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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

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), Springer-Verlag, 2008, 127-147. doi: 10.1007/978-3-540-74358-3_9.  Google Scholar

[2]

K. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design, Third Edition, Dover Books on Electrical Engineering Series, Dover Publications, 2011, URL http://books.google.de/books?id=9Y6D5vviqMgC. Google Scholar

[3]

S.-I. Azuma and T. Sugie, Dynamic quantization of nonlinear control systems, Automatic Control, IEEE Transactions on, 57 (2012), 875-888. doi: 10.1109/TAC.2011.2167824.  Google Scholar

[4]

D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 2., Belmont, MA: Athena Scientific, 1995. Google Scholar

[5]

R. S. Bucy, Stability and positive supermartingales, J. Differential Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1.  Google Scholar

[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), 4093-4098. Google Scholar

[7]

W. Fleming, The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9.  Google Scholar

[8]

D. Förstner, M. Jung and J. Lunze, A discrete-event model of asynchronous quantised systems, Automatica, 38 (2002), 1277-1286. doi: 10.1016/S0005-1098(02)00023-7.  Google Scholar

[9]

E. Fridman and M. Dambrine, Control under quantization, saturation and delay: An LMI approach, Automatica, 45 (2009), 2258-2264. doi: 10.1016/j.automatica.2009.05.020.  Google Scholar

[10]

L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property, in Proc. 46th IEEE CDC, (2007), 702-707. Google Scholar

[11]

L. Grüne and O. Junge, Global optimal control of perturbed systems, JOTA, 136 (2008), 411-429. doi: 10.1007/s10957-007-9312-z.  Google Scholar

[12]

L. Grüne and F. Müller, An algorithm for event-based optimal feedback control, in Proc. 48th IEEE CDC, 2009, 5311-5316. Google Scholar

[13]

J. P. Hespanha, P. Naghshtabrizi and Y. Xu, A survey of recent results in networked control systems, Proc. IEEE, 95 (2007), 138-162. doi: 10.1109/JPROC.2006.887288.  Google Scholar

[14]

O. Junge, Rigorous discretization of subdivision techniques, in EQUADIFF 99, Proceedings of the International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), vol. 2, 2000, 916-918.  Google Scholar

[15]

O. Junge and H. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270. doi: 10.1051/cocv:2004006.  Google Scholar

[16]

E. Kofman and J. Braslavsky, Level crossing sampling in feedback stabilization under data-rate constraints, in Proc. IEEE CDC, San Diego, USA, (2006), 4423-4428. doi: 10.1109/CDC.2006.377483.  Google Scholar

[17]

H. Kushner, On the stability of stochastic dynamical systems, Proc. Nat. Acad. Sci. USA, 53 (1965), 8-12. doi: 10.1073/pnas.53.1.8.  Google Scholar

[18]

H. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33, Academic Press, 1967.  Google Scholar

[19]

H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Applications of Mathematics, Vol. 35, Springer-Verlag, 1997.  Google Scholar

[20]

D. Liberzon, Quantization, time delays, and nonlinear stabilization, IEEE Transactions on Automatic Control, 51 (2006), 1190-1195. doi: 10.1109/TAC.2006.878780.  Google Scholar

[21]

D. Liberzon, Nonlinear control with limited information, Commun. Inf. Syst., 9 (2009), 41-57. doi: 10.4310/CIS.2009.v9.n1.a2.  Google Scholar

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L. Litz, T. Gabriel, M. Groß and O. Gabel, Networked Control Systems (NCS) - Stand und Ausblick, at - Automatisierungstechnik, 56 (2009), 4-19. doi: 10.1524/auto.2008.0682.  Google Scholar

[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, IEEE Transactions on, 57 (2012), 1326-1332. doi: 10.1109/TAC.2012.2191870.  Google Scholar

[24]

U. Lorenz and B. Monien, Error analysis in minimax trees, TCS, 313 (2004), 485-498. doi: 10.1016/j.tcs.2002.10.004.  Google Scholar

[25]

J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements, Automatica, 30 (1994), 417-431. doi: 10.1016/0005-1098(94)90119-8.  Google Scholar

[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), 2365-2370. doi: 10.1109/ACC.2005.1470320.  Google Scholar

[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), 732-747. doi: 10.1109/TAC.2009.2014930.  Google Scholar

[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), 365-373. doi: 10.1016/j.sysconle.2010.04.001.  Google Scholar

[29]

R. Sailer and F. Wirth, Stabilization of nonlinear systems with delayed data-rate-limited feedback, in Proc. European Control Conference, ECC2009, (2009), 1734-1739. Google Scholar

[30]

J. Schroeder, Modeling, State Observation and Diagnosis of Quantized Systems, Springer, Berlin, 2003.  Google Scholar

[31]

P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, 52 (2007), 1680-1685. doi: 10.1109/TAC.2007.904277.  Google Scholar

[32]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comp. Math., 2 (2002), 53-117.  Google Scholar

[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, Zürich, Switzerland, 2007. Google Scholar

[34]

X. Wang and M. Lemmon, Event-triggering in distributed networked systems with data dropouts and delays, Hybrid Systems: Computation and control (HSCC), 366-380, Lecture Notes in Comput. Sci., 5469, Springer, Berlin, 2009. doi: 10.1007/978-3-642-00602-9_26.  Google Scholar

[35]

C. Zhang, K. Chen and G. Dullerud, Stabilization of markovian jump linear systems with limited information - a convex approach, in Proc. ACC, (2009), 4013-4019. doi: 10.1109/ACC.2009.5160685.  Google Scholar

[36]

C. Zhang and G. Dullerud, Uniform stabilization of markovian jump linear systems with logarithmic quantization, in Proc. IEEE CDC, (2009), 2094-2099. doi: 10.1109/CDC.2009.5400361.  Google Scholar

[37]

L. Zhang, H. Gao and O. Kaynak, Network-induced constraints in networked control systems-a survey, IEEE Transactions on Industrial Informatics, 9 (2013), 403-416. doi: 10.1109/TII.2012.2219540.  Google Scholar

show all references

References:
[1]

K. Aström, Event based control, in Analysis and Design of Nonlinear Control Systems (eds. A. Astolfi and L. Marconi), Springer-Verlag, 2008, 127-147. doi: 10.1007/978-3-540-74358-3_9.  Google Scholar

[2]

K. Åström and B. Wittenmark, Computer-Controlled Systems: Theory and Design, Third Edition, Dover Books on Electrical Engineering Series, Dover Publications, 2011, URL http://books.google.de/books?id=9Y6D5vviqMgC. Google Scholar

[3]

S.-I. Azuma and T. Sugie, Dynamic quantization of nonlinear control systems, Automatic Control, IEEE Transactions on, 57 (2012), 875-888. doi: 10.1109/TAC.2011.2167824.  Google Scholar

[4]

D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 2., Belmont, MA: Athena Scientific, 1995. Google Scholar

[5]

R. S. Bucy, Stability and positive supermartingales, J. Differential Equations, 1 (1965), 151-155. doi: 10.1016/0022-0396(65)90016-1.  Google Scholar

[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), 4093-4098. Google Scholar

[7]

W. Fleming, The convergence problem for differential games, Journal of Mathematical Analysis and Applications, 3 (1961), 102-116. doi: 10.1016/0022-247X(61)90009-9.  Google Scholar

[8]

D. Förstner, M. Jung and J. Lunze, A discrete-event model of asynchronous quantised systems, Automatica, 38 (2002), 1277-1286. doi: 10.1016/S0005-1098(02)00023-7.  Google Scholar

[9]

E. Fridman and M. Dambrine, Control under quantization, saturation and delay: An LMI approach, Automatica, 45 (2009), 2258-2264. doi: 10.1016/j.automatica.2009.05.020.  Google Scholar

[10]

L. Grüne and O. Junge, Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property, in Proc. 46th IEEE CDC, (2007), 702-707. Google Scholar

[11]

L. Grüne and O. Junge, Global optimal control of perturbed systems, JOTA, 136 (2008), 411-429. doi: 10.1007/s10957-007-9312-z.  Google Scholar

[12]

L. Grüne and F. Müller, An algorithm for event-based optimal feedback control, in Proc. 48th IEEE CDC, 2009, 5311-5316. Google Scholar

[13]

J. P. Hespanha, P. Naghshtabrizi and Y. Xu, A survey of recent results in networked control systems, Proc. IEEE, 95 (2007), 138-162. doi: 10.1109/JPROC.2006.887288.  Google Scholar

