2025
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Ahmed, Saeed; Bachmann, Patrick; Trenn, Stephan Lyapunov Characterization for ISS of Impulsive Switched Systems Unpublished 2025, (conditionally accepted in IEEE Transactions on Automatic Control as regular paper). @unpublished{AhmeBach25pp,
title = {Lyapunov Characterization for ISS of Impulsive Switched Systems},
author = {Saeed Ahmed and Patrick Bachmann and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2025/12/Preprint-ABT251201.pdf, Preprint},
year = {2025},
date = {2025-12-01},
urldate = {2025-12-01},
abstract = {In this study, we investigate input-to-state stability (ISS) of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is not necessarily decreasing along trajectories, which we call generalized, and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a strictification technique for constructing a decreasing ISS-Lyapunov function from a large class of generalized ones, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee ISS of a particular class of impulsive switched systems when the switching signal is unknown.},
note = {conditionally accepted in IEEE Transactions on Automatic Control as regular paper},
keywords = {impulsive-systems, ISS, Lyapunov, nonlinear, stability, switched-systems},
pubstate = {published},
tppubtype = {unpublished}
}
In this study, we investigate input-to-state stability (ISS) of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is not necessarily decreasing along trajectories, which we call generalized, and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a strictification technique for constructing a decreasing ISS-Lyapunov function from a large class of generalized ones, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee ISS of a particular class of impulsive switched systems when the switching signal is unknown. |
Cimpean, Radu; Sterk, Alef; Trenn, Stephan ISS for PWA systems Unpublished 2025, (submitted). @unpublished{CimpSter25pp,
title = {ISS for PWA systems},
author = {Radu Cimpean and Alef Sterk and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2025/12/Preprint-CST251114.pdf, Preprint},
year = {2025},
date = {2025-11-14},
urldate = {2025-11-14},
abstract = {We offer a way to construct an ISS type bound for a PWA system. Starting from an ISS Lyapunov function in each region of the state space (called regional ISS Lyapunov function), and under suitable assumptions, we patch these functions and obtain initially a practical ISS type bound. After that, depending on the input size, we have to consider a distinction between large and small inputs. The large inputs dominate the dynamics and may induce region transitions, while the small inputs are sufficiently attenuated to direct and keep the trajectory within the region containing the equilibrium. The threshold between these regimes plays a key role in determining the point at which the system decays towards the equilibrium. By unifying the analysis across these input regimes, we provide a complete ISS-type bound valid for all admissible inputs. Consequently, the global ISS property is ensured, and further refined analysis can be applied.},
note = {submitted},
keywords = {ISS, switched-systems},
pubstate = {published},
tppubtype = {unpublished}
}
We offer a way to construct an ISS type bound for a PWA system. Starting from an ISS Lyapunov function in each region of the state space (called regional ISS Lyapunov function), and under suitable assumptions, we patch these functions and obtain initially a practical ISS type bound. After that, depending on the input size, we have to consider a distinction between large and small inputs. The large inputs dominate the dynamics and may induce region transitions, while the small inputs are sufficiently attenuated to direct and keep the trajectory within the region containing the equilibrium. The threshold between these regimes plays a key role in determining the point at which the system decays towards the equilibrium. By unifying the analysis across these input regimes, we provide a complete ISS-type bound valid for all admissible inputs. Consequently, the global ISS property is ensured, and further refined analysis can be applied. |
Cimpean, Radu; Trenn, Stephan Utilizing discontinuous piecewise Lyapunov function for a biological PWA system and extending the analysis via input-to-state type stability Proceedings Article In: Proc. 23rd European Control Conference (ECC 2025), pp. 194-200, EUCA IEEE, Thessaloniki, Greece, 2025. @inproceedings{CimpTren25,
title = {Utilizing discontinuous piecewise Lyapunov function for a biological PWA system and extending the analysis via input-to-state type stability},
author = {Radu Cimpean and Stephan Trenn},
url = {https://www.paperhost.org/proceedings/controls/ECC25/files/0230.pdf, Paper
https://stephantrenn.net/wp-content/uploads/2024/08/Preprint-CT250515.pdf, Preprint},
