Abstract
This paper deals with the problem of robust \(H_{\infty }\) filtering for uncertain 2-D discrete systems, the parameter uncertainties are assumed to reside in a polytopic region. Firstly, a new \(H_{\infty }\) performance analysis condition for the filtering error system is derived by exploiting a new structure of the Lyapunov function, and some analysis techniques. Secondly, based on the obtained condition, both parameter-independent and parameter-dependent \(H_{\infty }\) filters that ensure the robust asymptotic stability and a prescribed \(H_{\infty }\) performance level of the corresponding filtering error systems are designed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are presented to show that our results are less conservative than some existing ones.
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Badie, K., Alfidi, M., Tadeo, F., & Chalh, Z. (2018a). Delay-dependent stability and \(H_ {\infty }\) performance of 2-D continuous systems with delays. Circuits, Systems, and Signal Processing, 37(12), 5333–5350.
Badie, K., Alfidi, M., & Chalh, Z. (2018b). New relaxed stability conditions for uncertain two-dimensional discrete systems. Journal of Control, Automation and Electrical Systems, 29(6), 661–669.
Badie, K., Alfidi, M., & Chalh, Z. (2019a). Robust \(H_{\infty }\) control for 2-D discrete state delayed systems with polytopic uncertainties. Multidimensional Systems and Signal Processing, 30(3), 1327–1343.
Badie, K., Alfidi, M., & Chalh, Z. (2019). Exponential stability analysis for 2D discrete switched systems with state delays. Optimal Control Applications and Methods, 40(1088), 1103. https://doi.org/10.1002/oca.2537.
Barbosa, K. A., De Souza, C. E., & Trofino, A. (2005). Robust \(H_{2}\) filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions. Systems & Control Letters, 54(3), 251–262.
Benhayoun, M., Mesquine, F., & Benzaouia, A. (2013). Delay-dependent stabilizability of 2D delayed continuous systems with saturating control. Circuits, Systems, and Signal Processing, 32(6), 2723–2743.
Boukili, B., Hmamed, A., & Tadeo, F. (2016). Robust \(H_ {\infty }\) Filtering for 2-D discrete roesser systems. Journal of Control, Automation and Electrical Systems, 27(5), 497–505.
Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory (Vol. 15). In SIAM.
Chang, X. H., Park, J. H., & Tang, Z. (2015). New approach to \(H_ {\infty }\) filtering for discrete-time systems with polytopic uncertainties. Signal Processing, 113, 147–158.
de Souza, C. E., Xie, L., & Coutinho, D. F. (2010). Robust filtering for 2-D discrete-time linear systems with convex-bounded parameter uncertainty. Automatica, 46(4), 673–681.
Du, C., Xie, L., & Soh, Y. C. (2000). \(H_ {\infty }\) filtering of 2-D discrete systems. IEEE Transactions on Signal Processing, 48(6), 1760–1768.
Du, C., & Xie, L. (2002). \(H_{\infty }\) Control and filtering of two-dimensional systems (Vol. 278). Springer, Berlin
Duan, Z., Shen, J., Ghous, I., & Fu, J. (2019). \(H_{\infty }\) filtering for discrete-time 2D TS fuzzy systems with finite frequency disturbances. IET Control Theory & Applications, 13(13), 1983–1994.
Elsayed, A., & Grimble, M. J. (1989). A new approach to the H8 design of optimal digital linear filters. IMA Journal of Mathematical Control and Information, 6(2), 233–251.
El-Kasri, C., Hmamed, A., Alvarez, T., & Tadeo, F. (2012). Robust \(H_{\infty }\) filtering of 2D Roesser discrete systems: A polynomial approach. Mathematical Problems in Engineering (2012).
El-Kasri, C., Hmamed, A., Tissir, E. H., & Tadeo, F. (2013). Robust \(H_ {\infty }\) filtering for uncertain two-dimensional continuous systems with time-varying delays. Multidimensional Systems and Signal Processing, 24(4), 685–706.
El-Kasri, C., Hmamed, A., & Tadeo, F. (2014). Reduced-order \(H_ {\infty }\) filters for uncertain 2-D continuous systems, via LMIs and polynomial matrices. Circuits, Systems, and Signal Processing, 33(4), 1189–1214.
Feng, Z. Y., Xu, L., Wu, M., & She, J. H. (2012). \(H_ {\infty }\) static output feedback control of 2-D discrete systems in FM second model. Asian Journal of Control, 14(6), 1505–1513.
