Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco A smooth model for periodically switched descriptor systems Unpublished 2021, (submitted). @unpublished{MostTren21pp, title = {A smooth model for periodically switched descriptor systems}, author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca}, url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-MTV210511.pdf, Preprint}, year = {2021}, date = {2021-05-11}, abstract = {Switched descriptor systems characterized by a repetitive finite sequence of modes can exhibit state discontinuities at the switching time instants. The amplitudes of these discontinuities depend on the consistency projectors of the modes. A switched ordinary differential equations model whose continuous state evolution approximates the state of the original system is proposed. Sufficient conditions based on linear matrix inequalities on the modes projectors ensure that the approximation error is of linear order of the switching period. The theoretical findings are applied to a switched capacitor circuit and numerical results illustrate the practical usefulness of the proposed model.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } Switched descriptor systems characterized by a repetitive finite sequence of modes can exhibit state discontinuities at the switching time instants. The amplitudes of these discontinuities depend on the consistency projectors of the modes. A switched ordinary differential equations model whose continuous state evolution approximates the state of the original system is proposed. Sufficient conditions based on linear matrix inequalities on the modes projectors ensure that the approximation error is of linear order of the switching period. The theoretical findings are applied to a switched capacitor circuit and numerical results illustrate the practical usefulness of the proposed model. |

Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2021, (submitted for publication). @unpublished{LeeTren21pp, title = {Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling}, author = {Jin Gyu Lee and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/2021/04/Preprint-LTS210420.pdf, Preprint https://arxiv.org/abs/2012.14580v2, Extended ArXiv-version}, year = {2021}, date = {2021-04-20}, 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 = {}, 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. |

Chen, Yahao; Trenn, Stephan; Respondek, Witold Normal forms and internal regularization of nonlinear differential-algebraic control systems Unpublished 2021, (submitted). @unpublished{ChenTren21pp, title = {Normal forms and internal regularization of nonlinear differential-algebraic control systems}, author = {Yahao Chen and Stephan Trenn and Witold Respondek}, url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-CTR210413.pdf, Preprint}, year = {2021}, date = {2021-04-13}, abstract = {In this paper, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, i.e., when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } In this paper, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, i.e., when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs. |

Wijnbergen, Paul; Trenn, Stephan Optimal control of DAEs with unconstrained terminal costs Unpublished 2021, (submitted). @unpublished{WijnTren21pp, title = {Optimal control of DAEs with unconstrained terminal costs}, author = {Paul Wijnbergen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2021/03/Preprint-WT210317.pdf, Preprint}, year = {2021}, date = {2021-03-17}, abstract = {This paper is concerned with the linear quadratic optimal control problem for impulse controllable differential algebraic equations on a bounded half open interval. With respect to the cost functional, a general positive semi-definite weight matrix is considered in the terminal cost. It is shown that for this problem, there generally does not exist an input that minimizes the cost functional. First it is shown that the problem can be reduced to finding an input to an index-1 DAE that minimizes a different quadratic cost functional. Second, necessary and sufficient conditions in terms of matrix equations are given for the existence of an optimal control are stated.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } This paper is concerned with the linear quadratic optimal control problem for impulse controllable differential algebraic equations on a bounded half open interval. With respect to the cost functional, a general positive semi-definite weight matrix is considered in the terminal cost. It is shown that for this problem, there generally does not exist an input that minimizes the cost functional. First it is shown that the problem can be reduced to finding an input to an index-1 DAE that minimizes a different quadratic cost functional. Second, necessary and sufficient conditions in terms of matrix equations are given for the existence of an optimal control are stated. |

Berger, Thomas; Ilchmann, Achim; Trenn, Stephan Quasi feedback forms for differential-algebraic systems Unpublished 2021, (submitted for publication). @unpublished{BergIlch21pp, title = {Quasi feedback forms for differential-algebraic systems}, author = {Thomas Berger and Achim Ilchmann and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2021/02/Preprint-BIT210225.pdf, Preprint https://arxiv.org/abs/2102.12713, arXiv:2102.12713}, year = {2021}, date = {2021-02-25}, abstract = {We investigate feedback forms for linear time-invariant systems described by differential-algebraic equa- tions. Feedback forms are representatives of certain equivalence classes. For example state space trans- formations, invertible transformations from the left, and proportional state feedback constitute an equiva- lence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } We investigate feedback forms for linear time-invariant systems described by differential-algebraic equa- tions. Feedback forms are representatives of certain equivalence classes. For example state space trans- formations, invertible transformations from the left, and proportional state feedback constitute an equiva- lence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold. |

Trenn, Stephan Distributional restriction impossible to define Unpublished 2020, (submitted). @unpublished{Tren20ppf, title = {Distributional restriction impossible to define}, author = {Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/09/Preprint-Tre200901.pdf, Preprint}, year = {2020}, date = {2020-09-01}, abstract = {A counterexample is presented showing that it is not possible to define a restriction for distributions.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } A counterexample is presented showing that it is not possible to define a restriction for distributions. |

Iervolino, Raffaele; Vasca, Francesco; Trenn, Stephan Discontinuous Lyapunov functions for discontinous piecewise-affine systems Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). @misc{IervTren20m, title = {Discontinuous Lyapunov functions for discontinous piecewise-affine systems}, author = {Raffaele Iervolino and Francesco Vasca and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-ITV200122.pdf, Extended Abstract}, year = {2020}, date = {2020-01-22}, 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. We first introduce the feasible Filippov solution concept 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 highlighted 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. }, howpublished = {Extended Abstract}, note = {accepted for cancelled MTNS 20/21}, keywords = {}, pubstate = {published}, tppubtype = {misc} } 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. We first introduce the feasible Filippov solution concept 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 highlighted 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. |

Trenn, Stephan The Laplace transform and inconsistent initial values Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). @misc{Tren20m, title = {The Laplace transform and inconsistent initial values}, author = {Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-Tre200122.pdf, Extended Abstract}, year = {2020}, date = {2020-01-22}, abstract = {Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions.}, howpublished = {Extended Abstract}, note = {accepted for cancelled MTNS 20/21}, keywords = {}, pubstate = {published}, tppubtype = {misc} } Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions. |

# Submitted

A smooth model for periodically switched descriptor systems Unpublished 2021, (submitted). |

Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2021, (submitted for publication). |

Normal forms and internal regularization of nonlinear differential-algebraic control systems Unpublished 2021, (submitted). |

Optimal control of DAEs with unconstrained terminal costs Unpublished 2021, (submitted). |

Quasi feedback forms for differential-algebraic systems Unpublished 2021, (submitted for publication). |

Distributional restriction impossible to define Unpublished 2020, (submitted). |

Discontinuous Lyapunov functions for discontinous piecewise-affine systems Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). |

The Laplace transform and inconsistent initial values Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). |