year = {2025},
date = {2025-06-25},
urldate = {2025-06-25},
booktitle = {Proc. 23rd European Control Conference (ECC 2025)},
pages = {194-200},
publisher = {IEEE},
address = {Thessaloniki, Greece},
organization = {EUCA},
abstract = {We study a piecewise affine (PWA) model of the spiking of neurons located in a subsystem of the olfactory system. Our longterm goal is to understand the stability properties of a network of such neurons and as a first step towards this goal we consider the simplest case of the interaction between two types of neurons (excitatory and inhibitory). Due to the discontinuous nature of the PWA model, it is challenging to find a continuous Lyapunov function, we therefore utilize a recently proposed constructive method to find a discontinuous Lyapunov function. In order to utilize this method, it is necessary to define a suitable polyhedral partition of the state-space and carefully investigate the dynamics at the boundaries. As an additional step, we propose extending the analysis by employing tools from the input-to-state (ISS) stability framework.},
keywords = {ISS, LMIs, Lyapunov, neural-networks, stability, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
We study a piecewise affine (PWA) model of the spiking of neurons located in a subsystem of the olfactory system. Our longterm goal is to understand the stability properties of a network of such neurons and as a first step towards this goal we consider the simplest case of the interaction between two types of neurons (excitatory and inhibitory). Due to the discontinuous nature of the PWA model, it is challenging to find a continuous Lyapunov function, we therefore utilize a recently proposed constructive method to find a discontinuous Lyapunov function. In order to utilize this method, it is necessary to define a suitable polyhedral partition of the state-space and carefully investigate the dynamics at the boundaries. As an additional step, we propose extending the analysis by employing tools from the input-to-state (ISS) stability framework. |
Sahan, Gökhan; Trenn, Stephan Comments on “Relaxed Conditions for the Input-to-State Stability of Switched Nonlinear Time-Varying Systems” Journal Article In: IEEE Transactions on Automatic Control, vol. 70, no. 6, pp. 4233-4236, 2025. @article{SahaTren25,
title = {Comments on “Relaxed Conditions for the Input-to-State Stability of Switched Nonlinear Time-Varying Systems”},
author = {Gökhan Sahan and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2025/04/Preprint-ST250401.pdf, Preprint},
doi = {10.1109/TAC.2025.3558940},
year = {2025},
date = {2025-05-30},
urldate = {2025-04-08},
journal = {IEEE Transactions on Automatic Control},
volume = {70},
number = {6},
pages = {4233-4236},
abstract = {This study addresses the deficiencies in the assumptions of the results in (Chen and Yang, 2017) due to the lack of uniformity. We first show the missing hypothesis by presenting a counterexample. Then, we prove why they are wrong in that form and show the errors in the proof of the main result of (Chen and Yang, 2017). Next, we compare the assumptions and related results of (Chen and Yang, 2017) with similar works in the literature. Lastly, we give suggestions to complement the shortcomings of the hypotheses and thus correct them.},
keywords = {ISS, stability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
This study addresses the deficiencies in the assumptions of the results in (Chen and Yang, 2017) due to the lack of uniformity. We first show the missing hypothesis by presenting a counterexample. Then, we prove why they are wrong in that form and show the errors in the proof of the main result of (Chen and Yang, 2017). Next, we compare the assumptions and related results of (Chen and Yang, 2017) with similar works in the literature. Lastly, we give suggestions to complement the shortcomings of the hypotheses and thus correct them. |
Cimpean, Radu; Trenn, Stephan; Sterk, Alef Utilizing discontinuous piecewise Lyapunov function for a biological PWA system and extending the analysis via input-to-state stability Miscellaneous Book of Abstracts - 44th Benelux Meeting on Systems and Control, 2025. @misc{CimpTren25m,
title = {Utilizing discontinuous piecewise Lyapunov function for a biological PWA system and extending the analysis via input-to-state stability},
author = {Radu Cimpean and Stephan Trenn and Alef Sterk},
url = {https://stephantrenn.net/wp-content/uploads/2025/09/CimpTren24m.pdf, Abstract
https://www.beneluxmeeting.nl/2025/uploads/papers/boa.pdf, Book of Abstracts},
year = {2025},
date = {2025-03-18},
urldate = {2025-09-18},
howpublished = {Book of Abstracts - 44th Benelux Meeting on Systems and Control},
keywords = {ISS, LMIs, neural-networks, stability, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
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