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 and conference publications.

## 2019 |

Tanwani, Aneel; Trenn, Stephan Detectability and observer design for switched differential algebraic equations Journal Article Automatica, 99 , pp. 289-300, 2019. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{TanwTren19, title = {Detectability and observer design for switched differential algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2018/09/Preprint-TT180917.pdf, Preprint}, doi = {10.1016/j.automatica.2018.10.043}, year = {2019}, date = {2019-01-01}, journal = {Automatica}, volume = {99}, pages = {289-300}, abstract = {This paper studies detectability for switched linear differential–algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption.}, keywords = {DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {article} } This paper studies detectability for switched linear differential–algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption. |

## 2018 |

Kausar, Rukhsana; Trenn, Stephan Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs Inproceedings Klingenberg, Christian; Westdickenberg, Michael (Ed.): Theory, Numerics and Applications of Hyperbolic Problems II, pp. 123-135, Springer, Cham, 2018, ISBN: 978-3-319-91548-7, (Presented at XVI International Conference on Hyperbolic Problems (HYPO2016), Aachen). Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @inproceedings{KausTren18, title = {Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs}, author = {Rukhsana Kausar and Stephan Trenn}, editor = {Christian Klingenberg and Michael Westdickenberg}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT170418.pdf, Preprint}, doi = {10.1007/978-3-319-91548-7_9}, isbn = {978-3-319-91548-7}, year = {2018}, date = {2018-06-27}, booktitle = {Theory, Numerics and Applications of Hyperbolic Problems II}, pages = {123-135}, publisher = {Springer}, address = {Cham}, abstract = {In water distribution network instantaneous changes in valve and pump settings introduce jumps and sometimes impulses. In particular, a particular impulsive phenomenon which occurs due to sudden closing of valve is the so called water hammer. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). We observed that under some suitable assumptions the PDEs usually used to describe water flows can be simplified to differential algebraic equations (DAEs). The idea is to model water hammer phenomenon in the switched DAEs framework due to its special feature of studying such impulsive effects. To compare these two modeling techniques, a system of hyperbolic PDE model and the switched DAE model for a simple set up consisting of two reservoirs, six pipes and three valve is presented. The aim of this contribution is to present results of both models as motivation for the claim that a switched DAE modeling framework is suitable for describing a water hammer.}, note = {Presented at XVI International Conference on Hyperbolic Problems (HYPO2016), Aachen}, keywords = {application, DAEs, nonlinear, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } In water distribution network instantaneous changes in valve and pump settings introduce jumps and sometimes impulses. In particular, a particular impulsive phenomenon which occurs due to sudden closing of valve is the so called water hammer. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). We observed that under some suitable assumptions the PDEs usually used to describe water flows can be simplified to differential algebraic equations (DAEs). The idea is to model water hammer phenomenon in the switched DAEs framework due to its special feature of studying such impulsive effects. To compare these two modeling techniques, a system of hyperbolic PDE model and the switched DAE model for a simple set up consisting of two reservoirs, six pipes and three valve is presented. The aim of this contribution is to present results of both models as motivation for the claim that a switched DAE modeling framework is suitable for describing a water hammer. |

