Skip to main content
SpringerLink
Log in

Stability and robust stability of time-delay systems: A guided tour

  • Chapter
  • First Online:
Stability and Control of Time-delay Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 228))

Abstract

In this chapter, some recent stability and robust stability results on linear time-delay systems are outlined. The goal of this guided tour is to give (without entering the details) a wide overview of the state of the art of the techniques encountered in time-delay system stability problems. In particular, two specific stability problems with respect to delay (delay-independent and respectively delay-dependent) are analyzed and some references where the reader can find more details and proofs are pointed out. The references list is not intended to give a complete literature survey, but rather to be a source for a more complete bibliography. In order to simplify the presentation several examples have been considered.

On leave from Laboratoire d'Automatique de Grenoble (France); Also with the Department of Automatic Control, University “Politehnica” Bucharest (Romania)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abdallah, G., Dorato, P., Benitez-Read, J. and Byrne, R.: Delayed positive feedback can stabilize oscillatory systems. Proc. American Contr. Conf. (1993) 3106–3107.

    Google Scholar 

  2. Agathoklis, P. and Foda, S.: Stability and matrix Lyapunov equation for delay differential systems. Int. J. Contr., 49 (1989) 417–432.

    Google Scholar 

  3. Ahlfors, L. V.: Complex Analysis, 3rd Ed., McGraw-Hill Book Company, New York, 1979.

    Google Scholar 

  4. Amemyia, T.: Delay-independent stability of higher-order systems. Int. J. Contr., 50 (1989) 139–149.

    Google Scholar 

  5. Amemyia, T.: On the delay-independent stability of a delayed differential equation of a 1st order, J. Math. Anal. Appl., 142 (1989) 13–25.

    Google Scholar 

  6. Barmish, B. R. and Shi, Z.: Robust stability of perturbed systems with time-delays. Automatica, 25 (1989) 371–381.

    Google Scholar 

  7. Barnea, D. I.: A method and new results for stability and instability of autonomous functional differential equations. SIAM J. Appl. Math., 17 (1969) 681–697.

    Google Scholar 

  8. Bartlett, A. C., Hollot, C. V. and Lin, H.: Root locations of an entire polytope of polynomials: it suffices to check the edges. Math. Contr., Sign. & Syst., 1 (1988) 61–71.

    Google Scholar 

  9. Bélair, J.: Stability in delayed neural networks. in Ordinary and delay differential equations, J. Wiener, J. K. Hale (Editors), Pitman Research Notes Math. Series, 272, John Wiley & Sons, (1992) 6–9.

    Google Scholar 

  10. Bélair, J., Campbell, S. A. and van den Driessche, P.: Frustration, stability and delay-induced oscillations in a neural network model SIAM J. Appl. Math., 56 (1996) 245–255.

    Google Scholar 

  11. Bellman, R. E.: Vector Lyapunov functions. SIAM J. Contr., Ser. A, 1 (1962) 33–34.

    Google Scholar 

  12. Bellman, R. E. and Cooke, K. L.: Differential-Difference Equations, Academic Press, New York, 1963.

    Google Scholar 

  13. Bensoussan, A., Da Prato, G., Defour, M. C. and Mitter, S. K.: Representation and control of infinite dimensional systems. Systems & Control: Foundation & Applications, 2 volumes, Birkhäuser, Boston, 1993.

    Google Scholar 

  14. Bhatt, S. J. and Hsu, C. S.: Stability criteria for second-order dynamical systems with time lag. J. Applied Mechanics (1966) 113–118.

    Google Scholar 

  15. Bilous, O. and Admundson, N.: Chemical reactor stability and sensitivity. AI ChE Journal, 1 (1955) 513–521.

    Google Scholar 

  16. Boese, F. G.: Stability conditions for the general linear difference-differential equation with constant coefficients and one constant delay. J. Math. Anal. Appl., 140 (1989) pp. 136–176.

    Google Scholar 

  17. Boese, F. G.: Stability in a special class of retarded difference-differential equations with interval-valued parameters. J. Math. Anal. Appl., 181 (1994) 227–247.

    Google Scholar 

  18. Boese, F. G.: Stability criteria for second-order dynamical systems involving several time delays SIAM J. Math. Anal. 5 (1995) 1306–1330.

    Google Scholar 

  19. Bourlès, H.: α-stability of systems governed by a functional differential equation — extension of results concerning linear delay systems. Int. J. Contr., 45 (1987) 2233–2234.

    Google Scholar 

  20. Boyd, S. and Desoer, C. A.: Subharmonic functions and performance bounds in linear time-invariant feedback systems. IMA J. Math. Contr. Information, 2 (1985) 153–170.

    Google Scholar 

  21. Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V.: Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, 15, 1994.

    Google Scholar 

  22. Brierley, S. D., Chiasson, J. N., Lee, E. B. and Zak, S. H.: On stability independent of delay for linear systems. IEEE Trans. Automat. Control AC-27 (1982) 252–254.

    Google Scholar 

  23. Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando, 178, 1985.

    Google Scholar 

  24. Buslowicz, M.: Sufficient conditions for instability of delay differential systems. Int. J. Contr. 37 (1983) 1311–1321.

    Google Scholar 

  25. Campbell, S. A. and Bélair, J.: Multiple-delayed differential equations as models for biological control systems. Proc. World Math. Conf. (1993) 3110–3117.

