6. | Berger, Thomas; Reis, Timo; Trenn, Stephan Observability of linear differential-algebraic systems: A survey Incollection Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations IV, pp. 161–219, Springer-Verlag, Berlin-Heidelberg, 2017. @incollection{BergReis17, title = {Observability of linear differential-algebraic systems: A survey}, author = {Thomas Berger and Timo Reis and Stephan Trenn}, editor = {Achim Ilchmann and Timo Reis}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-BRT150721.pdf, Preprint}, doi = {10.1007/978-3-319-46618-7_4}, year = {2017}, date = {2017-01-01}, booktitle = {Surveys in Differential-Algebraic Equations IV}, pages = {161--219}, publisher = {Springer-Verlag}, address = {Berlin-Heidelberg}, series = {Differential-Algebraic Equations Forum}, abstract = {We investigate different concepts related to observability of linear constant coefficient differential-algebraic equations. Regularity, which, loosely speaking, guarantees existence and uniqueness of solutions for any inhomogeneity, is not required in this article. Concepts like impulse observability, observability at infinity, behavioral observability, strong and complete observability are described and defined in the time-domain. Special emphasis is placed on a normal form under output injection, state space and output space transformation. This normal form together with duality is exploited to derive Hautus type criteria for observability. We also discuss geometric criteria, Kalman decompositions and detectability. Some new results on stabilization by output injection are proved.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We investigate different concepts related to observability of linear constant coefficient differential-algebraic equations. Regularity, which, loosely speaking, guarantees existence and uniqueness of solutions for any inhomogeneity, is not required in this article. Concepts like impulse observability, observability at infinity, behavioral observability, strong and complete observability are described and defined in the time-domain. Special emphasis is placed on a normal form under output injection, state space and output space transformation. This normal form together with duality is exploited to derive Hautus type criteria for observability. We also discuss geometric criteria, Kalman decompositions and detectability. Some new results on stabilization by output injection are proved. |

5. | Petreczky, Mihály; Tanwani, Aneel; Trenn, Stephan Observability of switched linear systems Incollection Djemai, Mohamed; Defoort, Michael (Ed.): Hybrid Dynamical Systems, 457 , pp. 205–240, 2015. @incollection{PetrTanw15, title = {Observability of switched linear systems}, author = {Mihály Petreczky and Aneel Tanwani and Stephan Trenn}, editor = {Mohamed Djemai and Michael Defoort}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-PTT140211.pdf, Preprint}, doi = {10.1007/978-3-319-10795-0_8}, year = {2015}, date = {2015-01-01}, booktitle = {Hybrid Dynamical Systems}, volume = {457}, pages = {205--240}, abstract = {Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. We will present these different approaches, with a focus on the recently introduced notion of large-time observability in the context of switched linear systems and its geometric characterization. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times, and it is shown that the proposed conditions have a denseness property with respect to switching times. To relieve the computation burden, some relaxed conditions that do not rely on the switching times are given; this allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential-algebraic systems is discussed. Detailed examples are included to illustrate the geometric conditions and to better understand the theoretical development.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. We will present these different approaches, with a focus on the recently introduced notion of large-time observability in the context of switched linear systems and its geometric characterization. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times, and it is shown that the proposed conditions have a denseness property with respect to switching times. To relieve the computation burden, some relaxed conditions that do not rely on the switching times are given; this allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential-algebraic systems is discussed. Detailed examples are included to illustrate the geometric conditions and to better understand the theoretical development. |

4. | Trenn, Stephan Stability of switched DAEs Incollection Daafouz, Jamal; Tarbouriech, Sophie; Sigalotti, Mario (Ed.): Hybrid Systems with Constraints, pp. 57–83, London, 2013. @incollection{Tren13b, title = {Stability of switched DAEs}, author = {Stephan Trenn}, editor = {Jamal Daafouz and Sophie Tarbouriech and Mario Sigalotti}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre130116.pdf, Preprint}, doi = {10.1002/9781118639856.ch3}, year = {2013}, date = {2013-04-01}, booktitle = {Hybrid Systems with Constraints}, pages = {57--83}, address = {London}, chapter = {3}, series = {Automation - Control and Industrial Engineering Series}, abstract = {Differential algebraic equations (DAEs) are used to model dynamical systems with constraints given by algebraic equations. In the presence of sudden structural changes (e.g. switching or faults) this leads to a switched DAE. A special feature of switched DAEs is the presence of induced jumps or even Dirac impulses in the solution. This chapter studies stability of switched DAEs taking into account the presence of these jumps and impulses. For a rigorous mathematical treatment it is first necessary to introduce a suitable solution space - the space of piecewise-smooth distributions. Within this distributional solution space the notion of stability encompasses impulse-freeness which is studied first. Afterwards stability under arbitrary and slow switching is investigated. A generalization to switched DAEs of a classical result concerning stability and commutativity is presented as well as a converse Lyapunov theorem. The theoretical results are illustrated with intuitive examples.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Differential algebraic equations (DAEs) are used to model dynamical systems with constraints given by algebraic equations. In the presence of sudden structural changes (e.g. switching or faults) this leads to a switched DAE. A special feature of switched DAEs is the presence of induced jumps or even Dirac impulses in the solution. This chapter studies stability of switched DAEs taking into account the presence of these jumps and impulses. For a rigorous mathematical treatment it is first necessary to introduce a suitable solution space - the space of piecewise-smooth distributions. Within this distributional solution space the notion of stability encompasses impulse-freeness which is studied first. Afterwards stability under arbitrary and slow switching is investigated. A generalization to switched DAEs of a classical result concerning stability and commutativity is presented as well as a converse Lyapunov theorem. The theoretical results are illustrated with intuitive examples. |

3. | Trenn, Stephan Solution concepts for linear DAEs: a survey Incollection Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations I, pp. 137–172, springer, Berlin-Heidelberg, 2013. @incollection{Tren13a, title = {Solution concepts for linear DAEs: a survey}, author = {Stephan Trenn}, editor = {Achim Ilchmann and Timo Reis}, url = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre121015.pdf, Preprint}, doi = {10.1007/978-3-642-34928-7_4}, year = {2013}, date = {2013-01-01}, booktitle = {Surveys in Differential-Algebraic Equations I}, pages = {137--172}, publisher = {springer}, address = {Berlin-Heidelberg}, series = {Differential-Algebraic Equations Forum}, abstract = {This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks. |

2. | 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, Springer, London, 2012. @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}, publisher = {Springer}, 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 = {}, 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. |

1. | Trenn, Stephan Distributional differential algebraic equations PhD Thesis Institut für Mathematik, Technische Universität Ilmenau, 2009. @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 = {}, 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 |

# Books and Book Chapters

6. | Observability of linear differential-algebraic systems: A survey Incollection Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations IV, pp. 161–219, Springer-Verlag, Berlin-Heidelberg, 2017. |

5. | Observability of switched linear systems Incollection Djemai, Mohamed; Defoort, Michael (Ed.): Hybrid Dynamical Systems, 457 , pp. 205–240, 2015. |

4. | Stability of switched DAEs Incollection Daafouz, Jamal; Tarbouriech, Sophie; Sigalotti, Mario (Ed.): Hybrid Systems with Constraints, pp. 57–83, London, 2013. |

3. | Solution concepts for linear DAEs: a survey Incollection Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations I, pp. 137–172, springer, Berlin-Heidelberg, 2013. |

2. | 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, Springer, London, 2012. |

1. | Distributional differential algebraic equations PhD Thesis Institut für Mathematik, Technische Universität Ilmenau, 2009. |