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 fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.
Highlights
Many problems in industrial applications originate from instability phenomena, e.g., stick-slip vibrations, flutter, shimmy of vehicles and feedback instabilities in control systems
We consider fractional derivatives with an infinite lower bound of integration, which leads to an interpretation of fractionally damped systems as autonomous functional differential equation (FDE) with infinite delay and allows for the use of a generalized invariance principle which we develop in this paper
We introduce the necessary content on fractional calculus (Sect. 2) and summarize the theory on FDEs and the related method of Lyapunov functionals
Summary
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. 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. A finite-dimensional mechanical system with additional springpots has to be considered as a functional differential equation (FDE), for which a related Lyapunov theory exists [3,4,9,19,21]. Results regarding Lyapunov functions and special stability concepts exist [7,20,23] They cannot be used for ODE systems containing springpots, as a (generally) irrational differentiation order leads to an incommensurate fractional system. We study the example of a two degree-of-freedom mechanical archetype system with regularized Coulomb friction and noncollocated control and prove tracking of a desired solution (Sect. 5.3)
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