# 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)
Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds. (English) Zbl 1202.93102
Summary: The robust absolute stability problem for norm uncertain and structured uncertain discrete Lur’e systems is considered in this paper by using the Lyapunov function method. A sufficient condition of absolute stability for discrete Lur’e systems is established in terms of Linear Matrix Inequalities (LMIs) or the equivalent frequency-domain condition. We compare the result with the Popov-like criterion (Tsypkin criterion) and Extended Strictly Positive Real (ESPR) lemma. Furthermore, sufficient conditions on absolute stability for discrete Lur’e systems with norm and structured uncertainties are also presented based on linear matrix inequalities. Estimates of the maximum bounds of all admissible perturbations are given by generalized eigenvalue problems. Finally, several numerical examples are worked out to illustrate the efficiency of the main results.
##### MSC:
 93D09 Robust stability of control systems 93C55 Discrete-time control systems 93D20 Asymptotic stability of control systems 93D30 Scalar and vector Lyapunov functions
##### References:
 [1] Boyd, S.; Ghaoui, L. E.; Feron, E.: Linear matrix inequalities in system and control theory, (1994) [2] Ghaoui, L. El.; Lebret, H.: Robust solutions to least squares problems with uncertain data, SIAM J. Matrix anal. Appl. 18, No. 4, 1035-1064 (1997) · Zbl 0891.65039 · doi:10.1137/S0895479896298130 [3] Haddad, W. M.; Kapila, V.: Absolute stability criteria for multiple slope-restricted monotonic nonilinearities, IEEE trans. Automat. control 40, No. 2, 361-365 (1995) · Zbl 0825.93618 · doi:10.1109/9.341811 [4] Haddad, W. M.; Berstein, D. S.: Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis, IEEE trans. Automat. control 40, No. 3, 536-543 (1995) · Zbl 0821.93063 · doi:10.1109/9.376077 [5] F. Hao, T. Chu, L. Huang, L. Wang, Disturbances rejection analysis for structured uncertain Lur’e systems, in: IFAC 15th World Congress (CD-ROM), Session T-We-M15, Paper No. 706, Barcelona, Spain, July, 2002. [6] Hao, F.; Chu, T.; Huang, L.: Analysis of disturbances rejection for Lur’e system, Appl. math. Mech. 24, No. 3, 318-325 (2003) · Zbl 1039.93051 · doi:10.1007/BF02438269 [7] Hao, F.; Wang, L.; Yu, M.; Chu, T.: Robust stability and performance of uncertain lurie systems with state delays, Circuits syst. Signal process. 23, No. 4, 299-316 (2004) · Zbl 1051.93092 · doi:10.1007/s00034-004-3092-0 [8] Hao, F.: New conditions on absolute stability of uncertain Lur’e systems and the maximum admissible perturbed bound, IMA J. Math. control inf. 24, No. 3, 425-433 (2007) · Zbl 1131.93039 · doi:10.1093/imamci/dnl034 [9] Impram, S. T.; Munro, N.: A note on absolute stability of uncertain systems, Automatica 37, No. 4, 605-610 (2001) · Zbl 0972.93510 · doi:10.1016/S0005-1098(00)00194-1 [10] Kapila, V.; Haddad, W. M.: A multivariable extension of the tsypkin criterion using a Lyapunov function approach, IEEE trans. Automat. control 41, No. 1, 149-152 (1996) · Zbl 0842.93058 · doi:10.1109/9.481622 [11] Konishi, K.; Kokame, H.: Robust stability of Lur’e systems with time-varying uncertainties: a linear matrix inequality approach, Int. J. Syst. sci. 30, No. 1, 3-9 (1999) · Zbl 1065.93540 · doi:10.1080/002077299292605 [12] Lee, S. M.; Park, J. H.: Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities, Appl. math. Comput. 200, No. 1, 429-436 (2008) · Zbl 1146.93017 · doi:10.1016/j.amc.2007.11.031 [13] Marquez, H. J.; Diduch, P.: Absolute stability of systems with parametric uncertainty and nonlinear feedback, IEEE trans. Automat. control 39, No. 3, 664-668 (1994) · Zbl 0814.93055 · doi:10.1109/9.280784 [14] Montagner, V. F.; Oliveira, R. C. L.F.; Calliero, T. R.; Borges, R. A.; Peres, P. L. D.; Prieur, C.: Robust absolute stability and nonlinear state feedback stabilization based on polynomial lure functions, Nonlinear anal. 70, No. 5, 1803-1812 (2009) · Zbl 1155.93408 · doi:10.1016/j.na.2008.02.081 [15] Petersen, I. R.; Ugrinovskii, V. A.; Savkin, A. V.: Robust control design using H$\infty$ methods, (2000) [16] Popov, V. M.: Absolute stability of nonlinear systems of automatic control, Automat. remote control 21, 857-875 (1962) · Zbl 0107.29601 [17] Rantzer, A.: On the Kalman–yakubovich–Popov lemma, Syst. control lett. 28, No. 1, 7-10 (1996) · Zbl 0866.93052 · doi:10.1016/0167-6911(95)00063-1 [18] Tsypkin, Y. Z.: A criterion for absolute stability of automatic pulse systems with monotonic characteristics of the nonlinear element, Sov. phys. Doklady 9, 263-266 (1964) · Zbl 0134.30903 [19] Xie, H. M.: Absolute stability theory and its application, (1986)