Skip to main content
SpringerLink
Log in

New Criterion for the Realization of 2-D Interfered Digital Filters Described by the Fornasini–Marchesini Second Local State-Space Model

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

Abstract

The \(H_{\infty }\) stability problem for interfered 2-dimensional (2-D) digital filters (DFs) represented by the Fornasini–Marchesini second local state-space (FMSLSS) model with state saturation is investigated. By utilizing 2-D Lyapunov theory and the ‘passivity property’ of multiple saturation nonlinearities, a novel linear matrix inequality (LMI)-based \(H_{\infty }\) stability criterion is developed. The criterion confirms that the underlying 2-D system is asymptotically stable and has a specified \(H_{\infty }\) performance against external disturbance. The new criterion turns out to be less conservative than an existing \(H_{\infty }\) criterion. Finally, several examples are given to illustrate the efficacy of the criterion.

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

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. T.T. Aboulnasr, M.M. Fahmy, Finite-Word-Length Effects in Two-Dimensional Digital Systems, in Multidimensional Systems: Techniques and Applications (Marcel Dekker, New York, 1986)

    MATH  Google Scholar 

  2. N. Agarwal, H. Kar, A note on stability analysis of 2-D linear discrete systems based on the Fornasini-Marchesini second model: stability with asymmetric Lyapunov matrix. Digit. Signal Process. 37, 109–112 (2015)

    Google Scholar 

  3. N. Agarwal, H. Kar, New results on saturation overflow stability of 2-D state-space digital filters. J. Frankl. Inst. 353(12), 2743–2760 (2016)

    MathSciNet  MATH  Google Scholar 

  4. N. Agarwal, H. Kar, New results on saturation overflow stability of 2-D state-space digital filters described by the Fornasini–Marchesini second model. Signal Process. 128, 504–511 (2016)

    Google Scholar 

  5. C.K. Ahn, \(l_2-l_{\infty}\) elimination of overflow oscillations in 2-D digital filters described by Roesser model with external interference. IEEE Trans. Circuits Syst. II 60(6), 361–365 (2013)

    Google Scholar 

  6. C.K. Ahn, Two-dimensional digital filters described by Roesser model with interference attenuation. Digit. Signal Process. 23(4), 1296–1302 (2013)

    MathSciNet  Google Scholar 

  7. C.K. Ahn, \(l_2-l_{\infty}\) suppression of limit cycles in interfered two-dimensional digital filters: a Fornasini-Marchesini model case. IEEE Trans. Circuits Syst. II 61(8), 614–618 (2014)

    Google Scholar 

  8. C.K. Ahn, New passivity criterion for limit cycle oscillation removal of interfered 2D digital filters in the Roesser form with saturation nonlinearity. Nonlinear Dyn. 78(1), 409–420 (2014)

    MathSciNet  MATH  Google Scholar 

  9. C.K. Ahn, A new realization criterion for 2-D digital filters in the Fornasini-Marchesini second model with interference. Signal Process. 104, 225–231 (2014)

    Google Scholar 

  10. C.K. Ahn, P. Shi, M.V. Basin, Two-dimensional dissipative control and filtering for Roesser model. IEEE Trans. Autom. Control 60(7), 1745–1759 (2015)

    MathSciNet  MATH  Google Scholar 

  11. C.K. Ahn, P. Shi, R. Yang, Two-dimensional Hankel norm performance of Roesser-type filters. IEEE Trans. Circuits Syst. II (2017). https://doi.org/10.1109/TCSII.2017.2700887

    Article  Google Scholar 

  12. C.K. Ahn, L. Wu, P. Shi, Stochastic stability analysis for 2-D Roesser systems with multiplicative noise. Automatica 69, 356–363 (2016)

    MathSciNet  MATH  Google Scholar 

  13. D. Bors, S. Walczak, Application of 2D systems to investigation of a process of gas filtration. Multidim. Syst. Sign. Process. 23(1), 119–130 (2012)

    MathSciNet  MATH  Google Scholar 

  14. T. Bose, D.A. Trautman, Two’s complement quantization in two-dimensional state-space digital filters. IEEE Trans. Signal Process. 40(10), 2589–2592 (1992)

