# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\Bbb R^n$ 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 [{\it E. Fridman, M. Dambrine} and {\it 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 systems governed by ODE 93C05 Linear control systems
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 · doi:10.1109/9.867021 [4] Bullo, F.; Liberzon, D.: Quantized control via location optimization, IEEE transactions on automatic control 51, No. 1, 2-13 (2006) [5] Corradini, M.; Orlando, G.: Robust quantized feedback stabilization of linear systems, Automatica 44, 2458-2462 (2008) · Zbl 1153.93492 · doi:10.1016/j.automatica.2008.01.027 [6] Fagnani, F.; Zampieri, S.: Stability analysis and synthesis for scalar linear systems with a quantized feedback, IEEE transactions on automatic control 48, No. 9, 1569-1584 (2003) [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 · doi:10.1016/j.automatica.2008.01.012 [9] Fridman, E.; Shaked, U.: An improved stabilization method for linear time-delay systems, IEEE transactions on automatic control 47, 1931-1937 (2002) [10] Fu, M.; Xie, L.: The sector bound approach to quantized feedback control, IEEE transactions on automatic control 50, 1698-1711 (2005) [11] Haimovich, H.; Kofman, E.; Seron, M.: Systematic ultimate bound computation for sampled-data systems with quantization, Automatica 43, No. 6, 1117-1123 (2007) · Zbl 1282.93169 [12] Hale, J.; Verduyn-Lunel, S.: Introduction to functional differential equations, (1993) · Zbl 0787.34002 [13] Hu, T.; Lin, Z.: Control systems with actuator saturation: analysis and design, (2001) · 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 · doi:10.1016/S0005-1098(01)00209-6 [15] Ishii, H.; Francis, B.: Quadratic stabilization of sampled-data systems with quantization, Automatica 39, 1793-1800 (2003) · Zbl 1054.93035 · doi:10.1016/S0005-1098(03)00179-1 [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) · Zbl 1003.34002 [18] Kofman, E.; Seron, M.; Haimovich, H.: Control design with guaranteed ultimate bound for perturbed systems, Automatica 44, No. 7, 1815-1821 (2008) · Zbl 1149.93016 · doi:10.1016/j.automatica.2007.10.022 [19] Liberzon, D.: Hybrid feedback stabilization of systems with quantized signals, Automatica 39, 1543-1554 (2003) · Zbl 1030.93042 · doi:10.1016/S0005-1098(03)00151-1 [20] Liberzon, D.: Quantization, time delays and nonlinear stabilization, IEEE transactions on automatic control 51, No. 7, 1190-1195 (2006) [21] Mondie, S.; Kharitonov, V.: Exponential estimates for retarded time-delay systems, IEEE transactions on automatic control 50, No. 5, 268-273 (2005) [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 · doi:10.1016/j.automatica.2006.05.023 [23] Pepe, P.; Jiang, Z. P.: A Lyapunov-Krasovskiĭ methodology for ISS of iiss of time-delay systems, Systems & control letters 55, No. 12, 1006-1014 (2006) · Zbl 1120.93361 · doi:10.1016/j.sysconle.2006.06.013 [24] Sontag, E.; Wang, Y.: On characterizations of the input-to-state stability property, Systems & control letters 24, 351-359 (1995) · Zbl 0877.93121 · doi:10.1016/0167-6911(94)00050-6 [25] Tarbouriech, S.; Da Silva Jr., J. M. Gomes: Synthesis of controllers for continuous-time delay systems with saturating controls via lmis, IEEE transactions on automatic control 45, No. 1, 105-111 (2000) · Zbl 0978.93062 · doi:10.1109/9.827364 [26] Teel, A.: Connections between razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE transactions on automatic control 43, No. 7, 960-964 (1998) · Zbl 0952.93121 · doi:10.1109/9.701099