Advertisement

Discrete Event Dynamic Systems

, Volume 24, Issue 2, pp 199–218 | Cite as

Minimum attention control for linear systems

A linear programming approach
  • M. C. F. DonkersEmail author
  • P. Tabuada
  • W. P. M. H. Heemels
Article

Abstract

In this paper, we present a novel solution to the minimum attention control problem for linear systems. In minimum attention control, the objective is to minimise the ‘attention’ that a control task requires, given certain performance requirements. Here, we interpret ‘attention’ as the inverse of the interexecution time, i.e., the inverse of the time between two consecutive executions. Instrumental for our approach is a particular extension of the notion of a control Lyapunov function and the fact that we allow for only a finite number of possible interexecution times. By choosing this extended control Lyapunov function to be an ∞-norm-based function, the minimum attention control problem can be formulated as a linear program, which can be solved efficiently online. Furthermore, we provide a technique to construct a suitable ∞-norm-based (extended) control Lyapunov function. Finally, we illustrate the theory using a numerical example, which shows that minimum attention control outperforms an alternative ‘attention-aware’ control law available in the literature.

Keywords

Self-triggered control Attention-aware control Infinity-norm based Lyapunov functions Linear programmingx 

References

  1. Anta A, Tabuada P (2010) On the minimum attention and anytime attention problems for nonlinear systems. In: Proc. conf. decision & control, pp 3234–3239Google Scholar
  2. Anta A, Tabuada P (2010) To sample or not to sample: self-triggered control for nonlinear systems. IEEE Trans Automat Contr 55:2030–2042CrossRefMathSciNetGoogle Scholar
  3. Åström KJ, Wittenmark B (1997) Computer controlled systems. Prentice HallGoogle Scholar
  4. Brockett RW (1997) Minimum attention control. In: Proc. conf. decision & control, pp 2628–2632Google Scholar
  5. Chen T, Francis BA (1995) Optimal sampled-data control systems. SpringerGoogle Scholar
  6. Donkers MCF, Heemels WPMH (2012) Output-based event-triggered control with guaranteed \(\mathcal{L}_\infty\)-gain and improved and decentralised event-triggering. IEEE Trans Automat Contr 1362–1376Google Scholar
  7. Heemels WPMH, Sandee JH, van den Bosch PPJ (2008) Analysis of event-driven controllers for linear systems. Int J Control 81:571–590CrossRefzbMATHGoogle Scholar
  8. Henningsson T, Johannesson E, Cervin A (2008) Sporadic event-based control of first-order linear stochastic systems. Automatica 44:2890–2895CrossRefzbMATHMathSciNetGoogle Scholar
  9. Kellett CM, Teel AR (2004) Discrete-time asymptotic controllability implies smooth control-lyapunov function. Syst Control Lett 51:349–359CrossRefMathSciNetGoogle Scholar
  10. Khalil HK (1996) Nonlinear systems. Prentice HallGoogle Scholar
  11. Kiendl H, Adamy J, Stelzner P (1992) Vector norms as Lyapunov function for linear systems. IEEE Trans Automat Contr 37(6):839–842CrossRefzbMATHMathSciNetGoogle Scholar
  12. Kvasnica M, Grieder P, Baotić M (2004) Multi-parametric toolbox (MPT). http://control.ee.ethz.ch/~mpt/
  13. Lunze J, Lehmann D (2010) A state-feedback approach to event-based control. Automatica 46:211–215CrossRefzbMATHMathSciNetGoogle Scholar
  14. Mazo M Jr, Anta A, Tabuada P (2010) An ISS self-triggered implementation of linear controllers. Automatica 46:1310–1314CrossRefzbMATHMathSciNetGoogle Scholar
  15. Polański A (1995) On infinity norms as Lyapunov functions for linear systems. IEEE Trans Automat Contr 40(7):1270–1274CrossRefzbMATHGoogle Scholar
  16. Sontag E (1983) A Lyapunov-like characterization of asymptotic controllability. SIAM J Control Optim 21(3):462–471CrossRefzbMATHMathSciNetGoogle Scholar
  17. Tabuada P (2007) Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans Automat Contr 52:1680–1685CrossRefMathSciNetGoogle Scholar
  18. Velasco M, Fuertes JM, Martí P (2003) The self triggered task model for real-time control systems. In: Proc. IEEE real-time systems symposium, pp 67–70Google Scholar
  19. Walsh G, Ye H (2001) Scheduling of networked control systems. IEEE Control Syst Mag 21(1):57–65CrossRefGoogle Scholar
  20. Wang X, Lemmon M (2009) Self-triggered feedback control systems with finite-gain \(\mathcal{L}_2\) stability. IEEE Trans Automat Contr 45:452–467CrossRefMathSciNetGoogle Scholar
  21. Yook JK, Tilbury DM, Soparkar NR (2002) Trading computation for bandwidth: reducing communication in distributed control systems using state estimators. IEEE Trans Control Syst Technol 10(4):503–518CrossRefGoogle Scholar
  22. Yépez J, Velasco M, Martí P, Martín EX, Fuertes JM (2011) One-step finite horizon boundary with varying control gain for event-driven networked control systems. In: 37th annual conference of the IEEE industrial electronics societyGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • M. C. F. Donkers
    • 1
    Email author
  • P. Tabuada
    • 2
  • W. P. M. H. Heemels
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenNetherlands
  2. 2.University of CaliforniaLos AngelesUSA

Personalised recommendations