Global optimal feedbacks for stochastic quantized nonlinear event systems

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January 2014, 1(1): 163-176. doi: 10.3934/jcd.2014.1.163

Global optimal feedbacks for stochastic quantized nonlinear event systems

Stefan Jerg ^1, , Oliver Junge ^1, and Marcus Post ^1,

1.	Technische Universität München, Boltzmannstr. 3, 85747 Garching, Germany, Germany, Germany

Received January 2012 Revised February 2014 Published April 2014

Abstract
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We consider nonlinear control systems for which only quantized and event-triggered state information is available and which are subject to random delays and losses in the transmission of the state to the controller. We present an optimization based approach for computing globally stabilizing controllers for such systems. Our method is based on recently developed set oriented techniques for transforming the problem into a shortest path problem on a weighted hypergraph. We show how to extend this approach to a system subject to a stochastic parameter and propose a corresponding model for dealing with transmission delays.

Keywords: global feedback, event system., quantized system, Dynamic programming, set oriented numerics.

Mathematics Subject Classification: Primary: 93E20, 49L20; Secondary: 49M2.

Citation: Stefan Jerg, Oliver Junge, Marcus Post. Global optimal feedbacks for stochastic quantized nonlinear event systems. Journal of Computational Dynamics, 2014, 1 (1) : 163-176. doi: 10.3934/jcd.2014.1.163

References:

[1]	K. Aström, Event based control,, in Analysis and Design of Nonlinear Control Systems* (eds. A. Astolfi and L. Marconi)*, (2008), 127. doi: 10.1007/978-3-540-74358-3_9.
[2]	K. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design,, Third Edition, (2011).
[3]	S.-I. Azuma and T. Sugie, Dynamic quantization of nonlinear control systems,, Automatic Control, 57 (2012), 875. doi: 10.1109/TAC.2011.2167824.
[4]	D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 2.,, Belmont, (1995).
[5]	R. S. Bucy, Stability and positive supermartingales,, J. Differential Equations, 1 (1965), 151. doi: 10.1016/0022-0396(65)90016-1.
[6]	C. De Persis and F. Mazenc, Stability of quantized time-delay nonlinear systems: A Lyapunov-Krasowskii-functional approach,, in Proc. of IEEE Conf. on Decision and Control, (2009), 4093.
[7]	W. Fleming, The convergence problem for differential games,, Journal of Mathematical Analysis and Applications, 3 (1961), 102. doi: 10.1016/0022-247X(61)90009-9.
[8]	D. Förstner, M. Jung and J. Lunze, A discrete-event model of asynchronous quantised systems,, Automatica, 38 (2002), 1277. doi: 10.1016/S0005-1098(02)00023-7.
[9]	E. Fridman and M. Dambrine, Control under quantization, saturation and delay: An LMI approach,, Automatica, 45 (2009), 2258. doi: 10.1016/j.automatica.2009.05.020.
[10]	L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property,, in Proc. 46th IEEE CDC, (2007), 702.
[11]	L. Grüne and O. Junge, Global optimal control of perturbed systems,, JOTA, 136 (2008), 411. doi: 10.1007/s10957-007-9312-z.
[12]	L. Grüne and F. Müller, An algorithm for event-based optimal feedback control,, in Proc. 48th IEEE CDC, (2009), 5311.
[13]	J. P. Hespanha, P. Naghshtabrizi and Y. Xu, A survey of recent results in networked control systems,, Proc. IEEE, 95 (2007), 138. doi: 10.1109/JPROC.2006.887288.
[14]	O. Junge, Rigorous discretization of subdivision techniques,, in EQUADIFF 99, (2000), 916.
[15]	O. Junge and H. Osinga, A set oriented approach to global optimal control,, ESAIM Control Optim. Calc. Var., 10 (2004), 259. doi: 10.1051/cocv:2004006.