## 2017 |

Kausar, Rukhsana; Trenn, Stephan Impulses in structured nonlinear switched DAEs Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 3181 - 3186, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: application, CDC, DAEs, nonlinear, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @inproceedings{KausTren17b, title = {Impulses in structured nonlinear switched DAEs}, author = {Rukhsana Kausar and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT170920.pdf, Preprint}, doi = {10.1109/CDC.2017.8264125}, year = {2017}, date = {2017-12-14}, booktitle = {Proc. 56th IEEE Conf. Decis. Control}, pages = {3181 - 3186}, address = {Melbourne, Australia}, abstract = { Switched nonlinear differential algebraic equations (DAEs) occur in mathematical modeling of sudden transients in various physical phenomenons. Hence, it is important to investigate them with respect to the nature of their solutions. The few existing solvability results for switched nonlinear DAEs exclude Dirac impulses by definition; however, in many cases this is too restrictive. For example, in water distribution networks the water hammer effect can only be studied when allowing Dirac impulses in a nonlinear switched DAE description. We investigate existence and uniqueness of solutions with impulses for a general class of nonlinear switched DAEs, where we exploit a certain sparse structure of the nonlinearity.}, keywords = {application, CDC, DAEs, nonlinear, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } Switched nonlinear differential algebraic equations (DAEs) occur in mathematical modeling of sudden transients in various physical phenomenons. Hence, it is important to investigate them with respect to the nature of their solutions. The few existing solvability results for switched nonlinear DAEs exclude Dirac impulses by definition; however, in many cases this is too restrictive. For example, in water distribution networks the water hammer effect can only be studied when allowing Dirac impulses in a nonlinear switched DAE description. We investigate existence and uniqueness of solutions with impulses for a general class of nonlinear switched DAEs, where we exploit a certain sparse structure of the nonlinearity. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch observability for homogeneous switched DAEs Inproceedings Proc. 20th IFAC World Congress 2017, pp. 9355 - 9360, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{KustTren17a, title = {Switch observability for homogeneous switched DAEs}, author = {Ferdinand Küsters and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KTW170315.pdf, Preprint}, doi = {10.1016/j.ifacol.2017.08.1434}, issn = {2405-8963}, year = {2017}, date = {2017-03-25}, booktitle = {Proc. 20th IFAC World Congress 2017}, journal = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {9355 - 9360}, address = {Toulouse, France}, abstract = {We introduce the notions of switching time observability and switch observability for homogeneous switched differential-algebraic equations (DAEs). In contrast to mode detection, they do not require observability of the individual modes and are thus more suitable for fault detection and identification. Based on results in (Küsters and Trenn, 2017) for switched ordinary differential equations (ODEs), we characterize these notions for homogeneous switched DAEs and propose an observer for switch observable systems.}, keywords = {observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We introduce the notions of switching time observability and switch observability for homogeneous switched differential-algebraic equations (DAEs). In contrast to mode detection, they do not require observability of the individual modes and are thus more suitable for fault detection and identification. Based on results in (Küsters and Trenn, 2017) for switched ordinary differential equations (ODEs), we characterize these notions for homogeneous switched DAEs and propose an observer for switch observable systems. |

Tanwani, Aneel; Trenn, Stephan Observer design for detectable switched differential-algebraic equations Inproceedings Proc. 20th IFAC World Congress 2017, pp. 2953 - 2958, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems @inproceedings{TanwTren17b, title = {Observer design for detectable switched differential-algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT170320.pdf, Preprint}, doi = {10.1016/j.ifacol.2017.08.659}, issn = {2405-8963}, year = {2017}, date = {2017-03-22}, booktitle = {Proc. 20th IFAC World Congress 2017}, journal = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {2953 - 2958}, address = {Toulouse, France}, abstract = {This paper studies detectability for switched linear differential-algebraic equations (DAEs) and its application in synthesis of observers. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor over that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component in the estimator is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption.}, keywords = {DAEs, observability, observer, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } This paper studies detectability for switched linear differential-algebraic equations (DAEs) and its application in synthesis of observers. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor over that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component in the estimator is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption. |

Tanwani, Aneel; Trenn, Stephan Determinability and state estimation for switched differential–algebraic equations Journal Article Automatica, 76 , pp. 17–31, 2017, ISSN: 0005-1098. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{TanwTren17, title = {Determinability and state estimation for switched differential–algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT160919.pdf, Preprint}, doi = {10.1016/j.automatica.2016.10.024}, issn = {0005-1098}, year = {2017}, date = {2017-02-01}, journal = {Automatica}, volume = {76}, pages = {17--31}, abstract = {The problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential–algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appropriate notion of observability is presented which accommodates these phenomena. Based on this notion, we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state. In practice, such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system.}, keywords = {DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {article} } The problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential–algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appropriate notion of observability is presented which accommodates these phenomena. Based on this notion, we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state. In practice, such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system. |

