Below you find an interactive list of all my publications, which can be filtered by keywords, year, publication type and coauthors. There are also static lists of my books/book-chapters as well as journal-, conference-, and submitted publications.

## 2020 |

Iervolino, Raffaele; Vasca, Francesco; Trenn, Stephan Discontinuous Lyapunov functions for discontinous piecewise-affine systems Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). Abstract | Links | BibTeX | Tags: Lyapunov, misc, stability, switched-systems @misc{IervTren20m, title = {Discontinuous Lyapunov functions for discontinous piecewise-affine systems}, author = {Raffaele Iervolino and Francesco Vasca and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-ITV200122.pdf, Extended Abstract}, year = {2020}, date = {2020-01-22}, abstract = {Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. We first introduce the feasible Filippov solution concept by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is highlighted that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. }, howpublished = {Extended Abstract}, note = {accepted for cancelled MTNS 20/21}, keywords = {Lyapunov, misc, stability, switched-systems}, pubstate = {published}, tppubtype = {misc} } Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. We first introduce the feasible Filippov solution concept by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is highlighted that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. |

## 2016 |

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Solvability and stability of a power system DAE model Journal Article Syst. Control Lett., 97 , pp. 12–17, 2016. Abstract | Links | BibTeX | Tags: application, DAEs, Lyapunov, networks, solution-theory, stability @article{GrosTren16, title = {Solvability and stability of a power system DAE model}, author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-GTW160816.pdf, Preprint}, doi = {10.1016/j.sysconle.2016.08.003}, year = {2016}, date = {2016-11-01}, journal = {Syst. Control Lett.}, volume = {97}, pages = {12--17}, abstract = {The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization for the linearized power system DAE solely in terms of the network topology. As an extension to previous result we allow for higher order generator dynamics. Furthermore, we show that any solvable power system DAE is automatically of index one, which means that it is also numerically well posed. Finally, we show that any solvable power system DAE is stable but not asymptotically stable.}, keywords = {application, DAEs, Lyapunov, networks, solution-theory, stability}, pubstate = {published}, tppubtype = {article} } The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization for the linearized power system DAE solely in terms of the network topology. As an extension to previous result we allow for higher order generator dynamics. Furthermore, we show that any solvable power system DAE is automatically of index one, which means that it is also numerically well posed. Finally, we show that any solvable power system DAE is stable but not asymptotically stable. |

## 2014 |

Defoort, Michael; Djemai, Mohamed; Trenn, Stephan Nondecreasing Lyapunov functions Inproceedings Proc. 21st Int. Symposium Math. Theory Networks Systems (MTNS), pp. 1038–1043, 2014. Abstract | Links | BibTeX | Tags: Lyapunov, nonlinear, stability, switched-systems @inproceedings{DefoDjem14, title = {Nondecreasing Lyapunov functions}, author = {Michael Defoort and Mohamed Djemai and Stephan Trenn}, url = {http://fwn06.housing.rug.nl/mtns2014-papers/fullPapers/0067.pdf, Paper http://fwn06.housing.rug.nl/mtns/?page_id=38, Proceedings Website}, year = {2014}, date = {2014-07-01}, booktitle = {Proc. 21st Int. Symposium Math. Theory Networks Systems (MTNS)}, pages = {1038--1043}, abstract = {We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switched systems for which no piecewise-quadratic (classical) Lyapunov function exists.}, keywords = {Lyapunov, nonlinear, stability, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switched systems for which no piecewise-quadratic (classical) Lyapunov function exists. |

## 2012 |

Trenn, Stephan; Wirth, Fabian Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms Inproceedings Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 2666–2671, 2012, ISSN: 0191-2216. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{TrenWirt12b, title = {Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms}, author = {Stephan Trenn and Fabian Wirth}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TW120901.pdf, Preprint}, doi = {10.1109/CDC.2012.6426245}, issn = {0191-2216}, year = {2012}, date = {2012-12-12}, booktitle = {Proc. 51st IEEE Conf. Decis. Control, Maui, USA}, pages = {2666--2671}, abstract = {For linear switched differential algebraic equations (DAEs) we consider the problem of characterizing the maximal exponential growth rate of solutions. It is shown that a finite exponential growth rate exists if and only if the set of consistency projectors associated to the family of DAEs is product bounded. This result may be used to derive a converse Lyapunov theorem for switched DAEs. Under the assumption of irreducibility we show that a construction reminiscent of the construction of Barabanov norms is feasible as well.}, keywords = {CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } For linear switched differential algebraic equations (DAEs) we consider the problem of characterizing the maximal exponential growth rate of solutions. It is shown that a finite exponential growth rate exists if and only if the set of consistency projectors associated to the family of DAEs is product bounded. This result may be used to derive a converse Lyapunov theorem for switched DAEs. Under the assumption of irreducibility we show that a construction reminiscent of the construction of Barabanov norms is feasible as well. |

Trenn, Stephan; Wirth, Fabian A converse Lyapunov theorem for switched DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 789–792, WILEY-VCH Verlag, 2012, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{TrenWirt12a, title = {A converse Lyapunov theorem for switched DAEs}, author = {Stephan Trenn and Fabian Wirth}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TW120508.pdf, Preprint}, doi = {10.1002/pamm.201210381}, issn = {1617-7061}, year = {2012}, date = {2012-03-02}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {12}, number = {1}, pages = {789--792}, publisher = {WILEY-VCH Verlag}, abstract = {For switched ordinary differential equations (ODEs) it is well known that exponential stability under arbitrary switching yields the existence of a common Lyapunov function. The result is known as a “converse Lyapunov Theorem”. In this note we will present a converse Lyapunov theorem for switched differential algebraic equations (DAEs) as well as the construction of a Barabanov norm for irreducible switched DAEs.}, keywords = {DAEs, Lyapunov, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } For switched ordinary differential equations (ODEs) it is well known that exponential stability under arbitrary switching yields the existence of a common Lyapunov function. The result is known as a “converse Lyapunov Theorem”. In this note we will present a converse Lyapunov theorem for switched differential algebraic equations (DAEs) as well as the construction of a Barabanov norm for irreducible switched DAEs. |

## 2011 |

Liberzon, Daniel; Trenn, Stephan; Wirth, Fabian Commutativity and asymptotic stability for linear switched DAEs Inproceedings Proc. 50th IEEE Conf. Decis. Control and European Control Conf. ECC 2011, Orlando, USA, pp. 417–422, 2011. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{LibeTren11, title = {Commutativity and asymptotic stability for linear switched DAEs}, author = {Daniel Liberzon and Stephan Trenn and Fabian Wirth}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LTW110816.pdf, Preprint}, doi = {10.1109/CDC.2011.6160335}, year = {2011}, date = {2011-12-01}, booktitle = {Proc. 50th IEEE Conf. Decis. Control and European Control Conf. ECC 2011, Orlando, USA}, pages = {417--422}, abstract = {For linear switched ordinary differential equations with asymptotically stable constituent systems, it is well known that commutativity of the coefficient matrices implies asymptotic stability of the switched system under arbitrary switching. This result is generalized to linear switched differential algebraic equations (DAEs). Although the solutions of a switched DAE can exhibit jumps it turns out that it suffices to check commutativity of the “flow” matrices. As in the ODE case we are also able to construct a common quadratic Lyapunov function.}, keywords = {CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } For linear switched ordinary differential equations with asymptotically stable constituent systems, it is well known that commutativity of the coefficient matrices implies asymptotic stability of the switched system under arbitrary switching. This result is generalized to linear switched differential algebraic equations (DAEs). Although the solutions of a switched DAE can exhibit jumps it turns out that it suffices to check commutativity of the “flow” matrices. As in the ODE case we are also able to construct a common quadratic Lyapunov function. |

## 2009 |

Liberzon, Daniel; Trenn, Stephan On stability of linear switched differential algebraic equations Inproceedings Proc. Joint 48th IEEE Conf. Decis. Control and 28th Chinese Control Conf., pp. 2156–2161, 2009. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{LibeTren09, title = {On stability of linear switched differential algebraic equations}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT090903.pdf, Preprint}, doi = {10.1109/CDC.2009.5400076}, year = {2009}, date = {2009-12-01}, booktitle = {Proc. Joint 48th IEEE Conf. Decis. Control and 28th Chinese Control Conf.}, pages = {2156--2161}, abstract = {This paper studies linear switched differential algebraic equations (DAEs), i.e., systems defined by a finite family of linear DAE subsystems and a switching signal that governs the switching between them. We show by examples that switching between stable subsystems may lead to instability, and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms compared to those observed in switched systems described by ordinary differential equations (ODEs). We prove two sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The first result states that a common Lyapunov function guarantees stability under arbitrary switching when an additional condition involving consistency projectors holds (this extra condition is not needed when there are no jumps, as in the case of switched ODEs). The second result shows that stability is preserved under switching with sufficiently large dwell time.}, keywords = {CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } This paper studies linear switched differential algebraic equations (DAEs), i.e., systems defined by a finite family of linear DAE subsystems and a switching signal that governs the switching between them. We show by examples that switching between stable subsystems may lead to instability, and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms compared to those observed in switched systems described by ordinary differential equations (ODEs). We prove two sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The first result states that a common Lyapunov function guarantees stability under arbitrary switching when an additional condition involving consistency projectors holds (this extra condition is not needed when there are no jumps, as in the case of switched ODEs). The second result shows that stability is preserved under switching with sufficiently large dwell time. |