[16]	E. Kofman and J. Braslavsky, Level crossing sampling in feedback stabilization under data-rate constraints,, in Proc. IEEE CDC, (2006), 4423. doi: 10.1109/CDC.2006.377483.
[17]	H. Kushner, On the stability of stochastic dynamical systems,, Proc. Nat. Acad. Sci. USA, 53 (1965), 8. doi: 10.1073/pnas.53.1.8.
[18]	H. Kushner, Stochastic Stability and Control,, Mathematics in Science and Engineering, (1967).
[19]	H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications,, Applications of Mathematics, (1997).
[20]	D. Liberzon, Quantization, time delays, and nonlinear stabilization,, IEEE Transactions on Automatic Control, 51 (2006), 1190. doi: 10.1109/TAC.2006.878780.
[21]	D. Liberzon, Nonlinear control with limited information,, Commun. Inf. Syst., 9 (2009), 41. doi: 10.4310/CIS.2009.v9.n1.a2.
[22]	L. Litz, T. Gabriel, M. Groß and O. Gabel, Networked Control Systems (NCS) - Stand und Ausblick,, at - Automatisierungstechnik, 56 (2009), 4. doi: 10.1524/auto.2008.0682.
[23]	T. Liu, Z.-P. Jiang and D. J. Hill, Small-gain based output-feedback controller design for a class of nonlinear systems with actuator dynamic quantization,, Automatic Control, 57 (2012), 1326. doi: 10.1109/TAC.2012.2191870.
[24]	U. Lorenz and B. Monien, Error analysis in minimax trees,, TCS, 313 (2004), 485. doi: 10.1016/j.tcs.2002.10.004.
[25]	J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements,, Automatica, 30 (1994), 417. doi: 10.1016/0005-1098(94)90119-8.
[26]	S. Mastellone, C. Abdallah and P. Dorato, Model-based networked control for nonlinear systems with stochastic packet dropout,, in Proc. American Control Conference., 4 (2005), 2365. doi: 10.1109/ACC.2005.1470320.
[27]	D. Nesic and D. Liberzon, A unified framework for design and analysis of networked and quantized control systems,, IEEE Transactions on Automatic Control, 54 (2009), 732. doi: 10.1109/TAC.2009.2014930.
[28]	G. Pola, P. Pepe, M. D. D. Benedetto and P. Tabuada, Symbolic models for nonlinear time-delay systems using approximate bisimulations,, Systems and Control Letters, 59 (2010), 365. doi: 10.1016/j.sysconle.2010.04.001.
[29]	R. Sailer and F. Wirth, Stabilization of nonlinear systems with delayed data-rate-limited feedback,, in Proc. European Control Conference, (2009), 1734.
[30]	J. Schroeder, Modeling, State Observation and Diagnosis of Quantized Systems,, Springer, (2003).
[31]	P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks,, IEEE Transactions on Automatic Control, 52 (2007), 1680. doi: 10.1109/TAC.2007.904277.
[32]	W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comp. Math., 2 (2002), 53.
[33]	M. von Lossow, A min-man version of Dijkstra's algorithm with application to perturbed optimal control problems.,, in In: Proceedings of the GAMM Annual Meeting, (2007).
[34]	X. Wang and M. Lemmon, Event-triggering in distributed networked systems with data dropouts and delays,, Hybrid Systems: Computation and control (HSCC), (5469), 366. doi: 10.1007/978-3-642-00602-9_26.
[35]	C. Zhang, K. Chen and G. Dullerud, Stabilization of markovian jump linear systems with limited information - a convex approach,, in Proc. ACC, (2009), 4013. doi: 10.1109/ACC.2009.5160685.
[36]	C. Zhang and G. Dullerud, Uniform stabilization of markovian jump linear systems with logarithmic quantization,, in Proc. IEEE CDC, (2009), 2094. doi: 10.1109/CDC.2009.5400361.
[37]	L. Zhang, H. Gao and O. Kaynak, Network-induced constraints in networked control systems-a survey,, IEEE Transactions on Industrial Informatics, 9 (2013), 403. doi: 10.1109/TII.2012.2219540.

