Optimal Relation Between Quantization Precision and Sampling Rates

  • Arben Çela
  • Mongi Ben Gaid
  • Xu-Guang Li
  • Silviu-Iulian Niculescu
Part of the Communications and Control Engineering book series (CCE)

Abstract

In this chapter, the problem of the control over limited bandwidth communication channels is addressed. A finely grained model is adopted. Such a model ensures the respect of the bandwidth constraints and allows the influence characterization of update frequency and quantization precision on the control performance. A simple static strategy is first proposed, and its (W,V)-stability properties are studied. An efficient approach for the improvement of disturbance rejection capabilities and steady state precision are then proposed. This approach dynamically assigns the quantization precision of the control signals in order to improve the control performance by taking into account the communication and computation requirements of the introduced dynamic protocol. It naturally allows handling linear time invariant systems (LTI) with multiple inputs. Sufficient conditions for ensuring the practical stability of this approach are stated. Finally, the proposed approach is evaluated and illustrated through a numerical example.

References

  1. 21.
    A. Bemporad, M. Morari, Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407–427 (1999) MathSciNetCrossRefMATHGoogle Scholar
  2. 24.
    M.-M. Ben Gaid, A. Çela, Trading quantization precision for update rate in systems with limited communication: a model predictive approach. Automatica 46(7), 1210–1214 (2010) MathSciNetCrossRefMATHGoogle Scholar
  3. 27.
    M.-M. Ben Gaid, A. Çela, Y. Hamam, Optimal integrated control and scheduling of networked control systems with communication constraints: application to a car suspension system. IEEE Trans. Control Syst. Technol. 14(4), 776–787 (2006) CrossRefGoogle Scholar
  4. 29.
    M.-M. Ben Gaid, A. Çela, Y. Hamam, C. Ionete, Optimal scheduling of control tasks with state feedback resource allocation, in American Control Conference, Minneapolis, Minnesota, USA, June 2006 Google Scholar
  5. 36.
    F. Blanchini, Set invariance in control. Automatica 35(11), 1747–1767 (1999) MathSciNetCrossRefMATHGoogle Scholar
  6. 42.
    R.W. Brockett, D. Liberzon, Quantized feedback stabilization of linear systems. IEEE Transactions on Automatic Control 45(7), 1279–1289 (2000) MathSciNetCrossRefMATHGoogle Scholar
  7. 43.
    R.W. Brockett, Stabilization of motor networks, in 34th IEEE Conference on Decision and Control, New Orleans, LA, USA, December 1995 Google Scholar
  8. 44.
    R.W. Brockett, Minimum attention control, in 36th IEEE Conference on Decision and Control, San Diego, California, USA, December 1997 Google Scholar
  9. 59.
    A. Chaillet, A. Bicchi, Delay compensation in packet-switching networked controlled systems, in 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 2008 Google Scholar
  10. 69.
    C. De Paersis, F. Mazenc, Stability of quantized time-delay nonlinear systems: a Lyapunov-Krasowskii-functional approach. Math. Control Signals Syst. 21(4), 337–370 (2010) CrossRefGoogle Scholar
  11. 72.
    D.F. Delchamps, Stabilizing a linear system with quantized state feedback. IEEE Transactions on Automatic Control 35(8), 916–924 (1990) MathSciNetCrossRefMATHGoogle Scholar
  12. 80.
    N. Elia, S.K. Mitter, Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control 46(9), 1384–1400 (2001) MathSciNetCrossRefMATHGoogle Scholar
  13. 82.
    F. Fagnani, S. Zampieri, Quantized stabilization of linear systems: complexity versus performance. IEEE Transactions on Automatic Control 49(9), 1534–1548 (2004) MathSciNetCrossRefGoogle Scholar
  14. 87.
    A. Franci, A. Chaillet, Quantized control of nonlinear systems: a robust approach. Int. J. Control 83(12), 2453–2462 (2010) MathSciNetCrossRefMATHGoogle Scholar
  15. 99.
    G.C. Goodwin, H. Haimovich, D.E. Quevedo, J.S. Welsh, A moving horizon approach to networked control systems design. IEEE Transactions on Automatic Control 49(9), 1427–1445 (2004) MathSciNetCrossRefGoogle Scholar
  16. 122.
    D. Hristu-Varsakelis, Optimal control with limited communication. PhD thesis, Division of Engineering and Applied Sciences, Harvard University, June 1999 Google Scholar
  17. 140.
    I. Kolmanovsky, E.G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems. Math. Probl. Eng. 4(4), 317–367 (1998) CrossRefMATHGoogle Scholar
  18. 182.
    G.N. Nair, R.J. Evans, Stabilization with data-rate-limited feedback: tightest attainable bounds. Systems and Control Letters 41(1), 49–56 (2000) MathSciNetCrossRefMATHGoogle Scholar
  19. 184.
    D. Nešić, D. Liberzon, A united framework for design and analysis of networked and quantized control systems. IEEE Transactions on Automatic Control 54(4), 732–747 (2009) CrossRefGoogle Scholar
  20. 197.
    S.V. Rakovic, E.C. Kerrigan, K.I. Kouramas, D.Q. Mayne, Invariant approximations of the minimal robust positively invariant set. IEEE Transactions on Automatic Control 50(3), 406–410 (2005) MathSciNetCrossRefGoogle Scholar
  21. 227.
    S. Tatikonda, S. Mitter, Control under communication constraints. IEEE Transactions on Automatic Control 49(7), 1056–1068 (2004) MathSciNetCrossRefGoogle Scholar
  22. 237.
    K. Tsumura, H. Ishii, H. Hoshina, Tradeoffs between quantization and packet loss in networked control of linear systems. Automatica 45(12), 2963–2970 (2009) MathSciNetCrossRefMATHGoogle Scholar
  23. 253.
    W.S. Wong, R.W. Brockett, Systems with finite communication bandwidth constraints—part I: state estimation problems. IEEE Transactions on Automatic Control 42(9), 1294–1299 (1997) MathSciNetCrossRefMATHGoogle Scholar
  24. 254.
    W.S. Wong, R.W. Brockett, Systems with finite communication bandwidth constraints—part II: stabilization with limited information feedback. IEEE Transactions on Automatic Control 44(5), 1049–1053 (1999) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Arben Çela
    • 1
  • Mongi Ben Gaid
    • 2
  • Xu-Guang Li
    • 3
  • Silviu-Iulian Niculescu
    • 4
  1. 1.Department of Computer Science and TelecommunicationUniversité Paris-Est, ESIEE ParisNoisy-le-GrandFrance
  2. 2.Electronic and Real-Time Systems DepartmentIFP New EnergyRueil-MalmaisonFrance
  3. 3.School of Information Science and EngineeringNortheastern UniversityShenyangPeople’s Republic of China
  4. 4.L2S—Laboratoire des signaux et systèmesSupélecGif-sur-YvetteFrance

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