## 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 |

Trenn, Stephan Distributional averaging of switched DAEs with two modes Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 3616–3620, 2015. Abstract | Links | BibTeX | Tags: averaging, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{Tren15, title = {Distributional averaging of switched DAEs with two modes}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre150812.pdf, Preprint}, doi = {10.1109/CDC.2015.7402779}, year = {2015}, date = {2015-12-04}, booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan}, pages = {3616--3620}, abstract = {The averaging technique is a powerful tool for the analysis and control of switched systems. Recently, classical averaging results were generalized to the class of switched differential algebraic equations (switched DAEs). These results did not consider the possible Dirac impulses in the solutions of switched DAEs and it was believed that the presence of Dirac impulses does not prevent convergence towards an average model and can therefore be neglected. It turns out that the first claim (convergence) is indeed true, but nevertheless the Dirac impulses cannot be neglected, they play an important role for the resulting limit. This note first shows with a simple example how the presence of Dirac impulses effects the convergence towards an averaged model and then a formal proof of convergence in the distributional sense for switched DAEs with two modes is given.}, keywords = {averaging, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } The averaging technique is a powerful tool for the analysis and control of switched systems. Recently, classical averaging results were generalized to the class of switched differential algebraic equations (switched DAEs). These results did not consider the possible Dirac impulses in the solutions of switched DAEs and it was believed that the presence of Dirac impulses does not prevent convergence towards an average model and can therefore be neglected. It turns out that the first claim (convergence) is indeed true, but nevertheless the Dirac impulses cannot be neglected, they play an important role for the resulting limit. This note first shows with a simple example how the presence of Dirac impulses effects the convergence towards an averaged model and then a formal proof of convergence in the distributional sense for switched DAEs with two modes is given. |

## 2013 |

Tanwani, Aneel; Trenn, Stephan An observer for switched differential-algebraic equations based on geometric characterization of observability Inproceedings Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 5981–5986, 2013. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TanwTren13, title = {An observer for switched differential-algebraic equations based on geometric characterization of observability}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT130909.pdf, Preprint}, doi = {10.1109/CDC.2013.6760833}, year = {2013}, date = {2013-12-12}, booktitle = {Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy}, pages = {5981--5986}, abstract = {Based on our previous work dealing with geometric characterization of observability for switched differential-algebraic equations (switched DAEs), we propose an observer design for switched DAEs that generates an asymptotically convergent state estimate. Without assuming the observability of individual modes, the central idea in constructing the observer is to filter out the maximal information from the output of each of the active subsystems and combine it with the previously extracted information to obtain a good estimate of the state after a certain time has passed. In general, observability only holds when impulses in the output are taken into account, hence our observer incorporates the knowledge of impulses in the output. This is a distinguishing feature of our observer design compared to observers for switched ordinary differential equations.}, keywords = {CDC, DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } Based on our previous work dealing with geometric characterization of observability for switched differential-algebraic equations (switched DAEs), we propose an observer design for switched DAEs that generates an asymptotically convergent state estimate. Without assuming the observability of individual modes, the central idea in constructing the observer is to filter out the maximal information from the output of each of the active subsystems and combine it with the previously extracted information to obtain a good estimate of the state after a certain time has passed. In general, observability only holds when impulses in the output are taken into account, hence our observer incorporates the knowledge of impulses in the output. This is a distinguishing feature of our observer design compared to observers for switched ordinary differential equations. |

