# zbMATH — the first resource for mathematics

$$H^{\infty }$$ control with limited communication and message losses. (English) Zbl 1133.93016
Summary: We propose an $$H^{\infty }$$ approach to a remote control problem where the communication is constrained due to the use of a shared channel. The controller employs a periodic time sequencing scheme for message transmissions from multiple sensors and to multiple actuators of the system. It further takes into account the information on the random message losses that occur in the channel. An exact characterization for controller synthesis is obtained and is stated in terms of linear matrix inequalities. Furthermore, an analysis on the loss probabilities of the messages to accomplish stabilization is carried out. The results are illustrated through a numerical example.

##### MSC:
 93B36 $$H^\infty$$-control 93E03 Stochastic systems in control theory (general) 93B50 Synthesis problems
Full Text:
##### References:
 [1] Başar, T., Minimax control of switching systems under sampling, Systems control lett., 25, 315-325, (1995) · Zbl 0877.93041 [2] Bittanti, S.; Cuzzola, F.A., An LMI approach to periodic discrete-time unbiased filtering, Systems control lett., 42, 21-35, (2001) · Zbl 0985.93057 [3] Brockett, R.W., Stabilization of motor networks, (), 1484-1488 [4] Costa, O.L.V.; Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters, J. math. anal. appl., 179, 154-178, (1993) · Zbl 0790.93108 [5] Elia, N., Remote stabilization over fading channels, Systems control lett., 54, 238-249, (2005) · Zbl 1129.93498 [6] Hadjicostis, C.N.; Touri, R., Feedback control utilizing packet dropping network links, (), 1205-1210 [7] Hristu, D.; Morgansen, K., Limited communication control, Systems control lett., 37, 193-205, (1999) · Zbl 0948.93034 [8] D. Hristu-Varsakelis, W.S. Levine, (Eds.), Handbook of Networked and Embedded Control Systems, Birkhäuser, Boston, 2005. · Zbl 1094.93005 [9] Imer, O.Ç.; Yüksel, S.; Başar, T., Optimal control of LTI systems over unreliable communication links, Automatica, 42, 1429-1439, (2006) · Zbl 1128.93368 [10] H. Ishii, B.A. Francis, Limited Data Rate in Control Systems with Networks, Lecture Notes in Control and Information Sciences, vol. 275, Springer, Berlin, 2002. · Zbl 1001.93001 [11] Ishii, H.; Francis, B.A., Stabilization with control networks, Automatica, 38, 1745-1751, (2002) · Zbl 1011.93502 [12] H. Ishii, S. Hara, A subband coding approach to networked control, in: Proceedings of MTNS, 2006, pp. 2906-2911. Also Automatica (2008) to appear. [13] Ji, Y.; Chizeck, H.J., Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control theory adv. tech., 6, 289-319, (1990) [14] Lu, L.; Xie, L.; Fu, M., Optimal control of networked systems with limited communication: a combined heuristic and convex optimization approach, (), 1194-1199 [15] Masubuchi, I.; Ohara, A.; Suda, N., LMI-based controller synthesis: a unified formulation and solutions, Int. J. robust nonlinear control, 8, 669-686, (1998) · Zbl 0921.93012 [16] Scherer, C.; Gahinet, P.; Chilali, M., Multiobjective output-feedback control via LMI optimization, IEEE trans. automat. control, 42, 896-911, (1997) · Zbl 0883.93024 [17] Seiler, P.; Sengupta, R., A bounded real lemma for jump systems, IEEE trans. automat. control, 48, 1651-1654, (2003) · Zbl 1364.93223 [18] Seiler, P.; Sengupta, R., An $$H^\infty$$ approach to networked control, IEEE trans. automat. control, 50, 356-364, (2005) · Zbl 1365.93147 [19] Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.I.; Sastry, S.S., Kalman filtering with intermittent observations, IEEE trans. automat. control, 49, 1453-1464, (2004) · Zbl 1365.93512 [20] Xiao, L.; Hassibi, A.; How, J.P., Control with random communication delays via a discrete-time jump system approach, (), 2199-2204 [21] Zhang, L.; Hristu-Varsakelis, D., LQG control under limited communication, (), 185-190 [22] Zhang, L.; Shi, Y.; Chen, T.; Huang, B., A new method for stabilization of networked control systems with random delays, IEEE trans. automat. control, 50, 1177-1181, (2005) · Zbl 1365.93421
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.