Skip to main content
SpringerLink
Log in

Mean Square Limitations of Spatially Invariant Networked Systems

  • Chapter
Control of Cyber-Physical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 449))

Abstract

In this chapter, we summarize some of our recent results on spatially invariant networked systems. We study the effects of unreliable communication on these systems’ stability and performance. Because of their special structure, the quantities that characterize the limitations of these systems can be computed more easily. In particular, we focus on Mean Square stability and performance, and investigate network architectures, which are more robust and better performing. We consider multi-agent networked systems where the communication links are unreliable and stochastically dropout. Spatial invariance leads to a simplified computation of the MS stability limitation, and allows us to derive an uncertainty conservation law enjoyed by such systems. We then focus on distributed averaging systems, for which the loss of Mean Square stability leads to the emergence of certain complex behavior related to Lévy flights. We present closed form formulae characterizing Mean Square stability and performance in the presence of unreliable communication among the nodes. Finally, we study Mean square performance in the presence of unreliable links. Our results allow to characterize the interplay between Mean Square performance and stability of torus networks of different dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Tsitsiklis, J.N.: Problems in decentralized decision making and computation, Ph.D. Thesis, Massachusetts Institute of Technology (1984)

    Google Scholar 

  2. Tsitsiklis, J.N., Bertsekas, D.P., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE TAC 31(9), 803–812 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Contr. 48(6), 988–1001 (2003)

    Article  MathSciNet  Google Scholar 

  4. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49(9), 1520–1533 (2004)

    Article  MathSciNet  Google Scholar 

  5. Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control Special Issue on Networked Control 49(9), 1465–1476 (2004)

    Article  MathSciNet  Google Scholar 

  6. Spanos, D.P., Olfati-Saber, R., Murray, R.M.: Dynamic consensus on mobile networks. Presented at the 16th IFAC World Congr., Prague, Czech Republic (July 2005)

    Google Scholar 

  7. Reynolds, A.M., Smith, A.D., Menzel, R., Greggers, U., Reynolds, D.R., Riley, J.R.: Displaced honey bees perform optimal scale-free search flights. Ecology 88(8), 1955–1961 (2007)

    Article  Google Scholar 

  8. Bartumeus, F., Daluz, M.G.E., Viswanathan, G.M., Catalan, J.: Animal search strategies: A quantitative random-walk analysis. Ecology 86(11), 3078–3087 (2005)

    Article  Google Scholar 

  9. Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439, 462–465 (2006)

    Article  Google Scholar 

  10. Mandelbrot, B.: The variation of certain speculative prices. Journal of Business 36, 394–419 (1963)

    Article  Google Scholar 

  11. Mantegna, R., Stanley, H.: Scaling behavior in the dynamics of an economic index. Nature 376, 46–49 (1995)

    Article  Google Scholar 

  12. Davis, P.J.: Circulant Matrices. A Wiley-Interscience Publication, Pure and Applied Mathematics. John Wiley & Sons, New York (1979)

    MATH  Google Scholar 

  13. Bamieh, B., Paganini, F., Dahleh, M.: Distributed control of spatially invariant systems. IEEE Trans. Automat. Contr. 47, 1091–1118 (2002)

    Article  MathSciNet  Google Scholar 

  14. Wang, J., Elia, N.: Distributed agreement in the presence of noise. In: 47th Annual Allerton Conference, September 30-October 2. Allerton House, UIUC, Illinois (2009)

    Google Scholar 

  15. Wang, J., Elia, N.: Dynamic average consensus over random networks with additive noise. In: IEEE Conference on Decision and Control, pp. 4789–4794 (2010)

    Google Scholar 

  16. Wang, J., Elia, N.: Control approach to distributed optimization. In: Allerton Conference on Communication, Control and Computing, September 29-October 1, pp. 557–561 (2010)

    Google Scholar 

  17. Wang, J., Elia, N.: A Control perspective to centralized and distributed optimization. Finalist for Best Student Paper Award in Proc. 50th IEEE CDC-ECC (2011)

