Below you find an interactive list of all my publications, which can be filtered by keywords, year, publication type and coauthors. There are also static lists of my books/book-chapters as well as journal and conference publications.

## 2019 |

Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan Stability analysis for switched discrete-time linear singular systems Unpublished 2019, (submitted for publication). Abstract | BibTeX | Tags: stability, switched-DAEs, switched-systems, vidi @unpublished{AnhLinh19bpp, title = {Stability analysis for switched discrete-time linear singular systems}, author = {Pham Ky Anh and Pham Thi Linh and Do Duc Thuan and Stephan Trenn}, year = {2019}, date = {2019-04-18}, abstract = {The stability of arbitrarily switched discrete-time linear singular (SDLS) systems is studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1. Based on the joint spectral radius of a finite set of matrix pencils some necessary and sufficient conditions are established for exponential stability. Furthermore, sufficient conditions for exponential stability in terms of quadratic Lyapunov functions as well as certain commutativity conditions are presented. The theoretical findings are illustrated by several examples.}, note = {submitted for publication}, keywords = {stability, switched-DAEs, switched-systems, vidi}, pubstate = {published}, tppubtype = {unpublished} } The stability of arbitrarily switched discrete-time linear singular (SDLS) systems is studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1. Based on the joint spectral radius of a finite set of matrix pencils some necessary and sufficient conditions are established for exponential stability. Furthermore, sufficient conditions for exponential stability in terms of quadratic Lyapunov functions as well as certain commutativity conditions are presented. The theoretical findings are illustrated by several examples. |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco 2019, (submitted for publication). Abstract | Links | BibTeX | Tags: nonlinear, solution-theory, stability, switched-systems, vidi @unpublished{IervTren19pp, title = {Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions}, author = {Raffaele Iervolino and Stephan Trenn and Francesco Vasca}, url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-ITV190315.pdf, Preprint}, year = {2019}, date = {2019-03-15}, abstract = {Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In the first part of the paper the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result.}, note = {submitted for publication}, keywords = {nonlinear, solution-theory, stability, switched-systems, vidi}, pubstate = {published}, tppubtype = {unpublished} } Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In the first part of the paper the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result. |

## 2018 |

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Switch induced instabilities for stable power system DAE models Inproceedings IFAC-PapersOnLine, pp. 127-132, 2018, (Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)). Abstract | Links | BibTeX | Tags: application, stability, switched-DAEs, switched-systems, vidi @inproceedings{GrosTren18, title = {Switch induced instabilities for stable power system DAE models}, author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen}, url = {https://stephantrenn.net/wp-content/uploads/2018/04/Preprint-GTW180413.pdf, Preprint}, doi = {10.1016/j.ifacol.2018.08.022}, year = {2018}, date = {2018-07-11}, booktitle = {IFAC-PapersOnLine}, journal = {IFAC-PapersOnLine}, volume = {51}, number = {16}, pages = {127-132}, abstract = {It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenoma can indeed occur for a linearized DAE model of power grids. By making certain topological assumptions on the power grid, we can ensure stability under arbitrary switching.}, note = {Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)}, keywords = {application, stability, switched-DAEs, switched-systems, vidi}, pubstate = {published}, tppubtype = {inproceedings} } It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenoma can indeed occur for a linearized DAE model of power grids. By making certain topological assumptions on the power grid, we can ensure stability under arbitrary switching. |

## 2017 |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions Inproceedings Proc. 56th IEEE Conf. Decis. Control, pp. 5894 - 5899, Melbourne, Australia, 2017. 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 = {https://stephantrenn.net/wp-content/uploads/2017/09/Preprint-ITV170909.pdf, Preprint}, doi = {10.1109/CDC.2017.8264551}, year = {2017}, date = {2017-12-15}, booktitle = {Proc. 56th IEEE Conf. Decis. Control}, pages = {5894 - 5899}, address = {Melbourne, Australia}, 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.}, 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. |

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. |

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

## 2016 |

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Solvability and stability of a power system DAE model Journal Article Syst. Control Lett., 97 , 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 = {97}, 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. |

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. |

## 2015 |

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 |

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, synchronization @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, synchronization}, 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. |

## 2014 |

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. |

## 2013 |

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. |

## 2012 |

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. |

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. |

## 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 Conf. 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 Conf. 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. |

## 2009 |

Liberzon, Daniel; Trenn, Stephan On stability of linear switched differential algebraic equations Inproceedings Proc. Joint 48th IEEE Conf. Decis. Control and 28th Chinese Control Conf., 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 Conf. Decis. Control and 28th Chinese Control Conf.}, 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. |

## 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. |

## 2004 |

Ilchmann, Achim; Ryan, Eugene P; Trenn, Stephan Adaptive tracking within prescribed funnels Inproceedings Proc. 2004 IEEE Int. Conf. Control Appl., 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 = {Proc. 2004 IEEE Int. Conf. Control Appl.}, 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. |