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July  2021, 17(4): 1825-1843. doi: 10.3934/jimo.2020048

## The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

 1 School of Management, University of Shanghai for Science and Technology Shanghai, 200093, China 2 School of Science, Inner Mongolia University of Science and Technology Baotou, 014010, China

* Corresponding author: Yan Gao

Received  May 2019 Revised  November 2019 Published  July 2021 Early access  March 2020

In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

Citation: Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1825-1843. doi: 10.3934/jimo.2020048
##### References:
 [1] J. P. Aubin, Viability Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-0-8176-4910-4. [2] J. P. Aubin, Dynamic Economic Theory: A Viability Approach, Studies in Economic Theory, 5. Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60756-1. [3] J. P. Aubin, D. Pujal and P. Saint-Pierre, Dynamic management of portfolios with transaction costs under tychastic uncertainty, Numerical Methods in Finance, GERAD 25th Anniv. Ser., Springer, New York, 9 (2005), 59-89.  doi: 10.1007/0-387-25118-9_3. [4] A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM: Control, Optimization and Calculus of Variations, 4 (1999), 361-376.  doi: 10.1051/cocv:1999113. [5] A. Bacciotti and L. Mazzi, From Artstein-Sontag theorem to the min-projection strategy, Transactions of the Institute of Measurement and Control, 32 (2010), 571-581.  doi: 10.1177/0142331208095427. [6] F. Blanchini, Set invariance in control, Automatica J. IFAC, 35 (1999), 1747-1767.  doi: 10.1016/S0005-1098(99)00113-2. [7] M. S. Branicky, Multiple Lyapunov function and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic and Control, 43 (1998), 475-482.  doi: 10.1109/9.664150. [8] Z. Chen and Y. Gao, Determining the viable unbounded polyhedron under linear control systems, Asian Journal of Control, 16 (2014), 1561-1567.  doi: 10.1002/asjc.849. [9] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. doi: 10.1007/b97650. [10] J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Transactions on Automatic and Control, 47 (2002), 1883-1887.  doi: 10.1109/TAC.2002.804474. [11] Y. Gao, Viability criteria for differential inclusions, Journal of Systems Science and Complexity, 24 (2011), 825-834.  doi: 10.1007/s11424-011-9056-6. [12] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. [13] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042. [14] J. F. He, W. Xu, Z. G. Feng and X. S. Yang, On the global optimal solution for linear quadratic problems of switched system, Journal of Industrial and Management Optimization, 15 (2019), 817-832.  doi: 10.3934/jimo.2018072. [15] T. S. Hu, L. Q. Ma and Z. L. Lin, Stabilization of switched systems via composite quadratic functions, IEEE Transactions on Automatic and Control, 53 (2008), 2571-2585.  doi: 10.1109/TAC.2008.2006933. [16] D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8. [17] L. Liu, Y. Gao and F. C. Wang, Road safety analysis for high-speed vehicle in complex environments based on the viability kernel, IET Intelligent Transport Systems, 12 (2018), 495-503.  