An approximation for nonlinear differential-algebraic equations via singular perturbation theory. Issue 5 (2021)

• Record Type: Journal Article
• Authors: Chen, Yahao; Trenn, Stephan
• Is Part Of: IFAC-PapersOnLine. Volume 54:Issue 5(2021)
• Page Start: 187
• Page End: 192
• Publication Date: 2021
• Subjects: differential-algebraic equations -- singular perturbation -- jumps -- index-1 -- nonlinear Weierstrass form -- inconsistent initial values; Automatic control -- Periodicals; 629.805
• DOI: 10.1016/j.ifacol.2021.08.496
• Languages: English
• ISSNs: 2405-8963

Abstract

Abstract: In this paper, we study the jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector introduced in Liberzon and Trenn (2009) for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods (one is from the paper Liberzon and Trenn (2012) and the other is given by a MATLAB function) to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter and we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the C 1 -solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs.