# American Institute of Mathematical Sciences

## Observer-based control for a class of hybrid linear and nonlinear systems

 1 University of Genoa (DIME), Via Opera Pia 15, 16145 Genova, Italy 2 Laboratoire de Mathématiques Pures et Appliquées, University Mouloud Mammeri of Tizi-Ouzou, B.P. No. 17 RP, 15000 Tizi-Ouzou, Algeria 3 Centre de Recherche en Automatique de Nancy, University of Lorraine, CNRS UMR 7039, F-54400 Cosnes et Romain, France

* Corresponding author: A. Alessandri

Received  November 2019 Revised  February 2020 Published  May 2020

An approach to output feedback control for hybrid discrete-time systems subject to uncertain mode transitions is proposed. The system dynamics may assume different modes upon the occurrence of a switching that is not directly measurable. Since the current system mode is unknown, a regulation scheme is proposed by combining a Luenberger observer to estimate the continuous state, a mode estimator, and a controller fed with the estimates of both continuous state variables and mode. The closed-loop stability is ensured under suitable conditions given in terms of linear matrix inequalities. Since complexity and conservativeness grow with the increase of the modes, we address the problem of reducing the number of linear matrix inequalities by providing more easily tractable stability conditions. Such conditions are extended to deal with systems having also Lipschitz nonlinearities and affected by disturbances. The effectiveness of the proposed approach is shown by means of simulations.

Citation: A. Alessandri, F. Bedouhene, D. Bouhadjra, A. Zemouche, P. Bagnerini. Observer-based control for a class of hybrid linear and nonlinear systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020376
##### References:
 [1] A. Alessandri, M. Baglietto and G. Battistelli, Receding-horizon estimation for switching discrete-time linear systems, IEEE Trans. on Automatic Control, 50 (2005), 1736-1748.  doi: 10.1109/TAC.2005.858684.  Google Scholar [2] A. Alessandri, M. Baglietto and G. Battistelli, Luenberger observers for switching discrete-time linear systems, International Journal of Control, 80 (2007), 1931-1943.  doi: 10.1080/00207170701481683.  Google Scholar [3] A. Alessandri and F. Boem, State observers for systems subject to bounded disturbances using quadratic boundedness, IEEE Trans. on Automatic Control, to appear. Google Scholar [4] F. Blanchini, S. Miani and F. Mesquine, A separation principle for linear switching systems and parametrization of all stabilizing controllers, IEEE Trans. on Automatic Control, 54 (2009), 279-292.  doi: 10.1109/TAC.2008.2010896.  Google Scholar [5] S. Court, K. Kunisch and L. Pfeiffer, Hybrid optimal control problems for a class of semilinear parabolic equations, Discrete and Continuous Dynamical Systems - Series S, 11 (2018), 1031-1060.  doi: 10.3934/dcdss.2018060.  Google Scholar [6] M. de Oliveira, J. Bernussou and J. Geromel, A new discrete-time robust stability condition, Systems & Control Letters, 37 (1999), 261-265.  doi: 10.1016/S0167-6911(99)00035-3.  Google Scholar [7] R. Essick, J.-W. Lee and G. Dullerud, Control of linear switched systems with receding horizon modal information, IEEE Trans. on Automatic Control, 59 (2014), 2340-2352.  doi: 10.1109/TAC.2014.2321251.  