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.

## 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; Trenn, Stephan; Wirsen, Andreas Switch-observer for switched linear systems Inproceedings Proc. 56th IEEE Conf. Decision Control, 2017, (to appear). 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}, year = {2017}, date = {2017-12-15}, booktitle = {Proc. 56th IEEE Conf. Decision Control}, 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.}, note = {to appear}, 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; Patil, Deepak; Trenn, Stephan Switch observability for a class of inhomogeneous switched DAEs Inproceedings Proc. 56th IEEE Conf. Decision Control, 2017, (to appear). 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}, year = {2017}, date = {2017-12-14}, booktitle = {Proc. 56th IEEE Conf. Decision Control}, 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.}, note = {to appear}, 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. |

Kausar, Rukhsana; Trenn, Stephan Impulses in structured nonlinear switched DAEs Inproceedings Proc. 56th IEEE Conf. Decision Control, 2017, (to appear). 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}, year = {2017}, date = {2017-12-13}, booktitle = {Proc. 56th IEEE Conf. Decision Control}, 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.}, note = {to appear}, 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. |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions Inproceedings Proc. 56th IEEE Conf. Decision Control, 2017, (to appear). Abstract | Links | BibTeX | Tags: CDC, stability, switched systems @inproceedings{IervTren17, title = {Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions}, author = {Raffaele Iervolino and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-ITV170909.pdf, Preprint}, year = {2017}, date = {2017-12-12}, booktitle = {Proc. 56th IEEE Conf. Decision Control}, abstract = {State-dependent switched systems characterized by piecewise affine (PWA) dynamics in a polyhedral partition of the state space are considered. Sufficient conditions on the vectors fields such that the solution crosses the common boundaries of the polyhedra are expressed in terms of quadratic inequalities constrained to the polyhedra intersections. A piece- wise quadratic (PWQ) function, not necessarily continuous, is proposed as a candidate Lyapunov function (LF). The sign conditions and the negative jumps at the boundaries are expressed in terms of linear matrix inequalities (LMIs) via cone- copositivity. A sufficient condition for the asymptotic stability of the PWA system is then obtained by finding a PWQ-LF through the solution of a set LMIs. Numerical results with a conewise linear system and an opinion dynamics model show the effectiveness of the proposed approach.}, note = {to appear}, keywords = {CDC, stability, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } State-dependent switched systems characterized by piecewise affine (PWA) dynamics in a polyhedral partition of the state space are considered. Sufficient conditions on the vectors fields such that the solution crosses the common boundaries of the polyhedra are expressed in terms of quadratic inequalities constrained to the polyhedra intersections. A piece- wise quadratic (PWQ) function, not necessarily continuous, is proposed as a candidate Lyapunov function (LF). The sign conditions and the negative jumps at the boundaries are expressed in terms of linear matrix inequalities (LMIs) via cone- copositivity. A sufficient condition for the asymptotic stability of the PWA system is then obtained by finding a PWQ-LF through the solution of a set LMIs. Numerical results with a conewise linear system and an opinion dynamics model show the effectiveness of the proposed approach. |

Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks Unpublished 2017. Abstract | Links | BibTeX | Tags: DAEs, networks, observability, submitted @unpublished{PatiTesi17pp, title = {Indiscernible topological variations in DAE networks}, author = {Deepak Patil and Pietro Tesi and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/11/Preprint-PTT171123.pdf, Preprint}, year = {2017}, date = {2017-11-23}, 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 conditions simplifies to 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, submitted}, pubstate = {published}, tppubtype = {unpublished} } 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 conditions simplifies to 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 Unpublished 2017. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth distributions, submitted, switched DAEs, switched systems @unpublished{TanwTren17pp, title = {Detectability and Observer Design for Switched Differential Algebraic Equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/11/Preprint-TT171116.pdf, Preprint}, year = {2017}, date = {2017-11-16}, 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, submitted, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {unpublished} } 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. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched DAEs: convergence, partial averaging and stability Journal Article Automatica, 82 , pp. 145–157, 2017. Abstract | Links | BibTeX | Tags: averaging, DAEs, stability, switched DAEs, switched systems @article{MostTren17, title = {Averaging for switched DAEs: convergence, partial averaging and stability}, author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MTV170407.pdf, Preprint}, doi = {10.1016/j.automatica.2017.04.036}, year = {2017}, date = {2017-08-01}, journal = {Automatica}, volume = {82}, pages = {145--157}, abstract = {Averaging is a useful technique to simplify the analysis of switched systems. In this paper we present averaging results for the class of systems described by switched differential algebraic equations (DAEs). Conditions on the consistency projectors are given which guarantee convergence towards a non-switched averaged system. A consequence of this result is the possibility to stabilize switched DAEs via fast switching. We also study partial averaging in case the consistency projectors do not satisfy the conditions for convergence; the averaged system is then still a switched system, but is simpler than the original. The practical interest of the theoretical averaging results is demonstrated through the analysis of the dynamics of a switched electrical circuit.}, keywords = {averaging, DAEs, stability, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {article} } Averaging is a useful technique to simplify the analysis of switched systems. In this paper we present averaging results for the class of systems described by switched differential algebraic equations (DAEs). Conditions on the consistency projectors are given which guarantee convergence towards a non-switched averaged system. A consequence of this result is the possibility to stabilize switched DAEs via fast switching. We also study partial averaging in case the consistency projectors do not satisfy the conditions for convergence; the averaged system is then still a switched system, but is simpler than the original. The practical interest of the theoretical averaging results is demonstrated through the analysis of the dynamics of a switched electrical circuit. |