    Google Scholar 

  26. Chen, J.: On computing the maximal delay intervals for stability of linear delay systems. IEEE Trans. Automat. Contr. 40 (1995) 1087–1093.

    Google Scholar 

  27. Chen, J. and Latchman, H. A.: Frequency sweeping tests for stability independent of delay. IEEE Trans. Automat. Contr., 40 (1995) 1640–1645.

    Google Scholar 

  28. Chen, J. Gu, G. and Nett, C. N.: A new method for computing delay margins for stability of linear delay systems. Proc. 33rd IEEE CDC, Lake Buena Vista, Florida, U.S.A., (1994) 433–437.

    Google Scholar 

  29. Chen, J., Xu, D. and Shafai, B.: On sufficient conditions for stability independent of delay. Proc. 1994 American Contr. Conf., Baltimore, Maryland (1994) 1929–1933.

    Google Scholar 

  30. Cheres, E., Gutman, S. and Palmor, Z. J.: Quantitative measures of robustness for systems including delayed perturbations. IEEE Trans. Automat. Contr., 34 (1989) 1203–1204.

    Google Scholar 

  31. Chiasson, J.: A method for computing the interval of delay values for which a differential-delay system is stable. IEEE Trans. Automat. Contr. 33 (1988) 1176–1178.

    Google Scholar 

  32. Chiasson, J. N., Brierley, S. D. and Lee, E. B.: A simplified derivation of the Zeheb-Walach 2-D stability test with applications to time-delay systems. IEEE Trans. Automat. Contr., AC-30 (1985) 411–414; corrections in IEEE Trans. Automat. Contr., AC-31 (1986) 91–92.

    Google Scholar 

  33. Cooke, K. L. and Ferreira, J. M.: Stability conditions for linear retarded functional differential equations. J. Math. Annal. Appl., 96 (1983) 480–504.

    Google Scholar 

  34. Cooke, K. L. and van den Driessche, P.: On zeroes of some transcendental equations Funkcialaj Ekvacioj 29 (1986) 77–90.

    Google Scholar 

  35. Curtain, R. F.: A synthesis of time and frequency domain methods for the control of infinite-dimensional systems: A system theoretic approach. In H. T. Banks (Ed.), Control and estimation in distributed parameter system (1992) 171–224.

    Google Scholar 

  36. Curtain, R. F. and Pritchard, A. J.: Infinite-dimensional linear systems theory. Lecture Notes in Contr. and Inf. Sciences, 8, Springer-Verlag, Berlin, 1978.

    Google Scholar 

  37. Dambrine, M. and Richard, J. P.: Stability analysis on time-delay systems. Dynamic Syst. Appl., 2 (1993) 405–414.

    Google Scholar 

  38. Dambrine, M.: Contributions à l'étude de la stabilité des systèmes à retard. Ph.D. Thesis, LAIL URA CNRS D1440, Ecole Centrale de Lille, 1994.

    Google Scholar 

  39. Datko, R.: A procedure for determination of the exponential stability of certain differential-difference equations. Quart. Appl. Math. 36 (1978) 279–292.

    Google Scholar 

  40. Datko, R.: Remarks concerning the asymptotic stability and stabilization of linear delay differential equations. J. Math. Anal. Appl., 111 (1985) 571–584.

    Google Scholar 

  41. Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Contr. Optimization, 26 (1988) 697–713.

    Google Scholar 

  42. Desoer, C. A. and Vidyasagar, M.: Feedback System: Input-Output Properties. Academic Press, New York, 1975.

    Google Scholar 

  43. Devanathan, R.: A lower bound for limiting time delay for closed-loop stability of an arbitrary SISO plant. IEEE Trans. Automat. Contr., 40 (1995) 717–721.

    Google Scholar 

  44. Diekmann, O., von Gils, S. A., Verduyn Lunel, S. M. and Walther, H.-O.: Delay equations, Functional-, Complex-and Nonlinear Analysis. Appl. Math. Sciences Series, 110, Springer-Verlag, New York, 1995.

    Google Scholar 

  45. Driver, R. D.: Existence and stability of a delay-differential system. Arch. Rational Mech. Anal., 10 (1962) 401–426.

    Google Scholar 

  46. El Sakkary: Estimating robust dead time for closed loop stability. IEEE Trans. Automat. Contr., 35 (1990) 209–210.

    Google Scholar 

  47. El'sgol'ts, L. E. and Norkin, S. B.: Introduction to the theory and applications of differential equations with deviating arguments. Mathematics in Science and Eng., 105, Academic Press, New York, 1973.

    Google Scholar 

  48. Feron, E., Balakrishnan, V. and Boyd, S.: A design of stabilizing state feedback for delay systems via convex optimization. Proc. 31st IEEE Conf. Dec. Contr., Tuscon, Arizona, USA, (1992) 147–148.

    Google Scholar 

  49. Fiagbedzi, Y. A. and Pearson, A. E.: Feedback stabilization of linear autonomous time lag systems. IEEE Trans. Automat. Contr., AC-31 (1986) 847–855.

    Google Scholar 

  50. Fiala, J. and Lumia, R.: The effect of time delay and discrete control on the contact stability of simple position controllers. IEEE Trans. Automat. Contr., 39 (1994) 870–873.

    Google Scholar 

  51. Fliess, M. and Mounier, H.: Quelques propriétés structurelles des systèmes linéaires à retards constants. C.R. Acad. Sci. Paris, I-319 (1994) 289–294.

