The Annals of Applied Probability

When multiplicative noise stymies control

Jian Ding, Yuval Peres, Gireeja Ranade, and Alex Zhai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Our proof technique recursively bounds the conditional density of the system state to bound the progress the controller can make. This sidesteps the difficulty encountered in using the standard data-rate theorem style approach; that approach does not work because the mutual information per round between the system state and the observation is potentially unbounded.

It was known that a system with multiplicative observation noise can be stabilized using a simple memoryless linear strategy if the system growth is suitably bounded. The second main result in this paper shows that while memory cannot improve the performance of a linear scheme, a simple nonlinear scheme that uses one-step memory can do better than the best linear scheme.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 1963-1992.

Dates
Received: December 2016
Revised: August 2017
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869034

Digital Object Identifier
doi:10.1214/18-AAP1415

Mathematical Reviews number (MathSciNet)
MR3984251

Subjects
Primary: 93E03: Stochastic systems, general

Keywords
Control stability multiplicative noise

Citation

Ding, Jian; Peres, Yuval; Ranade, Gireeja; Zhai, Alex. When multiplicative noise stymies control. Ann. Appl. Probab. 29 (2019), no. 4, 1963--1992. doi:10.1214/18-AAP1415. https://projecteuclid.org/euclid.aoap/1563869034


