## Control under quantization, saturation and delay: an LMI approach.(English)Zbl 1179.93089

Summary: This paper studies quantized and delayed state-feedback control of linear systems with given constant bounds on the quantization error and on the time-varying delay. The quantizer is supposed to be saturated. We consider two types of quantizations: quantized control input and quantized state. The controller is designed with the following property: all the states of the closed-loop system starting from a neighborhood of the origin exponentially converge to some bounded region (both, in $$\mathbb R^n$$ and in some infinite-dimensional state space). Under suitable conditions the attractive region is inside the initial one. We propose decomposition of the quantization into a sum of a saturation and of a uniformly bounded (by the quantization error bound) disturbance. A Linear Matrix Inequalities (LMIs) approach via Lyapunov-Krasovskii method originating in the earlier work [E. Fridman, M. Dambrine and N. Yeganefar, Automatica 44, No. 9, 2364–2369 (2008; Zbl 1153.93502)] is extended to the case of saturated quantizer and of quantized state and is based on the simplified and improved Lyapunov-Krasovskii technique.

### MSC:

 93B52 Feedback control 93C15 Control/observation systems governed by ordinary differential equations 93C05 Linear systems in control theory

Zbl 1153.93502
Full Text:

### References:

 [1] Ariba, Y., & Gouaisbaut, F. (2007). Delay-dependent stability analysis of linear systems with time-varying delay. In 46-th IEEE Conference on Decision and Control (pp. 2053-2058) [2] Boyd, L., Ghaoui, L.El., Feron, E., & Balakrishnan, V. (1994). SIAM Frontier Series. Linear matrix inequalities in system and control theory. Philadelphia · Zbl 0816.93004 [3] Brocket, R.; Liberzon, D., Quantized feedback stabilization of linear systems, IEEE transactions on automatic control, 45, 1279-1289, (2000) · Zbl 0988.93069 [4] Bullo, F.; Liberzon, D., Quantized control via location optimization, IEEE transactions on automatic control, 51, 1, 2-13, (2006) · Zbl 1366.93333 [5] Corradini, M.; Orlando, G., Robust quantized feedback stabilization of linear systems, Automatica, 44, 2458-2462, (2008) · Zbl 1153.93492 [6] Fagnani, F.; Zampieri, S., Stability analysis and synthesis for scalar linear systems with a quantized feedback, IEEE transactions on automatic control, 48, 9, 1569-1584, (2003) · Zbl 1364.93561 [7] Fridman, E., & Dambrine, M. (2008). Control under quantization, saturation and delay: An LMI approach. In 17th IFAC World Congress, Seoul (pp. 3787-3792) [8] Fridman, E.; Dambrine, M.; Yeganefar, N., On input-to-state stability of systems with time-delay: A matrix inequalities approach, Automatica, 44, 2364-2369, (2008) · Zbl 1153.93502 [9] Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE transactions on automatic control, 47, 1931-1937, (2002) · Zbl 1364.93564 [10] Fu, M.; Xie, L., The sector bound approach to quantized feedback control, IEEE transactions on automatic control, 50, 1698-1711, (2005) · Zbl 1365.81064 [11] Haimovich, H.; Kofman, E.; Seron, M., Systematic ultimate bound computation for sampled-data systems with quantization, Automatica, 43, 6, 1117-1123, (2007) · Zbl 1282.93169 [12] Hale, J.; Verduyn-Lunel, S., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002 [13] Hu, T.; Lin, Z., Control systems with actuator saturation: analysis and design, (2001), Birkhauser Boston · Zbl 1061.93003 [14] Hu, T.; Lin, Z.; Chen, B., An analysis and design method for linear systems subject to actuator saturation and disturbance, Automatica, 38, 351-359, (2002) · Zbl 0991.93044 [15] Ishii, H.; Francis, B., Quadratic stabilization of sampled-data systems with quantization, Automatica, 39, 1793-1800, (2003) · Zbl 1054.93035 [16] Kalman, R.E. (1956). Nonlinear aspects of sampled-data control systems. In Proceedings of the symposium on nonlinear circuit theory Brooklyn, NY · Zbl 0081.34901 [17] Khalil, H., Nonlinear systems, (2002), Prentice Hall Englewood Cliffs, NJ [18] Kofman, E.; Seron, M.; Haimovich, H., Control design with guaranteed ultimate bound for perturbed systems, Automatica, 44, 7, 1815-1821, (2008) · Zbl 1149.93016 [19] Liberzon, D., Hybrid feedback stabilization of systems with quantized signals, Automatica, 39, 1543-1554, (2003) · Zbl 1030.93042 [20] Liberzon, D., Quantization, time delays and nonlinear stabilization, IEEE transactions on automatic control, 51, 7, 1190-1195, (2006) · Zbl 1366.93509 [21] Mondie, S.; Kharitonov, V., Exponential estimates for retarded time-delay systems, IEEE transactions on automatic control, 50, 5, 268-273, (2005) · Zbl 1365.93351 [22] Oucheriah, S., Robust exponential convergence of a class of linear delayed systems with bounded controllers and disturbances, Automatica, 42, 1863-1867, (2006) · Zbl 1130.93410 [23] Pepe, P.; Jiang, Z.P., A Lyapunov-krasovskii methodology for ISS of iiss of time-delay systems, Systems & control letters, 55, 12, 1006-1014, (2006) · Zbl 1120.93361 [24] Sontag, E.; Wang, Y., On characterizations of the input-to-state stability property, Systems & control letters, 24, 351-359, (1995) · Zbl 0877.93121 [25] Tarbouriech, S.; Gomes da Silva, J.M., Synthesis of controllers for continuous-time delay systems with saturating controls via lmis, IEEE transactions on automatic control, 45, 1, 105-111, (2000) · Zbl 0978.93062 [26] Teel, A., Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE transactions on automatic control, 43, 7, 960-964, (1998) · Zbl 0952.93121
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.