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Stabilization and Control over Gaussian Networks

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Information and Control in Networks

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 450))

Abstract

We provide an overview and some recent results on real-time communication and control over Gaussian channels. In particular, the problem of remote stabilization of linear systems driven by Gaussian noise over Gaussian relay channels is considered. Necessary and sufficient conditions for the mean-square stabilization are presented, which reveal signal-to-noise ratio requirements for stabilization which are tight in a certain class of settings. Optimal linear policies are constructed, global optimality and sub-optimality of such policies are investigated in a variety of settings. We also consider the design of low-delay sensing and transmit schemes for real-time communication.

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References

  1. Akyol, E., Viswanatha, K., Rose, K.: On conditions for linearity of optimal estimation. IEEE Trans. Inf. Theory 58(6), 3497–3508 (2012)

    Article  MathSciNet  Google Scholar 

  2. Andersson, M., Zaidi, A.A., Wernersson, N., Skoglund, M.: Nonlinear distributed sensing for closed-loop control over Gaussian channels. In: IEEE Swe–CTW, pp. 19–23 (2011)

    Google Scholar 

  3. Bansal, R., Başar, T.: Solutions to a class of linear-quadratic-Gaussian LQG stochastic team problems with nonclassical information. Syst. Control Lett. 9(2), 125–130 (1987)

    Article  MATH  Google Scholar 

  4. Bansal, R., Başar, T.: Stochastic teams with nonclassical information revisited: when is an affine law optimal? IEEE Trans. Autom. Control 32(6), 554–559 (1987)

    Article  MATH  Google Scholar 

  5. Bansal, R., Başar, T.: Simultaneous design of measurement and control strategies for stochastic systems with feedback. Automatica 25(5), 679–694 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Başar, T.: A trace minimization problem with applications in joint estimation and control under nonclassical information. J. Optim. Theory Appl. 31(3), 343–359 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Başar, T., Bansal, R.: Optimum design of measurement channels and control policies for linear-quadratic stochastic systems. Eur. J. Oper. Res. 73(2), 226–236 (1994)

    Article  MATH  Google Scholar 

  8. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Feedback stabilization over signal-to-noise ratio constrained channels. IEEE Trans. Autom. Control 52(8), 1391–1403 (2007)

    Article  MathSciNet  Google Scholar 

  9. Charalambous, C.D., Farhadi, A., Denic, S.Z.: Control of continuous-time linear Gaussian systems over additive Gaussian wireless fading channels: a separation principle. IEEE Trans. Autom. Control 53(4), 1013–1019 (2008)

    Article  MathSciNet  Google Scholar 

  10. Cover, T., Thomas, J.: Elements of Information Theory. Wiley, New York (2006)

    MATH  Google Scholar 

  11. Elia, N.: When Bode meets Shannon: control-oriented feedback communication schemes. IEEE Trans. Autom. Control 49(9), 1477–1488 (2004)

    Article  MathSciNet  Google Scholar 

  12. Farhadi, A., Ahmed, N.U.: Suboptimal decentralized control over noisy communication channels. Syst. Control Lett. 60(4), 285–293 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Freudenberg, J.S., Middleton, R.H., Solo, V.: Stabilization and disturbance attenuation over a Gaussian communication channel. IEEE Trans. Autom. Control 55(3), 795–799 (2010)

    Article  MathSciNet  Google Scholar 

  14. Gastpar, M., Vetterli, M.: On the capacity of large Gaussian relay networks. IEEE Trans. Inf. Theory 51(3), 765–779 (2005)

    Article  MathSciNet  Google Scholar 

  15. Gastpar, M., Rimoldi, B., Vetterli, M.: To code, or not to code: lossy source-channel communication revisited. IEEE Trans. Inf. Theory 49(5), 1147–1158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ho, Y.C.: Team decision theory and information structures. Proc. IEEE 68(6), 644–654 (1980)