    Google Scholar 

  18. Elia, N.: Remote stabilization over fading channels. Systems and Control Letters 54(3), 237–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Elia, N., Eisenbeis, J.N.: Limitations of linear control over packet drop networks

    Google Scholar 

  20. Arnold, L.: Stochastic differential equations: theory and applications. Krieger, Malabar (1992)

    Google Scholar 

  21. Kushner, H.J.: Stochastic Stability and Control. Academic Press, New York (1967)

    MATH  Google Scholar 

  22. Meyn, S.P., Tweedie, R.L.: Markov chains and stochastic stability. Springer, London (1993)

    Book  MATH  Google Scholar 

  23. Costa, O.L.V., Fragoso, M.D.: Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters. Journal of Mathematic Analysis Applications 179, 154–178 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Costa, O.L.V., Fragoso, M.D., Marques, R.P.: Discrete-Time Markov Jump Linear Systems. Springer (2005)

    Google Scholar 

  25. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1987)

    Google Scholar 

  26. Elia, N.: Emergence of Power Laws in Networked Control Systems. In: Proc. IEEE CDC 2006, pp. 490–495 (2006)

    Google Scholar 

  27. Wang, J., Elia, N.: Distributed averaging under constraints on information exchange: emergence of Lévy flights. IEEE Trans. on Automat. Control 57(10), 2435–2449 (2012)

    Article  MathSciNet  Google Scholar 

  28. Wang, J., Elia, N.: Mean square stability of consensus over fading networks with nonhomogeneous communication delays. In: Proc. of the 47th IEEE Conf. on Decision and Control, pp. 4614–4619 (December 2008)

    Google Scholar 

  29. Wang, J., Elia, N.: Mitigation of Complex Behavior over Networked Systems: Analysis of Spatially Invariant Structures. To appear in Automatica (2013)

    Google Scholar 

  30. Padmasola, P., Elia, N.: Bode Integral Limitations of Spatially Invariant Systems. In: IEEE CDC 2006, pp. 4327–4332 (2006)

    Google Scholar 

  31. Padmasola, P., Elia, N.: Mean Square Stabilization of LTI Systems Over Packet-Drop Networks with Imperfect Side Information. In: Proc. American Control Conference, pp. 5550–5555 (2006)

    Google Scholar 

  32. Napoli, M., Bamieh, B., Dahleh, M.: H 2 Norm Minimization for Distributed Continuous Time Systems: an Input/ Output Approach. In: Proc. 1998 Mathematical Theory of Networks and Systems, Padova, Italy (July 1998)

    Google Scholar 

  33. Bamieh, B., Jovanovic, M.R., Mitra, P., Paterson, S.: Coherence in large-scale networks: dimension dependent limitations of local feedback. IEEE Trans. on Automatic Control 57(9), 2235–2249 (2012)

    Article  Google Scholar 

  34. Ma, X., Elia, N.: Mean Square Performance Analysis for Torus-Networked Distributed Consensus under Stochastic Disturbances. Submitted to IEEE CDC (2013)

    Google Scholar 

  35. Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall, New Jersey (1996)

    MATH  Google Scholar 

  36. Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE Trans. Inform. Theory 52(6), 2508–2530 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Electrical and Computer Engineering, Iowa State University, Ames, U.S.A.

    Nicola Elia, Jing Wang & Xu Ma

Authors
  1. Nicola Elia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xu Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicola Elia .

Editor information

Editors and Affiliations

  1. , Department of Electrical and, The Johns Hopkins University, Baltimore, 21218, Maryland, USA

    Danielle C. Tarraf

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Elia, N., Wang, J., Ma, X. (2013). Mean Square Limitations of Spatially Invariant Networked Systems. In: Tarraf, D. (eds) Control of Cyber-Physical Systems. Lecture Notes in Control and Information Sciences, vol 449. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-01159-2_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-01159-2_19

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-01158-5

  • Online ISBN: 978-3-319-01159-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation