2025
|
Ahmed, Saeed; Bachmann, Patrick; Trenn, Stephan Lyapunov Characterization for ISS of Impulsive Switched Systems Unpublished 2025, (conditionally accepted in IEEE Transactions on Automatic Control as regular paper). @unpublished{AhmeBach25pp,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tppubtype = {unpublished}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. |
Karimi-Pour, Atiyeh; Trenn, Stephan; Ayati, Moosa; Zakerzadeh, Mohammad Reza Funnel-based output tracking control for nonlinear impulsive switched systems Journal Article In: Nonlinear Analysis: Hybrid Systems, vol. 60, no. 101661, pp. 1-25, 2025, (open access). @article{KariTren25,
title = {Funnel-based output tracking control for nonlinear impulsive switched systems},
author = {Atiyeh Karimi-Pour and Stephan Trenn and Moosa Ayati and Mohammad Reza Zakerzadeh},
url = {https://stephantrenn.net/wp-content/uploads/2025/12/KariTren25.pdf, Paper},
doi = {10.1016/j.nahs.2025.101661},
year = {2025},
date = {2025-12-01},
urldate = {2025-12-01},
journal = {Nonlinear Analysis: Hybrid Systems},
volume = {60},
number = {101661},
pages = {1-25},
abstract = {We study output tracking for nonlinear impulsive switched systems with global relative degree one under prescribed performance requirements. Classical funnel control is not directly applicable in this setting, since output jumps can cause violations of funnel constraints. To address this, we design an adjusted funnel boundary that contracts prior to jumps and expands afterward, computed offline based on the stability of the internal dynamics and bounded jump maps. We also derive sufficient conditions ensuring bounded control input. To obtain tighter bounds, practical ISS is employed in place of BIBO stability, yielding smaller input requirements. Additional refinements include asymmetric jump bounds, level-set adjustments, and real-time funnel adaptation, which further improve performance. Numerical examples confirm stability and practical tracking under disturbance impulses and switching.},
note = {open access},
keywords = {funnel-control, impulsive-systems, stability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
We study output tracking for nonlinear impulsive switched systems with global relative degree one under prescribed performance requirements. Classical funnel control is not directly applicable in this setting, since output jumps can cause violations of funnel constraints. To address this, we design an adjusted funnel boundary that contracts prior to jumps and expands afterward, computed offline based on the stability of the internal dynamics and bounded jump maps. We also derive sufficient conditions ensuring bounded control input. To obtain tighter bounds, practical ISS is employed in place of BIBO stability, yielding smaller input requirements. Additional refinements include asymmetric jump bounds, level-set adjustments, and real-time funnel adaptation, which further improve performance. Numerical examples confirm stability and practical tracking under disturbance impulses and switching. |
2024
|
Karimi-Pour, Atiyeh; Trenn, Stephan Funnel control for impulsive switched systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), pp. 7810-7815, IEEE Milan, Italy, 2024. @inproceedings{KariTren24,
title = {Funnel control for impulsive switched systems},
author = {Atiyeh Karimi-Pour and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/09/Preprint-KT240913.pdf, Preprint},
doi = {10.1109/CDC56724.2024.10886425},
year = {2024},
date = {2024-12-16},
urldate = {2024-12-16},
booktitle = {Proc. 63rd IEEE Conf. Decision Control (CDC 2024)},
pages = {7810-7815},
address = {Milan, Italy},
organization = {IEEE},
abstract = {Impulsive switched systems encompass various modes, each exhibiting distinct behaviours. Typically, a switching sequence orchestrates transitions between these modes, where state jumps may occur, potentially undermining output tracking performance or system stability. This work introduces a funnel controller tailored for relative degree one nonlinear impulsive switched systems. Notably, this controller operates solely based on system output without necessitating knowledge of system dynamics. Unlike classical funnel controllers with fixed boundaries, the proposed method dynamically adjusts the funnel boundary for each approaching jump, aiming to preserve adherence to the original boundary. No precise knowledge of jump instances or maps is required; approximate jump intervals and an upper bound for maximum jump height suffice. Theoretical analysis establishes that the error remains within the funnel, facilitating successful reference signal tracking. Performance validation is demonstrated via numerical simulation.},
keywords = {funnel-control, impulsive-systems, nonlinear, relative-degree, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
