zbMATH — the first resource for mathematics

Stabilization of Markov jump linear systems using quantized state feedback. (English) Zbl 1204.93127
Summary: This paper addresses the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback. Given a measure of quantization coarseness, a mode-dependent logarithmic quantizer and a mode-dependent linear state feedback law can achieve optimal coarseness for mean square quadratic stabilization of a Markov jump linear system, similar to existing results for linear time-invariant systems. The sector bound approach is shown to be non-conservative in investigating the corresponding quantized state feedback problem, and then a method of optimal quantizer/controller design in terms of linear matrix inequalities is presented. Moreover, when the mode process is not observed by the controller and quantizer, a mode estimation algorithm obtained by maximizing a certain probability criterion is given. Finally, an application to networked control systems further demonstrates the usefulness of the results.

93E15 Stochastic stability in control theory
93C05 Linear systems in control theory
93D15 Stabilization of systems by feedback
60J75 Jump processes (MSC2010)
93B52 Feedback control
Full Text: DOI
[1] Boukas, E.; Liu, Z., Robust \(H_\infty\) control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE transactions on automatic control, 46, 12, 1918-1924, (2001) · Zbl 1005.93050
[2] Costa, O.; Fragoso, M.; Marques, R., Discrete-time Markov jump linear systems, (2005), Springer London · Zbl 1081.93001
[3] de Souza, C., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE transactions on automatic control, 51, 5, 836-841, (2006) · Zbl 1366.93479
[4] Elia, N., Remote stabilization over fading channels, Systems and control letters, 54, 3, 237-249, (2005) · Zbl 1129.93498
[5] Elia, N.; Mitter, S., Stabilization of linear systems with limited information, IEEE transactions on automatic control, 46, 9, 1384-1400, (2001) · Zbl 1059.93521
[6] Elliott, R.; Aggoun, L.; Moore, J., Hidden Markov models estimation and control, (1995), Springer New York · Zbl 0819.60045
[7] Elliott, R.; Dufour, F.; Malcolm, W., State and mode estimation for discrete-time jump Markov systems, SIAM journal on control and optimization, 44, 1081-1104, (2005) · Zbl 1130.93423
[8] Fu, M.; Xie, L., The sector bound approach to quantized feedback control, IEEE transactions on automatic control, 50, 11, 1698-1711, (2005) · Zbl 1365.81064
[9] Gao, H.; Chen, T., A new approach to quantized feedback control systems, Automatica, 44, 2, 534-542, (2008) · Zbl 1283.93131
[10] Ho, T.; Chen, B., Novel extended viterbi-based multiple-model algorithms for state estimation of discrete-time systems with Markov jump parameters, IEEE transactions on signal processing, 54, 2, 393-404, (2006) · Zbl 1373.94611
[11] Hoshina, H., Tsumura, K., & Ishii, H. (2007). The coarsest logarithmic quantizers for stabilization of linear systems with packet losses. In Proceedings of the 46th IEEE conference on decision and control. (pp. 2235-2240) New Orleans, USA.
[12] Huang, M.; Dey, S., Stability of Kalman filtering with Markovian packet losses, Automatica, 43, 4, 598-607, (2007) · Zbl 1261.93083
[13] Hu, S.; Yan, W., Stability robustness of networked control systems with respect to packet loss, Automatica, 43, 7, 1243-1248, (2007) · Zbl 1123.93075
[14] Imer, O.; Yüksel, S.; Başar, T., Optimal control of LTI systems over unreliable communication links, Automatica, 42, 9, 1429-1439, (2006) · Zbl 1128.93368
[15] Matei, I., Martins, N., & Baras, J. (2008). Optimal linear quadratic regulator for Markovian jump linear systems, in the presence of one time-step delayed mode observations, In Proceedings of the 17th IFAC world congress. (pp. 8056-8061) Seoul, Korea.
[16] Rabiner, L., A tutorial on hidden Markov models and selected applications in speech recognition, Proceedings of the IEEE, 77, 2, 257-286, (1989)
[17] Schenato, L.; Sinopoli, B.; Franceschetti, M.; Poolla, K.; Sastry, S., Foundations of control and estimation over lossy networks, Proceedings of the IEEE, 95, 1, 163-187, (2007)
[18] Seiler, P.; Sengupta, R., An \(H_\infty\) approach to networked control, IEEE transactions on automatic control, 50, 3, 356-364, (2005) · Zbl 1365.93147
[19] Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.; Sastry, S., Kalman filtering with intermittent observations, IEEE transactions on automatic control, 49, 9, 1453-1464, (2004) · Zbl 1365.93512
[20] Viterbi, A., Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE transactions on information theory, 13, 2, 260-269, (1967) · Zbl 0148.40501
[21] Xiao, N., Xie, L., & Fu, M. (2009). Quantized stabilization of Markov jump linear systems via state feedback. In Proceedings of American control conference. (pp. 4020-4025), St. Louis, USA.
[22] Xiong, J.; Lam, J., Stabilization of linear systems over networks with bounded packet loss, Automatica, 43, 1, 80-87, (2007) · Zbl 1140.93383
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.