Export citation

References

  • [1] Athans, M., Ku, R. and Gershwin, S. B. (1977). The uncertainty threshold principle: Some fundamental limitations of optimal decision making under dynamic uncertainty. IEEE Trans. Automat. Control AC-22 491–495.
  • [2] Bao, L., Skoglund, M. and Johansson, K. H. (2011). Iterative encoder-controller design for feedback control over noisy channels. IEEE Trans. Automat. Control 56 265–278.
  • [3] Braslavsky, J. H., Middleton, R. H. and Freudenberg, J. S. (2007). Feedback stabilization over signal-to-noise ratio constrained channels. IEEE Trans. Automat. Control 52 1391–1403.
  • [4] Chiuso, A., Laurenti, N., Schenato, L. and Zanella, A. (2014). LQG-like control of scalar systems over communication channels: The role of data losses, delays and SNR limitations. Automatica J. IFAC 50 3155–3163.
  • [5] Dey, S., Chiuso, A. and Schenato, L. (2017). Feedback control over lossy SNR-limited channels: Linear encoder-decoder-controller design. IEEE Trans. Automat. Control 62 3054–3061.
  • [6] Ding, J., Peres, Y. and Ranade, G. (2016). A tiger by the tail: When multiplicative noise stymies control. In IEEE International Symposium on Information Theory.
  • [7] Elia, N. (2005). Remote stabilization over fading channels. Systems Control Lett. 54 237–249.
  • [8] Fang, S., Chen, J. and Ishii, H. (2016). Towards Integrating Control and Information Theories. Lecture Notes in Control and Information Sciences 465. Springer, Cham.
  • [9] Freudenberg, J. S., Middleton, R. H. and Braslavsky, J. H. (2011). Minimum variance control over a Gaussian communication channel. IEEE Trans. Automat. Control 56 1751–1765.
  • [10] Garone, E., Sinopoli, B., Goldsmith, A. and Casavola, A. (2012). LQG control for MIMO systems over multiple erasure channels with perfect acknowledgment. IEEE Trans. Automat. Control 57 450–456.
  • [11] Hespanha, J. P., Naghshtabrizi, P. and Xu, Y. (2007). A survey of recent results in networked control systems. Proc. IEEE 95 138–162.
  • [12] Kostina, V. and Hassibi, B. (2016). Rate cost tradeoffs in control. part 1: Lower bounds. arXiv:1612.02126 [cs.IT].
  • [13] Kostina, V., Peres, Y., Rácz, M. and Ranade, G. (2016). Rate-limited control of systems with uncertain gain. In Allerton Conference on Communication, Control, and Computing.
  • [14] Lapidoth, A. and Moser, S. M. (2003). Capacity bounds via duality with applications to multiple-antenna systems on flat-fading channels. IEEE Trans. Inform. Theory 49 2426–2467.
  • [15] Martins, N. C., Dahleh, M. A. and Elia, N. (2006). Feedback stabilization of uncertain systems in the presence of a direct link. IEEE Trans. Automat. Control 51 438–447.
  • [16] Matveev, A. S. and Savkin, A. V. (2004). The problem of LQG optimal control via a limited capacity communication channel. Systems Control Lett. 53 51–64.
  • [17] Matveev, A. S. and Savkin, A. V. (2009). Estimation and Control over Communication Networks. Control Engineering. Birkh\Hauser, Boston, MA.
  • [18] Meyr, H., Moeneclaey, M. and Fechtel, S. A. (1998). Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. Wiley, New York.
  • [19] Minero, P., Coviello, L. and Franceschetti, M. (2013). Stabilization over Markov feedback channels: The general case. IEEE Trans. Automat. Control 58 349–362.
  • [20] Minero, P., Franceschetti, M., Dey, S. and Nair, G. N. (2009). Data rate theorem for stabilization over time-varying feedback channels. IEEE Trans. Automat. Control 54 243–255.
  • [21] Nair, G. N. and Evans, R. J. (2004). Stabilizability of stochastic linear systems with finite feedback data rates. SIAM J. Control Optim. 43 413–436.
  • [22] Nair, G. N., Fagnani, F., Zampieri, S. andEvans, R. J. (2007). Feedback control under data rate constraints: An overview. Proc. IEEE 95 108–137.
  • [23] Okano, K. and Ishii, H. (2014). Minimum data rate for stabilization of linear systems with parametric uncertainties. arXiv preprint, arXiv:1405.5932.
  • [24] Park, S., Ranade, G. and Sahai, A. (2012). Carry-free models and beyond. In IEEE International Symposium on Information Theory 1927–1931. IEEE.
  • [25] Park, S. and Sahai, A. (2011). Intermittent Kalman filtering: Eigenvalue cycles and nonuniform sampling. In American Control Conference (ACC) 3692–3697.
  • [26] Rajasekaran, P. K., Satyanarayana, N. and Srinath, M. D. (1971). Optimum linear estimation of stochastic signals in the presence of multiplicative noise. IEEE Trans. Aerosp. Electron. Syst. AES-7 462–468.
  • [27] Ranade, G. (2014). Active systems with uncertain parameters: An information-theoretic perspective. Ph.D. thesis, Univ. California, Berkeley.
  • [28] Ranade, G. and Sahai, A. (2013). Non-coherence in estimation and control. In 51st Annual Allerton Conference on Communication, Control, and Computing.
  • [29] Ranade, G. and Sahai, A. (2015). Control capacity. In IEEE International Symposium on Information Theory. IEEE, New York.
  • [30] Ranade, G. and Sahai, A. (2017). Control capacity. arXiv:1701.04187 [cs.IT].
  • [31] Sahai, A. and Mitter, S. (2006). The necessity and sufficiency of anytime capacity for stabilization of a linear system over a noisy communication link. I. Scalar systems. IEEE Trans. Inform. Theory 52 3369–3395.
  • [32] Schenato, L., Sinopoli, B., Franceschetti, M., Poolla, K. and Sastry, S. S. (2007). Foundations of control and estimation over lossy networks. Proc. IEEE 95 163–187.
  • [33] Silva, E. I., Goodwin, G. C. and Quevedo, D. E. (2010). Control system design subject to SNR constraints. Automatica J. IFAC 46 428–436.
  • [34] Silva, E. I. and Pulgar, S. A. (2013). Performance limitations for single-input LTI plants controlled over SNR constrained channels with feedback. Automatica J. IFAC 49 540–547.
  • [35] Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M. I. and Sastry, S. S. (2004). Kalman filtering with intermittent observations. IEEE Trans. Automat. Control 49 1453–1464.
  • [36] Tatikonda, S. and Mitter, S. (2004). Control under communication constraints. IEEE Trans. Automat. Control 49 1056–1068.
  • [37] Tugnait, J. K. (1981). Stability of optimum linear estimators of stochastic signals in white multiplicative noise. IEEE Trans. Automat. Control 26 757–761.
  • [38] Wong, W. S. and Brockett, R. W. (1997). Systems with finite communication bandwidth constraints. I. State estimation problems. IEEE Trans. Automat. Control 42 1294–1299.
  • [39] Xiao, N., Xie, L. and Qiu, L. (2012). Feedback stabilization of discrete-time networked systems over fading channels. IEEE Trans. Automat. Control 57 2176–2189.
  • [40] Xu, L., Mo, Y., Xie, L. and Xiao, N. (2017). Mean square stabilization of linear discrete-time systems over power-constrained fading channels. IEEE Trans. Automat. Control 62 6505–6512.
  • [41] You, K. and Xie, L. (2011). Minimum data rate for mean square stabilizability of linear systems with Markovian packet losses. IEEE Trans. Automat. Control 56 772–785.
  • [42] Yüksel, S. and Başar, T. (2013). Stochastic Networked Control Systems: Stabilization and Optimization Under Information Constraints. Birkhäuser/Springer, New York.