    Article  Google Scholar 

  17. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  18. Johannesson, E., Rantzer, A., Bernhardsson, B.: Optimal linear control for channels with signal-to-noise ratio constraints. In: ACC, pp. 521–526 (2011)

    Google Scholar 

  19. Karlsson, J., Skoglund, M.: Analog distributed source-channel coding using sinusoids. In: IEEE ISWCS, pp. 279–282 (2009)

    Google Scholar 

  20. Karlsson, J., Skoglund, M.: Optimized low-delay source-channel-relay mapping. IEEE Trans. Commun. 58(5), 1397–1404 (2010)

    Article  Google Scholar 

  21. Kramer, G.: Feedback strategies for white Gaussian interference networks. IEEE Trans. Inf. Theory 48(6), 1423–1438 (2002)

    Article  MATH  Google Scholar 

  22. Kramer, G., Maric, I., Yates, R.: Cooperative communications. Found. Trends Netw. 1(3–4), 271–425 (2006)

    Article  Google Scholar 

  23. Kumar, U., Gupta, V., Laneman, J.N.: Sufficient conditions for stabilizability over Gaussian relay channel and cascade channels. In: IEEE CDC, pp. 4765–4770 (2010)

    Google Scholar 

  24. Kumar, U., Gupta, V., Laneman, J.N.: On stability across a Gaussian product channel. In: IEEE CDC, pp. 3142–3147 (2011)

    Google Scholar 

  25. Lee, K.H., Petersen, D.P.: Optimal linear coding for vector channels. IEEE Trans. Commun. 24(12), 1283–1290 (1976)

    Article  MATH  Google Scholar 

  26. Leong, A.S., Dey, S., Anand, J.: Optimal LQG control over continuous fading channels. In: 18th IFAC World Congress (2011)

    Google Scholar 

  27. Lipsa, G.M., Martins, N.C.: Optimal memoryless control in Gaussian noise: a simple counterexample. Automatica 47, 552–558 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Martins, N.C., Dahleh, M.A.: Feedback control in the presence of noisy channels: “Bode-like” fundamental limitations of performance. IEEE Trans. Autom. Control 53(7), 1604–1615 (2008)

    Article  MathSciNet  Google Scholar 

  29. Massey, J.L.: Causality, feedback and directed information. In: IEEE ISITA (1990)

    Google Scholar 

  30. Matveev, A.S., Savkin, A.V.: Estimation and Control over Communication Networks. Birkhäuser, Boston (2008)

    Google Scholar 

  31. Middleton, R.H., Rojas, A.J., Freudenberg, J.S., Braslavsky, J.H.: Feedback stabilization over a first order moving average Gaussian noise channel. IEEE Trans. Autom. Control 54(1), 163–167 (2009)

    Article  MathSciNet  Google Scholar 

  32. Ozarow, L.H.: The capacity of the white Gaussian multiple-access channel with feed-back. IEEE Trans. Inf. Theory 30(4), 623–629 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ozarow, L., Leung-Yan-Cheong, S.: An achievable region and outer bound for the Gaussian broadcast channel with feedback. IEEE Trans. Inf. Theory 30(4), 667–671 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pilc, R.J.: The optimum linear modulator for a Gaussian source used with a Gaussian channel. Bell Syst. Tech. J. 3075–3089 (1969)

    Google Scholar 

  35. Radner, R.: Team decision problems. Ann. Math. Stat. 33, 857–881 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sahai, A., Mitter, S.: The necessity and sufficiency of anytime capacity for stabilization of a linear system over noisy communication links—part I: scalar systems. IEEE Trans. Inf. Theory 52(8), 3369–3395 (2006)

    Article  MathSciNet  Google Scholar 

  37. Schalkwijk Kailath, T.: A coding scheme for additive noise channels with feedback—I: no bandwidth constraint. IEEE Trans. Inf. Theory 12(2), 172–182 (1966)

