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\(\ell_1\)-induced norm and controller synthesis for positive 2D systems with multiple delays. (English) Zbl 1447.93154
Summary: This paper addresses the problem of \(\ell_1\)-induced controller design for positive two-dimensional (2D) systems with multiple delays. An analytical method is presented to calculate the exact value of \(\ell_1\)-induced norm for positive 2D delayed systems. Necessary and sufficient conditions for asymptotical stability and \(\ell_1\)-induced performance are first derived for the addressed system. Then, based on the singular value decomposition (SVD) technique, controllers are designed to guarantee the asymptotic stability with a prescribed \(\ell_1\)-gain performance level. Those characterizations are formulated in the form of linear programming (LP). Finally, two examples are provided to show the effectiveness and correctness of the theoretical results.
93C28 Positive control/observation systems
93D20 Asymptotic stability in control theory
93C43 Delay control/observation systems
90C05 Linear programming
Full Text: DOI
[1] Fornasini, E.; Marchesini, G., State-space realization theory of two-dimensional filters, IEEE Trans. Automat. Control, 21, 4, 484-492 (1976) · Zbl 0332.93072
[2] Bracewell, R. N., Two-Dimensional Imaging, Prentice-Hall, 284, 71-81 (1995)
[3] Kaczorek, T., Two-dimensional linear systems (1985), Berlin, Springer-Verlag · Zbl 0593.93031
[4] Roesser, R. P., A discrete state-space model for linear image processing, IEEE Trans. Automat. Control, 20, 1, 1-10 (1975) · Zbl 0304.68099
[5] Fornasini, E.; Marchesini, G., Doubly-indexed dynamical systems: State-space models and structural properties, Math. Syst. Theory, 12, 1, 59-71 (1978) · Zbl 0392.93034
[6] Ghous, I.; Duan, Z.; Akhtara, J.; Jawad, M., Robust stabilization of uncertain 2D discrete-time delayed systems using sliding mode control, J. Frankl. Inst., 356, 16, 9407-9431 (2019) · Zbl 1423.93338
[7] Badie, K.; Alfidi, M.; Chalh, Z., Exponential stability analysis for 2D discrete switched systems with state delays, Opt. Control Appl. Methods, 40, 1088-1103 (2019)
[8] Duan, Z.; Jun, Z.; Jian, S., Finite frequency filter design for nonlinear 2D continuous systems in T-S form, J. Frankl. Inst., 354, 18, 8606-8625 (2017) · Zbl 1380.93157
[9] Duan, Z.; Xiang, Z.; Karimi, H. R., Delay-dependent H_∞ control for 2-D switched delay systems in the second FM model, J. Frankl. Inst., 350, 7, 1697-1718 (2013) · Zbl 1392.93009
[10] Ghous, I.; Xiang, Z., Reliable H_∞ control of 2-D continuous nonlinear systems with time varying delays, J. Frankl. Inst., 352, 12, 5758-5778 (2015) · Zbl 1395.93193
[11] Duan, Z.; Xiang, Z., State feedback H_∞ control for discrete 2D switched systems, J. Frankl. Inst., 350, 6, 1513-1530 (2013) · Zbl 1293.93608
[12] Valcher, M. E., Reachability properties of continuous-time positive systems, IEEE Trans. Autom. Control, 54, 7, 1586-1590 (2009) · Zbl 1367.93065
[13] Farina, L.; Rinaldi, S., Positive Linear Systems, Theory and Applications (2000), John Wiley & Sons, INC: John Wiley & Sons, INC New York · Zbl 0988.93002
[14] Benvenuti, L.; Santis, A.; Farina, L., Positive Systems, Lecture Notes in Control and Information Sciences (2003), Berlin, Germany: Springer-Verlag
[15] Fornasini, E.; Valcher, M., Linear copositive Lyapunov functions for continuous-time positive switched systems, IEEE Trans. Autom. Control, 55, 1933-1937 (2010) · Zbl 1368.93593
[16] S. Li, Z. Xiang, Positivity, exponential stability and disturbance attenuation performance for singular switched positive systems with time-varying distributed delays, Appl. Math. Comput.. 10.1016/j.amc.2019.124981 · Zbl 1433.93083
[17] Chen, X.; Chen, M.; Wang, L., Static Output-feedback Controller Synthesis for Positive Systems under ℓ_∞ Performance, 17, 2871-2880 (2019)
[18] Hu, M.; Wang, Y.; Xiao, J., l_1-gain analysis and control of impulsive positive systems with interval uncertainty and time delay, J. Frankl. Inst., 356, 16, 9180-9205 (2019) · Zbl 1423.93294
[19] Liang, J.; Wang, J.; Huang, T., ℓ_1 filtering for continuous-discrete T-S fuzzy positive roesser model, J. Frankl. Inst., 355, 15, 7281-7305 (2018) · Zbl 1398.93336
[20] Kaczorek, T., Positive 1-D and 2-D systems. Industrial Robot (2002), Springer
[21] Kaczorek, T., The choice of the forms of lyapunov functions for a positive 2-D roesser model, Int. J. Appl. Math. Comput. Sci., 17, 4, 471-475 (2008) · Zbl 1234.93089
[22] Duan, Z.; Xiang, Z.; Karimi, H. R., Stability and ℓ_1-gain analysis for positive 2-D T-S fuzzy state-delayed systems in the second F-M model, Neurocomputing, 142, 209-215 (2014)
[23] Kaczorek, T., Asymptotic stability of positive 2-D linear systems with delays, Bull. Pol. Acad. Sci. Tech. Sci., 57, 2, 133-138 (2009)
[24] Kaczorek, T., LMI approach to stability of 2-D positive systems, Multidimens. Syst. Signal Process., 20, 1, 39-54 (2009) · Zbl 1169.93022
[25] Kaczorek, T., Reachability and minimum energy control of positive 2D systems with delays, Control Cybern., 34, 2, 411-423 (2005) · Zbl 1167.93359
[26] Kaczorek, T., Realization problem for positive 2D systems with delays, Mach. Intell. Robot. Control, 6, 2, 61-68 (2004)
[27] J. Wang, J. Liang, Stability analysis and synthesis of uncertain two-dimensional switched positive systems, Proceedings of the Asian Control Conference, Gold Coast (2017) 735-740.