Gao, C. Y., Duan, G. R., & Meng, X. Y. (2008). Robust \(H_{\infty }\) filter design for 2D discrete systems in Roesser model. International Journal of Automation and Computing, 5(4), 413–418.
Gao, H., & Li, X. (2014). Robust filtering for uncertain systems: A parameter-dependent approach. Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-05903-7.
Ghous, I., & Xiang, Z. (2016). Robust state feedback \(H_{\infty }\) control for uncertain 2-D continuous state delayed systems in the Roesser model. Multidimensional Systems and Signal Processing, 27(2), 297–319.
Hmamed, A., Alfidi, M., Benzaouia, A., & Tadeo, F. (2008). LMI conditions for robust stability of 2D linear discrete-time systems. Mathematical Problems in Engineering,. https://doi.org/10.1155/2008/356124.
Hmamed, A., Kasri, C. E., Tissir, E. H., Alvarez, T., & Tadeo, F. (2013). Robust \(H_ {\infty }\) filtering for uncertain 2-D continuous systems with delays. International Journal of Innovative Computing, Information and Control, 9(5), 2167–2183.
Kaczorek, T. (1985). Two-dimensional linear systems. Berlin: Springer.
Kaczorek, T. (2009). LMI approach to stability of 2D positive systems. Multidimensional Systems and Signal Processing, 20(1), 39–54.
Li, L., Wang, W., & Li, X. (2013). New approach to \(H_{\infty }\) filtering of two-dimensional T-S fuzzy systems. International Journal of Robust and Nonlinear Control, 23(17), 1990–2012.
Li, X., & Gao, H. (2012). Robust finite frequency \(H_{\infty }\) filtering for uncertain 2-D Roesser systems. Automatica, 48(6), 1163–1170.
Li, X., & Gao, H. (2013). Robust finite frequency \(H_{\infty }\) filtering for uncertain 2-D systems: The FM model case. Automatica, 49(8), 2446–2452.
Lu, W. S. (Ed.). (1992). Two-dimensional digital filters (Vol. 80). CRC Press, Boca Raton.
Peng, D., & Guan, X. (2009). \(H_{\infty }\) filtering of 2-D discrete state-delayed systems. Multidimensional Systems and Signal Processing, 20(3), 265–284.
Petersen, I. R., & Savkin, A. V. (1999). Robust Kalman filtering for signals and systems with large uncertainties. Berlin: Springer.
Wu, Z. G., Shi, P., Su, H., & Chu, J. (2014). Asynchronous \(l_{2}-l_{\infty }\) filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities. Automatica, 50(1), 180–186.
Xu, S., Lam, J., Zou, Y., Lin, Z., & Paszke, W. (2005). Robust \(H_{\infty }\) filtering for uncertain 2-D continuous systems. IEEE Transactions on Signal Processing, 53(5), 1731–1738.
Xu, J., & Yu, L. (2009). Delay-dependent \(H_{\infty }\) control for 2-D discrete state delay systems in the second FM model. Multidimensional Systems and Signal Processing, 20(4), 333–349.
Yang, R., & Zheng, W. X. (2018). \(H_{\infty }\) filtering for discrete-time 2-D switched systems: An extended average dwell time approach. Automatica, 98, 302–313.
Zhang, J., Xia, Y., & Shi, P. (2009). Parameter-dependent robust \(H_{\infty }\) filtering for uncertain discrete-time systems. Automatica, 45(2), 560–565.
Zoulagh, T., El Haiek, B., Hmamed, A., & El Hajjaji, A. (2018). Homogenous polynomial \(H_{\infty }\) filtering for uncertain discrete-time systems: A descriptor approach. The International Journal of Adaptive Control and Signal Processing, 32(2), 378–389.
Acknowledgements
K. Badie acknowledges financial support for this research from the Centre National pour la Recherche Scientifique et Technique CNRST, Morocco (Pre-Doctoral Grants 9USMBA2017).
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Badie, K., Alfidi, M. & Chalh, Z. Further results on \(H_{\infty }\) filtering for uncertain 2-D discrete systems. Multidim Syst Sign Process 31, 1469–1490 (2020). https://doi.org/10.1007/s11045-020-00715-2
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DOI: https://doi.org/10.1007/s11045-020-00715-2