The direct method of Lyapunov for nonlinear dynamical systems with fractional damping
- PDF / 808,749 Bytes
- 21 Pages / 547.087 x 737.008 pts Page_size
- 96 Downloads / 196 Views
ORIGINAL PAPER
The direct method of Lyapunov for nonlinear dynamical systems with fractional damping Matthias Hinze
· André Schmidt · Remco I. Leine
Received: 12 August 2020 / Accepted: 15 September 2020 © The Author(s) 2020
Abstract In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractionalorder derivatives of state variables. The application of the related method is illustrated on a fractionally M. Hinze (B)· A. Schmidt · R. I. Leine Institute for Nonlinear Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany e-mail: [email protected] A. Schmidt e-mail: [email protected] R. I. Leine e-mail: [email protected]
damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control. Keywords Fractional damping · Springpot · Lyapunov functional · Functional differential equation · Invariance principle · Tracking control Mathematics Subject Classification 34K20 · 34K35 · 34K37 · 37N05
1 Introduction Many problems in industrial applications originate from (dynamic) instability phenomena, e.g., stick-slip vibrations, flutter, shimmy of vehicles and feedback instabilities in control systems. Methods to rigorously prove stability of linear and nonlinear systems are quintessential for the mitigation of instability-induced vibration through improved design, vibration observers or feedback control. The Lyapunov stability framework, which encompasses the direct method of Lyapunov, forms a central element in the research fields Nonlinear Dynamics and Control Theory [17]. The Lyapunov approach is classically formulated for ordinary differential equations (ODEs). It is the aim of this paper to provide a generalization of the direct method of Lyapunov for ODEs that contain additional fractional derivatives of system states. The term fractional refers to the mathematical theory of fractional calculus dealing with derivatives and
123
M. Hinze et al.
integrals of arbitrary (non-integer) order, see, e.g., [6,29,33] for an introduction. The theory h
Data Loading...
