×

Design of finite-time stabilizing controllers for nonlinear dynamical systems. (English) Zbl 1166.93013

Summary: Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using Hölder continuous Lyapunov functions. In this paper, we extend the finite-time stability theory to revisit time-invariant dynamical systems and to address time-varying dynamical systems. Specifically, we develop a Lyapunov-based stability and control design framework for finite-time stability as well as finite-time tracking for time-varying nonlinear dynamical systems. Furthermore, we use the vector Lyapunov function approach to study finite-time stabilization of compact sets for large-scale dynamical systems.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93A15 Large-scale systems
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations (1993) · Zbl 0785.34003
[2] Filippov, Differential Equations with Discontinuous Right-Hand Sides (1988) · Zbl 0664.34001
[3] Kawski, Stabilization of nonlinear systems in the plane, Systems and Control Letters 12 pp 169– (1989) · Zbl 0666.93103
[4] Yoshizawa, Stability Theory by Liapunov’s Second Method (1966) · Zbl 0144.10802
[5] Coddington, Theory of Ordinary Differential Equations (1955) · Zbl 0064.33002
[6] Bhat, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization 38 (3) pp 751– (2000) · Zbl 0945.34039
[7] Bhat, Geometric homogeneity with applications to finite-time stability, Mathematics of Control Signals and Systems 17 pp 101– (2005) · Zbl 1110.34033
[8] Haimo, Finite-time controllers, SIAM Journal on Control and Optimization 24 pp 760– (1986) · Zbl 0603.93005
[9] Bhat, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Transactions on Automatic Control 43 pp 678– (1998) · Zbl 0925.93821
[10] Hong, Finite-time stabilization and stabilizability of a class of controllable systems, Systems and Control Letters 46 pp 231– (2002) · Zbl 0994.93049
[11] Hong, On an output feedback finite-time stabilization problem, IEEE Transactions on Automatic Control 46 pp 305– (2001) · Zbl 0992.93075
[12] Moulay E, Perruquetti W. Finite-time stability of nonlinear systems. Proceedings of IEEE Conference on Decision and Control, Maui, HI, 2003; 3641-3646.
[13] Kurzweil, On the investigation of Lyapunov’s second theorem on the stability of motion, The American Mathematical Society Translations 24 pp 19– (1956) · Zbl 0127.30703
[14] Qian, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Transactions on Automatic Control 46 pp 1061– (2001) · Zbl 1012.93053
[15] Bellman, Vector Lyapunov functions, SIAM Journal on Control 1 pp 32– (1962) · Zbl 0144.10901
[16] Matrosov, Method of vector Liapunov functions of interconnected systems with distributed parameters (survey), Avtomatika i Telemekhanika 33 pp 63– (1972)
[17] Šiljak, Large-Scale Dynamic Systems: Stability and Structure (1978)
[18] Lakshmikantham, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems (1991) · Zbl 0721.34054
[19] Nersesov, On the stability and control of nonlinear dynamical systems via vector Lyapunov functions, IEEE Transactions on Automatic Control 51 (2) pp 203– (2006) · Zbl 1366.93553
[20] Kamke, Zur Theorie der Systeme gewöhnlicher Differential-Gleichungen. II, Acta Mathematica 58 pp 57– (1931)
[21] Wažewski, Systèmes des équations et des inégalités différentielles ordinaires aux deuxiémes membres monotones et leurs applications, Annales de la Société polonaise de mathématique 23 pp 112– (1950)
[22] Hale, Ordinary Differential Equations (1980) · Zbl 0433.34003
[23] Khalil, Nonlinear Systems (2002)
[24] Hahn, Stability of Motion (1967)
[25] Haddad, Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems, Nonlinear Analysis: Real World Applications 6 pp 35– (2005) · Zbl 1074.93030
[26] Awad, On the existence and stability of limit cycles for longitudinal acoustic modes in a combustion chamber, Combustion Science and Technology 9 pp 195– (1986)
[27] Culick, Nonlinear behavior of acoustic waves in combustion chambers I, Acta Astronautica 3 pp 715– (1976) · Zbl 0346.76062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.