[28] Duan, Z.; Xiang, Z.; Karimi, H. R., Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the roesser model, Inf. Sci., 272, 173-184 (2014) · Zbl 1341.93063
[29] Kaczorek, T., Asymptotic stability of positive 2D linear systems with delays, Bull. Pol. Acad. Sci. Techn. Sci., 57, 2, 133-138 (2009)
[30] Kaczorek, T., Independence of asymptotic stability of positive 2D linear systems with delays of their delays, Int. J. Appl. Math. Comput. Sci., 19, 2, 255-261 (2009) · Zbl 1167.93023
[31] Liu, X.; Yu, W.; Wang, L., Necessary and sufficient asymptotic stability criterion for 2-D positive systems with time-varying state delays described by roesser model, IET Control Theory Appl., 5, 5, 663-668 (2011)
[32] Shen, J.; Wang, W., Stability and positive observer design for positive 2D discrete-time system with multiple delays, Int. J. Syst. Sci., 48, 6, 1-10 (2016)
[33] Duan, Z.; Ghous, I.; Wang, B.; Shen, J., Necessary and sufficient stability criterion and stabilization for positive 2-D continuous-time systems with multiple delays, Asian J. Control, 21, 3, 1355-1366 (2019) · Zbl 1432.93279
[34] 476-470
[35] Wang, J.; Liang, J.; Dobaie, A. M., Dynamic output-feedback control for positive roesser system under the switched and T-S fuzzy rules, Inf. Sci., 422, 1-20 (2018) · Zbl 1447.93115
[36] Ghous, I.; Huang, S.; Xiang, Z., State feedback l_1-gain control of positive 2-D continuous switched delayed systems via state-dependent switching, Circu. Syst. Signal Process., 2432-2449 (2016) · Zbl 1346.93201
[37] Chen, X.; Chen, M.; Shen, J., A novel approach to ℓ_1-induced controller synthesis for positive systems with interval uncertainties, J. Frankl. Inst., 354, 8, 3364-3377 (2017) · Zbl 1364.93255
[38] Shen, J.; Lam, J., On ℓ_∞ and ℓ_∞ gains for positive systems with bounded time-varying delays, Int. J. Syst. Sci., 46, 11, 1953-1960 (2015) · Zbl 1332.93245
[39] Li, S.; Xiang, Z., Exponential stability analysis and l_1-gain control synthesis for positive switched T-S fuzzy systems, Nonlinear Anal. Hybrid Syst., 27, 77-91 (2018) · Zbl 1378.93108
[40] Briat, C., Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: l_1-gain and l_∞-gain characterization, Int. J. Robust Nonlinear Control, 23, 17, 1932-1954 (2012) · Zbl 1278.93188
[41] Chen, X.; Lam, J.; Li, P., ℓ_1-induced norm and controller synthesis of positive systems, Automatica, 49, 5, 1377-1385 (2013) · Zbl 1319.93024
[42] Chen, Y.; Zhao, C.; Lam, J.; Cui, Y.; Kwok, K., Stability and ℓ_1-gain analysis for positive 2-D Markov jump systems, Int. J. Syst. Sci., 50, 11, 2077-2087 (2019)
[43] Wang, J.; Liang, J.; Wang, L., Switched mechanisms for stability and ℓ_1-gain analysis of T-S fuzzy positive systems described by the F-M second model, J. Frankl. Inst., 355, 3, 1351-1372 (2018) · Zbl 1393.93074
[44] (in Press)
[45] Duan, Z.; Karimi, H. R.; Xiang, Z., Stability and l_1-gain analysis for positive 2D systems with state delays in the roesser model, Math. Probl. Eng., 2013, 1-10 (2013) · Zbl 1296.93149
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