[14]

O. Junge, Rigorous discretization of subdivision techniques, in EQUADIFF 99, Proceedings of the International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), vol. 2, 2000, 916-918.  Google Scholar

[15]

O. Junge and H. Osinga, A set oriented approach to global optimal control, ESAIM Control Optim. Calc. Var., 10 (2004), 259-270. doi: 10.1051/cocv:2004006.  Google Scholar

[16]

E. Kofman and J. Braslavsky, Level crossing sampling in feedback stabilization under data-rate constraints, in Proc. IEEE CDC, San Diego, USA, (2006), 4423-4428. doi: 10.1109/CDC.2006.377483.  Google Scholar

[17]

H. Kushner, On the stability of stochastic dynamical systems, Proc. Nat. Acad. Sci. USA, 53 (1965), 8-12. doi: 10.1073/pnas.53.1.8.  Google Scholar

[18]

H. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33, Academic Press, 1967.  Google Scholar

[19]

H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, Applications of Mathematics, Vol. 35, Springer-Verlag, 1997.  Google Scholar

[20]

D. Liberzon, Quantization, time delays, and nonlinear stabilization, IEEE Transactions on Automatic Control, 51 (2006), 1190-1195. doi: 10.1109/TAC.2006.878780.  Google Scholar

[21]

D. Liberzon, Nonlinear control with limited information, Commun. Inf. Syst., 9 (2009), 41-57. doi: 10.4310/CIS.2009.v9.n1.a2.  Google Scholar

[22]

L. Litz, T. Gabriel, M. Groß and O. Gabel, Networked Control Systems (NCS) - Stand und Ausblick, at - Automatisierungstechnik, 56 (2009), 4-19. doi: 10.1524/auto.2008.0682.  Google Scholar

[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, IEEE Transactions on, 57 (2012), 1326-1332. doi: 10.1109/TAC.2012.2191870.  Google Scholar

[24]

U. Lorenz and B. Monien, Error analysis in minimax trees, TCS, 313 (2004), 485-498. doi: 10.1016/j.tcs.2002.10.004.  Google Scholar

[25]

J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements, Automatica, 30 (1994), 417-431. doi: 10.1016/0005-1098(94)90119-8.  Google Scholar

[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), 2365-2370. doi: 10.1109/ACC.2005.1470320.  Google Scholar

[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), 732-747. doi: 10.1109/TAC.2009.2014930.  Google Scholar

[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), 365-373. doi: 10.1016/j.sysconle.2010.04.001.  Google Scholar

[29]

R. Sailer and F. Wirth, Stabilization of nonlinear systems with delayed data-rate-limited feedback, in Proc. European Control Conference, ECC2009, (2009), 1734-1739. Google Scholar

[30]

J. Schroeder, Modeling, State Observation and Diagnosis of Quantized Systems, Springer, Berlin, 2003.  Google Scholar

[31]

P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, 52 (2007), 1680-1685. doi: 10.1109/TAC.2007.904277.  Google Scholar

[32]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comp. Math., 2 (2002), 53-117.  Google Scholar

[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, Zürich, Switzerland, 2007. Google Scholar

[34]

X. Wang and M. Lemmon, Event-triggering in distributed networked systems with data dropouts and delays, Hybrid Systems: Computation and control (HSCC), 366-380, Lecture Notes in Comput. Sci., 5469, Springer, Berlin, 2009. doi: 10.1007/978-3-642-00602-9_26.  Google Scholar

[35]

C. Zhang, K. Chen and G. Dullerud, Stabilization of markovian jump linear systems with limited information - a convex approach, in Proc. ACC, (2009), 4013-4019. doi: 10.1109/ACC.2009.5160685.  Google Scholar

[36]

C. Zhang and G. Dullerud, Uniform stabilization of markovian jump linear systems with logarithmic quantization, in Proc. IEEE CDC, (2009), 2094-2099. doi: 10.1109/CDC.2009.5400361.  Google Scholar

[37]

L. Zhang, H. Gao and O. Kaynak, Network-induced constraints in networked control systems-a survey, IEEE Transactions on Industrial Informatics, 9 (2013), 403-416. doi: 10.1109/TII.2012.2219540.  Google Scholar

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