    Article  MATH  Google Scholar 

  38. Shamai, S., Verdu, S., Zamir, R.: Systematic lossy source/channel coding. IEEE Trans. Inf. Theory 44(2), 564–579 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shu, Z., Middleton, R.H.: Stabilization over power-constrained parallel Gaussian channels. IEEE Trans. Autom. Control 56(7), 1718–1724 (2011)

    Article  MathSciNet  Google Scholar 

  40. Silva, E.I., Goodwin, G.C., Quevedo, D.E.: Control system design subject to SNR constraints. Automatica 46(2), 428–436 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Silva, E.I., Derpich, M.S., Ostergaard, J.: A framework for control system design subject to average data-rate constraints. IEEE Trans. Autom. Control 56(8), 1886–1899 (2011)

    Article  MathSciNet  Google Scholar 

  42. Skoglund, N.W.M., Ramstad, T.: Polynomial based analog source-channel codes. IEEE Trans. Commun. 57(9), 2600–2606 (2009)

    Article  Google Scholar 

  43. Striebel, C.: Sufficient statistics in the optimum control of stochastic systems. J. Math. Anal. Appl. 12, 576–592 (1962)

    Article  MathSciNet  Google Scholar 

  44. Tatikonda, S., Mitter, S.: Control over noisy channels. IEEE Trans. Autom. Control 49(7), 1196–1201 (2004)

    Article  MathSciNet  Google Scholar 

  45. Tatikonda, S., Sahai, A., Mitter, S.: Stochastic linear control over a communication channel. IEEE Trans. Autom. Control 49(9), 1549–1561 (2004)

    Article  MathSciNet  Google Scholar 

  46. Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  47. Wernersson, N., Skoglund, M.: Nonlinear coding and estimation for correlated data in wireless sensor networks. IEEE Trans. Commun. 57(10), 2932–2939 (2009)

    Article  Google Scholar 

  48. Witsenhausen, H.S.: A counterexample in stochastic optimum control. SIAM J. Control 6, 131–147 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  49. Witsenhausen, H.S.: Separation of estimation and control for discrete time systems. Proc. IEEE 59(11), 1557–1566 (1971)

    Article  MathSciNet  Google Scholar 

  50. Wonham, W.M.: On the separation theorem of stochastic control. SIAM J. Control 6, 312–326 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wu, Y., Verdú, S.: Functional properties of MMSE. In: IEEE ISIT, pp. 1453–1457 (2010)

    Google Scholar 

  52. Yüksel, S.: Stochastic nestedness and the belief sharing information pattern. IEEE Trans. Autom. Control 54(12), 2773–2786 (2009)

    Article  Google Scholar 

  53. Yüksel, S.: On optimal causal coding of partially observed Markov sources under classical and non-classical information structures. In: IEEE ISIT, pp. 81–85 (2010)

    Google Scholar 

  54. Yüksel, S.: Characterization of information channels for asymptotic mean stationarity and stochastic stability of non-stationary/unstable linear systems. IEEE Trans. Inf. Theory 58(10), 6332–6354 (2012)

    Article  Google Scholar 

  55. Yüksel, S.: On optimal causal coding of partially observed Markov sources in single and multi-terminal settings. IEEE Trans. Inf. Theory 59, 424–437 (2013)

    Article  Google Scholar 

  56. Yüksel, S., Başar, T.: Control over noisy forward and reverse channels. IEEE Trans. Autom. Control 56, 1014–1029 (2011)

    Article  Google Scholar 

  57. Yüksel, S., Başar, T.: Stochastic Networked Control Systems: Stabilization and Optimization Under Information Constraints. Birkhäuser, Boston (2013)

    Book  Google Scholar 

  58. Yüksel, S., Tatikonda, S.: A counterexample in distributed optimal sensing and control. IEEE Trans. Autom. Control 54(4) (2009)

