Koopman-based stability analysis of differential-algebraic equations with applications to frictional multibody systems

Abstract

Periodic solutions of differential-algebraic equations (DAEs) and ordinary differential equations (ODEs) can be determined using the harmonic balance method (HBM), which is a frequency-domain approach that approximates the solution by its truncated Fourier series. The Koopman-Hill method, a method to determine the stability of periodic solutions found by HBM, and originally developed for ODEs, is generalized to DAEs in this work. Analogously to the ODE case, the core idea of the proposed Koopman-Hill method for DAEs is to establish a linear time-invariant but high-dimensional DAE which approximately governs the dynamics of small admissible perturbations around the periodic solution. The crucial difference to the ODE case is the fact that the evolution of this linear time-invariant DAE is not simply given by a matrix exponential, but by a more complicated expression involving a Drazin inverse, rendering the resulting monodromy matrix singular. Still, even in the DAE case, this novel relationship between the monodromy matrix and the Hill matrix is essentially given by one single formula, which is the main result of this work. Two academic mechanical systems, a mathematical pendulum formulated as an index-3 DAE and a nonsmooth frictional two-mass oscillator with switching index, demonstrate the applicability of the proposed method and its blindness to the DAE's index.