Control of distributed delay systems with uncertainties: a generalized Popov theory approach

Ivanescu, Dan ; Niculescu, Silviu-Iulian ; Dion, Jean-Michel ; Dugard, Luc

Kybernetika, Tome 37 (2001) no. 3, pp. 325-343

Abstract (VO)

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.

Détail

Classification : 93C23, 93C41, 93D10, 93D30
Keywords: Popov theory; time-delay system; uncertainty

Bibliographie

[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 | MR

[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

[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 | Zbl

[4] Hale J. K., Lunel S. M. Verduyn: Introduction to Functional Differential Equations. (Applied Mathematical Sciences 99.) Springer–Verlag, Berlin 1991 | MR

[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

[8] Ionescu V., Weiss M.: Continuous and discrete-time Riccati theory: a Popov function approach. Linear Algebra Appl. 193 (1993), 173–209 | MR

[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 | MR | Zbl

[11] Kolmanovskii V. B., Nosov V. R.: Stability of Functional Differential Equations. (Mathematics in Science and Engineering 180.) Academic Press, New York 1986 | MR | Zbl

[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 | MR

[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 | MR | Zbl

[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

[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 | MR

[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