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2013 | OriginalPaper | Chapter

1. General Theory

Author : Vladimir L. Kharitonov

Published in: Time-Delay Systems

Publisher: Birkhäuser Boston

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Abstract

This chapter serves as a brief introduction to the theory of the retarded type time-delay system. It starts with a discussion of such basic notions as solutions, initial conditions, and the state of a time-delay system. Then some results on the existence and uniqueness of an initial value problem are presented. Continuity properties of the solutions are discussed as well. The main part of the chapter is devoted to stability analysis. Here we define concepts of stability, asymptotic stability, and exponential stability of the trivial solution of a time-delay system. Classical stability results, obtained using the Lyapunov–Krasovskii approach, are given in the form of necessary and sufficient conditions. A short section with historical comments concludes the chapter.

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Literature
1.
go back to reference Ahlfors, L.V.: Complex Analysis. McGraw-Hill, New York (1979) Ahlfors, L.V.: Complex Analysis. McGraw-Hill, New York (1979)
2.
go back to reference Arnold, V.I.: Differential Equations. MIT Press, Cambridge, MA (1978) Arnold, V.I.: Differential Equations. MIT Press, Cambridge, MA (1978)
3.
go back to reference Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic, New York (1963) Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic, New York (1963)
4.
go back to reference Castelan, W.B., Infante, E.F.: On a functional equation arising in the stability theory of difference-differential equations. Q. Appl. Math. 35, 311–319 (1977) Castelan, W.B., Infante, E.F.: On a functional equation arising in the stability theory of difference-differential equations. Q. Appl. Math. 35, 311–319 (1977)
5.
go back to reference Castelan, W.B., Infante, E.F.: A Liapunov functional for a matrix neutral difference-differential equation with one delay. J. Math. Anal. Appl. 71, 105–130 (1979) Castelan, W.B., Infante, E.F.: A Liapunov functional for a matrix neutral difference-differential equation with one delay. J. Math. Anal. Appl. 71, 105–130 (1979)
6.
go back to reference Curtain, R.F., Pritchard, A.J.: Infinite-dimensional linear systems theory. In: Lecture Notes in Control and Information Sciences, vol. 8. Springer, Berlin (1978) Curtain, R.F., Pritchard, A.J.: Infinite-dimensional linear systems theory. In: Lecture Notes in Control and Information Sciences, vol. 8. Springer, Berlin (1978)
7.
go back to reference Datko, R.: An algorithm for computing Liapunov functionals for some differential difference equations. In: Weiss, L. (ed.) Ordinary Differential Equations, pp. 387–398. Academic, New York (1972) Datko, R.: An algorithm for computing Liapunov functionals for some differential difference equations. In: Weiss, L. (ed.) Ordinary Differential Equations, pp. 387–398. Academic, New York (1972)
8.
go back to reference Datko, R.: A procedure for determination of the exponential stability of certain diferential-difference equations. Q. Appl. Math. 36, 279–292 (1978) Datko, R.: A procedure for determination of the exponential stability of certain diferential-difference equations. Q. Appl. Math. 36, 279–292 (1978)
9.
go back to reference Datko, R.: Lyapunov functionals for certain linear delay-differential equations in a Hilbert space. J. Math. Anal. Appl. 76, 37–57 (1980) Datko, R.: Lyapunov functionals for certain linear delay-differential equations in a Hilbert space. J. Math. Anal. Appl. 76, 37–57 (1980)
10.
go back to reference Diekmann, O., von Gils, A.A., Verduyn-Lunel, S.M., Walther, H.-O.: Delay equations, functional-, complex- and nonlinear analysis. Applied Mathematics Sciences Series, vol. 110. Springer, New York (1995) Diekmann, O., von Gils, A.A., Verduyn-Lunel, S.M., Walther, H.-O.: Delay equations, functional-, complex- and nonlinear analysis. Applied Mathematics Sciences Series, vol. 110. Springer, New York (1995)
11.
go back to reference Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977) Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977)
12.
go back to reference Garcia-Lozano, H., Kharitonov, V.L.: Numerical computation of time delay Lyapunov matrices. In: 6th IFAC Workshop on Time Delay Systems, L’Aquila, Italy, 10–12 July 2006 Garcia-Lozano, H., Kharitonov, V.L.: Numerical computation of time delay Lyapunov matrices. In: 6th IFAC Workshop on Time Delay Systems, L’Aquila, Italy, 10–12 July 2006
13.
go back to reference Golub, G.H., van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore, MD (1983) Golub, G.H., van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore, MD (1983)
14.
go back to reference Gorecki, H., Fuksa, S., Grabowski, P., Korytowski, A.: Analysis and Synthesis of Time-Delay Systems. Polish Scientific Publishers, Warsaw (1989) Gorecki, H., Fuksa, S., Grabowski, P., Korytowski, A.: Analysis and Synthesis of Time-Delay Systems. Polish Scientific Publishers, Warsaw (1989)
15.
go back to reference Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK (1981) Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK (1981)
16.
go back to reference Gu, K.: Discretized Lyapunov functional for uncertain systems with multiple time-delay. Int. J. Contr. 72, 1436–1445 (1999) Gu, K.: Discretized Lyapunov functional for uncertain systems with multiple time-delay. Int. J. Contr. 72, 1436–1445 (1999)
17.
go back to reference Gu, K., Han, Q.-L., Luo, A.C.J., Niculescu, S.-I.: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74, 737–744 (2001) Gu, K., Han, Q.-L., Luo, A.C.J., Niculescu, S.-I.: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74, 737–744 (2001)
18.
go back to reference Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time Delay Systems. Birkhauser, Boston, MA (2003) Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time Delay Systems. Birkhauser, Boston, MA (2003)
19.
go back to reference Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic, New York (1966) Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic, New York (1966)
20.
go back to reference Halanay, A., Yorke, J.A.: Some new results and problems in the theory of differential-delay equatons. SIAM Rev. 31, 55–80 (1971) Halanay, A., Yorke, J.A.: Some new results and problems in the theory of differential-delay equatons. SIAM Rev. 31, 55–80 (1971)
21.
go back to reference Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1971) Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1971)
22.
go back to reference Hale, J.K., Infante, E.F., Tsen, F.S.P.: Stability in linear delay equations. J. Math. Anal. Appl. 105, 533–555 (1985) Hale, J.K., Infante, E.F., Tsen, F.S.P.: Stability in linear delay equations. J. Math. Anal. Appl. 105, 533–555 (1985)
23.
go back to reference Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
24.
go back to reference Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory 1: Modelling, State Space Analysis, Stability and Robustness. Springer, Heidelberg (2005) Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory 1: Modelling, State Space Analysis, Stability and Robustness. Springer, Heidelberg (2005)
25.
go back to reference Horn, R.A., Johnson, C.A.: Matrix Analysis. Cambridge University Press, Cambridge, UK (1985) Horn, R.A., Johnson, C.A.: Matrix Analysis. Cambridge University Press, Cambridge, UK (1985)
26.
go back to reference Huang, W.: Generalization of Liapunov’s theorem in a linear delay system. J. Math. Anal. Appl. 142, 83–94 (1989) Huang, W.: Generalization of Liapunov’s theorem in a linear delay system. J. Math. Anal. Appl. 142, 83–94 (1989)
27.
go back to reference Infante, E.F.: Some results on the Lyapunov stability of functional equations. In: Hannsgen, K.B., Herdmn, T.L., Stech, H.W., Wheeler, R.L. (eds.) Volterra and Functional Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 81, pp. 51–60. Marcel Dekker, New York (1982) Infante, E.F.: Some results on the Lyapunov stability of functional equations. In: Hannsgen, K.B., Herdmn, T.L., Stech, H.W., Wheeler, R.L. (eds.) Volterra and Functional Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 81, pp. 51–60. Marcel Dekker, New York (1982)
28.
go back to reference Infante, E.F., Castelan, W.V.: A Lyapunov functional for a matrix difference-differential equation. J. Differ. Equat. 29, 439–451 (1978) Infante, E.F., Castelan, W.V.: A Lyapunov functional for a matrix difference-differential equation. J. Differ. Equat. 29, 439–451 (1978)
29.
go back to reference Jarlebring, E., Vanbiervliet, J., Michiels, W.: Characterizing and computing the \({\mathcal{H}}_{2}\) norm of time delay systems by solwing the delay Lyapunov equation. In: Proceedings of the 49th IEEE Conference on Decision and Control (2010) Jarlebring, E., Vanbiervliet, J., Michiels, W.: Characterizing and computing the \({\mathcal{H}}_{2}\) norm of time delay systems by solwing the delay Lyapunov equation. In: Proceedings of the 49th IEEE Conference on Decision and Control (2010)
30.
go back to reference Kailath, T.: Linear Systems. Prentice-Hall, Engelewood Cliffs, NJ (1980) Kailath, T.: Linear Systems. Prentice-Hall, Engelewood Cliffs, NJ (1980)
31.
go back to reference Kharitonov, V.L.: Robust stability analysis of time delay systems: a survey. Annu. Rev. Control 23, 185–196 (1999) Kharitonov, V.L.: Robust stability analysis of time delay systems: a survey. Annu. Rev. Control 23, 185–196 (1999)
32.
go back to reference Kharitonov, V.L.: Lyapunov functionals and Lyapunov matrices for neutral type time-delay stystems: a single delay case. Int. J. Control 78, 783–800 (2005) Kharitonov, V.L.: Lyapunov functionals and Lyapunov matrices for neutral type time-delay stystems: a single delay case. Int. J. Control 78, 783–800 (2005)
33.
go back to reference Kharitonov, V.L.: Lyapunov matrices for a class of time delay systems. Syst. Control Lett. 55, 610–617 (2006) Kharitonov, V.L.: Lyapunov matrices for a class of time delay systems. Syst. Control Lett. 55, 610–617 (2006)
34.
go back to reference Kharitonov, V.L.: Lyapunov matrices for a class of neutral type time delay systems. Int. J. Control 81, 883–893 (2008) Kharitonov, V.L.: Lyapunov matrices for a class of neutral type time delay systems. Int. J. Control 81, 883–893 (2008)
35.
go back to reference Kharitonov, V.L.: Lyapunov matrices: Existence and uniqueness issues. Automatica 46, 1725–1729 (2010) Kharitonov, V.L.: Lyapunov matrices: Existence and uniqueness issues. Automatica 46, 1725–1729 (2010)
36.
go back to reference Kharitonov, V.L.: Lyapunov functionals and matrices. Ann. Rev. Control 34, 13–20 (2010) Kharitonov, V.L.: Lyapunov functionals and matrices. Ann. Rev. Control 34, 13–20 (2010)
37.
go back to reference Kharitonov, V.L.: On the uniqueness of Lyapunov matrices for a time-delay system. Syst. Control Lett. 61, 397–402 (2012) Kharitonov, V.L.: On the uniqueness of Lyapunov matrices for a time-delay system. Syst. Control Lett. 61, 397–402 (2012)
38.
go back to reference Kharitonov, V.L., Hinrichsen, D.: Exponential estimates for time delay systems. Syst. Control Lett. 53, 395–405 (2004) Kharitonov, V.L., Hinrichsen, D.: Exponential estimates for time delay systems. Syst. Control Lett. 53, 395–405 (2004)
39.
go back to reference Kharitonov, V.L., Mondie, S., Ochoa, G.: Frequency stability analysis of linear systems with general distributed delays. Lect. Notes Control Inf. Sci. 388, 61–71 (2009) Kharitonov, V.L., Mondie, S., Ochoa, G.: Frequency stability analysis of linear systems with general distributed delays. Lect. Notes Control Inf. Sci. 388, 61–71 (2009)
40.
go back to reference Kharitonov, V.L., Plischke, E.: Lyapunov matrices for time delay systems. Syst. Control Lett. 55, 697–706 (2006) Kharitonov, V.L., Plischke, E.: Lyapunov matrices for time delay systems. Syst. Control Lett. 55, 697–706 (2006)
41.
go back to reference Kharitonov, V.L., Zhabko, A.P.: Robust stability of time-delay systems. IEEE Trans. Auto.. Control 39, 2388–2397 (1994) Kharitonov, V.L., Zhabko, A.P.: Robust stability of time-delay systems. IEEE Trans. Auto.. Control 39, 2388–2397 (1994)
42.