Trenn, Stephan Edge-wise funnel synchronization Inproceedings PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2017, ISSN: 1617-7061, (to appear). Abstract | Links | BibTeX | Tags: funnel control, networks, nonlinear @inproceedings{Tren17, title = {Edge-wise funnel synchronization}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre170523.pdf, Preprint}, issn = {1617-7061}, year = {2017}, date = {2017-06-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {17}, number = {1}, publisher = {WILEY-VCH Verlag}, abstract = {Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory.}, note = {to appear}, keywords = {funnel control, networks, nonlinear}, pubstate = {published}, tppubtype = {inproceedings} } Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory. |

Kausar, Rukhsana; Trenn, Stephan Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs Inproceedings Proc. of the XVI International Conference on Hyperbolic Problems 2016, 2017, (to appear). Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, piecewise-smooth distributions, solution theory, switched DAEs, switched systems @inproceedings{KausTren17a, title = {Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs}, author = {Rukhsana Kausar and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT170418.pdf, Preprint}, year = {2017}, date = {2017-04-17}, booktitle = {Proc. of the XVI International Conference on Hyperbolic Problems 2016}, 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 = {to appear}, 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. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch observability for homogeneous switched DAEs Inproceedings IFAC-PapersOnLine, pp. 9355 - 9360, 2017, ISSN: 2405-8963, (Presented at 20th IFAC World Congress 2017 in Toulouse, France.). 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 = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {9355 - 9360}, 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.}, note = {Presented at 20th IFAC World Congress 2017 in Toulouse, France.}, 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 IFAC-PapersOnLine, pp. 7333 - 7338, 2017, ISSN: 2405-8963, (Presented at 20th IFAC World Congress 2017 in Toulouse, France.). 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 = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {7333 - 7338}, 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.}, note = {Presented at 20th IFAC World Congress 2017 in Toulouse, France.}, 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. |

Kall, Jochen; Kausar, Rukhsana; Trenn, Stephan Modeling water hammers via PDEs and switched DAEs with numerical justification Inproceedings IFAC-PapersOnLine, pp. 5349 - 5354, 2017, ISSN: 2405-8963, (Presented at 20th IFAC World Congress 2017 in Toulouse, France.). Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, solution theory, switched DAEs, switched systems @inproceedings{KallKaus17, title = {Modeling water hammers via PDEs and switched DAEs with numerical justification}, author = {Jochen Kall and Rukhsana Kausar and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KKT170324.pdf, Preprint}, doi = {10.1016/j.ifacol.2017.08.927}, issn = {2405-8963}, year = {2017}, date = {2017-03-23}, booktitle = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {5349 - 5354}, abstract = {In water distribution networks instantaneous changes in valve and pump settings may introduces jumps and peaks in the pressure. In particular, a well known phenomenon in response to the sudden closing of a valve is the so called water hammer, which (if not taken into account properly) may destroy parts of the water network. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). After discussing this PDE model we propose a simplified model using switched differential-algebraic equations (DAEs). Switched DAEs are known to be able to produce infinite peaks in response to sudden structural changes. These peaks (in the mathematical form of Dirac impulses) can easily be predicted and may allow for a simpler analysis of complex water networks in the future. As a first step toward that goal, we verify the novel modeling approach by comparing these two modeling techniques numerically for a simple set up consisting of two reservoirs, a pipe and a valve.}, note = {Presented at 20th IFAC World Congress 2017 in Toulouse, France.}, keywords = {application, DAEs, nonlinear, solution theory, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } In water distribution networks instantaneous changes in valve and pump settings may introduces jumps and peaks in the pressure. In particular, a well known phenomenon in response to the sudden closing of a valve is the so called water hammer, which (if not taken into account properly) may destroy parts of the water network. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). After discussing this PDE model we propose a simplified model using switched differential-algebraic equations (DAEs). Switched DAEs are known to be able to produce infinite peaks in response to sudden structural changes. These peaks (in the mathematical form of Dirac impulses) can easily be predicted and may allow for a simpler analysis of complex water networks in the future. As a first step toward that goal, we verify the novel modeling approach by comparing these two modeling techniques numerically for a simple set up consisting of two reservoirs, a pipe and a valve. |

Tanwani, Aneel; Trenn, Stephan Observer Design for Detectable Switched Differential-Algebraic Equations Inproceedings IFAC-PapersOnLine, pp. 2953 - 2958, 2017, ISSN: 2405-8963, (Presented at 20th IFAC World Congress 2017 in Toulouse, France.). 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 = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {2953 - 2958}, 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.}, note = {Presented at 20th IFAC World Congress 2017 in Toulouse, France.}, 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 |

