Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Unpublished 2022, (submitted). @unpublished{ChenTren22ppb,
title = {On impulse-free solutions and stability of switched nonlinear differential-algebraic equations},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/05/Preprint-CT220505.pdf, Preprint},
year = {2022},
date = {2022-05-05},
urldate = {2022-05-05},
abstract = {In this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [22, 23], for linear DAEs. Then we show that the common Lyapunov functions stability conditions proposed in our previous work [12] (which differ from the ones in [23]) can be applied to switched nonlinear DAEs with high-index models which are not equivalent to the nonlinear Weierstrass form. Moreover, we generalize the commutativity stability conditions [27] for switched nonlinear ordinary differential equations to the nonlinear DAEs case. Finally, some simulation results of switching electrical circuits and numerical examples are given to illustrate the usefulness of the proposed stability conditions.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
In this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [22, 23], for linear DAEs. Then we show that the common Lyapunov functions stability conditions proposed in our previous work [12] (which differ from the ones in [23]) can be applied to switched nonlinear DAEs with high-index models which are not equivalent to the nonlinear Weierstrass form. Moreover, we generalize the commutativity stability conditions [27] for switched nonlinear ordinary differential equations to the nonlinear DAEs case. Finally, some simulation results of switching electrical circuits and numerical examples are given to illustrate the usefulness of the proposed stability conditions. |
Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Unpublished 2022, (submitted). @unpublished{HossTren22ppc,
title = {Midpoint based balanced truncation for switched linear systems with known switching signal},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/05/Preprint-HT220422.pdf, Preprint},
year = {2022},
date = {2022-04-22},
abstract = {We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size. |
Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Unpublished 2022, (submitted). @unpublished{HossTren22ppb,
title = {Reduced realization for switched linear systems with known mode sequence},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/05/Preprint-HT220413.pdf, Preprint
https://doi.org/10.5281/zenodo.6410136, Matlab sources},
year = {2022},
date = {2022-04-13},
urldate = {2022-04-13},
abstract = {We consider switched linear systems with mode-dependent state-dimensions and/or state jumps and propose a method to obtain a switched system of reduced size with identical input-output behavior. Our approach is based in considering time-dependent reachability and unobservability spaces as well as suitable extended reachability and restricted unobservability spaces together with the notion of a weak Kalman decomposition. A key feature of our approach is that only the mode sequence of the switching signal needs to be known and not the exact switching times. However, the size of a minimal realization will in general depend on the mode durations, hence it cannot be expected that our method always leads to minimal realization. Nevertheless, we show that our method is optimal in the sense that a repeated application doesn’t lead to a further reduction and we also highlight a practically relevant special case, where minimality is achieved.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We consider switched linear systems with mode-dependent state-dimensions and/or state jumps and propose a method to obtain a switched system of reduced size with identical input-output behavior. Our approach is based in considering time-dependent reachability and unobservability spaces as well as suitable extended reachability and restricted unobservability spaces together with the notion of a weak Kalman decomposition. A key feature of our approach is that only the mode sequence of the switching signal needs to be known and not the exact switching times. However, the size of a minimal realization will in general depend on the mode durations, hence it cannot be expected that our method always leads to minimal realization. Nevertheless, we show that our method is optimal in the sense that a repeated application doesn’t lead to a further reduction and we also highlight a practically relevant special case, where minimality is achieved. |
Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time Unpublished 2022, (submitted). @unpublished{YinJaya22pp,
title = {On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time},
author = {Hao Yin and Bayu Jayawardhana and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/04/Preprint-YJT220315.pdf, Preprint},
year = {2022},
date = {2022-03-15},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
|
Chen, Yahao; Trenn, Stephan Impulse-free jump solutions of nonlinear differential-algebraic equations Unpublished 2022, (submitted). @unpublished{ChenTren22pp,
title = {Impulse-free jump solutions of nonlinear differential-algebraic equations},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-CT220312.pdf, Preprint},
year = {2022},
date = {2022-03-12},
urldate = {2021-05-17},
abstract = {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.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
Hossain, Sumon; Trenn, Stephan Reduced realization of switched linear systems with known switching sequence Unpublished 2021, (submitted). @unpublished{HossTren21pp,
title = {Reduced realization of switched linear systems with known switching sequence},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/10/Preprint-HT211016.pdf, Preprint},
year = {2021},
date = {2021-10-16},
abstract = {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.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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,
title = {Edge-wise funnel output synchronization of heterogeneous agents with relative degree one},
author = {Jin Gyu Lee and Thomas Berger and Stephan Trenn and Hyungbo Shim},
url = {https://stephantrenn.net/wp-content/uploads/2021/10/Preprint-LBTS211011.pdf, Preprint
https://arxiv.org/abs/2110.05330, ArXiV},
year = {2021},
date = {2021-10-11},
urldate = {2021-10-11},
abstract = {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.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