    Google Scholar 

  59. Zaidi, A.A., Oechtering, T.J., Skoglund, M.: Rate sufficient conditions for closed–loop control over AWGN relay channels. In: IEEE ICCA, pp. 602–607 (2010)

    Google Scholar 

  60. Zaidi, A.A., Oechtering, T.J., Skoglund, M.: Sufficient conditions for closed-loop control over multiple-access and broadcast channels. In: IEEE CDC (2010)

    Google Scholar 

  61. Zaidi, A.A., Oechtering, T.J., Yüksel, S., Skoglund, M.: Closed-loop control over half-duplex AWGN relay channels. In: Reglermöte (2010)

    Google Scholar 

  62. Zaidi, A.A., Khormuji, M.N., Skoglund, M.: Nonlinear transmission strategies for a general half-duplex AWGN relay channel. In: IEEE Swe–CTW, pp. 58–61 (2011)

    Google Scholar 

  63. Zaidi, A.A., Oechtering, T.J., Skoglund, M.: Closed-loop stabilization over Gaussian interference channel. In: 18th IFAC World Congress (2011)

    Google Scholar 

  64. Zaidi, A.A., Oechtering, T.J., Yüksel, S., Skoglund, M.: Sufficient conditions for closed-loop control over a Gaussian relay channel. In: IEEE ACC, pp. 2240–2245 (2011)

    Google Scholar 

  65. Zaidi, A.A., Yüksel, S., Oechtering, T.J., Skoglund, M.: On optimal policies for control and estimation over a Gaussian relay channel. In: IEEE CDC (2011)

    Google Scholar 

  66. Zaidi, A.A., Oechtering, T.J., Skoglund, M.: On stabilization over a Gaussian interference channel. In: IEEE ECC (2013)

    Google Scholar 

  67. Zaidi, A.A., Oechtering, T.J., Yüksel, S., Skoglund, M.: Stabilization of linear systems over Gaussian networks. IEEE Trans. Autom. Control (2012, accepted)

    Google Scholar 

  68. Zaidi, A.A., Yüksel, S., Oechtering, T.J., Skoglund, M.: On the tightness of linear policies for stabilization of linear systems over Gaussian networks. Syst. Control Lett. (2013, under review)

    Google Scholar 

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Acknowledgements

We would like to thank the editors, B. Bernhardsson, G. Como, and A. Rantzer, for giving us an opportunity to write this chapter. We are also very grateful to Johanness Kron (formerly Johanness Karlsson) for performing numerical simulations to generate the figures included in Sect. 2.6 of this chapter. Some of these results are part of his PhD thesis.

This research was supported by LCCC—Linnaeus Grant VR 2007-8646, Swedish Research Council.

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Authors and Affiliations

  1. KTH Royal Institute of Technology, Stockholm, Sweden

    Ali A. Zaidi, Tobias J. Oechtering & Mikael Skoglund

  2. Queen’s University, Kingston, Canada

    Serdar Yüksel

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  1. Ali A. Zaidi
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  2. Tobias J. Oechtering
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  3. Serdar Yüksel
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  4. Mikael Skoglund
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Corresponding author

Correspondence to Ali A. Zaidi .

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Editors and Affiliations

  1. Department of Automatic Control, Lund University, Lund, Sweden

    Giacomo Como

  2. Department of Automatic Control, Lund University, Lund, Sweden

    Bo Bernhardsson

  3. Department of Automatic Control, Lund University, Lund, Sweden

    Anders Rantzer

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Zaidi, A.A., Oechtering, T.J., Yüksel, S., Skoglund, M. (2014). Stabilization and Control over Gaussian Networks. In: Como, G., Bernhardsson, B., Rantzer, A. (eds) Information and Control in Networks. Lecture Notes in Control and Information Sciences, vol 450. Springer, Cham. https://doi.org/10.1007/978-3-319-02150-8_2

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  • DOI: https://doi.org/10.1007/978-3-319-02150-8_2

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  • Online ISBN: 978-3-319-02150-8

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