Impulsive switched systems encompass various modes, each exhibiting distinct behaviours. Typically, a switching sequence orchestrates transitions between these modes, where state jumps may occur, potentially undermining output tracking performance or system stability. This work introduces a funnel controller tailored for relative degree one nonlinear impulsive switched systems. Notably, this controller operates solely based on system output without necessitating knowledge of system dynamics. Unlike classical funnel controllers with fixed boundaries, the proposed method dynamically adjusts the funnel boundary for each approaching jump, aiming to preserve adherence to the original boundary. No precise knowledge of jump instances or maps is required; approximate jump intervals and an upper bound for maximum jump height suffice. Theoretical analysis establishes that the error remains within the funnel, facilitating successful reference signal tracking. Performance validation is demonstrated via numerical simulation. |
Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Journal Article In: Automatica, vol. 160, no. 111447, pp. 1-12, 2024, (open access). @article{MostTren24,
title = {Averaging for switched impulsive systems with pulse width modulation},
author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/MostTren24.pdf, Paper},
doi = {10.1016/j.automatica.2023.111447},
year = {2024},
date = {2024-02-01},
urldate = {2024-02-01},
journal = {Automatica},
volume = {160},
number = {111447},
pages = {1-12},
abstract = {Linear switched impulsive systems (SIS) are characterized by ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes nontrivial the application of classical averaging techniques. In this paper we consider SIS with pulse width modulation (PWM) and we propose an averaged model whose solution approximates the moving average of the SIS solution with an error which decreases with the multiple of the switching period and by decreasing the PWM period. The averaging result requires milder assumptions on the system matrices with respect to those needed by the previous averaging techniques for SIS. The interest of the proposed model is strengthened by the fact that it reduces to the classical averaged model for PWM systems when there are no jumps in the state. The theoretical results are verified through numerical results obtained by considering a switched capacitor electrical circuit.},
note = {open access},
keywords = {application, averaging, DAEs, impulsive-systems, LMIs, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
Linear switched impulsive systems (SIS) are characterized by ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes nontrivial the application of classical averaging techniques. In this paper we consider SIS with pulse width modulation (PWM) and we propose an averaged model whose solution approximates the moving average of the SIS solution with an error which decreases with the multiple of the switching period and by decreasing the PWM period. The averaging result requires milder assumptions on the system matrices with respect to those needed by the previous averaging techniques for SIS. The interest of the proposed model is strengthened by the fact that it reduces to the classical averaged model for PWM systems when there are no jumps in the state. The theoretical results are verified through numerical results obtained by considering a switched capacitor electrical circuit. |
Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Journal Article In: IEEE Transactions on Automatic Control, vol. 69, no. 1, pp. 535-542, 2024. @article{HossTren24,
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/2023/05/Preprint-HT230508.pdf, Preprint},
doi = {10.1109/TAC.2023.3269721},
year = {2024},
date = {2024-01-01},
urldate = {2024-01-01},
journal = {IEEE Transactions on Automatic Control},
volume = {69},
number = {1},
pages = {535-542},
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.},
keywords = {controllability, impulsive-systems, model-reduction, observability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
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. |
2023
|
Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, vol. 154, no. 111065, pp. 1-9, 2023, (open access). @article{HossTren23a,
title = {Reduced realization for switched linear systems with known mode sequence},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/HossTren23a.pdf, Paper
https://doi.org/10.5281/zenodo.6410136, Matlab sources},
doi = {10.1016/j.automatica.2023.111065},
year = {2023},
date = {2023-08-01},
urldate = {2023-08-01},
journal = {Automatica},
volume = {154},
number = {111065},
pages = {1-9},
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 = {open access},
keywords = {controllability, impulsive-systems, model-reduction, observability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
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. |
2022
|
Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco An averaged model for switched systems with state jumps applicable for PWM descriptor systems Proceedings Article In: Proceedings of the 2022 European Control Conference (ECC), pp. 1085-1090, London, 2022. @inproceedings{MostTren22b,
title = {An averaged model for switched systems with state jumps applicable for PWM descriptor systems},
author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-MTV220329.pdf, Preprint},
doi = {10.23919/ECC55457.2022.9838189},
year = {2022},
date = {2022-07-12},
urldate = {2022-07-12},
booktitle = {Proceedings of the 2022 European Control Conference (ECC)},
pages = {1085-1090},
address = {London},