## 2012 |

Trenn, Stephan; Willems, Jan C Switched behaviors with impulses - a unifying framework Inproceedings Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 3203-3208, 2012, ISSN: 0743-1546. Abstract | Links | BibTeX | Tags: CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TrenWill12, title = {Switched behaviors with impulses - a unifying framework}, author = {Stephan Trenn and Jan C. Willems}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TW120813.pdf, Preprint}, doi = {10.1109/CDC.2012.6426883}, issn = {0743-1546}, year = {2012}, date = {2012-12-13}, booktitle = {Proc. 51st IEEE Conf. Decis. Control, Maui, USA}, pages = {3203-3208}, abstract = {We present a new framework to describe and study switched behaviors. We allow for jumps and impulses in the trajectories induced either implicitly by the dynamics after the switch or explicitly by “impacts”. With some examples from electrical circuit we motivate that the dynamical equations before and after the switch already uniquely define the “dynamics” at the switch, i.e. jumps and impulses. On the other hand, we also allow for external impacts resulting in jumps and impulses not induced by the internal dynamics. As a first theoretical result in this new framework we present a characterization for autonomy of a switched behavior.}, keywords = {CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We present a new framework to describe and study switched behaviors. We allow for jumps and impulses in the trajectories induced either implicitly by the dynamics after the switch or explicitly by “impacts”. With some examples from electrical circuit we motivate that the dynamical equations before and after the switch already uniquely define the “dynamics” at the switch, i.e. jumps and impulses. On the other hand, we also allow for external impacts resulting in jumps and impulses not induced by the internal dynamics. As a first theoretical result in this new framework we present a characterization for autonomy of a switched behavior. |

Trenn, Stephan Switched differential algebraic equations Incollection Vasca, Francesco; Iannelli, Luigi (Ed.): Dynamics and Control of Switched Electronic Systems - Advanced Perspectives for Modeling, Simulation and Control of Power Converters, pp. 189–216, London, 2012. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @incollection{Tren12, title = {Switched differential algebraic equations}, author = {Stephan Trenn}, editor = {Francesco Vasca and Luigi Iannelli}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre110830.pdf, Preprint}, doi = {10.1007/978-1-4471-2885-4_6}, year = {2012}, date = {2012-01-01}, booktitle = {Dynamics and Control of Switched Electronic Systems - Advanced Perspectives for Modeling, Simulation and Control of Power Converters}, pages = {189--216}, address = {London}, chapter = {6}, abstract = {In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form $Ex'=Ax+Bu$ where $E$ is, in general, a singular matrix and $u$ is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments. However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the consistency projectors. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. In order to capture this impulsive behavior the space of piecewise-smooth distributions is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs (including the uniqueness of the jumps induced by the switches). With the help of the consistency projector a condition is formulated whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role.}, keywords = {DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {incollection} } In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form $Ex'=Ax+Bu$ where $E$ is, in general, a singular matrix and $u$ is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments. However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the consistency projectors. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. In order to capture this impulsive behavior the space of piecewise-smooth distributions is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs (including the uniqueness of the jumps induced by the switches). With the help of the consistency projector a condition is formulated whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role. |