    Google Scholar 

  52. Florchinger, P. and Verriest, E. I.: Stabilization of Nonlinear Stochastic Systems with Delay Feedback. Proc. 32nd IEEE Conf. Decision and Control San Antonio TX (1993) 859–860.

    Google Scholar 

  53. Fu, M., Olbrot, A. W. and Polis, M. P.: Robust stability for time-delay systems: The edge theorem and graphical tests. IEEE Trans. Automat. Contr., 34 (1989) 813–820.

    Google Scholar 

  54. Furumochi, T.: Stability and boundedness in functional differential equations. J. Math. Anal. Appl., 113 (1986) 473–489.

    Google Scholar 

  55. Garey, M. and Johnoson, D.: Computers and intractability: A guide to the theory of NP-completeness. Freeman, San Francisco, 1979.

    Google Scholar 

  56. Glader, C., Hognas, G., Makila, P. and Toivonen, H. T.: Approximation of delay systems — a case study. Int. J. Contr., 53 (1991) 369–390.

    Google Scholar 

  57. Gohberg, I., Lancaster, P. and Rodman, L.: Matrix Polynomials. Computer Science & Appl. Math., Academic Press, New York, 1982.

    Google Scholar 

  58. Golub, G. H. and van Loan, Ch. F.: Matrix computations. John Hopkins Univ. Press, Baltimore & London, 1989.

    Google Scholar 

  59. Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. Kluwer Academic Publishers, Math. Its Appl. Series, 74, 1992.

    Google Scholar 

  60. Górecki, H., Fuksa, S., Gabrowski, P. and Korytowski, A.: Analysis and Synthesis of Time Delay Systems. John Wiley & Sons (PWN), Warszawa Poland 1989.

    Google Scholar 

  61. Goubet, A., Dambrine, M. and Richard, J. P.: An extension of stability criteria for linear and nonlinear time-delay systems, Proc. IFAC Syst. Struct. Contr., Nantes, France (1995) 278–283.

    Google Scholar 

  62. Goubet-Bartholomeüs, A.: Sur la stabilité et la stabilisation des systèmes retardés: Conditions en fonction du retard (in French), Ph.D. Thesis, Univ. des Sciences et Technologies de Lille, 1996.

    Google Scholar 

  63. Gu, G. and Lee, E. B.: Stability testing of time-delay systems. Automatica, 25 (1989) 777–780.

    Google Scholar 

  64. Gu, G., Khargonekar, P. P., Lee, E. B. and Misra, P.: Finite dimensional approximations of unstable infinite-dimensional systems. SIAM J. Contr. Opt., 30 (1992) 704–716.

    Google Scholar 

  65. Habets, L.: Algebraic and computational aspects of time-delay systems. Ph. Thesis, Eindhoven Univ. Technology, 1994.

    Google Scholar 

  66. Haddock, J. R. and Terjeki, J.: Liapunov-Razumikhin functions and an invariance principle for functional differential equations. J. Diff. Eq., 48 (1983) 95–122.

    Google Scholar 

  67. Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York, 1966.

    Google Scholar 

  68. Hale, J. K.: Dynamics and delays. In S. Busenberg, M. Martelli (Editors) Delay Differential Equations and Dynamical Systems, Lecture Notes in Math., 1475 (1991) 16–30, Springer Verlag, Berlin.

    Google Scholar 

  69. Hale, J. K., Magalhaes, L. T. and Oliva, W. M.: An introduction to infinite dynamical systems — Geometric theory. Applied Math. Sciences, 47, Springer Verlag, New York, 1985.

    Google Scholar 

  70. Hale, J. K. and Verduyn Lunel, S. M.: Introduction to Functional Differential Equations. Applied Math. Sciences, 99, Springer-Verlag, New York, 1991.

    Google Scholar 

  71. Hale, J. K., Infante, E. F. and Tsen, F. S. P.: Stability in linear delay equations. J. Math. Anal. Appl., 105 (1985) 533–555.

    Google Scholar 

  72. Hale, J. K. and Huang, W.: Global geometry of the stable regions for two delay differential equations. J. Math. Anal. Appl., 178 (1993) 344–362.

    Google Scholar 

  73. Hale, J.K., Effects of delays on stability and control. Report CDSN97-270, Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 1997.

    Google Scholar 

  74. Hertz, D., Jury, E. I. and Zeheb, E.: Stability independent and dependent of delay for delay differential systems. J. Franklin Inst., 318 (1984) 143–150.

    Google Scholar 

  75. Hertz, D., Jury, E. I. and Zeheb, E.: Root exclusion from complex polydomains and some of its applications. Automatica, 23 (1987) 399–404.

    Google Scholar 

  76. Hmamed, A.: On the stability of time-delay systems: New results. Int. J. Contr., 43 (1986) 321–324.

    Google Scholar 

  77. Hocherman, J. and Zeheb, E.: Robust stability of time delay systems under uncertainty conditions. ECCTD'93-Circuit Theory and Design (1993) 409–414.

    Google Scholar 

  78. Hocherman, J., Kogan, J. and Zeheb, E.: On exponential stability of linear systems and Hurwitz stability of characteristic quasipolynomials. Syst. & Contr. Lett. 25 (1995) 1–7.

    Google Scholar 

  79. Hopfield, J. J.: Neural networks and physical systems with emergent collective computation abilities. Proc. National Acad. Science U.S.A., 79 (1982) 2554–2558.

    Google Scholar 

  80. Hsu, C. S.: Application of the τ-decomposition method to dynamical systems subjected to retarded follower forces. J. Appl. Mechanics, 37 (1970) 258–266.

    Google Scholar 

  81. Hsu, C. S. and Bhatt, S. J.: Stability charts for second-order dynamical systems with time lag. J. Appl. Mechanics (1966) 119–124.

    Google Scholar 

  82. Huang, W.: Generalization of Lyapunov's theorem in a linear delay system. J. Math. Anal. Appl., 142 (1989) 83–94.

    Google Scholar 

  83. Infante, E. F. and Castelan, W. B.: A Lyapunov functional for a matrix difference-differential equation. J. Diff. Eq., 29 (1978) 439–451.

    Google Scholar 

  84. Ivanov, A. F. and Verriest, E. I.: Robust Stability of Delay-Difference Equations. In Systems and Networks: Mathematical Theory and Applications, (U. Helmke, R. Mennicken, and J. Saurer, eds.) University of Regensburg (1994) 725–726.

    Google Scholar 

  85. Jacobson, C. A. and Nett, C. N.: Linear state-space systems in infinite-dimensional space: The role and characterization of joint stabilizability/detectability. IEEE Trans. Automat. Contr., 33 (1988) 541–549.

    Google Scholar 

  86. Kamen, E.W.: Lectures on Algebraic System Theory: Linear Systems over Rings. NASA Contractor Report 3016 1978.

    Google Scholar 

  87. Kamen, E. W.: On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations. IEEE Trans. Automat. Contr., AC-25 (1980) 983–984.

    Google Scholar 

  88. Kamen, E. W.: Linear systems with commensurate time delays: Stability and stabilization independent of delay. IEEE Trans. Automat. Contr., AC-27 (1982) 367–375; corrections in IEEE Trans. Automat. Contr., AC-28 (1983) 248–249.

    Google Scholar 

  89. Kato, J.: Liapunov's second method in functional differential equations. Tôhoku Math. Journ., 332 (1980) 487–492.

    Google Scholar 

  90. Kharitonov, V. L. and Zhabko, A. P.: Robust stability of time-delay systems. IEEE Trans. Automat. Contr., 39 (1994) 2388–2397.

    Google Scholar 

  91. Kogan, J.: Robust stability and convexity LNCIS, vol. 201, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  92. Kohonen, T.: Self organization and Associative Memory. Springer Verlag, Berlin, 1984.

    Google Scholar 

  93. Kolmanovskii, V. B. and Nosov, V. R.: Stability of Functional Differential Equations. Mathematics in Science and Eng., 180, Academic Press, New York, 1986.

    Google Scholar 

  94. Kolmanovskii, V. and Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht the Netherlands 1992.

    Google Scholar 

  95. Krasovskii, N. N.: Stability of motion. Stanford University Press, 1963.

    Google Scholar 

  96. Kuang, Y.: Delay differential equations with applications in population dynamics. Academic Press, Boston, 1993.

    Google Scholar 

  97. Lakshmikantam, V. and Leela, S.: Differential and integral inequalities. Academic Press, New York, 1969.

    Google Scholar 

  98. Lam, J.: Convergence of a class of Padé approximations for delay systems. Int. J. Contr., 52 (1990) 989–1008.

    Google Scholar 

  99. Lancaster, P. and Tismenetsky, M.: The theory of matrices (2nd Edition). Comp. Science Appl. Math. Series, Academic Press, Orlando, 1985.

    Google Scholar 

  100. Lee, E. B., Lu, W.-S. and Wu, N. E.: A Lyapunov theory for linear time-delay systems. IEEE Trans. Automat. Contr., AC-31 (1986) 259–261.

    Google Scholar 

  101. Lee, H. and Hsu, C.: On the τ-decomposition method of stability analysis for retarded dynamical systems. SIAM J. Contr., 7 (1969) 242–259.

    Google Scholar 

  102. Lehman, B.: Stability of chemical reactions in a CSTR with delayed recycle stream. Proc. 1994 Amer. Contr. Conf., Baltimore, Maryland, U.S.A., (1994) 3521–3522.

    Google Scholar 

  103. Lehman, B. and Verriest, E. I.: Stability of a continuous stirred reactor with delay in the recycle streams. Proc. 30th IEEE Conf. Dec. Contr., Brighton, England, (1991) 1875–1876.

    Google Scholar 

  104. Lehman, B. and Shujaee, K.: Delay independent stability conditions and decay estimates for time-varying functional differential equations. IEEE Trans. Automat. Contr., 39 (1994) 1673–1676.

    Google Scholar 

  105. Lehman, B., Bentsman, J., Verduyn-Lunel, S., and Verriest, E. I.: Vibrational control of nonlinear time-lag systems with arbitrarily large but bounded delay: Averaging theory, stabilizability, and transient behavior. IEEE Trans. Automatic Control, AC-39 5 (1994) 898–912.

    Google Scholar 

  106. Lewis, R. M. and Anderson, B. D. O.: Necessary and sufficient conditions for delay independent stability of linear autonomous systems. IEEE Trans. Automat. Control, AC-25 (1980) 735–739.

