Controllability of Delayed Discret Fornasini-Marchesini Model via Quantization and Random Packet Dropouts

Adnène Arbi ^{1 ,  2}

¹ Department of Advanced Sciences and Technologies at National School of Advanced Sciences and Technologies of Borj Cedria, University of Carthage, Carthage, Tunisia
² Laboratory of Engineering Mathematics (LR01ES13), Tunisia Polytechnic School, University of Carthage, Carthage, Tunisia

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 38

Résumé

This research is devoted to Fornasnisi-Marchesini model (FM). More precisely, the investigation of the control problem for the second model discrete-time FM. The model takes into account the random packet loss and quantization errors in the network environment. So our modelling method has the potential to achieve a better stabilization effects. Random packet dropouts, time delays and quantization are taken into consideration in the feedback control problem simultaneously. Measured signals are quantized before being communicated. A logarithmic quantizer is utilized and quantized signal measurements are handled by a sector bound method. The random packet dropouts are modeled as a Bernoulli process. A control law model which depends on packet dropouts and quantization is formulated. Notably, we lighten the assumptions by using the Schur complement. Besides, both a state feedback controller and an observer-based output feedback controller are designed to ensure corresponding closed-loop systems asymptotically stability. Sufficient conditions on mean square asymptotic stability in terms of LMIs have been obtained. Finally, two numerical example show the feasibility of our theoretical results.

Détail

DOI : 10.1051/mmnp/2022040

Bibliographie

[1] A. Arbi, J. Cao Pseudo-almost periodic solution on time-space scales for a novel class of competitive neutral-type neural networks with mixed time-varying delays and leakage delays Neural Process. Lett 2017 719 745

[2] A. Arbi Dynamics of BAM neural networks with mixed delays and leakage time-varying delays in the weighted pseudo-almost periodic on time-space scales Math. Methods Appl. Sci 2018 1230 1255

[3] A. Arbi Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term Chaos Solit. Fract 2021 111436

[4] A. Arbi, J. Cao, A. Alsaedi Improved synchronization analysis of competitive neural networks with time-varying delays Nonlinear Anal., Model. Control 2018 82 102

[5] A. Arbi, N. Tahri Almost anti-periodic solution of inertial neural networks model on time scales MATEC Web Conf 2022 1 6

[6] A. Arbi, J. Cao, M. Es-saiydy, M. Zarhouni and M. Zitane, Dynamics of delayed cellular neural networks in the Stepanov pseudo almost automorphic space. Discr. Continu. Dyn. Syst. In press (2022) 1–13. http://dx.doi.org/10.3934/dcdss.2022136.

[7] A. Arbi, N. Tahri, C. Jammazi, C. Huang, J. Cao Almost anti-periodic solution of inertial neural networks with leakage and time-varying delays on timescales Circ. Syst. Signal Process 2022 1940 1956

[8] D. Bors, S. Walczak Application of 2-D systems to investigation of a process of gas filtration Multidimens. Syst. Signal Process 2012 119 130

[9] X. Bu, H. Wang, Z. Hou, W. Qian H∞ control for a class of 2-D nonlinear systems with intermittent measurements Appl. Math. Comput 2014 651 662

[10] S. Chen, J. Shen State estimation of networked system with multiple quantizers and channel packet dropouts Circ. Syst. Signal Process 2017 1379 1392

[11] P. Dan, H. Xu A novel approach to delay-variation-dependent stability analysis of 2-D discrete-time systems with mixed delays IEEE Access 2019 99817 99829

[12] C. Du and L. Xie, H∞ control and filtering of two-dimensional systems. Springer, Berlin (2002).

[13] Z. Duan, Z. Xiang, H. Karimi Delay-dependent H∞ control for 2-D switched delay systems in the second FM model J. Franklin Inst 2013 1697 1718

[14] N. Elia, K. Mitter Stabilization of linear systems with limited information IEEE Trans. Autom. Control 2001 1384 1400

[15] Z. Feng, L. Xu, M. Wu, Y. He Delay-dependent robust stability and stabilisation of uncertain two-dimensional discrete systems with time-varying delays IET Control Theory Appl 2010 1959 1971

[16] M. Fu, L. Xie The sector bound approach to quantized feedback control IEEE Trans. Autom. Control 2005 1698 1711

[17] K. Galkowski, E. Rogers Control systems analysis for the Fornasini-Marchesini 2D systems model - progress after four decades Int. J. Control 2018 2801 2822