## 2010 |

Domínguez-García, Alejandro D; Trenn, Stephan Detection of impulsive effects in switched DAEs with applications to power electronics reliability analysis Inproceedings Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5662–5667, 2010. Abstract | Links | BibTeX | Tags: application, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{DomiTren10, title = {Detection of impulsive effects in switched DAEs with applications to power electronics reliability analysis}, author = {Alejandro D. Domínguez-García and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-DT100810.pdf, Preprint}, doi = {10.1109/CDC.2010.5717011}, year = {2010}, date = {2010-12-17}, booktitle = {Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA}, pages = {5662--5667}, abstract = {This paper presents an analytical framework for detecting the presence of jumps and impulses in the solutions of switched differential algebraic equations (switched DAEs). The framework can be applied in the early design stage of fault-tolerant power electronics systems to identify design flaws that could jeopardize its reliability. The system is described by a switched differential algebraic equation, accounting for both fault-free system configurations and the configurations that arise after component faults, where each configuration p is defined by a pair of matrices (Ep;Ap). For each configuration p, the so called consistency projector is obtained from the pair (Ep;Ap). Based on the consistency projectors of all possible configurations, conditions for impulse-free and jump-free solutions of the switched DAE are established. A case-study of a dual redundant buck converter is presented to illustrate the framework.}, keywords = {application, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } This paper presents an analytical framework for detecting the presence of jumps and impulses in the solutions of switched differential algebraic equations (switched DAEs). The framework can be applied in the early design stage of fault-tolerant power electronics systems to identify design flaws that could jeopardize its reliability. The system is described by a switched differential algebraic equation, accounting for both fault-free system configurations and the configurations that arise after component faults, where each configuration p is defined by a pair of matrices (Ep;Ap). For each configuration p, the so called consistency projector is obtained from the pair (Ep;Ap). Based on the consistency projectors of all possible configurations, conditions for impulse-free and jump-free solutions of the switched DAE are established. A case-study of a dual redundant buck converter is presented to illustrate the framework. |

Tanwani, Aneel; Trenn, Stephan On observability of switched differential-algebraic equations Inproceedings Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5656–5661, 2010. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TanwTren10, title = {On observability of switched differential-algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT100821.pdf, Preprint}, doi = {10.1109/CDC.2010.5717685}, year = {2010}, date = {2010-12-16}, booktitle = {Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA}, pages = {5656--5661}, abstract = {We investigate observability of switched differential algebraic equations. The article primarily focuses on a class of switched systems comprising of two modes and a switching signal with a single switching instant. We provide a necessary and sufficient condition under which it is possible to recover the value of state trajectory (globally in time) with the help of switching phenomenon, even though the constituent subsystems may not be observable. In case the switched system is not globally observable, we discuss the concept of forward observability which deals with the recovery of state trajectory after the switching. A necessary and sufficient condition that characterizes forward observability is presented.}, keywords = {CDC, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We investigate observability of switched differential algebraic equations. The article primarily focuses on a class of switched systems comprising of two modes and a switching signal with a single switching instant. We provide a necessary and sufficient condition under which it is possible to recover the value of state trajectory (globally in time) with the help of switching phenomenon, even though the constituent subsystems may not be observable. In case the switched system is not globally observable, we discuss the concept of forward observability which deals with the recovery of state trajectory after the switching. A necessary and sufficient condition that characterizes forward observability is presented. |

## 2009 |

Trenn, Stephan Regularity of distributional differential algebraic equations Journal Article Math. Control Signals Syst., 21 (3), pp. 229–264, 2009. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory @article{Tren09b, title = {Regularity of distributional differential algebraic equations}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre090806.pdf, Preprint}, doi = {10.1007/s00498-009-0045-4}, year = {2009}, date = {2009-12-01}, journal = {Math. Control Signals Syst.}, volume = {21}, number = {3}, pages = {229--264}, abstract = {Time-varying differential algebraic equations (DAEs) of the form E x' = A x + f are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.}, keywords = {DAEs, piecewise-smooth-distributions, solution-theory}, pubstate = {published}, tppubtype = {article} } Time-varying differential algebraic equations (DAEs) of the form E x' = A x + f are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given. |