    Google Scholar 

  15. S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)

    MATH  Google Scholar 

  16. R.N. Bracewell, Two-Dimensional Imaging (Prentice Hall, Englewood Cliffs, 1995)

    MATH  Google Scholar 

  17. H.J. Butterweck, J.H.F. Ritzerfeld, M.J. Werter, Finite Wordlength Effects in Digital Filters: A Review (EUT report 88-E-205 Eindhoven University of Technology, Eindhoven, The Netherlands, 1988)

    Google Scholar 

  18. Y. Chen, S. Fei, Y. Li, Robust stabilization for uncertain saturated time-delay systems: a distributed-delay-dependent polytopic approach. IEEE Trans. Autom. Control 62(7), 3455–3460 (2017)

    MathSciNet  MATH  Google Scholar 

  19. D.A. Dewasurendra, P.H. Bauer, A novel approach to grid sensor networks, in Proceedings 15th IEEE International Conference on Electronics, Circuits and Systems, Saint Julian's, Malta, (2008) pp. 1191–1194

  20. A. Dey, H. Kar, An LMI based criterion for the global asymptotic stability of 2-D discrete state-delayed systems with saturation nonlinearities. Digit. Signal Process. 22(4), 633–639 (2012)

    MathSciNet  Google Scholar 

  21. C. Du, L. Xie, \(H_{\infty}\) Control and Filtering of Two-Dimensional Systems (Springer-Verlag, New York, 2002)

    MATH  Google Scholar 

  22. E. Fornasini, A 2-D systems approach to river pollution modelling. Multidim. Syst. Sign. Process. 2(3), 233–265 (1991)

    MathSciNet  MATH  Google Scholar 

  23. E. Fornasini, G. Marchesini, Doubly-indexed dynamical systems: state-space models and structural properties. Math. Syst. Theory 12(1), 59–72 (1978)

    MathSciNet  MATH  Google Scholar 

  24. P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox For Use with MATLAB (The Mathworks Inc., Natick, 1995)

    Google Scholar 

  25. I. Ghous, Z. Xiang, H.R. Karimi, State feedback control for 2-D switched delay systems with actuator saturation in the second FM model. Circuits Syst. Signal Process. 34(7), 2167–2192 (2015)

    MathSciNet  MATH  Google Scholar 

  26. T. Hinamoto, Stability of 2-D discrete systems described by the Fornasini-Marchesini second model. IEEE Trans. Circuits Syst. I 44(3), 254–257 (1997)

    MathSciNet  Google Scholar 

  27. V.K.R. Kandanvli, H. Kar, Global asymptotic stability of 2-D digital filters with a saturation operator on the state-space. IEEE Trans. Circuits Syst. II 67(11), 2742–2746 (2020)

    Google Scholar 

  28. V.K.R. Kandanvli, H. Kar, Novel realizability criterion for saturation overflow oscillation-free 2-D digital filters based on the Fornasini-Marchesini second model. Circuits Syst. Signal Process. 40(10), 5220–5233 (2021)

    Google Scholar 

  29. H. Kar, V. Singh, Stability analysis of 2-D state-space digital filters with overflow nonlinearities. IEEE Trans. Circuits Syst. I 47(4), 598–601 (2000)

    Google Scholar 

  30. H. Kar, V. Singh, Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities. IEEE Trans. Signal Process. 49(5), 1097–1105 (2001)

    Google Scholar 

  31. H. Kar, V. Singh, Stability analysis of 2-D digital filters described by the Fornasini-Marchesini second model using overflow nonlinearities. IEEE Trans. Circuits Syst. I 48(5), 612–617 (2001)

    Google Scholar 

  32. H. Kar, V. Singh, Robust stability of 2-D discrete systems described by the Fornasini-Marchesini second model employing quantization/overflow nonlinearities. IEEE Trans. Circuits Syst. II 51(11), 598–602 (2004)

    Google Scholar 

  33. J. Liang, T. Huang, T. Hayat, F. Alsaadi, \(H_{\infty}\) filtering for two-dimensional systems with mixed time delays, randomly occurring saturations and nonlinearities. Int. J. Gen. Syst. 44(2), 226–239 (2015)

    MathSciNet  MATH  Google Scholar 

  34. D. Liu, Lyapunov stability of two-dimensional digital filters with overflow nonlinearities. IEEE Trans. Circuits Syst. I 45(5), 574–577 (1998)