Camlibel, Kanat; Iannelli, Luigi; Tanwani, Aneel; Trenn, Stephan Differential-algebraic inclusions with maximal monotone operators Inproceedings Proc. 55th IEEE Conf. Decis. Control, Las Vegas, USA, pp. 610–615, 2016. Abstract | Links | BibTeX | Tags: CDC, DAEs, nonlinear, solution theory @inproceedings{CamlIann16, title = {Differential-algebraic inclusions with maximal monotone operators}, author = {Kanat Camlibel and Luigi Iannelli and Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-CITT160923.pdf, Preprint}, doi = {10.1109/CDC.2016.7798336}, year = {2016}, date = {2016-12-01}, booktitle = {Proc. 55th IEEE Conf. Decis. Control, Las Vegas, USA}, pages = {610--615}, abstract = {The term differential-algebraic inclusions (DAIs) not only describes the dynamical relations using set-valued mappings, but also includes the static algebraic inclusions, and this paper considers the problem of existence of solutions for a class of such dynamical systems described by the inclusion ddt Px in -M(x) for a symmetric positive semi-definite matrix P in R^(n x n), and a maximal monotone operator M:R^n => R^n. The existence of solutions is proved using the tools from the theory of maximal monotone operators. The class of solutions that we study in the paper have the property that, instead of the whole state, only Px is absolutely continuous and unique. This framework, in particular, is useful for studying passive differential-algebraic equations (DAEs) coupled with maximal monotone relations. Certain class of irregular DAEs are also covered within the proposed general framework. Applications from electrical circuits are included to provide a practical motivation.}, keywords = {CDC, DAEs, nonlinear, solution theory}, pubstate = {published}, tppubtype = {inproceedings} } The term differential-algebraic inclusions (DAIs) not only describes the dynamical relations using set-valued mappings, but also includes the static algebraic inclusions, and this paper considers the problem of existence of solutions for a class of such dynamical systems described by the inclusion ddt Px in -M(x) for a symmetric positive semi-definite matrix P in R^(n x n), and a maximal monotone operator M:R^n => R^n. The existence of solutions is proved using the tools from the theory of maximal monotone operators. The class of solutions that we study in the paper have the property that, instead of the whole state, only Px is absolutely continuous and unique. This framework, in particular, is useful for studying passive differential-algebraic equations (DAEs) coupled with maximal monotone relations. Certain class of irregular DAEs are also covered within the proposed general framework. Applications from electrical circuits are included to provide a practical motivation. |

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Solvability and stability of a power system DAE model Journal Article Syst. Control Lett., 29 , pp. 12–17, 2016. Abstract | Links | BibTeX | Tags: application, DAEs, Lyapunov, networks, solution theory, stability @article{GrosTren16, title = {Solvability and stability of a power system DAE model}, author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-GTW160816.pdf, Preprint}, doi = {10.1016/j.sysconle.2016.08.003}, year = {2016}, date = {2016-11-01}, journal = {Syst. Control Lett.}, volume = {29}, pages = {12--17}, abstract = {The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization for the linearized power system DAE solely in terms of the network topology. As an extension to previous result we allow for higher order generator dynamics. Furthermore, we show that any solvable power system DAE is automatically of index one, which means that it is also numerically well posed. Finally, we show that any solvable power system DAE is stable but not asymptotically stable.}, keywords = {application, DAEs, Lyapunov, networks, solution theory, stability}, pubstate = {published}, tppubtype = {article} } The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization for the linearized power system DAE solely in terms of the network topology. As an extension to previous result we allow for higher order generator dynamics. Furthermore, we show that any solvable power system DAE is automatically of index one, which means that it is also numerically well posed. Finally, we show that any solvable power system DAE is stable but not asymptotically stable. |

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. |

Trenn, Stephan Stabilization of switched DAEs via fast switching Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 827–828, WILEY-VCH Verlag, 2016, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: averaging, DAEs, stability, switched DAEs, switched systems @inproceedings{Tren16, title = {Stabilization of switched DAEs via fast switching}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre160511.pdf, Preprint}, doi = {10.1002/pamm.201610402}, issn = {1617-7061}, year = {2016}, date = {2016-05-12}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {16}, number = {1}, pages = {827--828}, publisher = {WILEY-VCH Verlag}, abstract = {Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching.}, keywords = {averaging, DAEs, stability, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching. |

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. |

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. |

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 |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for non-homogeneous switched DAEs Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2951–2956, 2015. Abstract | Links | BibTeX | Tags: application, averaging, CDC, DAEs, switched DAEs, switched systems @inproceedings{MostTren15b, title = {Averaging for non-homogeneous switched DAEs}, author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MTV150901.pdf, Preprint}, doi = {10.1109/CDC.2015.7402665}, year = {2015}, date = {2015-12-02}, booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan}, pages = {2951--2956}, abstract = {Averaging is widely used for approximating the dynamics of switched systems. The validity of an averaged model typically depends on the switching frequency and on some technicalities regarding the switched system structure. For homogeneous linear switched differential algebraic equations it is known that an averaged model can be obtained. In this paper an averaging result for non-homogeneous switched systems is presented. A switched electrical circuit illustrates the practical interest of the result.}, keywords = {application, averaging, CDC, DAEs, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } Averaging is widely used for approximating the dynamics of switched systems. The validity of an averaged model typically depends on the switching frequency and on some technicalities regarding the switched system structure. For homogeneous linear switched differential algebraic equations it is known that an averaged model can be obtained. In this paper an averaging result for non-homogeneous switched systems is presented. A switched electrical circuit illustrates the practical interest of the result. |

