Discretization schemes for Lyapunov-Krasovskii functionals in time-delay systems

Gu, Keqin

Kybernetika, Tome 37 (2001) no. 4, pp. 479-504

Abstract (VO)

This article gives an overview of discretized Lyapunov functional methods for time-delay systems. Quadratic Lyapunov–Krasovskii functionals are discretized by choosing the kernel to be piecewise linear. As a result, the stability conditions may be written in the form of linear matrix inequalities. Conservatism may be reduced by choosing a finer mesh. Simplification techniques, including elimination of variables and using integral inequalities are also discussed. Systems with multiple delays and distributed delays are also treated. Finally, the treatment of uncertainties and input-output performance requirements are discussed.

Détail

Classification : 93B40, 93C23, 93C55, 93D05, 93D09
Keywords: time-delay system; Lyapunov-Krasovskii functional; multiple delays

Bibliographie

[1] Boyd S., Ghaoui L. El, Feron, E., Balakrishnan V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia 1994 | MR | Zbl

[2] Boyd S., Yang Q.: Structured and simultaneous Lyapunov functions for system stability problems. Internat. J. Control 49 (1989), 2215–2240 | DOI | MR | Zbl

[3] Souza C. E. De, Li X.: Delay-dependent robust $H_{\infty }$ control of uncertain linear state-delayed systems. Automatica 35 (1999), 1313–1321 | DOI | MR

[4] Doyle J. C., Wall, J., Stein G.: Performance and robustness analysis for structured uncertainty. In: IEEE Conference on Decision and Control 1982, pp. 629–636

[5] Gahinet P., Nemirovski A., Laub A. J., Chilali M.: LMI Control Toolbox for use with MATLAB. Natick, MA MathWorks, 1995

[6] Gu K.: Discretized LMI set in the stability problem of linear uncertain time-delay systems. Internat. J. Control 68 (1997), 923–934 | DOI | MR

[7] Gu K.: Stability of linear time-delay systems with block-diagonal uncertainty. In: 1998 American Control Conference, Philadelphia 1998, pp. 1943–1947

[8] Gu K.: Discretized Lyapunov functional for uncertain systems with multiple time-delay. Internat. J. Control 72 (1999), 16, 1436–1445 | DOI | MR | Zbl

[9] Gu K.: Partial solution of LMI in stability problem of time-delay systems. In: Proc. 38th IEEE Conference on Decision and Control 1999, pp. 227–232

[10] Gu K.: A generalized discretization scheme of Lyapunov functional in the stability problem of linear uncertain time-delay systems. Internat. J. Robust and Nonlinear Control 9 (1999), 1–14 | DOI | MR | Zbl

[11] Gu K.: An integral inequality in the stability problem of time-delay systems. In: Proc. 39th IEEE Conference on Decision and Control 2000

[12] Gu K.: A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. Internat. J. Control 74 (2001), 10, 967–976 | DOI | MR | Zbl

[13] Gu K., Han Q.-L.: Discretized Lyapunov functional for linear uncertain systems with time-varying delay. In: 2000 American Control Conference, Chicago 2000

[14] Gu K., Han Q.-L., Luo A. C. J., Niculescu S.-I.: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Internat. J. Control 74 (2001), 7, 737–744 | DOI | MR | Zbl

[15] Gu K., Luo A. C. J., Niculescu S.-I.: Discretized Lyapunov functional for systems with distributed delay. In: 1999 European Control Conference, Karlsruhe 1999

[16] Gu K., Niculescu S.-I.: Additional dynamics in transformed time-delay systems. IEEE Trans. Automat. Control 45 (2000), 572–575 | DOI | MR | Zbl

[17] Gu K., Niculescu S.-I.: Further remarks on additional dynamics in various model transformations of linear delay systems. In: 2000 American Control Conference, Chicago 2000 | MR | Zbl

[18] Hale J. K., Lunel S. M. Verduyn: Introduction to Functional Differential Equations. Springer–Verlag, New York 1993 | MR

[19] Han Q.-L., Gu K.: On robust stability of time-delay systems with norm-bounded uncertainty. IEEE Trans. Automat. Control, accepted | MR | Zbl

[20] Han Q.-L., Gu K.: Stability of linear systems with time-varying delay: a generalized discretized Lyapunov functional approach. Asian J. of Control, accepted

[21] Huang W.: Generalization of Liapunov’s theorem in a linear delay system. J. Math. Anal. Appl. 142 (1989), 83–94 | DOI | MR | Zbl

[22] Infante E. F., Castelan W. V.: A Lyapunov functional for a matrix difference-differential equation. J. Differential Equations 29 (1978), 439–451 | DOI | MR

[23] Kharitonov V.: Robust stability analysis of time delay systems: A survey. In: Proc. IFAC System Structure Control, Nantes 1998

[24] Kharitonov V. L., Melchor–Aguilar D. A.: Some remarks on model transformations used for stability and robust stability analysis of time-delay systems. In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999, pp. 1142–1147

[25] Kolmanovskii V. B., Niculescu S.-I., Gu K.: Delay effects on stability: a survey. In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999, pp. 1993–1998

[26] Kolmanovskii V. B., Richard J.-P.: Stability of some systems with distributed delays. In: JESA, special issue on “Analysis and Control of Time-delay Systems”, 31 (1997), 971–982

[27] Krasovskii N. N.: Stability of Motion. Stanford University Press, 1963 | MR | Zbl

[28] Li X., Souza C. E. de: Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica 33 (1997), 1657–1662 | DOI | MR

[29] Nesterov Y., Nemirovskii A.: Interior–Point Polynomial Algorithms in Convex Programming SIAM, Philadelphia 199. | MR

[30] Niculescu S. I., Dugard, L., Dion J. M.: Stabilité et stabilisation robustes des systèmes à retard. In: Proc. Journées Robustesse, Toulouse 1995

[31] Niculescu S. I., Souza C. E. de, Dion J. M., Dugard L.: Robust stability and stabilization of uncertain linear systems with state delay: single delay case (I), and Multiple delays case (II). In: Proc. IFAC Workshop Robust Control Design, Rio de Janeiro 1994, pp. 469–474 and 475–480

[32] Niculescu S. I., Verriest E. I., Dugard, L., Dion J. M.: Stability and robust stability of time-delay systems: A guided tour. In: Stability and Control of Time-Delay Systems (L. Dugard and E. I. Verriest, eds., Lecture Notes in Control and Information Sciences), Springer–Verlag, London 1997, pp. 1–71 | MR

[33] Packard A., Doyle J.: The complex structured singular value. Automatica 29 (1993), 71–109 | DOI | MR | Zbl

[34] Zhou K., Doyle J. C., Glover K.: Robust and Optimal Control. Prentice Hall, Englewood Cliffs, N.J. 1996 | Zbl