# American Institute of Mathematical Sciences

January  2015, 2(1): 65-81. doi: 10.3934/jcd.2015.2.65

## Numerical event-based ISS controller design via a dynamic game approach

 1 University of Bayreuth, Chair of Applied Mathematics, Universitätsstraße 30, 95440 Bayreuth 2 University of Bayreuth, Chair of Applied Mathematics, Universitãtsstraße 30, 95440 Bayreuth, Germany

Received  April 2014 Revised  January 2015 Published  August 2015

We present an event-based numerical design method for an input-to-state practically stabilizing (ISpS) state feedback controller for perturbed nonlinear discrete time systems. The controllers are designed to be constant on quantization regions which are not assumed to be small. A transition of the state from one quantization region to another triggers an event upon which the control value changes.
The controller construction relies on the conversion of the ISpS design problem into a robust controller design problem which is solved by a set oriented discretization technique followed by the solution of a dynamic game on a hypergraph. We present and analyze this approach with a particular focus on keeping track of the quantitative dependence of the resulting gain and the size of the exceptional region for practical stability from the design parameters of our event-based controller.
Citation: Lars Grüne, Manuela Sigurani. Numerical event-based ISS controller design via a dynamic game approach. Journal of Computational Dynamics, 2015, 2 (1) : 65-81. doi: 10.3934/jcd.2015.2.65
##### References:
 [1] K. Arzén, A simple event-based PID controller, in Proc. 14th IFAC World Congress, 1999, 423-428. [2] K. J. Åström and B. Bernhardsson, Comparison of periodic and event-based sampling for first-order stochastic systems, in Proc. 14th IFAC World Congress, 1999, 301-306. [3] M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications, Springer US, 2015, 1-14. doi: 10.1007/s13235-015-0156-0. [4] C. De Persis, R. Sailer and F. Wirth, On a small-gain approach to distributed event-triggered control, in Proc. 14th IFAC World Congress, 2011, 2401-2406. [5] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Philadephia, 2014. [6] P. J. Gawthrop and L. B. Wang, Event-driven intermittent control, International Journal of Control, 82 (2009), 2235-2248. doi: 10.1080/00207170902978115. [7] P. Giesl and S. Hafstein, Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565. doi: 10.3934/dcds.2012.32.3539. [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, vol. 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. [9] L. Grüne, S. Jerg, O. Junge, D. Lehmann, J. Lunze, F. Müller and M. Post, Two complementary approaches to event-based control, at-Automatisierungstechnik (Special Issue on Networked Control Systems), 58 (2010), 173-182. [10] L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization, Systems Control Lett., 54 (2005), 169-180. doi: 10.1016/j.sysconle.2004.08.005. [11] L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property, in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana, 2007, 702-707. [12] L. Grüne and O. Junge, Global optimal control of perturbed systems, J. Optim. Theory Appl., 136 (2008), 411-429. doi: 10.1007/s10957-007-9312-z. [13] L. Grüne and C. Kellet, ISS-Lyapunov functions for discontinuous discrete-time systems, IEEE Trans. Autom. Control, 59 (2014), 3098-3103. doi: 10.1109/TAC.2014.2321667. [14] L. Grüne and F. Müller, Set oriented optimal control using past information, in Proc. 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), CD-Rom, Paper 125.pdf, Blacksburg, Virginia, 2008. [15] L. Grüne and F. Müller, An algorithm for event-based optimal feedback control, in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, 2009, 5311-5316. [16] L. Grüne and M. Sigurani, Numerical ISS controller design via a dynamic game approach, in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, 1732-1737. [17] S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electronic Journal of Differential Equations. Monograph, Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007, Available electronically at http://ejde.math.txstate.edu/. [18] Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica, 37 (2001), 857-869. doi: 10.1016/S0005-1098(01)00028-0. [19] Z.-P. Jiang and Y. Wang, A converse Lyapunov theorem for discrete-time systems with disturbances, Systems Control Lett., 45 (2002), 49-58. doi: 10.1016/S0167-6911(01)00164-5. [20] O. Junge and H. M. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270 (electronic). doi: 10.1051/cocv:2004006. [21] J. Lunze (ed.), Control Theory of Digitally Networked Systems, Springer, 2014. doi: 10.1007/978-3-319-01131-8. [22] J. Lunze and D. Lehmann, A state-feedback approach to event-based control, Automatica, 46 (2010), 211-215. doi: 10.1016/j.automatica.2009.10.035. [23] M. Mazo and P. Tabuada, Decentralized event-triggered control over wireless sensor/actuator networks, IEEE Trans. Autom. Control, 56 (2010), 2456-2461. doi: 10.1109/TAC.2011.2164036. [24] M. Sigurani, C. Stöcker, L. Grüne and J. Lunze, Experimental evaluation of two complementary decentralized event-based control methods, Control Eng. Practice, 35 (2015), 22-34. doi: 10.1016/j.conengprac.2014.10.002. [25] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control, 52 (2007), 1680-1685. doi: 10.1109/TAC.2007.904277. [26] M. von Lossow, A min-max version of Dijkstra's algorithm with application to perturbed optimal control problems, in Proc. Appl. Math. Mech. (PAMM), 7 (2007), 4130027-4130028. doi: 10.1002/pamm.200700646. [27] X. Wang and M. D. Lemmon, Attentively efficient controllers for event-triggered feedback systems, in Proc. 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, 2011, 4698-4703. doi: 10.1109/CDC.2011.6160699. [28] X. Wang and M. D. Lemmon, On event design in event-triggered feedback systems, Automatica, 47 (2011), 2319-2322. doi: 10.1016/j.automatica.2011.05.027. [29] H. Yu and P. J. Antsaklis, Event-triggered real-time scheduling for stabilization of passive and output feedback passive systems, in Proc. American Control Conference, 2011, 1674-1679.

