Chen, Yahao; Trenn, Stephan An approximation for nonlinear differential-algebraic equations via singular perturbation theory Proceedings Article In: Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC-PapersOnLine, pp. 187-192, Brussels, Belgium, 2021, (open access). @inproceedings{ChenTren21c,
title = {An approximation for nonlinear differential-algebraic equations via singular perturbation theory},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/ChenTren21c.pdf, Paper
},
doi = {10.1016/j.ifacol.2021.08.496},
year = {2021},
date = {2021-03-26},
urldate = {2021-03-26},
booktitle = {Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC-PapersOnLine},
volume = {54},
number = {5},
pages = {187-192},
address = {Brussels, Belgium},
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 C1-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.},
note = {open access},
keywords = {DAEs, nonlinear, normal-forms, solution-theory},
pubstate = {published},
tppubtype = {inproceedings}
}
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 C1-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. |
Borsche, Raul; Kocoglu, Damla; Trenn, Stephan A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Journal Article In: Mathematics of Control, Signals, and Systems (MCSS), vol. 32, pp. 455-487, 2020, (Open Access). @article{BorsKoco20,
title = {A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs},
author = {Raul Borsche and Damla Kocoglu and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2020/11/23-MCSS2020.pdf, Paper},
doi = {10.1007/s00498-020-00267-7},
year = {2020},
date = {2020-11-18},
urldate = {2020-11-18},
journal = {Mathematics of Control, Signals, and Systems (MCSS)},
volume = {32},
pages = {455-487},
abstract = {A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.},
note = {Open Access},
keywords = {DAEs, delay, networks, PDEs, piecewise-smooth-distributions, solution-theory, switched-DAEs},
pubstate = {published},
tppubtype = {article}
}
A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions. |
Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential algebraic equations Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. @misc{ChenTren20m,
title = {On geometric and differentiation index of nonlinear differential algebraic equations},
author = {Yahao Chen and Stephan Trenn},
editor = {Raffaella Carloni and Bayu Jayawardhana and Erjen Lefeber},
url = {https://www.beneluxmeeting.nl/2020/uploads/papers/boa.pdf, Book of Abstracts
https://stephantrenn.net/wp-content/uploads/2021/03/ChenTren20.pdf, Extended Abstract},
year = {2020},
date = {2020-03-12},
howpublished = {Book of Abstracts - 39th Benelux Meeting on Systems and Control},
keywords = {DAEs, nonlinear, solution-theory},
pubstate = {published},
tppubtype = {misc}
}
|