    Google Scholar 

  107. Louisell, J.: A stability analysis for a class of differential-delay equations having time-varying delay. In S. Busenberg, M. Martelli (Eds.): Delay Differential Equations and Dynamical Systems, Lecture Notes in Math., 1475 (1991) 225–242, Springer Verlag, Berlin.

    Google Scholar 

  108. Louisell, J.: Absolute stability in linear delay-differential systems: Ill-possedness and robustness. IEEE Trans. Automat. Contr., 40 (1995) 1288–1291.

    Google Scholar 

  109. MacDonald, N.: Time lags in biological models. Lecture Notes in Biomathematics, 27, Springer Verlag, Berlin, 1978.

    Google Scholar 

  110. Malek-Zavarei, M. and Jamshidi, M.: Time Delay Systems: Analysis, Optimization and Applications. North-Holland Systems and Control Series, 9, Amsterdam, 1987.

    Google Scholar 

  111. Manitius, A.: Necessary and sufficient conditions of approximate controllability for linear retarded systems. SIAM J. Contr. Opt., 19 (1981) 516–532.

    Google Scholar 

  112. Manitius, A. and Triggiani, R.: Function space controllability of retarded systems: a derivation from abstract operator conditions. SIAM J. Opt. Contr., 16, (1978) 599–645.

    Google Scholar 

  113. Mao, X.: Stability of stochastic differential equations with respect to semimartingales. Pitman Research Notes in Mathematics Series 251 Longman Harlow UK 1991.

    Google Scholar 

  114. Mao, X.: Robustness of stability of nonlinear systems with stochastic delay perturbations. Systems & Control Lett. 19 (1992) 391–400.

    Google Scholar 

  115. Mao, X.: Exponential Stability of stochastic differential equations. Marcel Dekker New York 1994.

    Google Scholar 

  116. Marcus, C. M. and Westervelt, R. M.: Stability of analog neural networks with delay. Phys. Rev., A 39 (1989) 347–359.

    Google Scholar 

  117. Marcus, M.: Finite dimensional multiplinear algebra (vol. I). Marcel Dekker, New York, 1973.

    Google Scholar 

  118. Marshall, J. E., Górecki, H., Walton, K. and Korytowski, A.: Time-delay systems: Stability and performance criteria with applications. Ellis Horwood, New York, 1992.

    Google Scholar 

  119. Matrosov, V. M.: Comparison principle and vector Lyapunov functions. Diff. Urav. 4 (1968) 1374–1386.

    Google Scholar 

  120. Mikoljska, Z.: Une remarque sur des notes de Razumichin et Krasovskij sur la stabilité asymptotique. Annales Polonici Mathematici 22 (1969) 69–72.

    Google Scholar 

  121. Mohammed, S.-E. A.: Stochastic functional differential equations. Pitman Research Notes in Mathematics 99 London UK 1984.

    Google Scholar 

  122. Mohammed, S.-E. A.: Stability of linear delay equations under a small noise. Proc. Edinburgh Math. Soc. 29 (1986) 233–254.

    Google Scholar 

  123. Mori, T.: Criteria for asymptotic stability of linear time-delay systems. IEEE Trans. Automat. Contr., AC-30 (1985) 158–160.

    Google Scholar 

  124. Mori, T., Fukuma, N. and Kuwahara, M.: Simple stability criteria for single and composite linear systems with time delay Int. J. Contr., 34 (1981) 1175–1184.

    Google Scholar 

  125. Mori, T., Fukuma, N. and Kuwahara, M.: On an estimate of the decay rate for stable linear delay systems. Int. J. Contr., 36 (1982) 95–97.

    Google Scholar 

  126. Mori, T., and Kokame, H.: Stability of \(\dot x\)(t)=Ax(t)+Bx(t−τ) IEEE Trans. Automat. Contr., AC-34 (1989) 460–462.

    Google Scholar 

  127. Morse, A. S.: Ring models for delay differential systems Automatica, 12 (1976) 529–531.

    Google Scholar 

  128. Mounier, H.: Propriétés structurelles des systèmes linéaires à retards: Aspects théoriques et pratiques. Université Paris-Sud, Orsay, 1995.

    Google Scholar 

  129. Myshkis, A. D.: General theory of differential equations with delay. Uspehi, Mat. Nauk, 4 (1949) 99–141 (Engl. Transl. AMS, 55, 1–62, 1951).

    Google Scholar 

  130. Nechayeva, I. G. and Khusainov, D. Ya.: Exponential estimates of solutions of linear stochastic differential functional systems. Ukrainian Math. J. 42 (10) (1990).

    Google Scholar 

  131. Neimark, J.: D-subdivisions and spaces of quasi-polynomials. Prikl. Math. Mech., 13 (1949) 349–380.

    Google Scholar 

  132. Nemirovskii, A.: Several NP-hard problems arising in robust stability analysis. Math. Contr. Signals, Syst, 6 (1993) 99–105.

    Google Scholar 

  133. Niculescu, S.-I.: On the stability and stabilization of linear systems with delayed-state, (in French). Ph.D. Thesis, Laboratoire d'Automatique de Grenoble, INPG, February, 1996.

    Google Scholar 

  134. Niculescu, S.-I., de Souza, C. E., Dion, J.-M. and Dugard, L.: Robust stability and stabilization of uncertain linear systems with state delay: Single delay case (I). Proc. IFAC Symp. Robust Contr. Design, Rio de Janeiro, Brasil (1994) 469–474.

