Abstract
When feedback controls are used for time-delay control systems, the resulting closed-loop systems become homogeneous time-delay systems without external variables. Thus the stability of homogeneous time-delay systems is very important. This chapter is devoted to various stability and robust stability test methods for homogeneous time-delay systems. After general time-delay systems are briefly introduced, then linear time-delay systems are followed. The existence and uniqueness of solutions of general nonlinear time-delay systems are dealt with. The stability, uniform stability, asymptotic stability, and uniform asymptotic stability for time-delay systems are defined. Two important stability results such as the Krasovskii theorem and the Razumikhin theorem are introduced. Linear and bilinear matrix inequalities are introduced together with recent integral inequalities for quadratic functions such as in the integral inequality lemma and the Jensen inequality lemmas with and without reciprocal convexity, which are useful when the Krasovskii and Razumikhin theorems are utilized. Various stability test methods, either delay-independent or delay-dependent, for single state delayed systems are developed based on the Razumikhin theorem, the Krasovskii theorem, and the discretized state description. Some results are extended to systems with time-varying delays with and without bounded derivatives. Robust stability test methods for time-delay systems with model uncertainties are developed based on the Krasovskii theorem and the discretized state description. Some results based on the Krasovskii theorem are extended to systems with both model uncertainties and time-varying delays with and without bounded derivatives. Some stability and robust stability test methods for distributed state delayed systems are briefly introduced since distributed delays are often appeared in optimal controls in later chapters.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia
Driver RD (1962) Existence and stability of solutions of a delay differential system. Springer, Berlin
Fridman E (2001) New \(\text{L}\)yapunov-\(\text{ K }\)rasovskii functionals for stability of linear retarded and neutral type systems. Syst Control Lett 43(4):309–319
Fridman E, Shaked U (2002) A descriptor system approach to \(\cal{H}_\infty \) control of time-delay systems. IEEE Trans Autom Control 47(2):253–270
Fridman E, Shaked U (2002) An improved stabilization method for linear systems with time-delay. IEEE Trans Autom Control 47(11):1931–1937
Fridman E, Shaked U (2003) Delay-dependent stability and \(\cal{H}_\infty \) control: constant and time-varying delays. Int J Control 76(1):48–60
Gu K (1997) Discretized \(\text{ LMI }\) set in the stability problem of linear uncertain time-delay systems. Int J Control 68(4):923–934
Gu K (2000) An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE conference on decision and control. Sydney, Australia, pp 2805–2810
Gu K, Han QL (2000) Controller design for time-delay systems using discretized Lyapunov functional approach. In: Proceedings of the 39th IEEE conference on decision and control. Sydney, Australia, pp 2793–2798
Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems. Birkhäuser, Basel
Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Springer, Berlin
Ko JW, Park P (2009) Delay-dependent stability criteria for systems with time-varying delays: state discretization approach. IEICE Trans Fundam Electron Commun Comput Sci E92–A(4):1136–1141
Krasovskii NN (1956) On the application of the second method of Lyapunov for equations with time delays. Prikl Mat Mek 20:315–327
Lee WI, Lee SY, Park P (2016) A combined first- and second-order reciprocal convexity approach for stability analysis of systems with interval time-varying delays. J Franklin Inst 353(9):2104–2116
Lee SY, Lee WI, Park P (2016) Polynomials-based integral inequality for stability analysis of linear systems with time-varying delays. J Franklin Inst 354(4):2053–2067
Lee SY, Lee WI, Park P (2017) Improved stability criteria for linear systems with interval time-varying delays: generalized zero equalities approach. Appl Math Comput 292:336–348
Lee WI, Park P (2014) Second-order reciprocally convex approach to stability of systems with interval time-varying delays. Appl Math Comput 229:245–253
Lur’e AI (1957) Some nonlinear problems in the theory of automatic control. H. M. Stationery Office, London (In Russian, 1951)
Michiels W, Niculescu SI (2007) Stability and stabilization of time-delay systems. SIAM, Philadelphia
Moon YS, Park P, Kwon WH, Lee YS (2001) Delay-dependent robust stabilization of uncertain state-delayed systems. Int J Control 74(14):1447–1455
Niculescu SI, de Souza CE, Dion JM, Dugard L (1994) Robust stability and stabilization of uncertain linear systems with state delay: single delay case (i). In: Proceedings of the IFAC symposium on robust control design. Rio de Janeiro, pp 469–474
Niculescu SI, Verriest EI, Dugard L, Dion JM (1998) Stability and robust stability of time-delay systems: a guided tour. Stability and control of time-delay systems. Lecture notes in control and information sciences, vol 228. Springer, New York
Oǧuztöreli MN (1966) Time-lag control systems. Academic Press, New York
Park P (1999) A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans Autom Control 44(4):876–877
Park P, Ko JW (2007) Stability and robust stability for systems with a time-varying delay. Automatica 43(10):1855–1858
Park P, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1):235–238
Razumikhin BS (1956) On the stability of systems with a delay. Prikl Mat Meh 20:500–512
Razumikhin BS (1960) Application of Lyapunov’s method to problems in the stability of systems with a delay. Automat i Telemekh 21:740–749
Yakubovich VA (1962) The solution of certain matrix inequalities in automatic control theory. Soviet Math Dokl 3:620–623 (In Russian, 1961)
Yakubovich VA (1964) Solution of certain matrix inequalities encountered in nonlinear control theory. Soviet Math Dokl 5:652–656
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Kwon, W.H., Park, P. (2019). Stability of Time-Delay Systems. In: Stabilizing and Optimizing Control for Time-Delay Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-92704-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-92704-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92703-9
Online ISBN: 978-3-319-92704-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)