Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan Stability analysis for switched discrete-time linear singular systems Unpublished 2019, (submitted for publication). @unpublished{AnhLinh19ppb, title = {Stability analysis for switched discrete-time linear singular systems}, author = {Pham Ky Anh and Pham Thi Linh and Do Duc Thuan and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2019/10/Preprint-ALTT191016.pdf, Preprint}, year = {2019}, date = {2019-10-16}, abstract = {The stability of arbitrarily switched discrete-time linear singular (SDLS) systems is studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1. Based on the joint spectral radius of a finite set of matrix pencils some necessary and sufficient conditions are established for exponential stability. Furthermore, sufficient conditions for exponential stability in terms of quadratic Lyapunov functions as well as certain commutativity conditions are presented. The theoretical findings are illustrated by several examples.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } The stability of arbitrarily switched discrete-time linear singular (SDLS) systems is studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1. Based on the joint spectral radius of a finite set of matrix pencils some necessary and sufficient conditions are established for exponential stability. Furthermore, sufficient conditions for exponential stability in terms of quadratic Lyapunov functions as well as certain commutativity conditions are presented. The theoretical findings are illustrated by several examples. |

Wijnbergen, Paul; Trenn, Stephan Impulse controllability of switched differential-algebraic equations Unpublished 2019, (submitted for publication). @unpublished{WijnTren19pp, title = {Impulse controllability of switched differential-algebraic equations}, author = {Paul Wijnbergen and Stephan Trenn}, url = {https://stephantrenn.net/wp-content/uploads/2019/10/Preprint-WT191009.pdf, Preprint}, year = {2019}, date = {2019-10-09}, abstract = {This paper addressed impulse controllability of switched DAEs on a finite interval. We first present a forward approach where we define certain subspaces forward in time, which then are used to provide a sufficient condition for impulse controllability. In order to obtain a full characterization we present afterwards a backward approach, where a sequence of subspaces is defined backwards in time. With the help of the last element of this backward sequence, we are able to fully characterize impulse controllability. All results are geometric results and thus independent of a coordinate system.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } This paper addressed impulse controllability of switched DAEs on a finite interval. We first present a forward approach where we define certain subspaces forward in time, which then are used to provide a sufficient condition for impulse controllability. In order to obtain a full characterization we present afterwards a backward approach, where a sequence of subspaces is defined backwards in time. With the help of the last element of this backward sequence, we are able to fully characterize impulse controllability. All results are geometric results and thus independent of a coordinate system. |

Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Unpublished 2019, (submitted for publication). @unpublished{LeeBerg19pp, 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/2019/10/Preprint-LBTS191001.pdf, Preprint}, year = {2019}, date = {2019-10-01}, 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.}, note = {submitted for publication}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } 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. |

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

# Submitted

Stability analysis for switched discrete-time linear singular systems Unpublished 2019, (submitted for publication). |

Impulse controllability of switched differential-algebraic equations Unpublished 2019, (submitted for publication). |

Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Unpublished 2019, (submitted for publication). |

Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2019, (submitted for publication). |

2019, (submitted for publication). |