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Stephan Trenn
Stephan Trenn

Stephan Trenn

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Category: Submitted

Posted on 2019-12-062019-12-06

Paper on PDEs coupled to swDAEs submitted

Finally we have published our key theoretical paper four our DFG-project “Coupling hyperbolic PDEs with switched DAEs: Analysis, numerics and application to blood flow models”:

Borsche, Raul; Kocoglu, Damla; Trenn, Stephan

A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Unpublished

2019.

Abstract | Links | BibTeX

@unpublished{BorsKoco19pp,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},
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.},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}

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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.

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  • https://stephantrenn.net/wp-content/uploads/2019/12/Preprint-BKT191128.pdf

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In this paper, we provide a novel distributional solution framework to handle jumps and Dirac impulses on the boundaries of the domain of a hyperbolic PDE. It also includes a nice example of electrical power lines (modeled via the telegraph equation) with a switching transformer (modeled by a switched DAE). The simulations nicely show how a Dirac impulse (induced by a switch) is moving through time and space as predicted by the theory.

Posted on 2019-12-062019-12-06

Two submission to IFAC WC ’20

Together with my PhD-students we have prepared the following two submission to the IFAC World Congress 2020 to be held in Berlin next year:

Hossain, Sumon; Trenn, Stephan

A time-varying Gramian based model reduction approach for Linear Switched Systems Unpublished

2019.

Abstract | Links | BibTeX

@unpublished{HossTren19pp,
title = {A time-varying Gramian based model reduction approach for Linear Switched Systems},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/12/Preprint-HT191118.pdf, Preprint},
year = {2019},
date = {2019-11-18},
abstract = {Switched system can be considered as a special class of piecewise constantly time- varying systems. In this paper, a model reduction approach is proposed for piecewise constantly switched systems, based on balancing based model order reduction (MOR) method for linear time-varying systems. Time-varying controllability and observability Gramians are computed in finite time interval and then developed balancing theory for the linear time-varying systems. A low dimensional approximate system is computed by applying projection based balanced truncation (BT) method. Finally, the proposed approach is illustrated numerically.},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}

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Switched system can be considered as a special class of piecewise constantly time- varying systems. In this paper, a model reduction approach is proposed for piecewise constantly switched systems, based on balancing based model order reduction (MOR) method for linear time-varying systems. Time-varying controllability and observability Gramians are computed in finite time interval and then developed balancing theory for the linear time-varying systems. A low dimensional approximate system is computed by applying projection based balanced truncation (BT) method. Finally, the proposed approach is illustrated numerically.

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Wijnbergen, Paul; Jeeninga, Mark; Trenn, Stephan

On stabilizability of switched Differential Algebraic Equations Unpublished

2019.

Abstract | Links | BibTeX

@unpublished{WijnJeen19pp,
title = {On stabilizability of switched Differential Algebraic Equations},
author = {Paul Wijnbergen and Mark Jeeninga and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/12/Preprint-WJT191118.pdf, Preprint},
year = {2019},
date = {2019-11-18},
abstract = {This paper considers stabilizability of switched differential algebraic equations (DAEs). We first introduce the notion of interval stabilizability and show that under a certain uniformity assumption, stabilizability can be concluded from interval stabilizability. A geometric approach is taken to find necessary and sufficient conditions for interval stabilizability. Then the analysis is extended resulting in a characterization of controllability.},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}

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This paper considers stabilizability of switched differential algebraic equations (DAEs). We first introduce the notion of interval stabilizability and show that under a certain uniformity assumption, stabilizability can be concluded from interval stabilizability. A geometric approach is taken to find necessary and sufficient conditions for interval stabilizability. Then the analysis is extended resulting in a characterization of controllability.

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Posted on 2019-10-10

Two submission two ECC’20

We have submitted the following two papers to the ECC’20 taking place in St. Petersburg, Russia:

Wijnbergen, Paul; Trenn, Stephan

Impulse controllability of switched differential-algebraic equations Unpublished

2019, (submitted for publication).