Shim, Hyungbo; Trenn, Stephan A preliminary result on synchronization of heterogeneous agents via funnel control Inproceedings Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2229–2234, 2015. Abstract | Links | BibTeX | Tags: CDC, funnel control, networks, nonlinear, stability @inproceedings{ShimTren15, title = {A preliminary result on synchronization of heterogeneous agents via funnel control}, author = {Hyungbo Shim and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-ST150902.pdf, Preprint}, doi = {10.1109/CDC.2015.7402538}, year = {2015}, date = {2015-12-01}, booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan}, pages = {2229--2234}, abstract = {We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research.}, keywords = {CDC, funnel control, networks, nonlinear, stability}, pubstate = {published}, tppubtype = {inproceedings} } We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research. |

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

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Partial averaging for switched DAEs with two modes Inproceedings Proc. 2015 European Control Conf. (ECC), Linz, Austria, pp. 2896–2901, 2015. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched DAEs, switched systems @inproceedings{MostTren15a, title = {Partial averaging for switched DAEs with two modes}, author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MTV150316.pdf, Preprint}, doi = {10.1109/ECC.2015.7330977}, year = {2015}, date = {2015-03-01}, booktitle = {Proc. 2015 European Control Conf. (ECC), Linz, Austria}, pages = {2896--2901}, abstract = {In this paper an averaging result for switched systems whose modes are represented by means of differential algebraic equations (DAEs) is presented. Homogeneous switched DAEs with periodic switchings between two modes are considered. It is proved that a (switched) averaged system can be defined also in the presence of state jumps whose amplitude does not decrease with the increasing of the switching frequency. A switched capacitor electrical circuit is considered as an illustrative example.}, keywords = {averaging, DAEs, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } In this paper an averaging result for switched systems whose modes are represented by means of differential algebraic equations (DAEs) is presented. Homogeneous switched DAEs with periodic switchings between two modes are considered. It is proved that a (switched) averaged system can be defined also in the presence of state jumps whose amplitude does not decrease with the increasing of the switching frequency. A switched capacitor electrical circuit is considered as an illustrative example. |

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

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. |

## 2014 |

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Topological solvability and index characterizations for a common DAE power system model Inproceedings Proc. 2014 IEEE Conf. Control Applications (CCA), pp. 9–14, IEEE 2014. Abstract | Links | BibTeX | Tags: application, DAEs, networks, nonlinear, solution theory @inproceedings{GrosTren14, title = {Topological solvability and index characterizations for a common DAE power system model}, author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-GTW140904.pdf, Preprint}, doi = {10.1109/CCA.2014.6981321}, year = {2014}, date = {2014-10-10}, booktitle = {Proc. 2014 IEEE Conf. Control Applications (CCA)}, pages = {9--14}, organization = {IEEE}, abstract = {For the widely-used power system model consisting of the generator swing equations and the power flow equations resulting in a system of differential algebraic equations (DAEs), we introduce a sufficient and necessary solvability condition for the linearized model. This condition is based on the topological structure of the power system. Furthermore we show sufficient conditions for the linearized DAE-system and a nonlinear version of the model to have differentiation index equal to one.}, keywords = {application, DAEs, networks, nonlinear, solution theory}, pubstate = {published}, tppubtype = {inproceedings} } For the widely-used power system model consisting of the generator swing equations and the power flow equations resulting in a system of differential algebraic equations (DAEs), we introduce a sufficient and necessary solvability condition for the linearized model. This condition is based on the topological structure of the power system. Furthermore we show sufficient conditions for the linearized DAE-system and a nonlinear version of the model to have differentiation index equal to one. |

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

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

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

## 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. |

Costantini, Giuliano; Trenn, Stephan; Vasca, Francesco Regularity and passivity for jump rules in linear switched systems Inproceedings Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 4030–4035, 2013, ISSN: 0191-2216. Abstract | Links | BibTeX | Tags: CDC, DAEs, solution theory, switched DAEs, switched systems @inproceedings{CostTren13, title = {Regularity and passivity for jump rules in linear switched systems}, author = {Giuliano Costantini and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-CTV130906.pdf, Preprint}, doi = {10.1109/CDC.2013.6760506}, issn = {0191-2216}, year = {2013}, date = {2013-12-11}, booktitle = {Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy}, pages = {4030--4035}, abstract = {A wide class of linear switched systems (LSS) can be represented by a sequence of modes each one described by a set of differential algebraic equations (DAEs). LSS can exhibit discontinuities in the state evolution, also called jumps, when the state at the end of a mode is not consistent with the DAEs of the successive mode. Then the problem of defining a proper state jump rule arises when an inconsistent initial condition is given. Regularity and passivity conditions provide two conceptually different jump maps respectively. In this paper, after proving some preliminary result on the jump analysis within the regularity framework, it is shown the equivalence of regularity-based and passivity-based jump rules. A switched capacitor electrical circuit is used to numerically confirm the theoretical result.}, keywords = {CDC, DAEs, solution theory, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } A wide class of linear switched systems (LSS) can be represented by a sequence of modes each one described by a set of differential algebraic equations (DAEs). LSS can exhibit discontinuities in the state evolution, also called jumps, when the state at the end of a mode is not consistent with the DAEs of the successive mode. Then the problem of defining a proper state jump rule arises when an inconsistent initial condition is given. Regularity and passivity conditions provide two conceptually different jump maps respectively. In this paper, after proving some preliminary result on the jump analysis within the regularity framework, it is shown the equivalence of regularity-based and passivity-based jump rules. A switched capacitor electrical circuit is used to numerically confirm the theoretical result. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco An averaging result for switched DAEs with multiple modes Inproceedings Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 1378 - 1383, 2013. Abstract | Links | BibTeX | Tags: averaging, CDC, DAEs, switched DAEs, switched systems @inproceedings{IannPedi13b, title = {An averaging result for switched DAEs with multiple modes}, author = {Luigi Iannelli and Carmen Pedicini and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IPTV130911.pdf, Preprint}, doi = {10.1109/CDC.2013.6760075}, year = {2013}, date = {2013-12-10}, booktitle = {Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy}, pages = {1378 - 1383}, abstract = {The major motivation of the averaging technique for switched systems is the construction of a smooth average system whose state trajectory approximates in some sense the state trajectory of the switched system. Averaging of dynamic systems represented by switched ordinary differential equations (ODEs) has been widely analyzed in the literature. The averaging approach can be useful also for the analysis of switched differential algebraic equations (DAEs). Indeed by analyzing the evolution of the switched DAEs state it is possible to conjecture the existence of an average model. However a trivial generalization of the ODE case is not possible due to the presence of state jumps. In this paper we discuss the averaging approach for switched DAEs and an approximation result is derived for homogenous switched linear DAE with periodic switching signals commuting among several modes. This approximation result extends a recent averaging result for switched DAEs with only two modes. Numerical simulations confirm the validity of the averaging approach for switched DAEs.}, keywords = {averaging, CDC, DAEs, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } The major motivation of the averaging technique for switched systems is the construction of a smooth average system whose state trajectory approximates in some sense the state trajectory of the switched system. Averaging of dynamic systems represented by switched ordinary differential equations (ODEs) has been widely analyzed in the literature. The averaging approach can be useful also for the analysis of switched differential algebraic equations (DAEs). Indeed by analyzing the evolution of the switched DAEs state it is possible to conjecture the existence of an average model. However a trivial generalization of the ODE case is not possible due to the presence of state jumps. In this paper we discuss the averaging approach for switched DAEs and an approximation result is derived for homogenous switched linear DAE with periodic switching signals commuting among several modes. This approximation result extends a recent averaging result for switched DAEs with only two modes. Numerical simulations confirm the validity of the averaging approach for switched DAEs. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco Averaging for switched DAEs Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 489–490, WILEY-VCH Verlag, 2013, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched DAEs, switched systems @inproceedings{IannPedi13c, title = {Averaging for switched DAEs}, author = {Luigi Iannelli and Carmen Pedicini and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IPTV130527.pdf, Preprint}, doi = {10.1002/pamm.201310237}, issn = {1617-7061}, year = {2013}, date = {2013-10-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {13}, number = {1}, pages = {489--490}, publisher = {WILEY-VCH Verlag}, abstract = {Switched differential-algebraic equations (switched DAEs) E_sigma(t) x'(t) = A_sigma(t) x(t) are suitable for modeling many practical systems, e.g. electrical circuits. When the switching is periodic and of high frequency, the question arises whether the solutions of switched DAEs can be approximated by an average non-switching system. It is well known that for a quite general class of switched ordinary differential equations (ODEs) this is the case. For switched DAEs, due the presence of the so-called consistency projectors, it is possible that the limit of trajectories for faster and faster switching does not exist. Under certain assumptions on the consistency projectors a result concerning the averaging for switched DAEs is presented.}, keywords = {averaging, DAEs, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } Switched differential-algebraic equations (switched DAEs) E_sigma(t) x'(t) = A_sigma(t) x(t) are suitable for modeling many practical systems, e.g. electrical circuits. When the switching is periodic and of high frequency, the question arises whether the solutions of switched DAEs can be approximated by an average non-switching system. It is well known that for a quite general class of switched ordinary differential equations (ODEs) this is the case. For switched DAEs, due the presence of the so-called consistency projectors, it is possible that the limit of trajectories for faster and faster switching does not exist. Under certain assumptions on the consistency projectors a result concerning the averaging for switched DAEs is presented. |

Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article IEEE Trans. Autom. Control, 58 (12), pp. 3126–3141, 2013. Abstract | Links | BibTeX | Tags: funnel control, input constraints, nonlinear, relative degree @article{LibeTren13b, title = {The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130702.pdf, Preprint}, doi = {10.1109/TAC.2013.2277631}, year = {2013}, date = {2013-08-16}, journal = {IEEE Trans. Autom. Control}, volume = {58}, number = {12}, pages = {3126--3141}, abstract = {The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions.}, keywords = {funnel control, input constraints, nonlinear, relative degree}, pubstate = {published}, tppubtype = {article} } The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco On averaging for switched linear differential algebraic equations Inproceedings Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 2163 – 2168, 2013. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched DAEs, switched systems @inproceedings{IannPedi13a, title = {On averaging for switched linear differential algebraic equations}, author = {Luigi Iannelli and Carmen Pedicini and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IPTV130326.pdf, Preprint http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6669240, IEEE Xplore Article Number 6669240}, year = {2013}, date = {2013-07-02}, booktitle = {Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland}, pages = {2163 -- 2168}, abstract = {Averaging is an effective technique which allows the analysis and control design of nonsmooth switched systems through the use of corresponding simpler smooth averaged systems. Approximation results and stability analysis have been presented in the literature for dynamic systems described by switched ordinary differential equations. In this paper the averaging technique is shown to be useful also for the analysis of switched systems whose modes are represented by means of differential algebraic equations (DAEs). An approximation result is derived for a simple but representative homogenous switched DAE with periodic switching signals and two modes. Simulations based on a simple electric circuit model illustrate the theoretical result.}, keywords = {averaging, DAEs, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } Averaging is an effective technique which allows the analysis and control design of nonsmooth switched systems through the use of corresponding simpler smooth averaged systems. Approximation results and stability analysis have been presented in the literature for dynamic systems described by switched ordinary differential equations. In this paper the averaging technique is shown to be useful also for the analysis of switched systems whose modes are represented by means of differential algebraic equations (DAEs). An approximation result is derived for a simple but representative homogenous switched DAE with periodic switching signals and two modes. Simulations based on a simple electric circuit model illustrate the theoretical result. |

Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller: time delays and case study Inproceedings Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 1669–1674, 2013. Abstract | Links | BibTeX | Tags: application, funnel control, input constraints, nonlinear, relative degree @inproceedings{LibeTren13a, title = {The bang-bang funnel controller: time delays and case study}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130320.pdf, Preprint http://ieeexplore.ieee.org/document/6669120, IEEE Xplore Article Number 6669120}, year = {2013}, date = {2013-07-01}, booktitle = {Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland}, pages = {1669--1674}, abstract = {We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered.}, keywords = {application, funnel control, input constraints, nonlinear, relative degree}, pubstate = {published}, tppubtype = {inproceedings} } We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered. |

Trenn, Stephan Stability of switched DAEs Incollection Daafouz, Jamal; Tarbouriech, Sophie; Sigalotti, Mario (Ed.): Hybrid Systems with Constraints, pp. 57–83, London, 2013. Abstract | Links | BibTeX | Tags: DAEs, stability, switched DAEs, switched systems @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 = {DAEs, stability, switched DAEs, switched systems}, 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. |

Hackl, Christoph M; Hopfe, Norman; Ilchmann, Achim; Mueller, Markus; Trenn, Stephan Funnel control for systems with relative degree two Journal Article SIAM J. Control Optim., 51 (2), pp. 965–995, 2013. Abstract | Links | BibTeX | Tags: application, funnel control, input constraints, nonlinear, relative degree @article{HackHopf13, title = {Funnel control for systems with relative degree two}, author = {Christoph M. Hackl and Norman Hopfe and Achim Ilchmann and Markus Mueller and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/HackHopf13.pdf, Paper}, doi = {10.1137/100799903 }, year = {2013}, date = {2013-03-19}, journal = {SIAM J. Control Optim.}, volume = {51}, number = {2}, pages = {965--995}, abstract = {Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.}, keywords = {application, funnel control, input constraints, nonlinear, relative degree}, pubstate = {published}, tppubtype = {article} } Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. |

Berger, Thomas; Trenn, Stephan Addition to ``The quasi-Kronecker form for matrix pencils'' Journal Article SIAM J. Matrix Anal. & Appl., 34 (1), pp. 94–101, 2013. Abstract | Links | BibTeX | Tags: DAEs, normal forms, solution theory @article{BergTren13, title = {Addition to ``The quasi-Kronecker form for matrix pencils''}, author = {Thomas Berger and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/BergTren13.pdf, Paper}, doi = {10.1137/120883244}, year = {2013}, date = {2013-02-11}, journal = {SIAM J. Matrix Anal. & Appl.}, volume = {34}, number = {1}, pages = {94--101}, abstract = {We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences.}, keywords = {DAEs, normal forms, solution theory}, pubstate = {published}, tppubtype = {article} } We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences. |

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

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

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. |

Berger, Thomas; Trenn, Stephan The quasi-Kronecker form for matrix pencils Journal Article SIAM J. Matrix Anal. & Appl., 33 (2), pp. 336–368, 2012. Abstract | Links | BibTeX | Tags: DAEs, normal forms, solution theory @article{BergTren12, title = {The quasi-Kronecker form for matrix pencils}, author = {Thomas Berger and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/BergTren12.pdf, Paper}, doi = {10.1137/110826278}, year = {2012}, date = {2012-05-03}, journal = {SIAM J. Matrix Anal. & Appl.}, volume = {33}, number = {2}, pages = {336--368}, abstract = {We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit.}, keywords = {DAEs, normal forms, solution theory}, pubstate = {published}, tppubtype = {article} } We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit. |

Liberzon, Daniel; Trenn, Stephan Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability Journal Article Automatica, 48 (5), pp. 954–963, 2012. Abstract | Links | BibTeX | Tags: DAEs, nonlinear, solution theory, stability, switched DAEs, switched systems @article{LibeTren12, title = {Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT111011.pdf, Preprint}, doi = {10.1016/j.automatica.2012.02.041}, year = {2012}, date = {2012-05-01}, journal = {Automatica}, volume = {48}, number = {5}, pages = {954--963}, abstract = {We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov’s direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.}, keywords = {DAEs, nonlinear, solution theory, stability, switched DAEs, switched systems}, pubstate = {published}, tppubtype = {article} } We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov’s direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively. |