    Google Scholar 

  135. Niculescu, S.-I., de Souza, C. E., Dion, J.-M. and Dugard, L.: Robust stability and stabilization of uncertain linear systems with state delay: Multiple delays case (II). Proc. IFAC Symp. Robust Contr. Design, Rio de Janeiro, Brasil (1994) 475–480.

    Google Scholar 

  136. Niculescu, S.-I., de Souza, C. E., Dugard, L. and Dion, J.-M.: Robust exponential stability of uncertain linear systems with time-varying delays. Proc. 3rd European Contr. Conf., Rome, Italy (1995) 1802–1808.

    Google Scholar 

  137. Niculescu, S.-I., de Souza, C. E., Dion, J.-M. and Dugard, L.: Robust \(\mathcal{H}_\infty\) memoryless control for uncertain linear systems with time-varying delay. Proc. 3rd European Contr. Conf., Rome, Italy (1995) 1814–1819.

    Google Scholar 

  138. Niculescu, S.-I., Trofino-Neto, A., Dion, J.-M. and Dugard, L.: Delay-dependent stability of linear systems with delayed state: An LMI approach Proc. 34th IEEE Conf. Dec. Contr., New Orleans, United States (1995) 1495–1497.

    Google Scholar 

  139. Niculescu, S.-I., Dion, J.-M. and Dugard, L.: Delays-dependent stability for linear systems with several delays: An LMI approach Proc. 13th IFAC World Congr., San Francisco D (1996) 165–170.

    Google Scholar 

  140. Niculescu, S.-I., Dion, J.-M. and Dugard, L.: A matrix pencil approach for asymptotic stability of linear systems with delayed state. MITNS'96, Saint Louis, 1996.

    Google Scholar 

  141. Niculescu, S.-I. and Ionescu, V.: On delay-independent stability criteria: A matrix pencil approach. Internal Note LAG 95, to appear in IMA Journal Math. Contr. Information, 1997.

    Google Scholar 

  142. Niculescu, S.-I., Dion, J.-M. and Dugard, D.: On the stability of time-delay systems (in French). In J.-M. Dion, D. Popescu (Eds.) Commande optimale. Conception optimisée des systèmes, Diderot, Paris (1996) 249–283.

    Google Scholar 

  143. Niculescu, S.-I.: Stability and hyperbolicity of linear systems with delayed state: A matrix pencil approach. To appear in IMA J. Math. Contr. Information (1997).

    Google Scholar 

  144. Niculescu, S.-I.: Delay-interval stability and hyperbolicity of linear time-delay systems: A matrix pencil approach. 4th European Contr. Conf., Brussels, Belgium, July 1997.

    Google Scholar 

  145. Niculescu, S.-I. and Collado, J.: Stability and hyperbolicity of linear time-delay systems: A matrix pencil tensor product approach. to be presented at 4th IFAC Syst. Struct. Contr., Bucharest, Romania, October 1997.

    Google Scholar 

  146. Niculescu, S.-I.: \(\mathcal{H}_\infty\) memoryless control with an α-stability constraint for time delays systems: An LMI approach. accepted in IEEE Trans. Automat. Contr. (1997).

    Google Scholar 

  147. Nishioka, K., Adachi, N. and Takeuki, K.: Simple pivoting algorithm for root-locus method of linear systems with delay. Int. J. Contr., 53 (1991) 951–966.

    Google Scholar 

  148. Olbrot, A. W.: A sufficient large time-delay in feedback loop must destroy exponential stability of any decay rate. IEEE Trans. Automat. Contr., AC-29 (1984) 367–3687.

    Google Scholar 

  149. Olbrot, A. W. and Igwe, C. U. T.: Necessary and sufficient conditions for robust stability independent of delays and coefficient perturbations. 34th IEEE CDC, New Orleans, Louisiana, (1995).

    Google Scholar 

  150. Packard, A. and Doyle, J.: The complex structured singular value. Automatica 29 (1993) 71–109.

    Google Scholar 

  151. Partington, J. R.: Approximation of delay systems by Fourrier-Laguerre series. Automatica 27 (1991) 569–572.

    Google Scholar 

  152. Perlmutter, D.: Stability of chemical reactors, Prentice Hall, New Jersey, 1972.

    Google Scholar 

  153. Picard, P.: Sur l'observabilité et la commande des systèmes linéaires à retards modélisés sur un anneau (in French). Ph.D. Thesis,Ecole Centrale de Nantes, 1996.

    Google Scholar 

  154. RĂsvan, V.: Absolute stability of automatic control systems with delays (in Romanian). Eds. Academiei RSR, Bucharest, Romania, 1975.

    Google Scholar 

  155. Razumikhin, B. S.: On the stability of systems with a delay. Prikl. Math. Meh., 20 (1956) 500–512.

    Google Scholar 

  156. Repin, Yu. M.: On conditions for the stability of systems of differential equations for arbitrary delays. Uchen. Zap. Ural., 23 (1960) 31–34.

    Google Scholar 

  157. Rozkhov, V. I. and Popov, A. M.: Inequalities for solutions of certain systems of differential equations with large time-lag. Diff. Eq. 7 (1971) 271–278.

    Google Scholar 

  158. Salamon, D.: Structure and stability of finite dimensional approximations for functional differential equations. SIAM J. Contr. Opt., 23 (1985) 928–951.

