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.

## 2021 |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions Journal Article IEEE Transactions on Automatic Control, 2021, (to appear). Abstract | Links | BibTeX | Tags: nonlinear, solution-theory, stability, switched-systems @article{IervTren21, 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/2020/02/Preprint-ITV200204.pdf, Preprint}, doi = {10.1109/TAC.2020.2996597}, year = {2021}, date = {2021-04-01}, journal = {IEEE Transactions on Automatic Control}, 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 = {to appear}, keywords = {nonlinear, solution-theory, stability, switched-systems}, pubstate = {published}, tppubtype = {article} } 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. |

## 2020 |

Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Inproceedings Proc. European Control Conference (ECC 2020), pp. 911-916, Saint Petersburg, Russia, 2020. Abstract | Links | BibTeX | Tags: funnel-control, networks, nonlinear, synchronization @inproceedings{LeeBerg20, title = {Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization}, author = {Jin Gyu Lee and Thomas Berger and Stephan Trenn and Hyungbo Shim}, url = {https://stephantrenn.net/wp-content/uploads/2020/02/Preprint-LBTS200204.pdf, Preprint}, year = {2020}, date = {2020-05-14}, booktitle = {Proc. European Control Conference (ECC 2020)}, pages = {911-916}, address = {Saint Petersburg, Russia}, abstract = {A new approach to distributed consensus optimization is studied in this paper. The cost function to be minimized is a sum of local cost functions which are not necessarily convex as long as their sum is convex. This benefit is obtained from a recent observation that, with a large gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields. However, design of the large coupling gain requires global information such as network structure and individual agent dynamics. In this paper, we employ a nonlinear time-varying coupling of diffusive type, which we call `edge-wise funnel coupling.' This idea is borrowed from adaptive control, which enables decentralized design of distributed optimizers without knowledge of global information. Remarkably, without a common internal model, each agent achieves asymptotic consensus to the optimal solution of the global cost. We illustrate this result by a network that asymptotically finds the least-squares solution of a linear equation in a distributed manner.}, keywords = {funnel-control, networks, nonlinear, synchronization}, pubstate = {published}, tppubtype = {inproceedings} } A new approach to distributed consensus optimization is studied in this paper. The cost function to be minimized is a sum of local cost functions which are not necessarily convex as long as their sum is convex. This benefit is obtained from a recent observation that, with a large gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields. However, design of the large coupling gain requires global information such as network structure and individual agent dynamics. In this paper, we employ a nonlinear time-varying coupling of diffusive type, which we call `edge-wise funnel coupling.' This idea is borrowed from adaptive control, which enables decentralized design of distributed optimizers without knowledge of global information. Remarkably, without a common internal model, each agent achieves asymptotic consensus to the optimal solution of the global cost. We illustrate this result by a network that asymptotically finds the least-squares solution of a linear equation in a distributed manner. |

Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential-algebraic equations Unpublished 2020, (submitted for publication). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, solution-theory @unpublished{ChenTren20pp, title = {On geometric and differentiation index of nonlinear differential-algebraic equations}, author = {Yahao Chen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/02/Preprint-CT200204.pdf, Preprint}, year = {2020}, date = {2020-02-04}, abstract = {In this paper, we discuss two notions of index (the geometric index and the differentiation index), which appear in the studies of the solvability of nonlinear differential-algebraic equations DAEs. First, we analyze the solutions of nonlinear DAEs via a geometric method, then depending on the analysis of solutions, we show that although both of the two indices serve as a measure of the difficulties of solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. We also show that the two DAE indices have close relations with each other when some assumptions of smoothness and constant rankness are satisfied. An example of a pendulum system is used to illustrate our geometric method of solving DAEs and also our results of the relations of the two DAE indices.}, note = {submitted for publication}, keywords = {DAEs, nonlinear, solution-theory}, pubstate = {published}, tppubtype = {unpublished} } In this paper, we discuss two notions of index (the geometric index and the differentiation index), which appear in the studies of the solvability of nonlinear differential-algebraic equations DAEs. First, we analyze the solutions of nonlinear DAEs via a geometric method, then depending on the analysis of solutions, we show that although both of the two indices serve as a measure of the difficulties of solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. We also show that the two DAE indices have close relations with each other when some assumptions of smoothness and constant rankness are satisfied. An example of a pendulum system is used to illustrate our geometric method of solving DAEs and also our results of the relations of the two DAE indices. |

## 2019 |

Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2019, (submitted for publication). Abstract | Links | BibTeX | Tags: funnel-control, nonlinear, synchronization @unpublished{LeeTren19ppb, title = {Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling}, author = {Jin Gyu Lee and Stephan Trenn and Hyungbo Shim}, url = {https://stephantrenn.net/wp-content/uploads/2019/07/Preprint-LTS190719.pdf, Preprint}, year = {2019}, date = {2019-07-19}, abstract = {In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying graph structure. The proposed coupling law is motivated by the funnel control studied in adaptive controls under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by the high-gain coupling, and thus, we follow to call our coupling law as `(node-wise) funnel coupling.' By getting out of the conventional proof technique in the funnel control study, we now can obtain even asymptotic or finite-time synchronization with the same funnel coupling law. More interestingly, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, has been analyzed in this paper. In particular, we introduce a single scalar dynamics called `emergent dynamics' that is capable of illustrating the emergent synchronized behavior by its solution trajectory. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. A particular example illustrating the utility of the emergent dynamics is given also in the paper as a distributed median solver.}, note = {submitted for publication}, keywords = {funnel-control, nonlinear, synchronization}, pubstate = {published}, tppubtype = {unpublished} } In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying graph structure. The proposed coupling law is motivated by the funnel control studied in adaptive controls under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by the high-gain coupling, and thus, we follow to call our coupling law as `(node-wise) funnel coupling.' By getting out of the conventional proof technique in the funnel control study, we now can obtain even asymptotic or finite-time synchronization with the same funnel coupling law. More interestingly, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, has been analyzed in this paper. In particular, we introduce a single scalar dynamics called `emergent dynamics' that is capable of illustrating the emergent synchronized behavior by its solution trajectory. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. A particular example illustrating the utility of the emergent dynamics is given also in the paper as a distributed median solver. |

## 2018 |

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

## 2017 |

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

Trenn, Stephan Edge-wise funnel synchronization Inproceedings PAMM - Proc. Appl. Math. Mech., pp. 821 - 822, WILEY-VCH Verlag, 2017, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: funnel-control, networks, nonlinear, synchronization @inproceedings{Tren17, title = {Edge-wise funnel synchronization}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre170523.pdf, Preprint}, doi = {10.1002/pamm.201710378}, issn = {1617-7061}, year = {2017}, date = {2017-06-01}, booktitle = {PAMM - Proc. Appl. Math. Mech.}, volume = {17}, number = {1}, pages = {821 - 822}, 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.}, keywords = {funnel-control, networks, nonlinear, synchronization}, 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. |

Kall, Jochen; Kausar, Rukhsana; Trenn, Stephan Modeling water hammers via PDEs and switched DAEs with numerical justification Inproceedings Proc. 20th IFAC World Congress 2017, pp. 5349 - 5354, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, PDEs, 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 = {Proc. 20th IFAC World Congress 2017}, journal = {IFAC-PapersOnLine}, volume = {50}, number = {1}, pages = {5349 - 5354}, address = {Toulouse, France}, 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.}, keywords = {application, DAEs, nonlinear, PDEs, 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. |

## 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. |

## 2015 |

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 |

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. |

## 2013 |

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. |

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. |

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. |

## 2012 |

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. |

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. |

## 2010 |

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. |

## 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. |

## 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. |