Sutrisno,; Trenn, Stephan Observability of Switched Linear Singular System in Discrete Time: Single Switch Case Inproceedings In: Proc. European Control Conference (ECC21), Rotterdam, Netherlands, 2021, (to appear). @inproceedings{SutrTren21a,
title = {Observability of Switched Linear Singular System in Discrete Time: Single Switch Case},
author = {Sutrisno and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-ST210406.pdf, Preprint},
year = {2021},
date = {2021-06-29},
urldate = {2021-06-29},
booktitle = {Proc. European Control Conference (ECC21)},
address = {Rotterdam, Netherlands},
abstract = {In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is inde- pendent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is inde- pendent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability. |

Hossain, Sumon; Trenn, Stephan Minimal realization for linear switched systems with a single switch Inproceedings In: Proc. European Control Conference (ECC21), Rotterdam, Netherlands, 2021, (to appear). @inproceedings{HossTren21,
title = {Minimal realization for linear switched systems with a single switch},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-HT210406.pdf, Preprint},
year = {2021},
date = {2021-06-29},
booktitle = {Proc. European Control Conference (ECC21)},
address = {Rotterdam, Netherlands},
abstract = {We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple examples to illustrate the approach.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple examples to illustrate the approach. |

Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential-algebraic equations Inproceedings In: Proceedings of the MTNS 2020/21, 2021, (to appear). @inproceedings{ChenTren21b,
title = {On geometric and differentiation index of nonlinear differential-algebraic equations},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-CT210406.pdf, Preprint},
year = {2021},
date = {2021-04-06},
booktitle = {Proceedings of the MTNS 2020/21},
abstract = {We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices. |

Chen, Yahao; Trenn, Stephan An approximation for nonlinear differential-algebraic equations via singular perturbation theory Inproceedings In: Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC Brussels, Belgium, 2021, (to appear). @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/2021/03/Preprint-CT210326.pdf, Preprint},
year = {2021},
date = {2021-03-26},
booktitle = {Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21)},
address = {Brussels, Belgium},
organization = {IFAC},
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 = {to appear},
keywords = {},
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. |