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An elementary way to rigorously estimate convergence to equilibrium and escape rates
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 |
  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.
References:
[1] |
K. Arzén, A simple event-based PID controller,, in Proc. 14th IFAC World Congress, (1999), 423. Google Scholar |
[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. Google Scholar |
[3] |
M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications,, Springer US, (2015), 1.
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. Google Scholar |
[5] |
M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2014).
|
[6] |
P. J. Gawthrop and L. B. Wang, Event-driven intermittent control,, International Journal of Control, 82 (2009), 2235.
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.
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, (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. Google Scholar |
[10] |
L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization,, Systems Control Lett., 54 (2005), 169.
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, (2007), 702. Google Scholar |
[12] |
L. Grüne and O. Junge, Global optimal control of perturbed systems,, J. Optim. Theory Appl., 136 (2008), 411.
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.
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), (2008). Google Scholar |
[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, (2009), 5311. Google Scholar |
[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, (2013), 1732. Google Scholar |
[17] |
S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electronic Journal of Differential Equations. Monograph,, Texas State University-San Marcos, (2007).
|
[18] |
Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems,, Automatica, 37 (2001), 857.
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.
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.
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.
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.
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.
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.
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.
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, (2011), 4698.
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.
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. Google Scholar |
show all references
References:
[1] |
K. Arzén, A simple event-based PID controller,, in Proc. 14th IFAC World Congress, (1999), 423. Google Scholar |
[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. Google Scholar |
[3] |
M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications,, Springer US, (2015), 1.
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. Google Scholar |
[5] |
M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2014).
|
[6] |
P. J. Gawthrop and L. B. Wang, Event-driven intermittent control,, International Journal of Control, 82 (2009), 2235.
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.
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, (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. Google Scholar |
[10] |
L. Grüne and O. Junge, A set oriented approach to optimal feedback stabilization,, Systems Control Lett., 54 (2005), 169.
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, (2007), 702. Google Scholar |
[12] |
L. Grüne and O. Junge, Global optimal control of perturbed systems,, J. Optim. Theory Appl., 136 (2008), 411.
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.
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), (2008). Google Scholar |
[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, (2009), 5311. Google Scholar |
[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, (2013), 1732. Google Scholar |
[17] |
S. F. Hafstein, An Algorithm for Constructing Lyapunov Functions, vol. 8 of Electronic Journal of Differential Equations. Monograph,, Texas State University-San Marcos, (2007).
|
[18] |
Z.-P. Jiang and Y. Wang, Input-to-state stability for discrete-time nonlinear systems,, Automatica, 37 (2001), 857.
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.
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.
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.
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.
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.
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.
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.
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, (2011), 4698.
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.
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. Google Scholar |
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