Google Scholar [8] B. Grandvallet, A. Zemouche, H. Souley-Ali and M. Boutayeb, New LMI condition for observer-based $H_{\infty}$ stabilization of a class of nonlinear discrete-time systems, SIAM Journal on Control and Optimization, 51 (2013), 784-800.  doi: 10.1137/11085623X.  Google Scholar [9] R. Guo and L. Song, Optical chaotic secure algorithm based on space laser communication, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 1355-1369.  doi: 10.3934/dcdss.2019093.  Google Scholar [10] M. Halimi, G. Millerioux and J. Daafouz, Model-based modes detection and discernibility for switched affine discrete-time systems, IEEE Trans. on Automatic Control, 60 (2015), 1501-1514.  doi: 10.1109/TAC.2014.2383012.  Google Scholar [11] W. Heemels, J. Daafouz and G. Millerioux, Observer-based control of discrete-time LPV systems with uncertain parameters, IEEE Trans. on Automatic Control, 55 (2010), 2130-2135.  doi: 10.1109/TAC.2010.2051072.  Google Scholar [12] S. Ibrir, Static output feedback and guaranteed cost control of a class of discrete-time nonlinear systems with partial state measurements, Nonlinear Analysis, 68 (2008), 1784-1792.  doi: 10.1016/j.na.2007.01.011.  Google Scholar [13] X.-Q. Jiang and L.-C. Zhang, Stock price fluctuation prediction method based on time series analysis, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 915-927.  doi: 10.3934/dcdss.2019061.  Google Scholar [14] H. Kheloufi, F. Bedouhene, A. Zemouche and A. Alessandri, Observer-based stabilisation of linear systems with parameter uncertainties by using enhanced LMI conditions, Int. Journal of Control, 88 (2015), 1189-1200.  doi: 10.1080/00207179.2014.999258.  Google Scholar [15] J. Li and Y. Liu, Stabilization of a class of discrete-time switched systems via observer-based output feedback, Journal of Control Theory and Applications, 5 (2007), 307-311.  doi: 10.1007/s11768-006-6064-5.  Google Scholar [16] Z. Li, C. Wen and Y. Soh, Observer-based stabilization of switching linear systems, Automatica, 39 (2003), 517-524.  doi: 10.1016/S0005-1098(02)00267-4.  Google Scholar [17] W. Lv and S. Ji, Atmospheric environmental quality assessment method based on analytic hierarchy process, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 941-955.  doi: 10.3934/dcdss.2019063.  Google Scholar [18] Y. Qiu, W. Chen and Q. Nie, A hybrid method for stiff reaction-diffusion equations, Discrete and Continuous Dynamical Systems - Series B, 24 (2019), 6387-6417.  doi: 10.3934/dcdsb.2019144.  Google Scholar [19] R. Rajamani, W. Jeon, H. Movahedi and A. Zemouche, On the need for switched-gain observers for non-monotonic nonlinear systems, Automatica J. IFAC, 114 (2020), 108814, 12 pp. doi: 10.1016/j.automatica.2020.108814.  Google Scholar [20] Z. Song and J. Zhao, Observer-based robust $H_{\infty}$ control for uncertain switched systems, Journal of Control Theory and Applications, 5 (2007), 278-284.  doi: 10.1007/s11768-006-6053-8.  Google Scholar [21] W. Xiang, J. Xiao and M. Iqbal, Robust observer design for nonlinear uncertain switched systems under asynchronous switching, Nonlinear Analysis: Hybrid Systems, 6 (2012), 754-773.  doi: 10.1016/j.nahs.2011.08.001.  Google Scholar [22] A. Zemouche and M. Boutayeb, On LMI conditions to design observers for Lipschitz nonlinear systems, Automatica, 49 (2013), 585-591.  doi: 10.1016/j.automatica.2012.11.029.  Google Scholar

