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Two-Dimensional Singular Continuous Systems, Admissibility Analysis and \(H_\infty\) Control

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Abstract

This paper investigates the admissibility and \(H_\infty\) control of two-dimensional (2D) continuous singular systems described by Singular Roesser Models (SRMs). Our novel approach transforms any 2D SRM system into an equivalent, easier-to-analyze form while preserving its properties. We reformulate the control input to enable independent control of each dynamic, offering extensive control possibilities. The admissibility of the open-loop system, including acceptability, non-impulsiveness, and stability, is investigated. An \(H_\infty\) static state-feedback (SSF) controller is designed using Linear Matrix Inequalities (LMIs), solving a strict LMI condition to ensure closed-loop admissibility and specified performance. The effectiveness of the proposed method is demonstrated through two numerical examples, which highlight significant improvements over existing approaches. The results confirm both the robustness and practical feasibility of the method across a wide range of parameters, underscoring its broader applicability.

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References

  1. O. Bachelier, T. Cluzeau, A. Rigaud, F.J.S. Alvarez, N. Yeganefar, On exponential stability of a class of descriptor continuous linear 2D Roesser models. Int. J. Control 96(6), 1582–1593 (2023). https://doi.org/10.1080/00207179.2022.2057872

    Article  MathSciNet  Google Scholar 

  2. L. Benhayoun, M. Benhayoun, A. Benzaouia, Admissibility and stabilization of singular continuous 2-D systems described by Roesser model. Multidim. Syst. Sign. Process 31, 673–687 (2020). https://doi.org/10.1007/s11045-019-00681-4

    Article  MathSciNet  Google Scholar 

  3. A. Benzaouia, A. Hmamed, F. Tadeo, Two-Dimensional Systems: From Introduction to State of the Art Notes in Control and Information Science. (Springer, Cham, 2016). https://doi.org/10.1007/978-3-319-20116-0

    Book  Google Scholar 

  4. D. Bouagada, P.V. Dooren, On the stability of 2-D state-space models. Numer Linear Algebra Appl 20(2), 198–207. https://doi.org/10.1002/nla.836

  5. B. Boukili, A. Hmamed, A. Benzaouia, A. El Hajjaji, \(H_\infty\) state control for 2-D fuzzy FM systems with stochastic perturbation. J. Circuits Syst. Signal Process. 34(3), 779–796 (2015). https://doi.org/10.1007/s00034-014-9889-z

    Article  Google Scholar 

  6. B. Boukili, A. Hmamed, F. Tadeo, Robust \(H_\infty\) filtering for 2-D discrete Roesser system. J. Control Autom. Electr. Syst. 27(5), 497–505 (2016). https://doi.org/10.1007/s40313-016-0251-5

    Article  Google Scholar 

  7. B. Boukili, M. El Mallahi, A. El-Amrani, A. Hmamed, I. Boumhidi, \(H_\infty\) deconvolution filter for two-dimensional numerical systems using orthogonal moments. Optim. Control. Appl. Methods 42(5), 1337–1348 (2018). https://doi.org/10.1002/oca.2730

    Article  MathSciNet  Google Scholar 

  8. B. Boukili, M. ElMallahi, A. El-Amrani, A. Hmamed, I. Boumhidi, A new approach for \(H_\infty\) deconvolution filtering of 2-D systems described by the Fornasini–Marchesini and discrete moments. Pattern. Anal. Applic. 25, 63–76 (2022). https://doi.org/10.1007/s10044-021-01030-7

    Article  Google Scholar 

  9. M. Chaabane, F. Tadeo, D. Mehdi, M. Souissi, Robust admissibilization of descriptor systems by static output-feedback: an LMI approach. Math. Probl. Eng. (2011). https://doi.org/10.1155/2011/960981

    Article  MathSciNet  Google Scholar 

  10. S.F. Chen, Stability analysis and stabilization of 2-D singular Roesser models. Appl. Math. Comput. 250, 779–791 (2015). https://doi.org/10.1016/j.amc.2014.11.002

    Article  MathSciNet  Google Scholar 

  11. L. Dami, A. Benzaouia, \(H_\infty\) control for 2-D continuous singular systems described by the roesser model, in 11th International Conference on Systems and Control, (2023) p. 975–979 https://doi.org/10.1109/ICSC58660.2023.10449826