go back to reference Kharitonov, V.L., Zhabko, A.P.: Lyapunov-Krasovskii approach to robust stability analysis of time delay systems. Automatica 39, 15–20 (2003) Kharitonov, V.L., Zhabko, A.P.: Lyapunov-Krasovskii approach to robust stability analysis of time delay systems. Automatica 39, 15–20 (2003)
43.
go back to reference Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, the Netherlands (1992) Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, the Netherlands (1992)
44.
go back to reference Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic, New York (1986) Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic, New York (1986)
45.
go back to reference Kolmogorov, A., Fomin, S.: Elements of the Theory of Functions and Functional Analysis. Greylock, Rochester, NY (1961) Kolmogorov, A., Fomin, S.: Elements of the Theory of Functions and Functional Analysis. Greylock, Rochester, NY (1961)
46.
go back to reference Krasovskii, N.N.: Stability of Motion. [Russian], Moscow, 1959 [English translation]. Stanford University Press, Stanford, CA (1963) Krasovskii, N.N.: Stability of Motion. [Russian], Moscow, 1959 [English translation]. Stanford University Press, Stanford, CA (1963)
47.
go back to reference Krasovskii, N.N.: On using the Lyapunov second method for equations with time delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 315–327 (1956) Krasovskii, N.N.: On using the Lyapunov second method for equations with time delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 315–327 (1956)
48.
go back to reference Krasovskii, N.N.: On the asymptotic stability of systems with aftereffect [Russian]. Prikladnaya Matematika i Mekhanika. 20, 513–518 (1956) Krasovskii, N.N.: On the asymptotic stability of systems with aftereffect [Russian]. Prikladnaya Matematika i Mekhanika. 20, 513–518 (1956)
49.
go back to reference Lakshmikantam, V., Leela, S.: Differential and Integral Inequalities. Academic, New York (1969) Lakshmikantam, V., Leela, S.: Differential and Integral Inequalities. Academic, New York (1969)
50.
go back to reference Levinson, N., Redheffer, R.M.: Complex Variables. Holden-Day, Baltimore, MD (1970) Levinson, N., Redheffer, R.M.: Complex Variables. Holden-Day, Baltimore, MD (1970)
51.
go back to reference Louisel, J.: Growth estimates and asymptotic stability for a class of differential-delay equation having time-varying delay. J. Math. Anal. Appl. 164, 453–479 (1992) Louisel, J.: Growth estimates and asymptotic stability for a class of differential-delay equation having time-varying delay. J. Math. Anal. Appl. 164, 453–479 (1992)
52.
go back to reference Louisell, J.: Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system. In: Dugard, L., Verriest, E.I. (eds.) Stability and Control of Time-delay Systems. Lecture Notes in Control and Information Sciences, vol. 228, pp. 140–157. Springer, New York (1997) Louisell, J.: Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system. In: Dugard, L., Verriest, E.I. (eds.) Stability and Control of Time-delay Systems. Lecture Notes in Control and Information Sciences, vol. 228, pp. 140–157. Springer, New York (1997)
53.
go back to reference Louisell, J.: A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Trans. Autom. Control 46, 2008–2012 (2001) Louisell, J.: A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Trans. Autom. Control 46, 2008–2012 (2001)
54.
go back to reference Malek-Zavarei, M., Jamshidi, M.: Time delay systems: analysis, optimization and applications. North-Holland Systems and Control Series, vol. 9. North-Holland, Amsterdam (1987) Malek-Zavarei, M., Jamshidi, M.: Time delay systems: analysis, optimization and applications. North-Holland Systems and Control Series, vol. 9. North-Holland, Amsterdam (1987)
55.
go back to reference Marshall, J.E., Gorecki, H., Korytowski, A., Walton, K.: Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood, New York (1992) Marshall, J.E., Gorecki, H., Korytowski, A., Walton, K.: Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood, New York (1992)
56.
