American Institute of Mathematical Sciences

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

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

Lars Grüne 1, and  Manuela Sigurani 2,

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.
Keywords: dynamic game, set oriented numerics, input-to-state-stability, nonlinear systems., Event-based control.
Mathematics Subject Classification: 93B51, 93D09, 65P4.
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. 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, 301-306. Google Scholar

[3]

M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications, Springer US, 2015, 1-14. doi: 10.1007/s13235-015-0156-0.  Google Scholar

[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. Google Scholar

[5]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Philadephia, 2014.  Google Scholar

[6]

P. J. Gawthrop and L. B. Wang, Event-driven intermittent control, International Journal of Control, 82 (2009), 2235-2248. doi: 10.1080/00207170902978115.  Google Scholar

[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.  Google Scholar

[8]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, vol. 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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, Shanghai, China, 2009, 5311-5316. 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, Florence, Italy, 2013, 1732-1737. 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, Department of Mathematics, San Marcos, TX, 2007, Available electronically at http://ejde.math.txstate.edu/.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Lunze (ed.), Control Theory of Digitally Networked Systems, Springer, 2014. doi: 10.1007/978-3-319-01131-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

References:
[1]

K. Arzén, A simple event-based PID controller, in Proc. 14th IFAC World Congress, 1999, 423-428. 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, 301-306. Google Scholar

[3]

M. Bardi and J. P. Maldonado López, A Dijkstra-type algorithm for dynamic games, Dynamic Games and Applications, Springer US, 2015, 1-14. doi: 10.1007/s13235-015-0156-0.  Google Scholar

[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. Google Scholar

[5]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Philadephia, 2014.  Google Scholar

[6]

P. J. Gawthrop and L. B. Wang, Event-driven intermittent control, International Journal of Control, 82 (2009), 2235-2248. doi: 10.1080/00207170902978115.  Google Scholar

[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.  Google Scholar

[8]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, vol. 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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, Shanghai, China, 2009, 5311-5316. 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, Florence, Italy, 2013, 1732-1737. 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, Department of Mathematics, San Marcos, TX, 2007, Available electronically at http://ejde.math.txstate.edu/.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Lunze (ed.), Control Theory of Digitally Networked Systems, Springer, 2014. doi: 10.1007/978-3-319-01131-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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