Abstract | Links | BibTeX

@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}
}

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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.

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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).

Abstract | Links | BibTeX

@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}
}

Close

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.

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Posted on 2019-07-19

Journal paper on funnel synchronization submitted

In cooperation with colleagues from Seoul National University we have submitted the following paper

Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo

Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Unpublished

2019, (submitted for publication).

Abstract | Links | BibTeX

@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}
}

Close

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.

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  • Preprint

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Already in our 2015 CDC paper

Shim, Hyungbo; Trenn, Stephan

A preliminary result on synchronization of heterogeneous agents via funnel control Inproceedings

Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2229–2234, 2015.

Abstract | Links | BibTeX

@inproceedings{ShimTren15,
title = {A preliminary result on synchronization of heterogeneous agents via funnel control},
author = {Hyungbo Shim and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-ST150902.pdf, Preprint},
doi = {10.1109/CDC.2015.7402538},
year = {2015},
date = {2015-12-01},
booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan},
pages = {2229--2234},
abstract = {We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research.},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

Close

We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research.

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  • Preprint
  • doi:10.1109/CDC.2015.7402538

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we proposed the idea of funnel synchronization and the simulations looked very promising, but at that time we were not able to prove the synchronization property and also were not able to characterize the emergent behavior. Jin Gyu Lee was able to find the right proof techniques which is now the corner stone of our new submission.

Posted on 2019-04-18

Paper on stability of switched singular systems in discrete time submitted

We have submitted the manuscript

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).

Abstract | Links | BibTeX

@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}
}

Close

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.

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for publication.

My three Vietnames co-authors and I have analyzed the stability of switched linear singular system in discrete-time. The results build on our recently introduced notion of a one-step-map for index-1 switched singular systems:

Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan

The one-step-map for switched singular systems in discrete-time Inproceedings

Proc. 58th IEEE Conf. Decision Control (CDC) 2019, Nice, France, 2019, (to appear).

Abstract | Links | BibTeX

@inproceedings{AnhLinh19ppa,
title = {The one-step-map for switched singular systems in discrete-time},
author = {Pham Ky Anh and Pham Thi Linh and Do Duc Thuan and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-ALTT190910.pdf, Preprint},
year = {2019},
date = {2019-09-10},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
address = {Nice, France},
abstract = {We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-step- map which can be used to provide explicit solution formulas for general switching signals.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

Close

We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-step- map which can be used to provide explicit solution formulas for general switching signals.

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Posted on 2019-03-222019-06-17

Four CDC papers submitted

We have submitted the following four papers to the CDC 2019 in Nice, France.

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).

Abstract | Links | BibTeX

@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}
}

Close

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.

Close

  • Preprint

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Lee, Jin Gyu; Trenn, Stephan

Asymptotic tracking via funnel control Inproceedings

Proc. 58th IEEE Conf. Decision Control (CDC) 2019, Nice, France, 2019, (to appear).

Abstract | Links | BibTeX

@inproceedings{LeeTren19ppa,
title = {Asymptotic tracking via funnel control},
author = {Jin Gyu Lee and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-LT190910.pdf, Preprint},
year = {2019},
date = {2019-09-10},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
address = {Nice, France},
abstract = {Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowledge how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control – without any further assumptions – can achieve asymptotic tracking.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

Close

Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowledge how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control – without any further assumptions – can achieve asymptotic tracking.

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  • Preprint

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Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan

The one-step-map for switched singular systems in discrete-time Inproceedings

Proc. 58th IEEE Conf. Decision Control (CDC) 2019, Nice, France, 2019, (to appear).

Abstract | Links | BibTeX

@inproceedings{AnhLinh19ppa,
title = {The one-step-map for switched singular systems in discrete-time},
author = {Pham Ky Anh and Pham Thi Linh and Do Duc Thuan and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-ALTT190910.pdf, Preprint},
year = {2019},
date = {2019-09-10},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
address = {Nice, France},
abstract = {We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-step- map which can be used to provide explicit solution formulas for general switching signals.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

Close

We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-step- map which can be used to provide explicit solution formulas for general switching signals.

