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 |

Sutrisno, ; Trenn, Stephan Observability of Switched Linear Singular System in Discrete Time: Single Switch Case Inproceedings Proc. European Control Conference (ECC21), Rotterdam, Netherlands, 2021, (to appear). Abstract | Links | BibTeX | Tags: observability, switched-DAEs, switched-systems @inproceedings{SutrTren21, title = {Observability of Switched Linear Singular System in Discrete Time: Single Switch Case}, author = {Sutrisno and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-ST210406.pdf, Preprint}, year = {2021}, date = {2021-06-29}, booktitle = {Proc. European Control Conference (ECC21)}, address = {Rotterdam, Netherlands}, abstract = {In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is inde- pendent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability.}, note = {to appear}, keywords = {observability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is inde- pendent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability. |

Hossain, Sumon; Trenn, Stephan Minimal realization for linear switched systems with a single switch Inproceedings 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. |

## 2019 |

Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks Journal Article Automatica, 101 , pp. 280-289, 2019. Abstract | Links | BibTeX | Tags: DAEs, networks, observability @article{PatiTesi19, title = {Indiscernible topological variations in DAE networks}, author = {Deepak Patil and Pietro Tesi and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2019/01/Preprint-PTT181205.pdf, Preprint}, doi = {10.1016/j.automatica.2018.12.012}, year = {2019}, date = {2019-03-01}, journal = {Automatica}, volume = {101}, pages = {280-289}, abstract = {A problem of characterizing conditions under which a topological change in a network of differential algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogenous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network.}, keywords = {DAEs, networks, observability}, pubstate = {published}, tppubtype = {article} } A problem of characterizing conditions under which a topological change in a network of differential algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogenous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network. |

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 |

Küsters, Ferdinand; Trenn, Stephan Switch observability for switched linear systems Journal Article Automatica, 87 , pp. 121-127, 2018. Abstract | Links | BibTeX | Tags: observability, switched-systems @article{KustTren18, title = {Switch observability for switched linear systems}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/10/Preprint-KT170808.pdf, Preprint}, doi = {https://doi.org/10.1016/j.automatica.2017.09.024}, year = {2018}, date = {2018-01-01}, journal = {Automatica}, volume = {87}, pages = {121-127}, abstract = {Mode observability of switched systems requires observability of each individual mode. We consider other concepts of observability that do not have this requirement: Switching time observability and switch observability. The latter notion is based on the assumption that at least one switch occurs. These concepts are analyzed and characterized both for homogeneous and inhomogeneous systems.}, keywords = {observability, switched-systems}, pubstate = {published}, tppubtype = {article} } Mode observability of switched systems requires observability of each individual mode. We consider other concepts of observability that do not have this requirement: Switching time observability and switch observability. The latter notion is based on the assumption that at least one switch occurs. These concepts are analyzed and characterized both for homogeneous and inhomogeneous systems. |