doi: 10.1049/iet-its.2017.0168. [18] Y. Y. Lu and W. Zhang, A piecewise smooth control-Lyapunov function framework for switching stabilization, Automatica J. IFAC, 76 (2017), 258-265.  doi: 10.1016/j.automatica.2016.09.029. [19] J. F. Lv, Y. Gao and N. Zhao, Viability criteria for a switched system on bounded polyhedron, Asian Journal of Control, 20 (2018), 2380-2387.  doi: 10.1002/asjc.1719. [20] E. Moulay and R. Bourdais, Stabilization of nonlinear switched systems using control Lyapunov functions, Nonlinear Analysis Hybrid Systems, 1 (2007), 482-490.  doi: 10.1016/j.nahs.2005.12.001. [21] A. Oubraham and G. Zaccour, A survey of applications of viability theory to the sustainable exploitation of renewable resources, Ecological Economics, 145 (2018), 346-367.  doi: 10.1016/j.ecolecon.2017.11.008. [22] D. Panagou and K. J. Kyriakopoulos, Viability control for a class of underactuated systems, Automatica J. IFAC, 49 (2013), 17-29.  doi: 10.1016/j.automatica.2012.09.002. [23] A. Rapaport, J. P. Terreaux and L. Doyen, Viability analysis for the sustainable management of renewable resources, Mathematical and Computer Modelling, 43 (2006), 466-484.  doi: 10.1016/j.mcm.2005.12.014. [24] C. Rougé, J. D. Mathias and G. Deffuant, Extending the viability theory framework of resilience to uncertain dynamics and application to lake eutrophication, Ecological Indicators, 29 (2013), 420-433.  doi: 10.1016/j.ecolind.2012.12.032. [25] R. Sabatier, F. Joly and B. Hubert, Assessing both ecological and engineering resilience of a steppe agroecosystem using the viability theory, Agricultural Systems, 157 (2017), 146-156.  doi: 10.1016/j.agsy.2017.07.009. [26] A. I. Subbotin, Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-0847-1. [27] W. Tan and A. Packard, Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of squares programming, IEEE Transactions on Automatic and Control, 53 (2008), 565-571.  doi: 10.1109/TAC.2007.914221. [28] V. I. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic and Control, 22 (1977), 212-222.  doi: 10.1109/TAC.1977.1101446. [29] M. C. Valentino, F. A. Faria, V. A. Oliveira and L. F. C. Alberto, Ultimate boundedness sufficient conditions for nonlinear systems using TS fuzzy modelling, Fuzzy Sets and Systems, 361 (2018), 88–100, Available from: http://hdl.handle.net/11449/170819. doi: 10.1016/j.fss.2018.03.010. [30] M. C. Valentino, V. A. Oliveira, L. F. C. Alberto and D. A. Sant'Anna, An extension of the invariance principle for dwell-time switched nonlinear systems, Systems and Control Letters, 61 (2012), 580-586.  doi: 10.1016/j.sysconle.2012.02.007. [31] M. Wang and J. Feng, Stabilization of switched nonlinear systems using multiple Lyapunov function method, American Control Conference, St. Louis-Missouri, (2009), 1778–1782. [32] X. Zhang, Y. Gao and Z.-Q. Xia, Stabilization of switched systems with polytopic uncertainties via composite quadratic functions, Nonlinear Analysis Hybrid Systems, 11 (2014), 71-83.  doi: 10.1016/j.nahs.2013.06.002. [33] L. X. Zhang, S. L. Zhuang and R. D. Braatz, Switched model predictive control of switched linear systems: Feasibility, stability and robustness, Automatica J. IFAC, 67 (2016), 8-21.  doi: 10.1016/j.automatica.2016.01.010.