    Google Scholar 

  159. Salamon, D.: On controllability and observability of time-delay systems. IEEE Trans. Automat. Contr., AC-29 (1984) 432–439.

    Google Scholar 

  160. Schoen, G. M. and Geering, H. P.: Stability condition for a delay differential system. Int. J. Contr., 58 (1993) 247–252.

    Google Scholar 

  161. Sename, O.: Sur la commandabilité et le découplage des systèmes linéaires à retards, Thèse Université de Nantes — Ecole Centrale de Nantes, 1994.

    Google Scholar 

  162. Singh, T. and Vadali, S. R.: Robust time-delay control. J. Dynamical Syst., Meas. and Contr., 115 (1993) 303–306.

    Google Scholar 

  163. Shyu, K.-K. and Yan, J.-J.: Robust stability of uncertain time-delay systems and its stabilization by variable structure control. Int. J. Contr., 57 (1993) 237–246.

    Google Scholar 

  164. Sloss, J. M., Sadek, I. S., Bruch Jr. J. C. and Adali, S.: The effects of time delayed active displacement control of damped structures. Control Theory Advanced Tech. 10 (1995) 973–992.

    Google Scholar 

  165. Stépán, G.: Retarded dynamical systems: stability and characterisitc function, Research Notes in Math. Series, 210, John Wiley & Sons, 1989.

    Google Scholar 

  166. Sontag, E.D., Linear systems over commutative rings: a survey. Richerche Automat. 7 (1976), 1–34.

    Google Scholar 

  167. Su, J. H.: Further results on the robust stability of linear systems with a single delay. Syst. & Contr. Lett. 23 (1994) 375–379.

    Google Scholar 

  168. Su, J. H., Fong, I. K., and Tseng, C. L.: Stability analysis of linear systems with time delay. IEEE Trans. Automat. Contr. 39 (1994) 1341–1344.

    Google Scholar 

  169. Su, J. H.: The asymptotic stability of linear autonomous systems with commensurate delays. IEEE Trans. Automat. Contr. 40 (1995) 1114–1118.

    Google Scholar 

  170. Su, T. J. and Huang, C. G.: Robust stability of delay dependence for linear uncertain systems. IEEE Trans. Automat. Control 37 (1992) 1656–1659.

    Google Scholar 

  171. Su, T. J. and Liu, P.-L.: Robust stability for linear time-delay systems with delay-dependence. Int. J. Syst. Science 24 (1993) 1067–1080.

    Google Scholar 

  172. Suh, I. H. and Bien, Z.: A root-locus technique for linear systems with delay. IEEE Trans. Automat. Contr. AC-27 (1982) 205–208.

    Google Scholar 

  173. Thowsen, A.: Uniform ultimate boundness of the solutions of uncertain dynamic delay systems with state-dependent and memoryless feedback control. Int. J. Contr. 37 (1983) 1153–1143.

    Google Scholar 

  174. Toker, O. and Ozbay, H.: Complexity issues in robust stability of linear delay-differential systems. Math., Contr., Signals, Syst. 9 (1996) 386–400.

    Google Scholar 

  175. Tokumaru, H., Adachi, N. and Amemyian, T.: Macroscopic stability of interconnected systems. Proc. 6th IFAC Congress, paper ID 44.4, Academic Press, New York, 1966.

    Google Scholar 

  176. Townley, S. and Pritchard, A. J.: On problems of robust stability for uncertain systems with time-delay. Proc. 1st European Contr. Conf. Grenoble France (1991) 2078–2083.

    Google Scholar 

  177. Trinh, H. and Aldeen, M.: On the stability of linear systems with delayed perturbations. IEEE Trans. Automat. Contr. 39 (1994) 1948–1951.

    Google Scholar 

  178. Tsypkin, Ya. Z., and Fu, M.: Robust stability of time-delay systems with an uncertain time-delay constant. Int. J. Contr., 57 (1993) 865–879.

    Google Scholar 

  179. Verriest, E. I. and Ivanov, A. F.: Robust stabilization of systems with delayed feedback. Proc. Second Int'l Symposium on Implicit and Robust Systems Warszawa Poland (1991) 190–193.

    Google Scholar 

  180. Verriest, E. I.: Robust stability of time varying systems with unknown bounded delays. Proc. 33rd IEEE CDC Lake Buena Vista FL (1994) 417–422.

    Google Scholar 

  181. Verriest, E. I., Fan, M. K. H. and Kullstam, J.: Frequency domain robust stability criteria for linear delay systems. Proc. 32nd IEEE CDC San Antonio TX (1993) 3473–3478.

    Google Scholar 

  182. Verriest, E. I. and Ivanov, A. F.: Robust stability of systems with delayed feedback. Circ., Syst., Signal Proc. 13 (1994) 213–222.

    Google Scholar 

  183. Verriest, E. I.: Stabilization of deterministic and stochastic systems with uncertain time delays. Proc. 33rd IEEE CDC Orlando FL (1994) 3829–3834.

    Google Scholar 

  184. Verriest, E. I. and Florchinger, P.: Stability of stochastic systems with uncertain time delays. Systems & Control Letters 24 1 (1995) 41–47.

    Google Scholar 

  185. Verriest, E. I.: Stability of systems with distributed delays. Preprints of the IFAC System Structure and Control Nantes France (1995) 294–299.

