Wijnbergen, Paul; Trenn, Stephan Impulse-free interval-stabilization of switched differential algebraic equations Unpublished 2020, (submitted). @unpublished{WijnTren20pp, title = {Impulse-free interval-stabilization of switched differential algebraic equations}, author = {Paul Wijnbergen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/11/Preprint-WT201123.pdf, Preprint}, year = {2020}, date = {2020-11-23}, abstract = {In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. }, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. |

Sutrisno, ; Trenn, Stephan Observability of Switched Linear Singular System in Discrete Time: Single Switch Case Unpublished 2020, (submitted). @unpublished{SutrTren20pp, title = {Observability of Switched Linear Singular System in Discrete Time: Single Switch Case}, author = {Sutrisno and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/11/Preprint-ST201121.pdf, Preprint}, year = {2020}, date = {2020-11-21}, abstract = {In this paper, we investigate the observability of switched linear singular systems in discrete time. As a preliminary study, we restrict the systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that generally, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, for two-dimensional systems, it is fully independent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property i.e. swapping the mode sequence may destroy the observability.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } In this paper, we investigate the observability of switched linear singular systems in discrete time. As a preliminary study, we restrict the systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that generally, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, for two-dimensional systems, it is fully independent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property i.e. swapping the mode sequence may destroy the observability. |

Hossain, Sumon; Trenn, Stephan Minimal Realization for Linear Switched Systems Unpublished 2020, (submitted). @unpublished{HossTren20pp, title = {Minimal Realization for Linear Switched Systems}, author = {Sumon Hossain and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/11/Preprint-HT201120.pdf, Preprint}, year = {2020}, date = {2020-11-20}, abstract = {We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple example to illustrate the approach.}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple example to illustrate the approach. |

Chen, Yahao; Trenn, Stephan; Respondek, Witold Normal forms and internal regularization of nonlinear differential-algebraic control systems Unpublished 2020, (submitted). @unpublished{ChenTren20ppc, 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/2020/11/Preprint-CTR201113.pdf, Preprint}, year = {2020}, date = {2020-11-13}, 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 later 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 later 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. |

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}, note = {submitted}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } |

Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2020, (submitted for publication). @unpublished{LeeTren20pp, 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/2020/07/Preprint-LTS200702.pdf, Preprint}, year = {2020}, date = {2020-07-02}, 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. |

Iervolino, Raffaele; Vasca, Francesco; Trenn, Stephan Discontinuous Lyapunov functions for discontinous piecewise-affine systems Unpublished 2020, (extended abstract, submitted to MTNS). @unpublished{IervTren20pp, 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, Preprint}, 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. }, note = {extended abstract, submitted to MTNS}, keywords = {}, 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. 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 Unpublished 2020, (extended abstract, submitted to MTNS). @unpublished{Tren20b, title = {The Laplace transform and inconsistent initial values}, author = {Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-Tre200122.pdf, Preprint}, 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.}, note = {extended abstract, submitted to MTNS}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } 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

Impulse-free interval-stabilization of switched differential algebraic equations Unpublished 2020, (submitted). |

Observability of Switched Linear Singular System in Discrete Time: Single Switch Case Unpublished 2020, (submitted). |

Minimal Realization for Linear Switched Systems Unpublished 2020, (submitted). |

Normal forms and internal regularization of nonlinear differential-algebraic control systems Unpublished 2020, (submitted). |

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

Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2020, (submitted for publication). |

Discontinuous Lyapunov functions for discontinous piecewise-affine systems Unpublished 2020, (extended abstract, submitted to MTNS). |

The Laplace transform and inconsistent initial values Unpublished 2020, (extended abstract, submitted to MTNS). |