## 2017 |

Küsters, Ferdinand; Patil, Deepak; Trenn, Stephan Switch observability for a class of inhomogeneous switched DAEs Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 3175 - 3180, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, switched-DAEs, switched-systems @inproceedings{KustPati17b, title = {Switch observability for a class of inhomogeneous switched DAEs}, author = {Ferdinand Küsters and Deepak Patil and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KPT170919.pdf, Preprint}, doi = {10.1109/CDC.2017.8264124}, year = {2017}, date = {2017-12-13}, booktitle = {Proc. 56th IEEE Conf. Decis. Control}, pages = {3175 - 3180}, address = {Melbourne, Australia}, abstract = {Necessary and sufficient conditions for switching time and switch observability of a class of inhomogeneous switched differential algebraic equations (DAEs) are obtained. A characterization of initial states and inputs for which switched DAEs are switch unobservable is also provided by using the zeros of an augmented system obtained by combining the output of two modes suitably.}, keywords = {CDC, DAEs, observability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } Necessary and sufficient conditions for switching time and switch observability of a class of inhomogeneous switched differential algebraic equations (DAEs) are obtained. A characterization of initial states and inputs for which switched DAEs are switch unobservable is also provided by using the zeros of an augmented system obtained by combining the output of two modes suitably. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch-observer for switched linear systems Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 1749 - 1754, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: CDC, observability, observer, switched-systems @inproceedings{KustTren17b, title = {Switch-observer for switched linear systems}, author = {Ferdinand Küsters and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KTW170901.pdf, Preprint}, doi = {10.1109/CDC.2017.8263903}, year = {2017}, date = {2017-12-12}, booktitle = {Proc. 56th IEEE Conf. Decis. Control}, pages = {1749 - 1754}, address = {Melbourne, Australia}, abstract = {To determine the switching signal and the state of a switched linear system, one usually requires mode observability. This requires that all individual modes are observable and that the modes are distinguishable. In theory, it allows to determine the active mode in an arbitrarily short time. If one enlarges the observation to an interval that contains a switch, both assumptions (observability of each mode and clearly distinct dynamics) can be relaxed. In [Küsters and Trenn 2017] this concept, called switch observability, was formalized. It is of particular interest for fault identification. Based on switch observability, we propose an observer. This observer combines the information obtained before and after a switching instant to determine both the state and the switching signal. It is analyzed and illustrated in an example.}, keywords = {CDC, observability, observer, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } To determine the switching signal and the state of a switched linear system, one usually requires mode observability. This requires that all individual modes are observable and that the modes are distinguishable. In theory, it allows to determine the active mode in an arbitrarily short time. If one enlarges the observation to an interval that contains a switch, both assumptions (observability of each mode and clearly distinct dynamics) can be relaxed. In [Küsters and Trenn 2017] this concept, called switch observability, was formalized. It is of particular interest for fault identification. Based on switch observability, we propose an observer. This observer combines the information obtained before and after a switching instant to determine both the state and the switching signal. It is analyzed and illustrated in an example. |

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. |

Küsters, Ferdinand; Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks with applications to power grids Inproceedings Proc. 20th IFAC World Congress 2017, pp. 7333 - 7338, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: application, DAEs, networks, observability @inproceedings{KustPati17a, title = {Indiscernible topological variations in DAE networks with applications to power grids}, author = {Ferdinand Küsters and Deepak Patil and Pietro Tesi and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KPTT170320.pdf, Preprint}, doi = {10.1016/j.ifacol.2017.08.1478}, issn = {2405-8963}, year = {2017}, date = {2017-03-24}, booktitle = {Proc. 20th IFAC World Congress 2017}, journal = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {7333 - 7338}, address = {Toulouse, France}, abstract = {The ability to detect topology variations in dynamical networks defined by differential algebraic equations (DAEs) is considered. We characterize the existence of initial states, for which topological changes are indiscernible. A key feature of our characterization is the ability to verify indiscernibility just in terms of the nominal topology. We apply the results to a power grid model and also discuss the relationship to recent mode-detection results for switched DAEs.}, keywords = {application, DAEs, networks, observability}, pubstate = {published}, tppubtype = {inproceedings} } The ability to detect topology variations in dynamical networks defined by differential algebraic equations (DAEs) is considered. We characterize the existence of initial states, for which topological changes are indiscernible. A key feature of our characterization is the ability to verify indiscernibility just in terms of the nominal topology. We apply the results to a power grid model and also discuss the relationship to recent mode-detection results for switched DAEs. |

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. |

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. Abstract | Links | BibTeX | Tags: DAEs, observability, survey @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 = {DAEs, observability, survey}, 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. |

## 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. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Observer design based on constant-input observability for DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 813–814, WILEY-VCH Verlag, 2016, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: DAEs, observability, observer @inproceedings{KustTren16b, title = {Observer design based on constant-input observability for DAEs}, author = {Ferdinand Küsters and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KTW160511.pdf, Preprint}, doi = {10.1002/pamm.201610395}, issn = {1617-7061}, year = {2016}, date = {2016-01-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {16}, number = {1}, pages = {813--814}, publisher = {WILEY-VCH Verlag}, abstract = {For differential-algebraic equations (DAEs) an observability notion is considered which assumes the input to be unknown and constant. Based on this, an observer design is proposed.}, keywords = {DAEs, observability, observer}, pubstate = {published}, tppubtype = {inproceedings} } For differential-algebraic equations (DAEs) an observability notion is considered which assumes the input to be unknown and constant. Based on this, an observer design is proposed. |

