Hossain, Sumon; Trenn, Stephan Reduced realization of switched linear systems with known switching sequence Unpublished 2021, (submitted). @unpublished{HossTren21pp,We propose a novel reduction approach for switched linear systems with a fixed mode sequence based on subspaces related to the (time-varying) reachable and unobservable spaces. These subspaces are defined in such a way that they can be used to construct a switched weak Kalman decomposition, which is then in turn used to define a reduced switched linear system with an identical input-output behavior. The proposed method is illustrated with a low dimensional academic example. |
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Unpublished 2021, (submitted). @unpublished{LeeBerg21pp,In a recent work by three of the authors, in order to enforce synchronization for scalar heterogeneous multi-agent systems with some useful characteristics, a node-wise funnel coupling law was proposed. The emergent dynamics, to which each of the agents synchronizes, was characterized and it was studied how networks can be synthesized which exhibit these emergent dynamics. The advantage of this synthesis is its suitability for plug-and-play operation. However, the aforementioned emergent dynamics under node-wise funnel coupling are determined by an algebraic equation which does not admit an explicit solution in general, and even its pointwise solution proves rather difficult. Furthermore, the contractivity assumption on the emergent dynamics, required to establish the synchronization, is hard to be checked without solving the algebraic equation. To resolve these drawbacks, in the present paper we present a new funnel coupling law that uses edge-wise output differences. Under this novel coupling the benign properties of node-wise funnel coupling are retained, but the emergent dynamics are given explicitly by the blended dynamics of the multi-agent system, which already proved an advantageous tool in the analysis and design of such networks. Additionally, our results are not restricted to scalar systems and treat the case that neighboring agents only communicate their output information, and not their complete state. |
Chen, Yahao; Trenn, Stephan Impulse-free jump solutions of nonlinear differential-algebraic equations Unpublished 2021, (submitted). @unpublished{ChenTren21ppb,In this paper, we propose a novel notion called impulse-free jump solution for nonlinear differential-algebraic equations (DAEs) of the form E(x)x' = F(x) with inconsistent initial values. The term “impulse-free” means that there are no Dirac impulses caused by jumps from inconsistent initial values, i.e., the directions of jumps stay in ker E(x). We find that the existence and uniqueness of impulse-free jumps are closely related to the notion of geometric index-1 and the involutivity of the distribution defined by ker E(x). Moreover, a singular perturbed system approximation is proposed for nonlinear DAEs; we show that solutions of the perturbed system approximate both impulse-free jump solutions and C1-solutions of nonlinear DAEs. Finally, we show by some examples that our results of impulse-free jumps are useful for the problems like consistent initializations of nonlinear DAEs and transient behavior simulations of electric circuits. |
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,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. |
Submitted
Reduced realization of switched linear systems with known switching sequence Unpublished 2021, (submitted). |
Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Unpublished 2021, (submitted). |
Impulse-free jump solutions of nonlinear differential-algebraic equations Unpublished 2021, (submitted). |
Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished 2021, (submitted for publication). |