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

Hackl, Christoph M; Trenn, Stephan The bang-bang funnel controller: An experimental verification Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 735–736, GAMM Annual Meeting 2012, Darmstadt Wiley-VCH Verlag GmbH, Weinheim, 2012. Abstract | Links | BibTeX | Tags: application, funnel control, input constraints, nonlinear, relative degree @inproceedings{HackTren12, title = {The bang-bang funnel controller: An experimental verification}, author = {Christoph M. Hackl and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-HT120427.pdf, Preprint}, doi = {10.1002/pamm.201210356}, year = {2012}, date = {2012-03-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {12}, number = {1}, pages = {735--736}, publisher = {Wiley-VCH Verlag GmbH}, address = {Weinheim}, organization = {GAMM Annual Meeting 2012, Darmstadt}, abstract = {We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive.}, keywords = {application, funnel control, input constraints, nonlinear, relative degree}, pubstate = {published}, tppubtype = {inproceedings} } We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive. |

Berger, Thomas; Ilchmann, Achim; Trenn, Stephan The quasi-Weierstraß form for regular matrix pencils Journal Article Linear Algebra Appl., 436 (10), pp. 4052–4069, 2012, (published online February 2010). Abstract | Links | BibTeX | Tags: DAEs, normal forms, solution theory @article{BergIlch12a, title = {The quasi-Weierstraß form for regular matrix pencils}, author = {Thomas Berger and Achim Ilchmann and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-BIT091017.pdf, Preprint http://dx.doi.org/10.1016/S0024-3795(11)00688-4, Corrections (see Paragraph 6 of Note to Editors)}, doi = {10.1016/j.laa.2009.12.036}, year = {2012}, date = {2012-01-01}, journal = {Linear Algebra Appl.}, volume = {436}, number = {10}, pages = {4052--4069}, abstract = {Regular linear matrix pencils A- E d in K^{n x n}[d], where K=Q, R or C, and the associated differential algebraic equation (DAE) E x' = A x are studied. The Wong sequences of subspaces are investigate and invoked to decompose the K^n into V* + W*, where any bases of the linear spaces V* and W* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V* and ``pure'' inconsistent initial values W* - {0}. Furthermore, V* and W* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A- E d lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E x' = A x.}, note = {published online February 2010}, keywords = {DAEs, normal forms, solution theory}, pubstate = {published}, tppubtype = {article} } Regular linear matrix pencils A- E d in K^{n x n}[d], where K=Q, R or C, and the associated differential algebraic equation (DAE) E x' = A x are studied. The Wong sequences of subspaces are investigate and invoked to decompose the K^n into V* + W*, where any bases of the linear spaces V* and W* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V* and ``pure'' inconsistent initial values W* - {0}. Furthermore, V* and W* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A- E d lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E x' = A x. |

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. |

## 2011 |

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

## 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. |

Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller Inproceedings Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. Abstract | Links | BibTeX | Tags: CDC, funnel control, input constraints, nonlinear, relative degree @inproceedings{LibeTren10, title = {The bang-bang funnel controller}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806.pdf, Preprint http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806longVersion.pdf, Preprint (long version)}, doi = {10.1109/CDC.2010.5717742}, year = {2010}, date = {2010-12-15}, booktitle = {Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA}, pages = {690--695}, abstract = {A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough.}, keywords = {CDC, funnel control, input constraints, nonlinear, relative degree}, pubstate = {published}, tppubtype = {inproceedings} } A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. |

## 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. |

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

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. |

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 Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units Journal Article IEEE Transactions on Neural Networks, 19 (5), pp. 836–844, 2008, ISSN: 1045-9227. Abstract | Links | BibTeX | Tags: neural networks @article{Tren08a, title = {Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre071012.pdf, Preprint}, doi = {10.1109/TNN.2007.912306}, issn = {1045-9227}, year = {2008}, date = {2008-05-01}, journal = {IEEE Transactions on Neural Networks}, volume = {19}, number = {5}, pages = {836--844}, abstract = {This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function.}, keywords = {neural networks}, pubstate = {published}, tppubtype = {article} } This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function. |

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. |

## 2006 |

Mandaloju, Nagendra P; Trenn, Stephan Analogue Implementation of the Funnel Controller Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 823–824, WILEY-VCH Verlag, 2006, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: application, funnel control, nonlinear @inproceedings{MandTren06, title = {Analogue Implementation of the Funnel Controller}, author = {Nagendra P. Mandaloju and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MT060428.pdf, Preprint}, doi = {10.1002/pamm.200610391}, issn = {1617-7061}, year = {2006}, date = {2006-05-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {6}, number = {1}, pages = {823--824}, publisher = {WILEY-VCH Verlag}, abstract = {In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications.}, keywords = {application, funnel control, nonlinear}, pubstate = {published}, tppubtype = {inproceedings} } In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications. |