Close

  • Preprint

Close

Trenn, Stephan; Unger, Benjamin

Delay regularity of differential-algebraic equations Inproceedings

Proc. 58th IEEE Conf. Decision Control (CDC) 2019, Nice, France, 2019, (to appear).

Abstract | Links | BibTeX

@inproceedings{TrenUnge19pp,
title = {Delay regularity of differential-algebraic equations},
author = {Stephan Trenn and Benjamin Unger},
url = {https://stephantrenn.net/wp-content/uploads/2019/03/Preprint-TU190910.pdf, Preprint},
year = {2019},
date = {2019-09-10},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
address = {Nice, France},
abstract = {We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations can arise if a feedback controller is applied to a descriptor system and the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise-smooth distributions and an algorithm to determine whether a DDAE is delay-regular.},
note = {to appear},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

Close

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations can arise if a feedback controller is applied to a descriptor system and the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise-smooth distributions and an algorithm to determine whether a DDAE is delay-regular.

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Posted on 2019-03-162019-03-16

Paper on stability of discontinuous PWA submitted

Finally, we have finished our paper

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco

Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions Unpublished

2019, (submitted for publication).

Abstract | Links | BibTeX

@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}
}

Close

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.

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and submitted it for publication.

We have started to work on this topic almost four years ago when I visited Benevento. Inspired by the promising approach of using the cone-copositivity approach to find Lyapunov functions for piecewise-affine (PWA) systems in an automatic way, we were trying to extend this idea in two main directions: 1) We wanted to allow for discontinuous Lyapunov functions and 2) we wanted to cover also sliding and Zeno solutions. We obtained first results by focusing on the first point (i.e. we only considered classical solutions) and presented these ideas at the CDC 2017 in Melbourne, Australia. It turned out that the technicalities involved in dealing with general Filippov solutions were quite tricky and we still haven’t resolved all of them. In particular, we had to make two technical assumptions for PWA systems which we believe are always satisfied, because we could not construct counter examples which violates these assumptions, but we were also not able to prove them so far. Furthermore, classifying boundaries as crossing, non-reachable and sliding turned out to be harder as expected as well, in particular, due to the presence of Zeno-behavior. Nevertheless we have a very strong (i.e. not very conservative) Lyapunov stability theorem which is formulated in terms of pointwise-conditions. In order to use the cone-copositive approach it is necessary to make some uniformity assumption on the solution behavior along the boundaries, but these assumptions do not exclude Zeno and sliding behavior and allows discontinuities of the Lyapunov function on crossing boundaries.

We are now looking forward to receive constructive feedback from our peers to further improve on the manuscript for the final published version.

Posted on 2017-12-202019-09-11

Preprint on switched-induced instability in power grids

We have submitted the manuscript

Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas

Switch induced instabilities for stable power system DAE models Inproceedings

IFAC-PapersOnLine, pp. 127-132, 2018, (Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)).

Abstract | Links | BibTeX

@inproceedings{GrosTren18,
title = {Switch induced instabilities for stable power system DAE models},
author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen},
url = {https://stephantrenn.net/wp-content/uploads/2018/04/Preprint-GTW180413.pdf, Preprint},
doi = {10.1016/j.ifacol.2018.08.022},
year = {2018},
date = {2018-07-11},
booktitle = {IFAC-PapersOnLine},
journal = {IFAC-PapersOnLine},
volume = {51},
number = {16},
pages = {127-132},
abstract = {It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenoma can indeed occur for a linearized DAE model of power grids. By making certain topological assumptions on the power grid, we can ensure stability under arbitrary switching.},
note = {Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

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It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenoma can indeed occur for a linearized DAE model of power grids. By making certain topological assumptions on the power grid, we can ensure stability under arbitrary switching.