show all references

##### References:
 [1] J. P. Aubin, Viability Theory, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-0-8176-4910-4. [2] J. P. Aubin, Dynamic Economic Theory: A Viability Approach, Studies in Economic Theory, 5. Springer-Verlag, Berlin, 1997. doi: 10.1007/978-3-642-60756-1. [3] J. P. Aubin, D. Pujal and P. Saint-Pierre, Dynamic management of portfolios with transaction costs under tychastic uncertainty, Numerical Methods in Finance, GERAD 25th Anniv. Ser., Springer, New York, 9 (2005), 59-89.  doi: 10.1007/0-387-25118-9_3. [4] A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, ESAIM: Control, Optimization and Calculus of Variations, 4 (1999), 361-376.  doi: 10.1051/cocv:1999113. [5] A. Bacciotti and L. Mazzi, From Artstein-Sontag theorem to the min-projection strategy, Transactions of the Institute of Measurement and Control, 32 (2010), 571-581.  doi: 10.1177/0142331208095427. [6] F. Blanchini, Set invariance in control, Automatica J. IFAC, 35 (1999), 1747-1767.  doi: 10.1016/S0005-1098(99)00113-2. [7] M. S. Branicky, Multiple Lyapunov function and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic and Control, 43 (1998), 475-482.  doi: 10.1109/9.664150. [8] Z. Chen and Y. Gao, Determining the viable unbounded polyhedron under linear control systems, Asian Journal of Control, 16 (2014), 1561-1567.  doi: 10.1002/asjc.849. [9] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. doi: 10.1007/b97650. [10] J. Daafouz, P. Riedinger and C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Transactions on Automatic and Control, 47 (2002), 1883-1887.  doi: 10.1109/TAC.2002.804474. [11] Y. Gao, Viability criteria for differential inclusions, Journal of Systems Science and Complexity, 24 (2011), 825-834.  doi: 10.1007/s11424-011-9056-6. [12] Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. [13] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042. [14] J. F. He, W. Xu, Z. G. Feng and X. S. Yang, On the global optimal solution for linear quadratic problems of switched system, Journal of Industrial and Management Optimization, 15 (2019), 817-832.  doi: 10.3934/jimo.2018072. [15] T. S. Hu, L. Q. Ma and Z. L. Lin, Stabilization of switched systems via composite quadratic functions, IEEE Transactions on Automatic and Control, 53 (2008), 2571-2585.  doi: 10.1109/TAC.2008.2006933. [16] D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8. [17] L. Liu, Y. Gao and F. C. Wang, Road safety analysis for high-speed vehicle in complex environments based on the viability kernel, IET Intelligent Transport Systems, 12 (2018), 495-503.  doi: 10.1049/iet-its.2017.0168. [18] Y. Y. Lu and W. Zhang, A piecewise smooth control-Lyapunov function framework for switching stabilization, Automatica J. IFAC, 76 (2017), 258-265.  doi: 10.1016/j.automatica.2016.09.029. [19] J. F. Lv, Y. Gao and N. Zhao, Viability criteria for a switched system on bounded polyhedron, Asian Journal of Control, 20 (2018), 2380-2387.  doi: 10.1002/asjc.1719. [20] E. Moulay and R. Bourdais, Stabilization of nonlinear switched systems using control Lyapunov functions, Nonlinear Analysis Hybrid Systems, 1 (2007), 482-490.  doi: 10.1016/j.nahs.2005.12.001. [21] A. Oubraham and G. Zaccour, A survey of applications of viability theory to the sustainable exploitation of renewable resources, Ecological Economics, 145 (2018), 346-367.  doi: 10.1016/j.ecolecon.2017.11.008. [22] D. Panagou and K. J. Kyriakopoulos, Viability control for a class of underactuated systems, Automatica J. IFAC, 49 (2013), 17-29.  doi: 10.1016/j.automatica.2012.09.002. [23] A. Rapaport, J. P. Terreaux and L. Doyen, Viability analysis for the sustainable management of renewable resources, Mathematical and Computer Modelling, 43 (2006), 466-484.  doi: 10.1016/j.mcm.2005.12.014. [24] C. Rougé, J. D. Mathias and G. Deffuant, Extending the viability theory framework of resilience to uncertain dynamics and application to lake eutrophication, Ecological Indicators, 29 (2013), 420-433.  doi: 10.1016/j.ecolind.2012.12.032. [25] R. Sabatier, F. Joly and B. Hubert, Assessing both ecological and engineering resilience of a steppe agroecosystem using the viability theory, Agricultural Systems, 157 (2017), 146-156.  doi: 10.1016/j.agsy.2017.07.009. [26] A. I. Subbotin, Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-0847-1. [27] W. Tan and A. Packard, Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of squares programming, IEEE Transactions on Automatic and Control, 53 (2008), 565-571.  doi: 10.1109/TAC.2007.914221. [28] V. I. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic and Control, 22 (1977), 212-222.  doi: 10.1109/TAC.1977.1101446. [29] M. C. Valentino, F. A. Faria, V. A. Oliveira and L. F. C. Alberto, Ultimate boundedness sufficient conditions for nonlinear systems using TS fuzzy modelling, Fuzzy Sets and Systems, 361 (2018), 88–100, Available from: http://hdl.handle.net/11449/170819. doi: 10.1016/j.fss.2018.03.010. [30] M. C. Valentino, V. A. Oliveira, L. F. C. Alberto and D. A. Sant'Anna, An extension of the invariance principle for dwell-time switched nonlinear systems, Systems and Control Letters, 61 (2012), 580-586.  doi: 10.1016/j.sysconle.2012.02.007. [31] M. Wang and J. Feng, Stabilization of switched nonlinear systems using multiple Lyapunov function method, American Control Conference, St. Louis-Missouri, (2009), 1778–1782. [32] X. Zhang, Y. Gao and Z.-Q. Xia, Stabilization of switched systems with polytopic uncertainties via composite quadratic functions, Nonlinear Analysis Hybrid Systems, 11 (2014), 71-83.  doi: 10.1016/j.nahs.2013.06.002. [33] L. X. Zhang, S. L. Zhuang and R. D. Braatz, Switched model predictive control of switched linear systems: Feasibility, stability and robustness, Automatica J. IFAC, 67 (2016), 8-21.  doi: 10.1016/j.automatica.2016.01.010.
Trajectories of switched systems excluding sliding motions
Trajectories of switched systems including sliding motions
The phase portraits of subsystems 1 and 2
State responses under the switching law
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