Shannon Entropy And Tracking Dynamic Systems Over Noisy Channels

Document Type : Article

Author

Department of Electrical Engineering at Sharif University of Technology

Abstract

This paper is concerned with the estimation of state trajectory of linear discrete time
dynamic systems subject to parametric uncertainty over the compound erasure channel
that uses feedback channel intermittently. For this combined system and channel,
using the data processing inequality and a robust version of the Shannon lower bound,
a necessary condition on channel capacity for estimation of state trajectory at the
receiver giving almost sure asymptotically zero estimation error is presented. Then,
an estimation technique over the compound erasure channel that includes an encoder,
decoder and a sucient condition under which the estimation error at the receiver
is asymptotically zero almost surely is presented. This leads to the conclusion that
over the compound erasure channel, a condition on Shannon capacity in terms of
the rate of expansion of the Shannon entropy is a necessary and sucient condition
for estimation with uniform almost sure asymptotically zero estimation error. The
satisfactory performance of the proposed technique is illustrated using simulation.

Keywords

Main Subjects


References
1. Brinon Arranz, L., Seuret, A., and Canudas de Wit,
C. Translation control of a
eet circular formation
of AUVs under nite communication range", In Proc.
48th IEEE Conference on Decision and Control, pp.
8345-8350 (2009).
2. Brinon Arranz, L., Seuret, A., and Canudas de Wit,
C. Collaborative estimation of gradient direction by a
formation of AUVs under communication constraints",
In Proceedings of the 50th IEEE Conference on Decision
and Control, pp. 5583-5588 (2011).
3. Elia, N. When bode meets Shannon: control-oriented
feedback communication schemes", IEEE Trans. Automat.
Contr., 49(9), pp. 1477-1488 (2004).
4. Elia, N. and Eisenbeis, J.N. Limitations of linear
control over packet drop networks", IEEE Trans.
Automat. Contr., 56(4), pp. 826-841 (2011).
5. Martins, N.C., Dahleh, A., and Elia, N. Feedback
stabilization of uncertain systems in the presence of
a direct link", IEEE Trans. Automat. Contr., 51(3),
pp. 438-447 (2006).
6. Minero, P., Franceschetti, M., Dey, S., and Nair, N.
Data rate theorem for stabilization over time-varying
feedback channels", IEEE Trans. Automat. Contr.,
54(2), pp. 243-255 (Feb., 2009).
7. Minero, P., Coviello, L., and Franceschetti, M. Stabilization
over Markov feedback channels: the general
case", IEEE Trans. Automat. Contr., 58(2), pp. 349-
362 (2013).
8. Nair, G.N., Evans, R.J., Mareels, I.M.Y., and Moran,
W. Topological feedback entropy and nonlinear stabilization",
IEEE Trans. Automat. Contr., 49(9), pp.
1585-1597 (2004).
9. Nair, G.N. and Evans, R.J. Stabilizability of stochastic
linear systems with nite feedback data rates",
SIAM J. Control Optimization, 43(3), pp. 413-436
(2004).
10. Canudas de Wit, C., Gomez-Estern, F., and Rodrigues
Rubio, F. Delta-modulation coding redesign
for feedback-controlled systems", IEEE Transactions
on Industrial Electronics, 56(7), pp. 2684-2696 (2009).
11. Farhadi, A. and Ahmed, N.U. Tracking nonlinear
noisy dynamic systems over noisy communication
channels", IEEE Transactions on Communications,
59(4), pp. 955-961 (2011).
12. Charalambous, C.D. and Farhadi, A. A mathematical
framework for robust control over uncertain communication
channels", Proceedings of the 44th IEEE
Conference on Decision and Control, Seville, Spain,
pp. 2530-2535 (2005).
13. Charalambous, C.D. and Farhadi, A. Control of
feedback systems subject to the nite rate constraints
via Shannon lower bound", Proceedings of the 5th International
Symposium on Modeling and Optimization
in Mobile, Ad Hoc, and Wireless Networks, Cyprus,
