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Popov theory; time-delay system; uncertainty
The paper deals with the generalized Popov theory applied to uncertain systems with distributed time delay. Sufficient conditions for stabilizing this class of delayed systems as well as for $\gamma $-attenuation achievement are given in terms of algebraic properties of a Popov system via a Liapunov–Krasovskii functional. The considered approach is new in the context of distributed linear time-delay systems and gives some interesting interpretations of $H^\infty $ memoryless control problems in terms of Popov triplets and associated objects. The approach is illustrated via numerical examples. Dedicated to Acad. Vlad Ionescu, in memoriam.
[1] Cheres E., Gutman, S., Palmor Z. J.: Robust stabilization of uncertain dynamic system including state delay. IEEE Trans. Automat. Control 34 (1989), 1199–1203 DOI 10.1109/9.40753 | MR 1020938
[2] Dion J.-M., Dugard L., Ivanescu D., Niculescu S.-I., Ionescu V.: Robust $H_{\infty }$ control of time-delay systems: A generalized Popov theory approach. In: Perspectives in Control Theory and applications, Springer–Verlag, Berlin 1998, pp. 61–82 Zbl 0976.93066
[3] Dugard L., (eds.) E. I. Verriest: Stability and Control of Time–Delay Systems. (Lecture Notes in Computer Science 228.) Springer–Verlag, London 1997 MR 1482570 | Zbl 0901.00019
[4] Hale J. K., Lunel S. M. Verduyn: Introduction to Functional Differential Equations. (Applied Mathematical Sciences 99.) Springer–Verlag, Berlin 1991 MR 1243878
[5] Ionescu V., Niculescu S.-I., Dion J.-M., Dugard, L., Li H.: Generalized Popov theory applied to state-delayed systems. In: Proc. IFAC Conference on System Structure and Control, Nantes 1998, pp. 645–650
[6] Ionescu V., Niculescu S.-I., Woerdeman H.: On ${\mathcal L}_2$ memoryless control of time-delay systems. In: Proc. 36th IEEE Conference on Decision and Control, San Diego 1997, pp. 1344–1349
[7] Ionescu V., Oară, C., Weiss M.: Generalized Riccati Theory. Wiley, New York 1998 Zbl 0915.34024
[8] Ionescu V., Weiss M.: Continuous and discrete-time Riccati theory: a Popov function approach. Linear Algebra Appl. 193 (1993), 173–209 MR 1240278
[9] Ivanescu D., Niculescu S.-I., Dion J.-M., Dugard L.: Control of Distributed Varying Delay Systems Using Generalized Popov Theory. Internal Note, LAG–98
[10] Kojima A., Uchida, K., Shimemura E.: Robust stabilization of uncertain time delay systems via combined internal-external approach. IEEE Trans. Automat. Control 38 (1993), 373–378 DOI 10.1109/9.250497 | MR 1206835 | Zbl 0773.93066
[11] Kolmanovskii V. B., Nosov V. R.: Stability of Functional Differential Equations. (Mathematics in Science and Engineering 180.) Academic Press, New York 1986 MR 0860947 | Zbl 0593.34070
[12] Lee J. H., Kim S. W., Kwon W. H.: Memoryless $H_\infty $ controllers for state delayed systems. IEEE Trans. Automat. Control 39 (1994), 159–162 DOI 10.1109/9.273356 | MR 1258692
[13] Niculescu S.-I., Souza C. E. de, Dion J.-M., Dugard L.: Robust ${\mathcal H}_\infty $ memoryless control for uncertain linear systems with time-varying delay. In: 3rd European Control Conference, Rome 1995, pp. 1814–1818
[14] Niculescu S.-I., Ionescu V.: On delay-independent stability criteria: A matrix pencil approach. IMA J. Math. Control Inform. 14 (1997), 299–306 DOI 10.1093/imamci/14.3.299 | MR 1467584 | Zbl 0886.93056
[15] Niculescu S.-I., Ionescu V., Ivănescu D., Dion J.-M., Dugard L.: On generalized Popov theory for delay systems. In: 6th IEEE Mediteranean Conference, Sardaigne 1998 and Kybernetika 36 (2000), 2–20 MR 1760884
[16] Niculescu S.-I., Ionescu, V., Woerdeman H.: On the Popov theory for some classes of time-delay systems: A matrix pencil approach. In: MTNS’98, Padova 1998
[17] Oară C.: Proper deflating subspaces: properties, algorithmes and applications. Numer. Algorithms 7 (1994), 355–377 DOI 10.1007/BF02140690 | MR 1283105
[18] Xie L., Souza C. E. de: Robust stabilization and disturbance attenuation for uncertain delay system. In: Proc. 2nd European Control Conference, Groningen 1993, pp. 667–672
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