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##### References:
 [1] A. Alessandri, M. Baglietto and G. Battistelli, Receding-horizon estimation for switching discrete-time linear systems, IEEE Trans. on Automatic Control, 50 (2005), 1736-1748.  doi: 10.1109/TAC.2005.858684.  Google Scholar [2] A. Alessandri, M. Baglietto and G. Battistelli, Luenberger observers for switching discrete-time linear systems, International Journal of Control, 80 (2007), 1931-1943.  doi: 10.1080/00207170701481683.  Google Scholar [3] A. Alessandri and F. Boem, State observers for systems subject to bounded disturbances using quadratic boundedness, IEEE Trans. on Automatic Control, to appear. Google Scholar [4] F. Blanchini, S. Miani and F. Mesquine, A separation principle for linear switching systems and parametrization of all stabilizing controllers, IEEE Trans. on Automatic Control, 54 (2009), 279-292.  doi: 10.1109/TAC.2008.2010896.  Google Scholar [5] S. Court, K. Kunisch and L. Pfeiffer, Hybrid optimal control problems for a class of semilinear parabolic equations, Discrete and Continuous Dynamical Systems - Series S, 11 (2018), 1031-1060.  doi: 10.3934/dcdss.2018060.  Google Scholar [6] M. de Oliveira, J. Bernussou and J. Geromel, A new discrete-time robust stability condition, Systems & Control Letters, 37 (1999), 261-265.  doi: 10.1016/S0167-6911(99)00035-3.  Google Scholar [7] R. Essick, J.-W. Lee and G. Dullerud, Control of linear switched systems with receding horizon modal information, IEEE Trans. on Automatic Control, 59 (2014), 2340-2352.  doi: 10.1109/TAC.2014.2321251.  Google Scholar [8] B. Grandvallet, A. Zemouche, H. Souley-Ali and M. Boutayeb, New LMI condition for observer-based $H_{\infty}$ stabilization of a class of nonlinear discrete-time systems, SIAM Journal on Control and Optimization, 51 (2013), 784-800.  doi: 10.1137/11085623X.  Google Scholar [9] R. Guo and L. Song, Optical chaotic secure algorithm based on space laser communication, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 1355-1369.  doi: 10.3934/dcdss.2019093.  Google Scholar [10] M. Halimi, G. Millerioux and J. Daafouz, Model-based modes detection and discernibility for switched affine discrete-time systems, IEEE Trans. on Automatic Control, 60 (2015), 1501-1514.  doi: 10.1109/TAC.2014.2383012.  Google Scholar [11] W. Heemels, J. Daafouz and G. Millerioux, Observer-based control of discrete-time LPV systems with uncertain parameters, IEEE Trans. on Automatic Control, 55 (2010), 2130-2135.  doi: 10.1109/TAC.2010.2051072.  Google Scholar [12] S. Ibrir, Static output feedback and guaranteed cost control of a class of discrete-time nonlinear systems with partial state measurements, Nonlinear Analysis, 68 (2008), 1784-1792.  doi: 10.1016/j.na.2007.01.011.  Google Scholar [13] X.-Q. Jiang and L.-C. Zhang, Stock price fluctuation prediction method based on time series analysis, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 915-927.  doi: 10.3934/dcdss.2019061.  Google Scholar [14] H. Kheloufi, F. Bedouhene, A. Zemouche and A. Alessandri, Observer-based stabilisation of linear systems with parameter uncertainties by using enhanced LMI conditions, Int. Journal of Control, 88 (2015), 1189-1200.  doi: 10.1080/00207179.2014.999258.  Google Scholar [15] J. Li and Y. Liu, Stabilization of a class of discrete-time switched systems via observer-based output feedback, Journal of Control Theory and Applications, 5 (2007), 307-311.  doi: 10.1007/s11768-006-6064-5.  Google Scholar [16] Z. Li, C. Wen and Y. Soh, Observer-based stabilization of switching linear systems, Automatica, 39 (2003), 517-524.  doi: 10.1016/S0005-1098(02)00267-4.  Google Scholar [17] W. Lv and S. Ji, Atmospheric environmental quality assessment method based on analytic hierarchy process, Discrete and Continuous Dynamical Systems - Series S, 12 (2019), 941-955.  doi: 10.3934/dcdss.2019063.  Google Scholar [18] Y. Qiu, W. Chen and Q. Nie, A hybrid method for stiff reaction-diffusion equations, Discrete and Continuous Dynamical Systems - Series B, 24 (2019), 6387-6417.  doi: 10.3934/dcdsb.2019144.  Google Scholar [19] R. Rajamani, W. Jeon, H. Movahedi and A. Zemouche, On the need for switched-gain observers for non-monotonic nonlinear systems, Automatica J. IFAC, 114 (2020), 108814, 12 pp. doi: 10.1016/j.automatica.2020.108814.  Google Scholar [20] Z. Song and J. Zhao, Observer-based robust $H_{\infty}$ control for uncertain switched systems, Journal of Control Theory and Applications, 5 (2007), 278-284.  doi: 10.1007/s11768-006-6053-8.  Google Scholar [21] W. Xiang, J. Xiao and M. Iqbal, Robust observer design for nonlinear uncertain switched systems under asynchronous switching, Nonlinear Analysis: Hybrid Systems, 6 (2012), 754-773.  doi: 10.1016/j.nahs.2011.08.001.  Google Scholar [22] A. Zemouche and M. Boutayeb, On LMI conditions to design observers for Lipschitz nonlinear systems, Automatica, 49 (2013), 585-591.  doi: 10.1016/j.automatica.2012.11.029.  Google Scholar
Output feedback observer-based control scheme
State space piecewise regions
Finite state machine of the discrete state dynamics and tree of mode combinations over successive time instants
Case Study 1: state variables and their estimates
Case Study 2: state variables and their estimates
Case Study 3: state variables and their estimates
Case Study 3: switching signal and its estimate
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