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##### References:
 [1] K. Arzén, A simple event-based PID controller, in Proc. 14th IFAC World Congress, 1999, 423-428. [2] K. J. Åström and B. Bernhardsson, Comparison of periodic and event-based sampling for first-order stochastic systems, in Proc. 14th IFAC World Congress, 1999, 301-306. [3] M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications, Springer US, 2015, 1-14. doi: 10.1007/s13235-015-0156-0. [4] C. De Persis, R. Sailer and F. Wirth, On a small-gain approach to distributed event-triggered control, in Proc. 14th IFAC World Congress, 2011, 2401-2406. [5] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Philadephia, 2014. [6] P. J. Gawthrop and L. B. Wang, Event-driven intermittent control, International Journal of Control, 82 (2009), 2235-2248. doi: 10.1080/00207170902978115. [7] P. Giesl and S. Hafstein, Existence of piecewise linear Lyapunov functions in arbitrary dimensions, Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565. doi: 10.3934/dcds.2012.32.3539. [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, vol. 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. [9] L. Grüne, S. Jerg, O. Junge, D. Lehmann, J. Lunze, F. Müller and M. Post, Two complementary approaches to event-based control, at-Automatisierungstechnik (Special Issue on Networked Control Systems), 58 (2010), 173-182. [10] L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization, Systems Control Lett., 54 (2005), 169-180. doi: 10.1016/j.sysconle.2004.08.005. [11] L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property, in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana, 2007, 702-707. [12] L. Grüne and O. Junge, Global optimal control of perturbed systems, J. Optim. Theory Appl., 136 (2008), 411-429. doi: 10.1007/s10957-007-9312-z. [13] L. Grüne and C. Kellet, ISS-Lyapunov functions for discontinuous discrete-time systems, IEEE Trans. Autom. Control, 59 (2014), 3098-3103. doi: 10.1109/TAC.2014.2321667. [14] L. Grüne and F. Müller, Set oriented optimal control using past information, in Proc. 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), CD-Rom, Paper 125.pdf, Blacksburg, Virginia, 2008. [15] L. Grüne and F. Müller, An algorithm for event-based optimal feedback control, in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, 2009, 5311-5316. [16] L. Grüne and M. Sigurani, Numerical ISS controller design via a dynamic game approach, in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, 1732-1737. [17] S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electronic Journal of Differential Equations. Monograph, Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007, Available electronically at http://ejde.math.txstate.edu/. [18] Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica, 37 (2001), 857-869. doi: 10.1016/S0005-1098(01)00028-0. [19] Z.-P. Jiang and Y. Wang, A converse Lyapunov theorem for discrete-time systems with disturbances, Systems Control Lett., 45 (2002), 49-58. doi: 10.1016/S0167-6911(01)00164-5. [20] O. Junge and H. M. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270 (electronic). doi: 10.1051/cocv:2004006. [21] J. Lunze (ed.), Control Theory of Digitally Networked Systems, Springer, 2014. doi: 10.1007/978-3-319-01131-8. [22] J. Lunze and D. Lehmann, A state-feedback approach to event-based control, Automatica, 46 (2010), 211-215. doi: 10.1016/j.automatica.2009.10.035. [23] M. Mazo and P. Tabuada, Decentralized event-triggered control over wireless sensor/actuator networks, IEEE Trans. Autom. Control, 56 (2010), 2456-2461. doi: 10.1109/TAC.2011.2164036. [24] M. Sigurani, C. Stöcker, L. Grüne and J. Lunze, Experimental evaluation of two complementary decentralized event-based control methods, Control Eng. Practice, 35 (2015), 22-34. doi: 10.1016/j.conengprac.2014.10.002. [25] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control, 52 (2007), 1680-1685. doi: 10.1109/TAC.2007.904277. [26] M. von Lossow, A min-max version of Dijkstra's algorithm with application to perturbed optimal control problems, in Proc. Appl. Math. Mech. (PAMM), 7 (2007), 4130027-4130028. doi: 10.1002/pamm.200700646. [27] X. Wang and M. D. Lemmon, Attentively efficient controllers for event-triggered feedback systems, in Proc. 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, Florida, 2011, 4698-4703. doi: 10.1109/CDC.2011.6160699. [28] X. Wang and M. D. Lemmon, On event design in event-triggered feedback systems, Automatica, 47 (2011), 2319-2322. doi: 10.1016/j.automatica.2011.05.027. [29] H. Yu and P. J. Antsaklis, Event-triggered real-time scheduling for stabilization of passive and output feedback passive systems, in Proc. American Control Conference, 2011, 1674-1679.
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