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Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2020, (submitted for publication). @unpublished{LeeTren20pp, title = {Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling}, author = {Jin Gyu Lee and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a distributed median solver.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate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. |
Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential-algebraic equations Unpublished 2020, (submitted for publication). @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 = {}, 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. |
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. |
Borsche, Raul; Kocoglu, Damla; Trenn, Stephan A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Unpublished 2019, (submitted for publication). @unpublished{BorsKoco19pp, title = {A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs}, author = {Raul Borsche and Damla Kocoglu and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2019/12/Preprint-BKT191128.pdf, Preprint https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5884, Preprint (on TU-KL server)}, year = {2019}, date = {2019-11-28}, abstract = {A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions. |
Submitted
Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2020, (submitted for publication). |
On geometric and differentiation index of nonlinear differential-algebraic equations 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). |
A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Unpublished 2019, (submitted for publication). |