    Google Scholar 

  186. Verriest, E. I. and Ivanov, A. F.: Robust stability of delay-difference equations. Proc. 34th IEEE CDC New Orleans LA (1995) 386–391.

    Google Scholar 

  187. Verriest, E. I.: Stability and stabilization of stochastic systems with distributed delays. Proc. 34th IEEE CDC New Orleans LA (1995) 2205–2210.

    Google Scholar 

  188. Verriest, E. I. and Aggoune, W.: Stability of nonlinear differential delay systems. Proc. CESA-96 IMACS (1996) Lille France 790–795.

    Google Scholar 

  189. Verriest, E. I. and Fan, M. K. H.: Robust stability of nonlinearly perturbed delay systems. Proc. 35th IEEE CDC Kobe Japan (1996) 2090–2091.

    Google Scholar 

  190. Verriest, E. I. and Aggoune, W.: Stability of nonlinear differential delay systems. To appear in Mathematics and Computers in Simulation (1997).

    Google Scholar 

  191. Walton, K. and Marshall, J. E.: Direct method for TDS stability analysis. IEE Proc. 134 part D (1987) 101–107.

    Google Scholar 

  192. Wang, S. S.: Further results on stability of \(\dot x\) (t)=Ax(t)+Bx(t−τ). Syst. & Contr. Lett. 19 (1992) 165–168.

    Google Scholar 

  193. Wang, S. S., Chen, B. S. and Lin, T. P.: Robust stability of uncertain time-delay systems. Int. J. Control 46 (1987) 963–976.

    Google Scholar 

  194. Wang, W. J. and Wang, R. J.: New stability criteria for linear time-delay systems. Control — Theory and Advanced Tech. 10 (1995) 1213–1222.

    Google Scholar 

  195. Wu, H. and Mizukami, K.: Quantitative measures of robustness for uncertain time-delay dynamical systems. Proc. 32nd IEEE CDC San Antonio TX (1993) 2004–2005.

    Google Scholar 

  196. Xi, L. and de Souza, C. E.: LMI approach to delay-dependent robust stability and stabilization of uncertain linear delay systems. Proc. 34th IEEE Conf. Dec. Contr. New Orleans LA (1995) 3614–3619.

    Google Scholar 

  197. Xi, L. and de Souza, C. E.: Criteria for robust stability of uncertain linear systems with time-varying state delays. Proc. 13th IFAC World Congr. San Francisco CA H (1996) 137–142.

    Google Scholar 

  198. Xie, L. and de Souza, C. E.: Robust stabilization and disturbance attenuation for uncertain delay system. Proc. 2nd European Contr. Conf. Groningen The Netherlands (1993) 667–672.

    Google Scholar 

  199. Xu, B.: Comments on “Robust Stability of Delay Dependence for Linear Uncertain Systems”. IEEE Trans. Automat. Contr. AC-39 (1994) 2365.

    Google Scholar 

  200. Ye, H. Michel, A. M. and Wang, K.: Stability of nonlinear dynamical systems with parameter uncertainties with an application to neural networks. Proc. 1995 American Contr. Conf. Seattle WA (1995) 2772–2776.

    Google Scholar 

  201. Yoneyama, T.: On the 3/2 stability theorem for one-dimensional delay-differential equations. J. Math. Anal. Appl. 125 (1987) 161–173.

    Google Scholar 

  202. Yoneyama, T.: On the stability region of scalar delay-differential equations. J. Math. Anal. Appl. 134 (1988) 408–425.

    Google Scholar 

  203. Yu, W., Sobel, K. M. and Shapiro, E. Y.: A time domain approach to the robustness of time delay systems. Proc. 31st IEEE CDC Tucson, AZ (1992) 3726–3727.

    Google Scholar 

  204. Zhang, D.-N. Saeki, M. and Ando, K.: Stability margin calculation of systems with structured time-delay uncertainties. IEEE Trans. Automat. Contr. 37 (1992) 865–868.

    Google Scholar 

  205. Zheng, F., Cheng M. and Gao, W.: Feedback stabilization of linear systems with point delays in state and control variables Proc. 12th IFAC World Congr., Sydney, Australia 2 (1993) 375–378.

    Google Scholar 

  206. Zhou, K., Doyle, J. and Glover, K.: Robust and optimal control Prentice Hall, New Jersey, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Math. Laboratory, ENSTA, 32, boulevard Victor, 75739, Paris, Cedex 15, France

    Silviu-Iulian Niculescu

  2. School of Electrical and Computer Engineering, Georgia Institute of Technology, 30332-0250, Atlanta, GA, USA

    Erik I. Verriest

  3. Laboratoire d'Automatique de Grenoble, ENSIEG, BP 46, 38402, Saint Martin d'Hères, France

    Luc Dugard & Jean-Michel Dion

Authors
  1. Silviu-Iulian Niculescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erik I. Verriest
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luc Dugard
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jean-Michel Dion
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

L. Dugard E. I. Verriest

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag London Limited

About this chapter

Cite this chapter

Niculescu, SI., Verriest, E.I., Dugard, L., Dion, JM. (1998). Stability and robust stability of time-delay systems: A guided tour. In: Dugard, L., Verriest, E.I. (eds) Stability and Control of Time-delay Systems. Lecture Notes in Control and Information Sciences, vol 228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027479

Download citation

  • DOI: https://doi.org/10.1007/BFb0027479

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-76193-8

  • Online ISBN: 978-3-540-40935-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation