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.

## 2021 |

Hossain, Sumon; Trenn, Stephan Minimal realization for linear switched systems with a single switch Inproceedings Proc. European Control Conference (ECC21), Rotterdam, Netherlands, 2021, (to appear). Abstract | Links | BibTeX | Tags: controllability, normal-forms, observability, solution-theory, switched-systems @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 = {controllability, normal-forms, observability, solution-theory, switched-systems}, 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. |

Berger, Thomas; Ilchmann, Achim; Trenn, Stephan Quasi feedback forms for differential-algebraic systems Unpublished 2021, (submitted for publication). Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms @unpublished{BergIlch21pp, title = {Quasi feedback forms for differential-algebraic systems}, author = {Thomas Berger and Achim Ilchmann and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2021/07/Preprint-BIT210707.pdf, Preprint https://arxiv.org/abs/2102.12713, arXiv:2102.12713}, year = {2021}, date = {2021-02-25}, abstract = {We investigate feedback forms for linear time-invariant systems described by differential-algebraic equa- tions. Feedback forms are representatives of certain equivalence classes. For example state space trans- formations, invertible transformations from the left, and proportional state feedback constitute an equiva- lence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.}, note = {submitted for publication}, keywords = {controllability, DAEs, normal-forms}, pubstate = {published}, tppubtype = {unpublished} } We investigate feedback forms for linear time-invariant systems described by differential-algebraic equa- tions. Feedback forms are representatives of certain equivalence classes. For example state space trans- formations, invertible transformations from the left, and proportional state feedback constitute an equiva- lence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold. |

Wijnbergen, Paul; Trenn, Stephan Impulse-free interval-stabilization of switched differential algebraic equations Journal Article Systems & Control Letters, 149 , pp. 104870.1-10, 2021, (Open Access.). Abstract | Links | BibTeX | Tags: controllability, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems @article{WijnTren21, title = {Impulse-free interval-stabilization of switched differential algebraic equations}, author = {Paul Wijnbergen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2021/01/24-SCL149-104870.pdf, Paper}, doi = {10.1016/j.sysconle.2020.104870}, year = {2021}, date = {2021-01-23}, journal = {Systems & Control Letters}, volume = {149}, pages = {104870.1-10}, abstract = {In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. }, note = {Open Access.}, keywords = {controllability, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {article} } In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. |

## 2020 |

Wijnbergen, Paul; Trenn, Stephan Impulse controllability of switched differential-algebraic equations Inproceedings Proc. European Control Conference (ECC 2020), pp. 1561-1566, Saint Petersburg, Russia, 2020. Abstract | Links | BibTeX | Tags: controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{WijnTren20, title = {Impulse controllability of switched differential-algebraic equations}, author = {Paul Wijnbergen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/02/Preprint-WT200204.pdf, Preprint}, doi = {10.23919/ECC51009.2020.9143713}, year = {2020}, date = {2020-05-15}, booktitle = {Proc. European Control Conference (ECC 2020)}, pages = {1561-1566}, address = {Saint Petersburg, Russia}, abstract = {This paper addresses impulse controllability of switched DAEs on a finite interval. First we present a forward approach where we define certain subspaces forward in time. These subpsaces are then used to provide a sufficient condition for impulse controllability. In order to obtain a full characterization we present afterwards a backward approach, where a sequence of subspaces is defined backwards in time. With the help of the last element of this backward sequence, we are able to fully characterize impulse controllability. All results are geometric results and thus independent of a coordinate system.}, keywords = {controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } This paper addresses impulse controllability of switched DAEs on a finite interval. First we present a forward approach where we define certain subspaces forward in time. These subpsaces are then used to provide a sufficient condition for impulse controllability. In order to obtain a full characterization we present afterwards a backward approach, where a sequence of subspaces is defined backwards in time. With the help of the last element of this backward sequence, we are able to fully characterize impulse controllability. All results are geometric results and thus independent of a coordinate system. |

Wijnbergen, Paul; Trenn, Stephan A forward approach to controllability of switched DAEs Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. Links | BibTeX | Tags: controllability, misc, switched-DAEs, switched-systems @misc{WijnTren20m, title = {A forward approach to controllability of switched DAEs}, author = {Paul Wijnbergen 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/WijnTren20.pdf, Extended Abstract}, year = {2020}, date = {2020-03-12}, howpublished = {Book of Abstracts - 39th Benelux Meeting on Systems and Control}, keywords = {controllability, misc, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {misc} } |

## 2016 |

Küsters, Ferdinand; Trenn, Stephan Duality of switched DAEs Journal Article Math. Control Signals Syst., 28 (3), pp. 25, 2016. Abstract | Links | BibTeX | Tags: controllability, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{KustTren16a, title = {Duality of switched DAEs}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT160627.pdf, Preprint}, doi = {10.1007/s00498-016-0177-2}, year = {2016}, date = {2016-07-01}, journal = {Math. Control Signals Syst.}, volume = {28}, number = {3}, pages = {25}, abstract = {We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs.}, keywords = {controllability, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {article} } We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs. |

## 2015 |

Küsters, Ferdinand; Trenn, Stephan Duality of switched ODEs with jumps Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 4879–4884, 2015. Abstract | Links | BibTeX | Tags: CDC, controllability, observability, switched-systems @inproceedings{KustTren15b, title = {Duality of switched ODEs with jumps}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT150814.pdf, Preprint}, doi = {10.1109/CDC.2015.7402981}, year = {2015}, date = {2015-12-05}, booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan}, pages = {4879--4884}, abstract = {Duality between controllability/reachability and determinability/observability of switched systems with jumps is proven. The duality result is based on the recent characterization of controllability for switched differential-algebraic equations (DAEs) which share many properties with switched ordinary differential equations (ODEs) with jumps. Here we view the switching signal as given and fixed, which makes the overall switched system time-varying, in particular controllability and reachability do not coincide anymore.}, keywords = {CDC, controllability, observability, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } Duality between controllability/reachability and determinability/observability of switched systems with jumps is proven. The duality result is based on the recent characterization of controllability for switched differential-algebraic equations (DAEs) which share many properties with switched ordinary differential equations (ODEs) with jumps. Here we view the switching signal as given and fixed, which makes the overall switched system time-varying, in particular controllability and reachability do not coincide anymore. |

Küsters, Ferdinand; Trenn, Stephan Controllability characterization of switched DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 643–644, WILEY-VCH Verlag, 2015, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: controllability, DAEs, switched-DAEs, switched-systems @inproceedings{KustTren15a, title = {Controllability characterization of switched DAEs}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT150527.pdf, Preprint}, doi = {10.1002/pamm.201510311}, issn = {1617-7061}, year = {2015}, date = {2015-06-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {15}, number = {1}, pages = {643--644}, publisher = {WILEY-VCH Verlag}, abstract = {We study controllability of switched differential algebraic equations (switched DAEs) with fixed switching signal. Based on a behavioral definition of controllability we are able to establish a controllability characterization that takes into account possible jumps and impulses induced by the switches.}, keywords = {controllability, DAEs, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We study controllability of switched differential algebraic equations (switched DAEs) with fixed switching signal. Based on a behavioral definition of controllability we are able to establish a controllability characterization that takes into account possible jumps and impulses induced by the switches. |

Küsters, Ferdinand; Ruppert, Markus G -M; Trenn, Stephan Controllability of switched differential-algebraic equations Journal Article Syst. Control Lett., 78 (0), pp. 32 - 39, 2015, ISSN: 0167-6911. Abstract | Links | BibTeX | Tags: controllability, DAEs, switched-DAEs, switched-systems @article{KustRupp15, title = {Controllability of switched differential-algebraic equations}, author = {Ferdinand Küsters and Markus G.-M. Ruppert and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KRT150122.pdf, Preprint}, doi = {10.1016/j.sysconle.2015.01.011}, issn = {0167-6911}, year = {2015}, date = {2015-01-01}, journal = {Syst. Control Lett.}, volume = {78}, number = {0}, pages = {32 - 39}, abstract = {We study controllability of switched differential–algebraic equations. We are able to establish a controllability characterization where we assume that the switching signal is known. The characterization takes into account possible jumps induced by the switches. It turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times.}, keywords = {controllability, DAEs, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {article} } We study controllability of switched differential–algebraic equations. We are able to establish a controllability characterization where we assume that the switching signal is known. The characterization takes into account possible jumps induced by the switches. It turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times. |

## 2014 |

Ruppert, Markus G -M; Trenn, Stephan Controllability of switched DAEs: the single switch case Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 15–18, Wiley-VCH Verlag GmbH, 2014. Abstract | Links | BibTeX | Tags: controllability, switched-DAEs, switched-systems @inproceedings{RuppTren14, title = {Controllability of switched DAEs: the single switch case}, author = {Markus G.-M. Ruppert and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-RT140729.pdf, Preprint (contains some corrections w.r.t. the published version)}, doi = {10.1002/pamm.201410005}, year = {2014}, date = {2014-03-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {14}, number = {1}, pages = {15--18}, publisher = {Wiley-VCH Verlag GmbH}, abstract = {We study controllability of switched DAEs and formulate a definition of controllability in the behavioral sense. In order to characterize controllability for switched DAEs we first present new characterizations of controllability of non-switched DAEs based on the Wong-sequences. Afterwards a first result concerning the single-switch case is presented.}, keywords = {controllability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We study controllability of switched DAEs and formulate a definition of controllability in the behavioral sense. In order to characterize controllability for switched DAEs we first present new characterizations of controllability of non-switched DAEs based on the Wong-sequences. Afterwards a first result concerning the single-switch case is presented. |

Berger, Thomas; Trenn, Stephan Kalman controllability decompositions for differential-algebraic systems Journal Article Syst. Control Lett., 71 , pp. 54–61, 2014, ISSN: 0167-6911. Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms @article{BergTren14, title = {Kalman controllability decompositions for differential-algebraic systems}, author = {Thomas Berger and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-BT140603.pdf, Preprint}, doi = {10.1016/j.sysconle.2014.06.004}, issn = {0167-6911}, year = {2014}, date = {2014-01-01}, journal = {Syst. Control Lett.}, volume = {71}, pages = {54--61}, abstract = {We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition (KCD). The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part. This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems.}, keywords = {controllability, DAEs, normal-forms}, pubstate = {published}, tppubtype = {article} } We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition (KCD). The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part. This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems. |

## 2009 |

Trenn, Stephan A normal form for pure differential algebraic systems Journal Article Linear Algebra Appl., 430 (4), pp. 1070 – 1084, 2009. Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms, observability, relative-degree @article{Tren09a, title = {A normal form for pure differential algebraic systems}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre081215.pdf, Preprint}, doi = {10.1016/j.laa.2008.10.004}, year = {2009}, date = {2009-01-01}, journal = {Linear Algebra Appl.}, volume = {430}, number = {4}, pages = {1070 -- 1084}, abstract = {In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability.}, keywords = {controllability, DAEs, normal-forms, observability, relative-degree}, pubstate = {published}, tppubtype = {article} } In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability. |