  12. G.R. Duan, Analysis and Design of Descriptor Linear Systems, vol. 23 (Springer, Berlin, 2010). https://doi.org/10.1007/978-1-4419-6397-0

    Book  Google Scholar 

  13. A. El Asraoui, T. Zoulagh, B. Boukili, N. Chaibi, Admissibility analysis for 2-D SRM continuous systems, in 5th International Conference on Innovative Research in Applied Science, Engineering and Technology (IRASET) FEZ, Morocco, (2025), https://doi.org/10.1109/IRASET64571.2025.11007981

  14. A. El Asraoui, T. Zoulagh, B. Boukili, N. Chaibi, Static output feedback control for 2-D SRM continuous systems, in 33rd Mediterranean Conference on Control and Automation (MED), (2025), https://doi.org/10.1109/MED64031.2025.11073398

  15. A. El-Amrani, B. Boukili, A. El Hajjaji, A. Hmamed, I. Boumhidi, Robust finite frequency \(H_\infty\) model reduction for uncertain 2-D continuous systems, in 58th IEEE Conference on Decision and Control, CDC 2019, Nice, France (2019) p. 5629–5634. https://doi.org/10.1109/CDC40024.2019.9029526

  16. A. El-Amrani, B. Boukili, A. Hmamed, A. El Hajjaji, I. Boumhidi, Robust \(H_\infty\) filtering for 2-D continuous systems with finite frequency specifications. Int. J. Syst. Sci. 49(1), 43–57 (2018). https://doi.org/10.1080/00207721.2017.1391960

    Article  MathSciNet  Google Scholar 

  17. M. Elloumi, M. Ghamgui, D. Mehdi, F. Tadeo, M. Chaabane, Stability and stabilization of 2-D singular systems: a strict LMI approach. Circuits Syst. Signal Process 38(1), 3041–3057 (2019). https://doi.org/10.1007/s00034-018-01019-4

    Article  Google Scholar 

  18. K. Galkowski, LMI based stability analysis for 2-D continuous systems, in International Conference on Electronics Circuits and Systems3, 923–926 (2002). https://doi.org/10.1109/ICECS.2002.1046399

  19. K. Galkowski, A perspective on singularity in 2-D linear systems. Multidimens. Syst. Signal Process 11(1), 83–108 (2000). https://doi.org/10.1023/A:1008438730274

    Article  MathSciNet  Google Scholar 

  20. M. Ghamgui, M. Elloumi, M. Allouche, M. Chaabane, \(H_\infty\) control for 2D singular continuous systems. Appl. Sci. 14(10), 4225 (2024). https://doi.org/10.3390/app14104225

    Article  Google Scholar 

  21. P. Gokul, A. Kashkynbayev, M. Prakash, R. Rajan, Finite-time contractive stabilization for fractional-order switched systems via event-triggered impulse control. Commun. Nonlinear Sci. Numer. Simul. 143, 108656 (2025). https://doi.org/10.1016/j.cnsns.2025.108658

    Article  MathSciNet  Google Scholar 

  22. P. Gokul, G. Soundararajan, A. Kashkynbayev, R. Rakkiyappan, Finite-time contractive stability for fractional-order nonlinear systems with delayed impulses: applications to neural networks. Neurocomputing, 610(28), (2024). https://doi.org/10.1016/j.neucom.2024.128599

    Article  Google Scholar 

  23. T. Kaczorek, in Two-Dimensional Linear Systems, Notes in Control and Information Science (Heidelberg: Springer Berlin, 1985). https://doi.org/10.1007/BFb0005617 (2011). https://doi.org/10.1002/nla.836

  24. T. Kaczorek, Singular continuous 2-D linear systems. Bull. Acad. Sci. Tech. Sci. 42(4), 565–577 (1994)

    Google Scholar 

  25. S. Kririm, A. Hmamed, F. Tadeo, Robust \(H_\infty\) filtering for uncertain 2-D singular Roesser models. Circuits Syst. Signal Process. 34(7), 2213–2235 (2015). https://doi.org/10.1007/s00034-015-9967-x

    Article  MathSciNet  Google Scholar 

  26. S. Kririm, A. Hmamed, F. Tadeo, Analysis and design of \(H_\infty\) controllers for 2-D singular systems with delays. Circuits Syst. Signal Process. 35(5), 1579–1592 (2016). https://doi.org/10.1007/s00034-015-0139-9

    Article  Google Scholar 

  27. D. Liu, Lyapunov stability of two-dimensional digital filters with overflow nonlinearities. IEEE Trans. Circuits Syst. I Fundam. Theory Appl 45(5), 574–577 (1998). https://doi.org/10.1109/81.668870

    Article  MathSciNet  Google Scholar 

  28. D. Liu, A.N. Michel, Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities. IEEE Trans. Circuits Syst. I Fundam. Theory Appl 41, 127–137 (1994). https://doi.org/10.1109/81.269049

    Article  MathSciNet  Google Scholar 

  29. W.S. Lu, Two-Dimensional Digital Filters, vol. 80 (CRC Press, Boca Raton, 1992). https://doi.org/10.1201/9781003066637

    Book  Google Scholar 

  30. W. Marszalek, Two-dimensional state-space discrete models for hyperbolic partial differential equations. Appl. Math. Model. 8(1), 11–14 (1984). https://doi.org/10.1016/0307-904X(84)90170-7

    Article  MathSciNet  Google Scholar 

  31. N. Nafie, A. El-amrani, A. Elhajjaji, N. Chaibi, B. Boukili, Dissipativity-based model reduction for two-dimensional periodic systems. IET Control Theory Appl. 19(1), e70066 (2025), https://doi.org/10.1049/cth2.70066

    Article  MathSciNet  Google Scholar 

  32. H. Qobbi, T. Zoulagh, B. Boukili, K.A. , Barbosa, A. Hmamed, N. Chaibi, Further results on the R–R protocol based on \(H_\infty\) estimator for 2-D FMII systems with R–O uncertainties. Circuits Syst. Signal Process. (2024). https://doi.org/10.1007/s00034-024-02948-z

    Article  Google Scholar 

  33. F.A. Rihan, K. Udhayakumar, Optimal control of glucose-insulin dynamics via delay differential model with fractional-order. Alex. Eng. J. 114, 243 (2025). https://doi.org/10.1016/j.aej.2024.11.071

    Article  Google Scholar 

  34. S.M. Saadni, M. Chaabane, D. Mehdi, O. Bachelier, Robust stability and stabilization of a class of singular systems with multiple time-varying delays. IFAC Proc. 38(1), 161–166 (2005). https://doi.org/10.3182/20050703-6-CZ-1902.00971

    Article  Google Scholar 

  35. K. Udhayakumar, F.A. Rihan, X. Li, R. Rakkiyappan, Quasi-bipartite synchronisation of multiple inertial signed delayed neural networks under distributed event-triggered impulsive control strategy. IET Control Theory Appl. 15(2), 1615 (2021). https://doi.org/10.1049/cth2.12146

    Article  MathSciNet  Google Scholar 

  36. H. Xu., Y. Zou, \(H_\infty\) control for 2-D singular delayed systems. Int. J. Syst. Sci. 42(4), 609–619 (2011). https://doi.org/10.1080/00207720902974728

    Article  MathSciNet  Google Scholar 

  37. Y. Zou, S.L. Campbell, The jump behavior and stability analysis for 2-D singular systems. Multidimension. Syst. Signal Process. 11, 321–338 (2011). https://doi.org/10.1023/A:1008429611994

    Article  MathSciNet  Google Scholar 

  38. Y. Zou, H. Xu, Duality of 2-D singular systems of roesser models. J. Control Theory Appl. 5, 37–41 (2007). https://doi.org/10.1007/s11768-004-4189-y

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Sciences Dhar El Mehraz, LESSI-Lab, Sidi Mohamed Ben Abdellah University, Fès, Morocco

    Abderrahim El Asraoui, Abdelaziz Hmamed, Bensalem Boukili & Noreddine Chaibi

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  1. Abderrahim El Asraoui
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  2. Abdelaziz Hmamed
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  3. Bensalem Boukili
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Correspondence to Bensalem Boukili.

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El Asraoui, A., Hmamed, A., Boukili, B. et al. Two-Dimensional Singular Continuous Systems, Admissibility Analysis and \(H_\infty\) Control. Circuits Syst Signal Process (2025). https://doi.org/10.1007/s00034-025-03430-0

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