## 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. |

Tanwani, Aneel; Trenn, Stephan On detectability of switched linear differential-algebraic equations Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2957–2962, 2015. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, stability, switched-DAEs, switched-systems @inproceedings{TanwTren15, title = {On detectability of switched linear differential-algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT150904.pdf, Preprint}, doi = {10.1109/CDC.2015.7402666}, year = {2015}, date = {2015-12-03}, booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan}, pages = {2957--2962}, abstract = {This paper addresses the notion of detectability for continuous-time switched systems comprising linear differential-algebraic equations (DAEs). It relates to studying asymptotic stability of the set of state trajectories corresponding to zero input and zero output, with a fixed switching signal. Due to the nature of solutions of switched DAEs, the problem reduces to analyzing stability of the trajectories emanating from a non-vanishing unobservable subspace, for which we first derive a geometric expression. The stability of state trajectories starting from that subspace can then be checked in two possible ways. In the first case, detectability of switched DAE is shown to be equivalent to the asymptotic stability of a reduced order discrete-time switched system. In the second approach, the solutions from a non-vanishing unobservable subspace are mapped to the solutions of a reduced order continuous system with time-varying switching ordinary differential equations (ODEs). As a special case of the later approach, the reduced order switched system is time-invariant if the unobservable subspace is invariant for all subsystems}, keywords = {CDC, DAEs, observability, stability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } This paper addresses the notion of detectability for continuous-time switched systems comprising linear differential-algebraic equations (DAEs). It relates to studying asymptotic stability of the set of state trajectories corresponding to zero input and zero output, with a fixed switching signal. Due to the nature of solutions of switched DAEs, the problem reduces to analyzing stability of the trajectories emanating from a non-vanishing unobservable subspace, for which we first derive a geometric expression. The stability of state trajectories starting from that subspace can then be checked in two possible ways. In the first case, detectability of switched DAE is shown to be equivalent to the asymptotic stability of a reduced order discrete-time switched system. In the second approach, the solutions from a non-vanishing unobservable subspace are mapped to the solutions of a reduced order continuous system with time-varying switching ordinary differential equations (ODEs). As a special case of the later approach, the reduced order switched system is time-invariant if the unobservable subspace is invariant for all subsystems |

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. Abstract | Links | BibTeX | Tags: observability, switched-DAEs, switched-systems @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 = {observability, switched-DAEs, switched-systems}, 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. |

## 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 |

Tanwani, Aneel; Trenn, Stephan Observability of switched differential-algebraic equations for general switching signals Inproceedings Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 2648–2653, 2012. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, switched-DAEs, switched-systems @inproceedings{TanwTren12, title = {Observability of switched differential-algebraic equations for general switching signals}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT120822.pdf, Preprint}, doi = {10.1109/CDC.2012.6427087}, year = {2012}, date = {2012-12-11}, booktitle = {Proc. 51st IEEE Conf. Decis. Control, Maui, USA}, pages = {2648--2653}, abstract = {We study observability of switched differential-algebraic equations (DAEs) for arbitrary switching. We present a characterization of observability and a related property called determinability. These characterizations utilize the results for the single-switch case recently obtained by the authors. Furthermore, we study observability conditions when only the mode sequence of the switching signal (and not the switching times) are known. This leads to necessary and sufficient conditions for observability and determinability. We illustrate the results with illustrative examples.}, keywords = {CDC, DAEs, observability, switched-DAEs, switched-systems}, pubstate = {published}, tppubtype = {inproceedings} } We study observability of switched differential-algebraic equations (DAEs) for arbitrary switching. We present a characterization of observability and a related property called determinability. These characterizations utilize the results for the single-switch case recently obtained by the authors. Furthermore, we study observability conditions when only the mode sequence of the switching signal (and not the switching times) are known. This leads to necessary and sufficient conditions for observability and determinability. We illustrate the results with illustrative examples. |

## 2010 |

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 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. |