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
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show all references

References:
[1]

in Proc. 14th IFAC World Congress, 1999, 423-428. Google Scholar

[2]

in Proc. 14th IFAC World Congress, 1999, 301-306. Google Scholar

[3]

Springer US, 2015, 1-14. doi: 10.1007/s13235-015-0156-0.  Google Scholar

[4]

in Proc. 14th IFAC World Congress, 2011, 2401-2406. Google Scholar

[5]

SIAM, Philadephia, 2014.  Google Scholar

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International Journal of Control, 82 (2009), 2235-2248. doi: 10.1080/00207170902978115.  Google Scholar

[7]

Discrete Contin. Dyn. Syst., 32 (2012), 3539-3565. doi: 10.3934/dcds.2012.32.3539.  Google Scholar

[8]

Springer, Berlin, 2007.  Google Scholar

[9]

at-Automatisierungstechnik (Special Issue on Networked Control Systems), 58 (2010), 173-182. Google Scholar

[10]

Systems Control Lett., 54 (2005), 169-180. doi: 10.1016/j.sysconle.2004.08.005.  Google Scholar

[11]

in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana, 2007, 702-707. Google Scholar

[12]

J. Optim. Theory Appl., 136 (2008), 411-429. doi: 10.1007/s10957-007-9312-z.  Google Scholar

[13]

IEEE Trans. Autom. Control, 59 (2014), 3098-3103. doi: 10.1109/TAC.2014.2321667.  Google Scholar

[14]

in Proc. 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), CD-Rom, Paper 125.pdf, Blacksburg, Virginia, 2008. Google Scholar

[15]

in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, 2009, 5311-5316. Google Scholar

[16]

in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, 1732-1737. Google Scholar

[17]

Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007, Available electronically at http://ejde.math.txstate.edu/.  Google Scholar

[18]

Automatica, 37 (2001), 857-869. doi: 10.1016/S0005-1098(01)00028-0.  Google Scholar

[19]

Systems Control Lett., 45 (2002), 49-58. doi: 10.1016/S0167-6911(01)00164-5.  Google Scholar

[20]

ESAIM Control Optim. Calc. Var., 10 (2004), 259-270 (electronic). doi: 10.1051/cocv:2004006.  Google Scholar

[21]

Springer, 2014. doi: 10.1007/978-3-319-01131-8.  Google Scholar

[22]

Automatica, 46 (2010), 211-215. doi: 10.1016/j.automatica.2009.10.035.  Google Scholar

[23]

IEEE Trans. Autom. Control, 56 (2010), 2456-2461. doi: 10.1109/TAC.2011.2164036.  Google Scholar

[24]

Control Eng. Practice, 35 (2015), 22-34. doi: 10.1016/j.conengprac.2014.10.002.  Google Scholar

[25]

IEEE Trans. Autom. Control, 52 (2007), 1680-1685. doi: 10.1109/TAC.2007.904277.  Google Scholar

[26]

in Proc. Appl. Math. Mech. (PAMM), 7 (2007), 4130027-4130028. doi: 10.1002/pamm.200700646.  Google Scholar

[27]

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in Proc. American Control Conference, 2011, 1674-1679. Google Scholar

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