Abstract
A new type of two dimensional (2D) fractional order switched systems modeled with the Roesser model is introduced. Sufficient conditions for the stabilization by state feedback controllers are investigated for the positive 2D fractional order sub-systems. A numerical example is emloyed to show the usefulness of the proposed theoretical results.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Benzaouia, A., Hmamed, A., & Tadeo, F. (2016). Two-dimensional systems. Studies in Systems Decision and Control. 28.
Benzaouia, A., Hmamed, A., Mesquine, F., Benhayoun, M., & Tadeo, F. (2014). Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Transactions on Automatic Control, 59(8), 2203–2208.
Chen, L., He, Y., Chai, Y., & Wu, R. (2014). New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dynamics, 75, 633–641.
Dami, L., Benhayoun, M., & Benzaouia, A. (2017). Stabilization and positivity of 2d fractional order uncertain discrete-time systems. In 2017 14th International multi-conference on systems, signals & devices (SSD), pp. 545–548. IEEE.
Dami, L., Benhayoun, M., & Benzaouia, A. (2021). Robust stability and stabilization of singular 2d continuous systems with delays. In 2021 9th International Conference on Systems and Control (ICSC). pp. 24–31. IEEE.
Dami, L., Benzaouia, A., & Benhayoun, M. (2022). Asymptotic stability of switched \(2 d \) fractional order positive systems. In: 2022 10th International Conference on Systems and Control (ICSC), pp. 149–155. IEEE
Dercole, F., & Della Rossa, F. (2018). Tree-based algorithms for the stability of discrete-time switched linear systems under arbitrary and constrained switching. IEEE Transactions on Automatic Control, 64(9), 3823–3830.
Duan, Z., Xiang, Z., & Karimi, H. R. (2014). Delay-dependent exponential stabilization of positive 2d switched state-delayed systems in the roesser model. Information Sciences, 272, 173–184.
El-Sayed, A. M. (1996). Fractional-order diffusion-wave equation. International Journal of Theoretical Physics, 35, 311–322.
Fainshil, L., Margaliot, M., & Chigansky, P. (2009). On the stability of positive linear switched systems under arbitrary switching laws. IEEE Transactions on Automatic Control, 54(4), 897–899.
Fornasini, E., & Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21(4), 484–492.
Ghous, I., & Lu, J. (2020). Robust observer design for two-dimensional discrete positive switched systems with delays. IEEE Transactions on Circuits and Systems II: Express Briefs, 67(12), 3297–3301.
Gregorian, R., Martin, K. W., & Temes, G. C. (1983). Switched-capacitor circuit design. Proceedings of the IEEE, 71(8), 941–966.
Hamamci, S. E. (2007). An algorithm for stabilization of fractional-order time delay systems using fractional-order pid controllers. IEEE Transactions on Automatic Control, 52(10), 1964–1969.
Huang, S., & Xiang, Z. (2016). Stability of a class of fractional-order two-dimensional non-linear continuous-time systems. IET Control Theory & Applications, 10(18), 2559–2564.
Jahanshahi, H., Yousefpour, A., Munoz-Pacheco, J. M., Moroz, I., Wei, Z., & Castillo, O. (2020). A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method. Applied Soft Computing, 87, 105943.
Jia, J., Wang, F., & Zeng, Z. (2021). Global stabilization of fractional-order memristor-based neural networks with incommensurate orders and multiple time-varying delays: a positive-system-based approach. Nonlinear Dynamics, 104(3), 2303–2329.
Kaczorek, T. (2010). Positive linear systems with different fractional orders. Bulletin of the Polish Academy of Sciences. Technical Sciences, 58(3), 453–458.
Kaczorek, T., & Rogowski, K. (2010). Positivity and stabilization of fractional 2d linear systems described by the Roesser model. International Journal of Applied Mathematics and Computer Science, 20(1), 85–92.
Knorn, F., Mason, O., & Shorten, R. (2009). On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica, 45(8), 1943–1947.