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  • Preprint
  • doi:10.1016/j.ifacol.2018.08.022

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for publication. The results of this manuscript are based on Tjorben’s PhD-thesis.

Posted on 2017-11-232018-05-23

Paper on indiscernible topological changes submitted

We have submitted the manuscript

Patil, Deepak; Tesi, Pietro; Trenn, Stephan

Indiscernible topological variations in DAE networks Journal Article

Automatica, 101 , pp. 280-289, 2019.

Abstract | Links | BibTeX

@article{PatiTesi19,
title = {Indiscernible topological variations in DAE networks},
author = {Deepak Patil and Pietro Tesi and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/01/Preprint-PTT181205.pdf, Preprint},
doi = {10.1016/j.automatica.2018.12.012},
year = {2019},
date = {2019-03-01},
journal = {Automatica},
volume = {101},
pages = {280-289},
abstract = {A problem of characterizing conditions under which a topological change in a network of differential algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogenous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}

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A problem of characterizing conditions under which a topological change in a network of differential algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogenous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network.

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  • Preprint
  • doi:10.1016/j.automatica.2018.12.012

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for publication. It is an extension of our IFAC paper

Küsters, Ferdinand; Patil, Deepak; Tesi, Pietro; Trenn, Stephan

Indiscernible topological variations in DAE networks with applications to power grids Inproceedings

Proc. 20th IFAC World Congress 2017, pp. 7333 - 7338, Toulouse, France, 2017, ISSN: 2405-8963.

Abstract | Links | BibTeX

@inproceedings{KustPati17a,
title = {Indiscernible topological variations in DAE networks with applications to power grids},
author = {Ferdinand Küsters and Deepak Patil and Pietro Tesi and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KPTT170320.pdf, Preprint},
doi = {10.1016/j.ifacol.2017.08.1478},
issn = {2405-8963},
year = {2017},
date = {2017-03-24},
booktitle = {Proc. 20th IFAC World Congress 2017},
journal = {IFAC-PapersOnLine},
volume = {50},
number = {1},
pages = {7333 - 7338},
address = {Toulouse, France},
abstract = {The ability to detect topology variations in dynamical networks defined by differential algebraic equations (DAEs) is considered. We characterize the existence of initial states, for which topological changes are indiscernible. A key feature of our characterization is the ability to verify indiscernibility just in terms of the nominal topology. We apply the results to a power grid model and also discuss the relationship to recent mode-detection results for switched DAEs.},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}

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The ability to detect topology variations in dynamical networks defined by differential algebraic equations (DAEs) is considered. We characterize the existence of initial states, for which topological changes are indiscernible. A key feature of our characterization is the ability to verify indiscernibility just in terms of the nominal topology. We apply the results to a power grid model and also discuss the relationship to recent mode-detection results for switched DAEs.

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  • Preprint
  • doi:10.1016/j.ifacol.2017.08.1478

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where we extended the results to the MIMO case and also fully characterize the set of indiscernible initial states. Furthermore, we present sufficient conditions for regularity preserving topological changes.

Posted on 2017-11-162018-05-23

Preprint on observer design for detectable switched DAEs available

We have submitted our manuscript

Tanwani, Aneel; Trenn, Stephan

Detectability and observer design for switched differential algebraic equations Journal Article

Automatica, 99 , pp. 289-300, 2019.

Abstract | Links | BibTeX

@article{TanwTren19,
title = {Detectability and observer design for switched differential algebraic equations},
author = {Aneel Tanwani and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2018/09/Preprint-TT180917.pdf, Preprint},
doi = {10.1016/j.automatica.2018.10.043},
year = {2019},
date = {2019-01-01},
journal = {Automatica},
volume = {99},
pages = {289-300},
abstract = {This paper studies detectability for switched linear differential–algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}

Close

This paper studies detectability for switched linear differential–algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption.

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  • Preprint
  • doi:10.1016/j.automatica.2018.10.043

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for publication. We hope to receive constructive reviews soon.

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