Trenn, Stephan Distributional differential algebraic equations PhD Thesis Institut für Mathematik, Technische Universität Ilmenau, 2009. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @phdthesis{Tren09d, title = {Distributional differential algebraic equations}, author = {Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Diss090804.pdf, Download https://stephantrenn.net/wp-content/uploads/2017/09/Cover_Diss.jpg, Book Cover http://www.db-thueringen.de/servlets/DocumentServlet?id=13581, Publication-Website}, year = {2009}, date = {2009-01-01}, address = {Universitätsverlag Ilmenau, Germany}, school = {Institut für Mathematik, Technische Universität Ilmenau}, abstract = {Linear implicit differential equations of the form Ex'=Ax+f are studied. If the matrix E is not invertible, these equations contain differential as well as algebraic equations. Hence Ex'=Ax+f is called differential algebraic equation (DAE). A main goal of this dissertation is the consideration of certain distributions (or generalized functions) as solutions and studying time-varying DAEs, whose coefficient matrices have jumps. Therefore, a suitable solution space is derived. This solution space allows to study the important class of switched DAEs. The space of piecewise-smooth distributions is introduced as the solution space. For this space of distributions, it is possible to define a multiplication, hence DAEs can be studied whose coefficient matrices have also distributional entries. A distributional DAE is an equation of the form Ex'=Ax+f where the matrices E and A contain piecewise-smooth distributions as entries and the solutions x as well as the inhomogeneities f are also piecewise-smooth distributions. For distributional DAEs, existence and uniqueness of solutions are studied, therefore, the concept of regularity for distributional DAEs is introduced. Necessary and sufficient conditions for existence and uniqueness of solutions are derived. As special cases, the equations x'=Ax+f (distributional ODEs) and Nx'=x+f (pure distributional DAE) are studied and explicit solution formulae are given. Switched DAEs are distributional DAEs with piecewise constant coefficient matrices. Sufficient conditions are given which ensure that all solutions of a switched DAE are impulse free. Furthermore, it is studied which conditions ensure that arbitrary switching between stable subsystems yield a stable overall system. Finally, controllability and observability for distributional DAEs are studied. For this, it is accounted for the fact that input signals can contain impulses, hence an ``instantaneous'' control is theoretically possible. For a DAE of the form Nx'=x+bu}, keywords = {DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {phdthesis} } Linear implicit differential equations of the form Ex'=Ax+f are studied. If the matrix E is not invertible, these equations contain differential as well as algebraic equations. Hence Ex'=Ax+f is called differential algebraic equation (DAE). A main goal of this dissertation is the consideration of certain distributions (or generalized functions) as solutions and studying time-varying DAEs, whose coefficient matrices have jumps. Therefore, a suitable solution space is derived. This solution space allows to study the important class of switched DAEs. The space of piecewise-smooth distributions is introduced as the solution space. For this space of distributions, it is possible to define a multiplication, hence DAEs can be studied whose coefficient matrices have also distributional entries. A distributional DAE is an equation of the form Ex'=Ax+f where the matrices E and A contain piecewise-smooth distributions as entries and the solutions x as well as the inhomogeneities f are also piecewise-smooth distributions. For distributional DAEs, existence and uniqueness of solutions are studied, therefore, the concept of regularity for distributional DAEs is introduced. Necessary and sufficient conditions for existence and uniqueness of solutions are derived. As special cases, the equations x'=Ax+f (distributional ODEs) and Nx'=x+f (pure distributional DAE) are studied and explicit solution formulae are given. Switched DAEs are distributional DAEs with piecewise constant coefficient matrices. Sufficient conditions are given which ensure that all solutions of a switched DAE are impulse free. Furthermore, it is studied which conditions ensure that arbitrary switching between stable subsystems yield a stable overall system. Finally, controllability and observability for distributional DAEs are studied. For this, it is accounted for the fact that input signals can contain impulses, hence an ``instantaneous'' control is theoretically possible. For a DAE of the form Nx'=x+bu |

## 2008 |

Trenn, Stephan Distributional solution theory for linear DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 10077–10080, WILEY-VCH Verlag, 2008, ISSN: 1617--7061. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory @inproceedings{Tren08b, title = {Distributional solution theory for linear DAEs}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre080424.pdf, Preprint}, doi = {10.1002/pamm.200810077}, issn = {1617--7061}, year = {2008}, date = {2008-05-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {8}, number = {1}, pages = {10077--10080}, publisher = {WILEY-VCH Verlag}, abstract = {A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given.}, keywords = {DAEs, piecewise-smooth-distributions, solution-theory}, pubstate = {published}, tppubtype = {inproceedings} } A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. |