pp. 1-7 (2007).
14. Farhadi, A., Domun, J., and Canudas de Wit, C. A
supervisory control policy over an acoustic communication
network", International Journal of Control, 88(5),
pp. 946-958 (2015).
A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 25 (2018) 3517{3531 3527
15. Farhadi, A. Feedback channel in linear noiseless
dynamic systems controlled over the packet erasure
network", International Journal of Control, 88(8), pp.
1490-1503 (2015).
16. Farhadi, A. Stability of linear dynamic systems over
the packet erasure channel: a co-design approach",
International Journal of Control, 88(12), pp. 2488-
2498 (2015).
17. Farhadi, A. Stability of nonlinear uncertain Lipschitz
systems over the digital noiseless channel", Scientia
Iranica, Transactions on Computer Science and Engineering
and Electrical Engineering, 25(3), pp. 1523 -
1532 (2018).
18. Tatikonda, S. and Mitter, S. Control over noisy
channels", IEEE Transactions on Automatic Control,
49(7), pp. 1196-1201 (2004).
19. Jiang, X.W., Guan, Z.H., Feng, G., Wu, Y., and Yuan,
F.S. Optimal tracking performance of networked control
systems with channel input power constraint",
IET Control Theory and Applications, 6(11), pp. 1690-
1698 (2012).
20. Zhan, X.S., Wu, J., Jiang, T., and Jiang, X.W.
Optimal performance of networked control systems
under the packet dropouts and channel noise", ISA
Transactions, 58, pp. 214-221 (2015).
21. Jiang, X.W., Hu, B., Guan, Z.H., Zhang, X.H.,
and Yu, L. Best achievable tracking performance
for networked control systems with encoder-decoder",
Information Sciences, 305, pp. 184-195 (2015).
22. Matveev, A.S. and Savkin, A.V. Shannon zero error
capacity in the problems of state estimation and
stabilization via noisy communication channels", International
Journal of Control, 80, pp. 241-255 (2007).
23. Cover, T.M. and Thomas, J.A., Elements of Information
Theory, John Wiley and Sons, USA (1991).
24. Geromel, J.C. Optimal linear ltering under parameter
uncertainty", IEEE Transactions on Signal
Processing, 47(1), pp. 168-175 (1999).
25. Xu, S., Dooren, P.V., Stefan, R., and Lam, J. Robust
stability and stabilization for singular systems with
state delay and parameter uncertainty", IEEE Transactions
on Automatic Control, 47(7), pp. 1122-1128
(2002).
26. Gao, H., Meng, X., and Chen, T. A parameterdependent
approach to robust H1 ltering for timedelay
systems", IEEE Transactions on Automatic
Control, 53(10), pp. 2420-2425 (2008).
27. El Ghaoui, L. and Cala ore, G. Robust ltering for
discrete-time systems with bounded noise and parametric
uncertainty", IEEE Transactions on Automatic
Control, 46(7), pp. 1084-1089 (2001).
28. Wolfowitz, J. Simultaneous channels", Archive for
Rational Mechanics and Analysis, 4, pp. 371-386
(1959).
29. Lapidoth, A. and Narayan, P. Reliable communication
under channel uncertainty", IEEE Transactions
on Information Theory, 44(6), pp. 2148-2177 (1998).
30. Shrader, B. and Permuter, H. Feedback capacity
of the compound channel", IEEE Transactions on
Information Theory, 55(8), pp. 3629-3644 (2009).
31. Sakrison, D.J. The rate distortion function for a class
of sources", Information and Control, 15, pp. 165-195
(1969).
32. Linder, T. and Zamir, R. On the asymptotic tightness
of the Shannon lower bound", IEEE Transactions on
Information Theory, 40(6), pp. 2026-2031 (1994).
33. Charalambous, C.D. and Farhadi, A. LQG optimality
and separation principle for general discrete time partially
observed stochastic systems over nite capacity
communication channels", Automatica, 44(12), pp.
3181-3188 (2008).
34. Billingsley, P. Probability and Measure, John Wiley
and Sons, USA (1995).
Volume 25, Issue 6
Transactions on Computer Science & Engineering and Electrical Engineering (D)
November and December 2018
Pages 3517-3531