi Shankar Shukla University, India, in 2002 and an M.Tech. and Ph.D. degrees from Motilal Nehru National Institute of Technology (MNNIT), Allahabad, India, in 2009 and 2015, respectively. He has held the post of Assistant Professor, Associate Professor, and Professor at Bhilai Institute of Technology, Durg, India. He is currently working as an Assistant Professor in the Department of Electronics Engineering at National Institute of Technology, Uttarakhand, India. He is serving as an Associate Editor of the Journal of Control Automation and Electrical Systems. His current research interests include finite wordlength effects, multi-dimensional systems, and delayed systems and robotics.

Valter J. S. Leite úna (MG, Brazil). He received his Ph.D. degree in electrical engineering and in automatique et informatique industrielle from the University of Campinas (Brazil) and from the INSA de Toulouse (France), respectively, in 2005. He has been with CEFET-MG since 1997 and currently, he is an Associate Professor in the Department of Mechatronic Engineering at campus Divinópolis (MG, Brazil). V. J. S. Leite serves as an Associate Editor for the International Journal of Robust and Nonlinear Control, International Journal of Control, Automation and Electrical Systems, and Journal of Systems and Control Engineering. He authored more than 160 papers in journals, book chapters, and conferences. His main research interests include robust control, delay systems, control of LPV systems, fuzzy T-S systems, and constrained systems.

Rishi Nigam was born in Lucknow, India in 1990. He received his B.Tech. degree from National Institute of Technology (NIT), Srinagar, in 2012, and an M.Tech. degree from NIT Agartala, in 2017 and pursuing a Ph.D. degree from NIT Uttarakhand since 2019. He has worked as Assistant Professor under National Project Implementation Unit (NPIU), Director General of Training (DGT), Ministry of Skill Development and Entrepreneurship, Government of India, (TEQIP-3) from January 2018 till 2021. His current research interests include nonlinear systems, delayed systems, and one dimensional and multidimensional systems.

Surekha Bhusnur is a Professor and Head of Electrical and Electronics Engineering Department at Bhilai Institute of Technology, Durg, Chhattisgarh, India. She holds a Ph.D. degree for the research work on robust control using two loop control structure and coefficient diagram method. She owns membership of The Institution of Engineers, India. Her areas of interests are control systems, computer aided power system analysis, and fuzzy logic systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pandey, S., Tadepalli, S.K., Leite, V.J.S. et al. Stability of 2-D Discrete Systems in the Presence of Saturation Function and Delays. Int. J. Control Autom. Syst. 21, 788–799 (2023). https://doi.org/10.1007/s12555-021-1061-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-021-1061-5

Keywords

Search

Navigation