show all references

References:

[1]	K. Aström, Event based control,, in Analysis and Design of Nonlinear Control Systems* (eds. A. Astolfi and L. Marconi)*, (2008), 127. doi: 10.1007/978-3-540-74358-3_9.
[2]	K. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design,, Third Edition, (2011).
[3]	S.-I. Azuma and T. Sugie, Dynamic quantization of nonlinear control systems,, Automatic Control, 57 (2012), 875. doi: 10.1109/TAC.2011.2167824.
[4]	D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 2.,, Belmont, (1995).
[5]	R. S. Bucy, Stability and positive supermartingales,, J. Differential Equations, 1 (1965), 151. doi: 10.1016/0022-0396(65)90016-1.
[6]	C. De Persis and F. Mazenc, Stability of quantized time-delay nonlinear systems: A Lyapunov-Krasowskii-functional approach,, in Proc. of IEEE Conf. on Decision and Control, (2009), 4093.
[7]	W. Fleming, The convergence problem for differential games,, Journal of Mathematical Analysis and Applications, 3 (1961), 102. doi: 10.1016/0022-247X(61)90009-9.
[8]	D. Förstner, M. Jung and J. Lunze, A discrete-event model of asynchronous quantised systems,, Automatica, 38 (2002), 1277. doi: 10.1016/S0005-1098(02)00023-7.
[9]	E. Fridman and M. Dambrine, Control under quantization, saturation and delay: An LMI approach,, Automatica, 45 (2009), 2258. doi: 10.1016/j.automatica.2009.05.020.
[10]	L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property,, in Proc. 46th IEEE CDC, (2007), 702.
[11]	L. Grüne and O. Junge, Global optimal control of perturbed systems,, JOTA, 136 (2008), 411. doi: 10.1007/s10957-007-9312-z.
[12]	L. Grüne and F. Müller, An algorithm for event-based optimal feedback control,, in Proc. 48th IEEE CDC, (2009), 5311.
[13]	J. P. Hespanha, P. Naghshtabrizi and Y. Xu, A survey of recent results in networked control systems,, Proc. IEEE, 95 (2007), 138. doi: 10.1109/JPROC.2006.887288.
[14]	O. Junge, Rigorous discretization of subdivision techniques,, in EQUADIFF 99, (2000), 916.
[15]	O. Junge and H. Osinga, A set oriented approach to global optimal control,, ESAIM Control Optim. Calc. Var., 10 (2004), 259. doi: 10.1051/cocv:2004006.
[16]	E. Kofman and J. Braslavsky, Level crossing sampling in feedback stabilization under data-rate constraints,, in Proc. IEEE CDC, (2006), 4423. doi: 10.1109/CDC.2006.377483.
[17]	H. Kushner, On the stability of stochastic dynamical systems,, Proc. Nat. Acad. Sci. USA, 53 (1965), 8. doi: 10.1073/pnas.53.1.8.
[18]	H. Kushner, Stochastic Stability and Control,, Mathematics in Science and Engineering, (1967).
[19]	H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications,, Applications of Mathematics, (1997).
[20]	D. Liberzon, Quantization, time delays, and nonlinear stabilization,, IEEE Transactions on Automatic Control, 51 (2006), 1190. doi: 10.1109/TAC.2006.878780.
[21]	D. Liberzon, Nonlinear control with limited information,, Commun. Inf. Syst., 9 (2009), 41. doi: 10.4310/CIS.2009.v9.n1.a2.
[22]	L. Litz, T. Gabriel, M. Groß and O. Gabel, Networked Control Systems (NCS) - Stand und Ausblick,, at - Automatisierungstechnik, 56 (2009), 4. doi: 10.1524/auto.2008.0682.
[23]	T. Liu, Z.-P. Jiang and D. J. Hill, Small-gain based output-feedback controller design for a class of nonlinear systems with actuator dynamic quantization,, Automatic Control, 57 (2012), 1326. doi: 10.1109/TAC.2012.2191870.
[24]	U. Lorenz and B. Monien, Error analysis in minimax trees,, TCS, 313 (2004), 485. doi: 10.1016/j.tcs.2002.10.004.
[25]	J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements,, Automatica, 30 (1994), 417. doi: 10.1016/0005-1098(94)90119-8.
[26]	S. Mastellone, C. Abdallah and P. Dorato, Model-based networked control for nonlinear systems with stochastic packet dropout,, in Proc. American Control Conference., 4 (2005), 2365. doi: 10.1109/ACC.2005.1470320.
[27]	D. Nesic and D. Liberzon, A unified framework for design and analysis of networked and quantized control systems,, IEEE Transactions on Automatic Control, 54 (2009), 732. doi: 10.1109/TAC.2009.2014930.
[28]	G. Pola, P. Pepe, M. D. D. Benedetto and P. Tabuada, Symbolic models for nonlinear time-delay systems using approximate bisimulations,, Systems and Control Letters, 59 (2010), 365. doi: 10.1016/j.sysconle.2010.04.001.
[29]	R. Sailer and F. Wirth, Stabilization of nonlinear systems with delayed data-rate-limited feedback,, in Proc. European Control Conference, (2009), 1734.
[30]	J. Schroeder, Modeling, State Observation and Diagnosis of Quantized Systems,, Springer, (2003).
[31]	P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks,, IEEE Transactions on Automatic Control, 52 (2007), 1680. doi: 10.1109/TAC.2007.904277.
[32]	W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comp. Math., 2 (2002), 53.
[33]	M. von Lossow, A min-man version of Dijkstra's algorithm with application to perturbed optimal control problems.,, in In: Proceedings of the GAMM Annual Meeting, (2007).
[34]	X. Wang and M. Lemmon, Event-triggering in distributed networked systems with data dropouts and delays,, Hybrid Systems: Computation and control (HSCC), (5469), 366. doi: 10.1007/978-3-642-00602-9_26.
[35]	C. Zhang, K. Chen and G. Dullerud, Stabilization of markovian jump linear systems with limited information - a convex approach,, in Proc. ACC, (2009), 4013. doi: 10.1109/ACC.2009.5160685.
[36]	C. Zhang and G. Dullerud, Uniform stabilization of markovian jump linear systems with logarithmic quantization,, in Proc. IEEE CDC, (2009), 2094. doi: 10.1109/CDC.2009.5400361.
[37]	L. Zhang, H. Gao and O. Kaynak, Network-induced constraints in networked control systems-a survey,, IEEE Transactions on Industrial Informatics, 9 (2013), 403. doi: 10.1109/TII.2012.2219540.

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