abstract = {Switched descriptor systems with pulse width modulation are characterized by modes whose dynamics are described by differential algebraic equations; this type of models can be viewed as switched impulsive systems, i.e. switched systems with ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes the application of the classical averaging technique nontrivial. In this paper we propose an averaged model for switched impulsive systems. The state trajectory of the proposed averaged model is shown to approximate the one of the original system with an error of order of the switching period. The model reduces to the classical averaged model when there are no jumps in the state. The practical interest of the theoretical averaging result is demonstrated through numerical simulations of a switched capacitor electrical circuit.},
keywords = {averaging, DAEs, impulsive-systems, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
Switched descriptor systems with pulse width modulation are characterized by modes whose dynamics are described by differential algebraic equations; this type of models can be viewed as switched impulsive systems, i.e. switched systems with ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes the application of the classical averaging technique nontrivial. In this paper we propose an averaged model for switched impulsive systems. The state trajectory of the proposed averaged model is shown to approximate the one of the original system with an error of order of the switching period. The model reduces to the classical averaged model when there are no jumps in the state. The practical interest of the theoretical averaging result is demonstrated through numerical simulations of a switched capacitor electrical circuit. |
2015
|
Küsters, Ferdinand; Trenn, Stephan Duality of switched ODEs with jumps Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 4879–4884, 2015. @inproceedings{KustTren15b,
title = {Duality of switched ODEs with jumps},
author = {Ferdinand Küsters and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT150814.pdf, Preprint},
doi = {10.1109/CDC.2015.7402981},
year = {2015},
date = {2015-12-05},
urldate = {2015-12-05},
booktitle = {Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan},
pages = {4879--4884},
abstract = {Duality between controllability/reachability and determinability/observability of switched systems with jumps is proven. The duality result is based on the recent characterization of controllability for switched differential-algebraic equations (DAEs) which share many properties with switched ordinary differential equations (ODEs) with jumps. Here we view the switching signal as given and fixed, which makes the overall switched system time-varying, in particular controllability and reachability do not coincide anymore.},
keywords = {controllability, impulsive-systems, observability, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
Duality between controllability/reachability and determinability/observability of switched systems with jumps is proven. The duality result is based on the recent characterization of controllability for switched differential-algebraic equations (DAEs) which share many properties with switched ordinary differential equations (ODEs) with jumps. Here we view the switching signal as given and fixed, which makes the overall switched system time-varying, in particular controllability and reachability do not coincide anymore. |
2012
|
Trenn, Stephan; Willems, Jan C. Switched behaviors with impulses - a unifying framework Proceedings Article In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 3203-3208, 2012, ISSN: 0743-1546. @inproceedings{TrenWill12,
title = {Switched behaviors with impulses - a unifying framework},
author = {Stephan Trenn and Jan C. Willems},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TW120813.pdf, Preprint},
doi = {10.1109/CDC.2012.6426883},
issn = {0743-1546},
year = {2012},
date = {2012-12-13},
urldate = {2012-12-13},
booktitle = {Proc. 51st IEEE Conf. Decis. Control, Maui, USA},
pages = {3203-3208},
abstract = {We present a new framework to describe and study switched behaviors. We allow for jumps and impulses in the trajectories induced either implicitly by the dynamics after the switch or explicitly by “impacts”. With some examples from electrical circuit we motivate that the dynamical equations before and after the switch already uniquely define the “dynamics” at the switch, i.e. jumps and impulses. On the other hand, we also allow for external impacts resulting in jumps and impulses not induced by the internal dynamics. As a first theoretical result in this new framework we present a characterization for autonomy of a switched behavior.},
keywords = {DAEs, impulsive-systems, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
We present a new framework to describe and study switched behaviors. We allow for jumps and impulses in the trajectories induced either implicitly by the dynamics after the switch or explicitly by “impacts”. With some examples from electrical circuit we motivate that the dynamical equations before and after the switch already uniquely define the “dynamics” at the switch, i.e. jumps and impulses. On the other hand, we also allow for external impacts resulting in jumps and impulses not induced by the internal dynamics. As a first theoretical result in this new framework we present a characterization for autonomy of a switched behavior. |