go back to reference Mondie, S.: Assesing the exact stability region of the single delay scalar equation via its Lyapunov function. IMA J. Math. Control Inf. (2012). doi: ID:DNS004 Mondie, S.: Assesing the exact stability region of the single delay scalar equation via its Lyapunov function. IMA J. Math. Control Inf. (2012). doi: ID:DNS004
57.
go back to reference Myshkis, A.D.: General theory of differential equations with delay [Russian]. Uspekhi Matematicheskikh Nauk. 4, 99–141 (1949) Myshkis, A.D.: General theory of differential equations with delay [Russian]. Uspekhi Matematicheskikh Nauk. 4, 99–141 (1949)
58.
go back to reference Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. Springer, Heidelberg (2001) Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. Springer, Heidelberg (2001)
59.
go back to reference Ochoa, G., Mondie, S., Kharitonov, V.L.: Time delay systems with distributed delays: critical values. In: Proceedings of the 8th IFAC Workshop on Time Delay Systems, Sinaia, Romania, 1–3 Sept 2009 Ochoa, G., Mondie, S., Kharitonov, V.L.: Time delay systems with distributed delays: critical values. In: Proceedings of the 8th IFAC Workshop on Time Delay Systems, Sinaia, Romania, 1–3 Sept 2009
60.
go back to reference Plishke, E.: Transient effects of linear dynamical systems. Ph.D. thesis, University of Bremen, Bremen, Germany (2005) Plishke, E.: Transient effects of linear dynamical systems. Ph.D. thesis, University of Bremen, Bremen, Germany (2005)
61.
go back to reference Razumikhin, B.S.: On the stability of systems with a delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 500–512 (1956) Razumikhin, B.S.: On the stability of systems with a delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 500–512 (1956)
62.
go back to reference Razumikhin, B.S.: Application of Liapunov’s method to problems in the stability of systems with a delay [Russian]. Automatika i Telemekhanika. 21, 740–749 (1960) Razumikhin, B.S.: Application of Liapunov’s method to problems in the stability of systems with a delay [Russian]. Automatika i Telemekhanika. 21, 740–749 (1960)
63.
go back to reference Repin, Yu.M.: Quadratic Lyapunov functionals for systems with delay [Russian]. Prikladnaya Matematika i Mekhanika. 29, 564–566 (1965) Repin, Yu.M.: Quadratic Lyapunov functionals for systems with delay [Russian]. Prikladnaya Matematika i Mekhanika. 29, 564–566 (1965)
64.
go back to reference Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003) Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)
65.
go back to reference Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1973) Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1973)
66.
go back to reference Rudin, W.: Functional Analysis. McGraw-Hill, New York (1987) Rudin, W.: Functional Analysis. McGraw-Hill, New York (1987)
67.
go back to reference Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Function. Wiley, New York (1989) Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Function. Wiley, New York (1989)
68.
go back to reference Velazquez-Velazquez, J., Kharitonov, V.L.: Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Syst. Control Lett. 58, 17–25 (2009) Velazquez-Velazquez, J., Kharitonov, V.L.: Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Syst. Control Lett. 58, 17–25 (2009)
69.
go back to reference Volterra, V.: Sulle equazioni integrodifferenciali della teorie dell’elasticita. Atti. Accad. Lincei. 18, 295 (1909) Volterra, V.: Sulle equazioni integrodifferenciali della teorie dell’elasticita. Atti. Accad. Lincei. 18, 295 (1909)
70.
go back to reference Volterra, V.: Theorie mathematique de la lutte pour la vie [French]. Gauthier-Villars, Paris (1931) Volterra, V.: Theorie mathematique de la lutte pour la vie [French]. Gauthier-Villars, Paris (1931)
71.
go back to reference Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ (1996) Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ (1996)
72.
go back to reference Zubov, V.I.: The Methods of A.M. Lyapunov and Their Applications. Noordhoff, Groningen, the Netherlands (1964) Zubov, V.I.: The Methods of A.M. Lyapunov and Their Applications. Noordhoff, Groningen, the Netherlands (1964)
Metadata
Title
General Theory
Author
Vladimir L. Kharitonov
Copyright Year
2013
Publisher
Birkhäuser Boston
DOI
https://doi.org/10.1007/978-0-8176-8367-2_1