    MathSciNet  MATH  Google Scholar 

  35. J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei,Taiwan, (2004), pp. 284–289

  36. W.-S. Lu, On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Trans. Circuits Syst. I 41(10), 665–669 (1994)

    Google Scholar 

  37. J. Monteiro, R.V. Leuken, Integrated Circuit and System Design: Power and Timing Modeling, Optimization and Simulation (Springer, Berlin, 2010)

    Google Scholar 

  38. T. Ooba, On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities. IEEE Trans. Circuits Syst. I 47(8), 1263–1265 (2000)

    MathSciNet  Google Scholar 

  39. T. Ooba, Asymptotic stability of two-dimensional discrete systems with saturation nonlinearities. IEEE Trans. Circuits Syst. I 60(1), 178–188 (2013)

    MathSciNet  MATH  Google Scholar 

  40. M. Rehan, M. Tufail, M.T. Akhtar, On elimination of overflow oscillations in linear time-varying 2-D digital filters represented by a Roesser model. Signal Process. 127, 247–252 (2016)

    Google Scholar 

  41. R.P. Roesser, A discrete state-space model for linear image processing. IEEE Trans. Autom. Control 20(1), 1–10 (1975)

    MathSciNet  MATH  Google Scholar 

  42. E. Rogers, K. Galkowski, W. Paszke, K.L. Moore, P.H. Bauer, L. Hladowski, P. Dabkowski, Multidimensional control systems: case studies in design and evaluation. Multidim. Syst. Sign. Process. 26(4), 895–939 (2015)

    MathSciNet  MATH  Google Scholar 

  43. H. Shen, J. Wang, J.H. Park, Z.-G. Wu, Condition of the elimination of overflow oscillations in two-dimensional digital filters with external interference. IET Signal Process. 8(8), 885–890 (2014)

    Google Scholar 

  44. J.M.G.D. Silva, S. Tarbouriech, Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Trans. Autom. Control 50(1), 106–111 (2005)

    MathSciNet  MATH  Google Scholar 

  45. S. Singh, H. Kar, Realization of two’s complement overflow oscillation-free 2D Lipschitz nonlinear digital filters. J. Control Autom. Electr. Syst. 32, 1540–1552 (2021)

    Google Scholar 

  46. V. Singh, Stability analysis of 2-D discrete systems described by the Fornasini–Marchesini second model with state saturation. IEEE Trans. Circuits Syst. II 55(8), 793–796 (2008)

    Google Scholar 

  47. V. Singh, Stability analysis of 2-D linear discrete systems based on the Fornasini–Marchesini second model: stability with asymmetric Lyapunov matrix. Digit. Signal Process. 26, 183–186 (2014)

    Google Scholar 

  48. Y. Tsividis, Mixed Analog-Digital VLSI Devices and Technology (World Scientific Publishing, Singapore, 2002)

    Google Scholar 

  49. B. Wang, Q. Zhu, L. Xu, Expected power bound and stability of two-dimensional digital filters with multiplicative noise in the FMLSS model. J. Frankl. Inst. 358(4), 2500–2514 (2021)

    MathSciNet  MATH  Google Scholar 

  50. Z.-G. Wu, Y. Shen, P. Shi, Z. Shu, H. Su, \(H_{\infty}\) control for 2D Markov jump systems in Roesser model. IEEE Trans. Autom. Control 64(1), 427–432 (2019)

    MATH  Google Scholar 

  51. L. Wu-Sheng, E.B. Lee, Stability analysis for two-dimensional systems via a Lyapunov approach. IEEE Trans. Circuits Syst. 32(1), 61–68 (1985)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Communication Engineering, C. V. Raman Global University, Bhubaneswar, 752054, India

    Mani Kant Kumar

  2. Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, 211004, India

    Haranath Kar

Authors
  1. Mani Kant Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haranath Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mani Kant Kumar.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, M.K., Kar, H. New Criterion for the Realization of 2-D Interfered Digital Filters Described by the Fornasini–Marchesini Second Local State-Space Model. Circuits Syst Signal Process 42, 3117–3137 (2023). https://doi.org/10.1007/s00034-022-02248-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02248-4

Keywords

Search

Navigation