Ilchmann, Achim; Sawodny, Oliver; Trenn, Stephan Pneumatic cylinders: modelling and feedback force-control Journal Article Int. J. Control, 79 (6), pp. 650–661, 2006. Abstract | Links | BibTeX | Tags: application, input constraints @article{IlchSawo06, title = {Pneumatic cylinders: modelling and feedback force-control}, author = {Achim Ilchmann and Oliver Sawodny and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IST050502.pdf, Preprint}, doi = {10.1080/00207170600645875}, year = {2006}, date = {2006-01-01}, journal = {Int. J. Control}, volume = {79}, number = {6}, pages = {650--661}, abstract = {In this paper, we model, analyse, and control an experimental set-up of a servo pneumatic cylinder. The dynamic behaviour of pneumatic actuator systems is dominant by non-linear functions. First, a mathematical model for the pneumatic system is derived. Secondly, we investigate the mathematical properties of this model and show boundedness and positiveness of certain variables. Thirdly, we prove that a proportional output feedback controller with saturation achieves practical tracking a wide class of reference trajectories. We verify the theoretical results and the effectiveness of the control by experiments.}, keywords = {application, input constraints}, pubstate = {published}, tppubtype = {article} } In this paper, we model, analyse, and control an experimental set-up of a servo pneumatic cylinder. The dynamic behaviour of pneumatic actuator systems is dominant by non-linear functions. First, a mathematical model for the pneumatic system is derived. Secondly, we investigate the mathematical properties of this model and show boundedness and positiveness of certain variables. Thirdly, we prove that a proportional output feedback controller with saturation achieves practical tracking a wide class of reference trajectories. We verify the theoretical results and the effectiveness of the control by experiments. |

## 2005 |

French, Mark; Trenn, Stephan l Proc. 44th IEEE Conf. Decis. Control and European Control Conf. (ECC), pp. 2865–2870, 2005. Abstract | Links | BibTeX | Tags: CDC, stability, switched systems @inproceedings{FrenTren05, title = {l^{p} gain bounds for switched adaptive controllers}, author = {Mark French and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-FT050913.pdf, Preprint}, doi = {10.1109/CDC.2005.1582598}, year = {2005}, date = {2005-12-01}, booktitle = {Proc. 44th IEEE Conf. Decis. Control and European Control Conf. (ECC)}, pages = {2865--2870}, abstract = {A class of discrete plants controlled by a switching adaptive strategy is considered, and l^p bounds, 1 ≤ p ≤ ∞, are obtained for the closed loop gain relating input and output disturbances to internal signals.}, keywords = {CDC, stability, switched systems}, pubstate = {published}, tppubtype = {inproceedings} } A class of discrete plants controlled by a switching adaptive strategy is considered, and l^p bounds, 1 ≤ p ≤ ∞, are obtained for the closed loop gain relating input and output disturbances to internal signals. |

Ilchmann, Achim; Ryan, Eugene P; Trenn, Stephan Tracking control: Performance funnels and prescribed transient behaviour Journal Article Syst. Control Lett., 54 (7), pp. 655–670, 2005. Abstract | Links | BibTeX | Tags: funnel control @article{IlchRyan05, title = {Tracking control: Performance funnels and prescribed transient behaviour}, author = {Achim Ilchmann and Eugene P. Ryan and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IRT041112.pdf, Preprint}, doi = {10.1016/j.sysconle.2004.11.005}, year = {2005}, date = {2005-01-01}, journal = {Syst. Control Lett.}, volume = {54}, number = {7}, pages = {655--670}, publisher = {Elsevier}, abstract = {Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.}, keywords = {funnel control}, pubstate = {published}, tppubtype = {article} } Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary. |

## 2004 |

Ilchmann, Achim; Ryan, Eugene P; Trenn, Stephan Adaptive tracking within prescribed funnels Inproceedings Proceedings of the 2004 IEEE International Conference on Control Applications, 2004., pp. 1032–1036, 2004. Abstract | Links | BibTeX | Tags: funnel control, nonlinear, stability @inproceedings{IlchRyan04b, title = {Adaptive tracking within prescribed funnels}, author = {Achim Ilchmann and Eugene P. Ryan and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IRT040512.pdf, Preprint}, doi = {10.1109/CCA.2004.1387507}, year = {2004}, date = {2004-09-01}, booktitle = {Proceedings of the 2004 IEEE International Conference on Control Applications, 2004.}, volume = {2}, pages = {1032--1036}, abstract = {Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given lambda > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary.}, keywords = {funnel control, nonlinear, stability}, pubstate = {published}, tppubtype = {inproceedings} } Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given lambda > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary. |

Ilchmann, Achim; Trenn, Stephan Input constrained funnel control with applications to chemical reactor models Journal Article Syst. Control Lett., 53 (5), pp. 361–375, 2004. Abstract | Links | BibTeX | Tags: application, funnel control, input constraints @article{IlchTren04, title = {Input constrained funnel control with applications to chemical reactor models}, author = {Achim Ilchmann and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IT040715.pdf, Preprint}, doi = {10.1016/j.sysconle.2004.05.014}, year = {2004}, date = {2004-01-01}, journal = {Syst. Control Lett.}, volume = {53}, number = {5}, pages = {361--375}, publisher = {Elsevier}, abstract = {Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*.}, keywords = {application, funnel control, input constraints}, pubstate = {published}, tppubtype = {article} } Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*. |