Lin, Y., & Zhao, P. (2022). Stability analysis of nonlinear impulsive switched positive systems. International Journal of Nonlinear Sciences and Numerical Simulation. https://doi.org/10.1515/ijnsns-2020-0264
Lv, W., Geng, H., Zhou, B., Chen, H., Yuan, R., Chuanxin, M., Liu, R., Xing, B., & Wang, F. (2022). The behavior, transport, and positive regulation mechanism of zno nanoparticles in a plant-soil-microbe environment. Environmental Pollution, 7, 120368.
Radun, A. V. (1995). Design considerations for the switched reluctance motor. IEEE Transactions on Industry Applications, 31(5), 1079–1087.
Rakkiyappan, R., Chandrasekar, A., Lakshmanan, S., & Park, J. H. (2015). Exponential stability for markovian jumping stochastic bam neural networks with mode-dependent probabilistic time-varying delays and impulse control. Complexity, 20(3), 39–65.
Rakkiyappan, R., Premalatha, S., Chandrasekar, A., & Cao, J. (2016). Stability and synchronization analysis of inertial memristive neural networks with time delays. Cognitive neurodynamics, 10, 437–451.
Rakkiyappan, R., Zhu, Q., & Chandrasekar, A. (2014). Stability of stochastic neural networks of neutral type with markovian jumping parameters: A delay-fractioning approach. Journal of the Franklin Institute, 351(3), 1553–1570.
Rami, M. A., Tadeo, F., & Helmke, U. (2011). Positive observers for linear positive systems, and their implications. International Journal of Control, 84(4), 716–725.
Roesser, R. (1975). A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 20(1), 1–10.
Shi, S., Fei, Z., Sun, W., & Yang, X. (2017). Stabilization of 2-d switched systems with all modes unstable via switching signal regulation. IEEE Transactions on Automatic Control, 63(3), 857–863.
Stamova, I., & Stamov, G. (2014). Stability analysis of impulsive functional systems of fractional order. Communications in Nonlinear Science and Numerical Simulation, 19(3), 702-709.
Vanchinathan, K., & Selvaganesan, N. (2021). Adaptive fractional order pid controller tuning for brushless dc motor using artificial bee colony algorithm. Results in Control and Optimization, 4, 100032.
Vanchinathan, K., Valluvan, K., Gnanavel, C., & Gokul, C. (2022). Numerical simulation and experimental verification of fractional-order pi\(\lambda \) controller for solar pv fed sensorless brushless dc motor using whale optimization algorithm. Electric Power Components and Systems, 50(1–2), 64–80.
Wang, J., & Liang, J. (2017). Stability analysis and synthesis of uncertain two-dimensional switched positive systems. In 2017 11th Asian control conference (ASCC). pp. 735–740. IEEE.
Wang, J., Hou, Y., Jiang, L., & Zhang, L. (2021). Robust stability and stabilization of 2d positive system employing saturation. Circuits, Systems, and Signal Processing, 40, 1183–1206.
Wu, L., Yang, R., Shi, P., & Su, X. (2015). Stability analysis and stabilization of 2-d switched systems under arbitrary and restricted switchings. Automatica, 59, 206–215.
Xiang, Z., & Huang, S. (2013). Stability analysis and stabilization of discrete-time 2d switched systems. Circuits, Systems, and Signal Processing, 32, 401–414.
Zhang, X., Niu, P., Ma, Y., Wei, Y., & Li, G. (2017). Global Mittag-leffler stability analysis of fractional-order impulsive neural networks with one-side Lipschitz condition. Neural Networks, 94, 67–75.
Zhang, J., & Sun, Y. (2023). Practical exponential stability of two dimensional nonlinear switched positive systems in the Roesser model. Journal of Systems Science and Complexity, 36(3), 1103–1115.
Zhang, X., & Wang, Z. (2020). Stability and robust stabilization of uncertain switched fractional order systems. ISA Transactions, 103, 1–9.
Zhang, J., Zhao, X., & Chen, Y. (2016). Finite-time stability and stabilization of fractional order positive switched systems. Circuits, Systems, and Signal Processing, 35, 2450–2470.
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Dami, L., Benzaouia, A. Stabilization of Switched Two Dimensional Fractional Order Positive Systems Modeled by the Roesser Model. J Control Autom Electr Syst 34, 1136–1144 (2023). https://doi.org/10.1007/s40313-023-01037-x
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DOI: https://doi.org/10.1007/s40313-023-01037-x