[18] L. Ghaoui, F. Oustry, M. Aitrami A cone complementarity linearization algorithm for static output-feedback and related problems IEEE Trans. Autom. Control 1997 1171 1176

[19] I. Ghous, Z. Xiang Robust state feedback H∞ control for uncertain 2-D continuous state delayed systems in the Roesser model Multidimens. Syst. Signal Process 2016 297 319

[20] Y. Guo, S.S. Ge, A. Arbi Stability of traveling waves solutions for nonlinear cellular neural networks with distributed delays J. Syst. Sci. Complex 2022 18 31

[21] S. Huang, Z. Xiang Delay-dependent robust H∞ control for 2-D discrete non linear systems with state delays Multidimens. Syst. Signal Process 2014 775 794

[22] S. Huang, Z. Xiang Delay-dependent stability for discrete 2D switched systems with state delays in the Roesser model Circ. Syst. Signal Process 2013 2821 2837

[23] S. Jiang, H. Fang H∞ static output feedback control for nonlinear networked control systems with time delays and packet dropouts ISA Trans 2013 215 222

[24] S. Jiang, H. Fang Quantized stabilization of discrete-time systems in a networked environment Appl. Math. Model 2014 1685 1697

[25] Y. Niu, T. Jia, X. Wang, F. Yang Output-feedback control design for NCSs subject to quantization and dropout Inf. Sci 2009 3804 3813

[26] W. Paszke, J. Lam, K. Galkowski, S. Xu, Z. Lin Robust stability and stabilization of 2-D discrete state-delayed systems Syst. Control Lett 2004 277 291

[27] D. Peng, X. Guan Output feedback H∞ control for 2-D state-delayed systems Circ. Syst. Signal Process 2009 147 167

[28] D. Peng, C. Hua Delay-dependent stability and static output feedback control of 2-D discrete systems with interval time-varying delays Asian J. Control 2014 1726 1734

[29] F. Qu, Z. Guan, T. Li, F. Yuan Stability of wireless networked control systems with packet loss IET Control Theory Appl 2012 2362 2366

[30] V. Singh Stability analysis of 2-D discrete systems described by the Fornasini-Marchesini second model with state saturation IEEE Trans. Circ. Syst. II 2008 793 796

[31] C. Sowmiya, R. Raja, Q. Zhu, G. Rajchakit Further mean-square asymptotic stability of impulsive discrete-time stochastic BAM neural networks with Markovian jumping and multiple time-varying delays J. Franklin Inst 2019 561 591

[32] L. Su, G. Chesi Robust stability of uncertain linear systems with input and output quantization and packet loss Automatica 2018 267 273

[33] K. Wan, X.D. Li Iterative learning control for two-dimensional linear discrete systems with Fornasini-Marchesini model Int. J. Control Autom,. Syst 2017 1710 1719

[34] K. Wan, D. Xu, Y-S. Wei Iterative learning control for 2-D linear discrete Fornasini-Marchesini model with input saturation Int. J. Syst. Sci 2020 1482 1494

[35] Z. Wang, F. Yang, D. Ho, X. Liu Robust H∞ control for networked systems with random packet losses IEEE Trans. Syst. Man Cybern. Part B 2007 916 924

[36] Y. Wang, S. Yu An improved dynamic quantization scheme for uncertain linear networked control systems Automatica 2018 244 248

[37] B.Y. Zhu, Q.L. Zhang, X.F. Zhang Decentralized robust guaranteed cost control for uncertain TS fuzzy interconnected systems with time delays International Journal of Information and Systems Sciences 2005 73 88

[38] L. Xu, L. Wu, Q. Wu, Z. Lin, Y. Xiao On realization of 2D discrete systems by Fornasini-Marchesini model Int. J. Control Autom. Syst 2005 631 639

[39] J. Xu, L. Yu Delay-dependent H∞ control for 2-D discrete state delay systems in the second FM model Multidimens. Syst. Signal Process 2009 333 349

[40] R. Yang, P. Shi, G. Liu, H. Gao Network-based feedback control for systems with mixed delays based on quantization and dropout compensation Automatica 2011 2805 2809

[41] F. Yang, Z. Wang, Y. Huang, M. Gani Robust H∞ control with missing measurements and time delays IEEE Trans. Autom. Control 2007 1666 1672

[42] F. Yang, Z. Wang, Y. Huang, M. Gani H∞ control for networked systems with random communication delays IEEE Trans. Autom. Control 2006 511 518

[43] K. You, M. Fu, L. Xie Mean square stability for Kalman filtering with Markovian packet losses Automatica 2011 2647 2657