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

Global optimal feedbacks for stochastic quantized nonlinear event systems

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.
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.

[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.

[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.

[4]

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

[5]

R. S. Bucy, Stability and positive supermartingales, J. Differential Equations, 1 (1965), 151-155. 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), 4093-4098.

[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.

[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.

[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.

[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.

[11]

L. Grüne and O. Junge, Global optimal control of perturbed systems, JOTA, 136 (2008), 411-429. 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, 5311-5316.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

D. Liberzon, Nonlinear control with limited information, Commun. Inf. Syst., 9 (2009), 41-57. 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), 4-19. 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, IEEE Transactions on, 57 (2012), 1326-1332. doi: 10.1109/TAC.2012.2191870.

[24]

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

[25]

J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements, Automatica, 30 (1994), 417-431. 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), 2365-2370. 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), 732-747. 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), 365-373. 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, ECC2009, (2009), 1734-1739.

[30]

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

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[5]

R. S. Bucy, Stability and positive supermartingales, J. Differential Equations, 1 (1965), 151-155. 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), 4093-4098.

[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.

[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.

[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.

[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.

[11]

L. Grüne and O. Junge, Global optimal control of perturbed systems, JOTA, 136 (2008), 411-429. 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, 5311-5316.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[21]

D. Liberzon, Nonlinear control with limited information, Commun. Inf. Syst., 9 (2009), 41-57. 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), 4-19. 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, IEEE Transactions on, 57 (2012), 1326-1332. doi: 10.1109/TAC.2012.2191870.

[24]

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

[25]

J. Lunze, Qualitative modelling of linear dynamical systems with quantized state measurements, Automatica, 30 (1994), 417-431. 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), 2365-2370. 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), 732-747. 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), 365-373. 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, ECC2009, (2009), 1734-1739.

[30]

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

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[1]

Gary Froyland, Oliver Junge, Kathrin Padberg-Gehle. Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013). Journal of Computational Dynamics, 2015, 2 (1) : i-ii. doi: 10.3934/jcd.2015.2.1i

[2]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[3]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations and Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83

[4]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations and Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

[5]

Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial and Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188

[6]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443

[7]

Sang-Heon Lee. Development of concurrent structural decentralised discrete event system using bisimulation concept. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 305-317. doi: 10.3934/naco.2016013

[8]

Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229

[9]

Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004

[10]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6387-6403. doi: 10.3934/dcdsb.2021024

[11]

Wenbo Fu, Debnath Narayan. Optimization algorithm for embedded Linux remote video monitoring system oriented to the internet of things (IOT). Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1341-1354. doi: 10.3934/dcdss.2019092

[12]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19

[13]

Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144

[14]

Jing Zhang, Jiahui Qian, Han Zhang, Ling He, Bin Li, Jing Qin, Hongning Dai, Wei Tang, Weidong Tian. Maxillofacial surgical simulation system with haptic feedback. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3645-3657. doi: 10.3934/jimo.2020137

[15]

Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang, Yi-An Chen. On global boundedness of the Chen system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1673-1681. doi: 10.3934/dcdsb.2017080

[16]

Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361

[17]

Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689

[18]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134

[19]

Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177

[20]

Benedetta Lisena. Dynamic behaviour of a periodic competitive system with pulses. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 723-729. doi: 10.3934/dcdss.2013.6.723

 Impact Factor: 

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]