Матрицы Ляпунова для класса систем с экспоненциальным ядром

Алексей Николаевич Алисейко

doi:10.21638/11701/spbu10.2017.301

Authors

Алексей Николаевич Алисейко St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/11701/spbu10.2017.301

Abstract

The problem of computation of Lyapunov matrices arises when Lyapunov—Krasovskii functionals are applied for stability analysis of linear time-invariant delay systems. A Lyapunov matrix is a solution of a matrix time-delay differential equation that satisfies two special conditions. It was shown that there exists a unique Lyapunov matrix if and only if the Lyapunov condition is satisfied, i. e. time-delay system has no eigenvalues symmetric with respect to the origin. Nevertheless, computational methods for Lyapunov matrices exist only for several classes of time-delay systems. In this contribution, we study time-delay systems with distributed delay and exponential kernel. In a contribution of Kharitonov, the problem of finding a Lyapunov matrix for this class of time-delay systems was reduced to the computation of solutions to an auxiliary delay-free system of ordinary differential equations with boundary conditions. However, the boundary conditions that were proposed earlier are not sufficient for the uniqueness of solutions to the auxiliary system, and results reported in the paper by Kharitonov do not allow us to obtain the Lyapunov matrix from a solution to an auxiliary system. These substantial differences between this class of time-delay systems and the wellstudied class of linear systems with one delay arise from ambiguity in the choice of boundary conditions for the auxiliary system. In this paper we propose a new set of boundary conditions that allows us to develop a theory similar to that of systems with one delay. It is shown that a solution of the auxiliary system with the new boundary conditions allows us to obtain the Lyapunov matrix. It is established that the auxiliary system admits a unique solution if and only if the Lyapunov condition is satisfied. Thus, one can verify existence and uniqueness of the Lyapunov matrix during its construction. Refs 12.

Keywords:

time-delay systems, Lyapunov matrix

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References

Литература

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Infante E. F., Castelan W. B. A Liapunov functional for a matrix difference-differential equation // Journal of Differential Equations. 1978. Vol. 29, N 3. P. 439–451.

Huang W. Generalization of Liapunov’s theorem in a linear delay system // Journal of Mathematical Analysis and Applications. 1989. Vol. 142, N 1. P. 83–94.

Kharitonov V. L., Zhabko A. P. Lyapunov—Krasovskii approach to the robust stability analysis of time-delay systems // Automatica. 2003. Vol. 39, N 1. P. 15–20.

Kharitonov V. L. On the uniqueness of Lyapunov matrices for a time-delay system // Systems & Control Letters. 2012. Vol. 61, N 3. P. 397–402.

Kharitonov V. L. Time-delay systems: Lyapunov functionals and matrices. Basel: Birkhäuser, 2013. 311 p.

Kharitonov V. L., Plischke E. Lyapunov matrices for time-delay systems // Systems & Control Letters. 2006. Vol. 55, N 9. P. 697–706.

Garcia-Lozano H., Kharitonov V. L. Lyapunov matrices for time delay systems with commensurate delays // 2nd IFAC Symposium on System, Structure and Control / ed. S. Mondié. Oaxaca, Mexico: IFAC, 2004. Vol. 1. P. 102–106.

Алисейко А. Н. Матрицы Ляпунова для класса уравнений с распределенным запаздыванием // Процессы управления и устойчивость. 2016. Т. 3(19), № 1. С. 68–74.

Kharitonov V. L. Lyapunov matrices for a class of time delay systems // Systems & Control Letters. 2006. Vol. 55, N 7. P. 610–617.

Bellman R., Cooke K. L. Differential-difference equations. New York: Academic Press, 1963. 482 p.

References

Krasovskii N. N. Nekotorye zadachi teorii ustojchivosti dvizheniya [Some problems of the stability of motion]. Moscow, Fizmatgiz Publ., 1959, 211 p. (In Russian)

Repin Iu. M. Quadratic Liapunov functionals for systems with delay. Journal of Applied Mathematics and Mechanics, 1965, vol. 29, no. 3, pp. 669–672.

Infante E. F., Castelan W. B. A Liapunov functional for a matrix difference-differential equation. Journal of Differential Equations, 1978, vol. 29, no. 3, pp. 439–451.

Huang W. Generalization of Liapunov’s theorem in a linear delay system. Journal of Mathematical Analysis and Applications, 1989, vol. 142, no. 1, pp. 83–94.

Kharitonov V. L., Zhabko A. P. Lyapunov—Krasovskii approach to the robust stability analysis of time-delay systems. Automatica, 2003, vol. 39, no. 1, pp. 15–20.

Kharitonov V. L. On the uniqueness of Lyapunov matrices for a time-delay system. Systems & Control Letters, 2012, vol. 61, no. 3, pp. 397–402.

Kharitonov V. L. Time-delay systems: Lyapunov functionals and matrices. Basel, Birkhäuser Publ., 2013, 311 p.

Kharitonov V. L., Plischke E. Lyapunov matrices for time-delay systems. Systems & Control Letters, 2006, vol. 55, no. 9, pp. 697–706.

Garcia-Lozano H., Kharitonov V. L. Lyapunov matrices for time delay systems with commensurate delays. 2nd IFAC Symposium on System, Structure and Control. Ed. by S. Mondié. Oaxaca, Mexico, IFAC, 2004, vol. 1, pp. 102–106.

Aliseyko A. N. Matritsy Lyapunova dlya klassa uravnenij s raspredelyonnym zapazdyvaniem [Lyapunov matrices for a class of equations with distributed delay]. Processy upravleniya i ustojchivost’ [Control Processes and Stability], 2016, vol. 3(19), no. 1, pp. 68–74. (In Russian)

Kharitonov V. L. Lyapunov matrices for a class of time delay systems. Systems & Control Letters, 2006, vol. 55, no. 7, pp. 610–617.

Bellman R., Cooke K. L. Differential-difference equations. New York, Academic Press, 1963, 482 p.