2024
|
Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched differential algebraic equations Journal Article In: Mathematics of Control, Signals, and Systems, vol. 36, iss. 2, 2024, (to appear, open access). @article{WijnTren24,
title = {Impulse-controllability of system classes of switched differential algebraic equations},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/08/Preprint-WT220806.pdf, Preprint},
doi = {10.1007/s00498-023-00367-0},
year = {2024},
date = {2024-06-01},
urldate = {2024-06-01},
journal = {Mathematics of Control, Signals, and Systems},
volume = {36},
issue = {2},
abstract = {In this paper impulse controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. In the case of a system class generated by switching signals with a fixed mode sequence it is shown that either all or almost all systems are impulse-controllable, or that all or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that although all systems are impulse-controllable, the input achieving impulse-free solutions might still depend on the switching times in the future, which causes some causality issues. Therefore, the concept of (quasi-) causal impulse-controllability is introduced and system classes which are (quasi-) causal are characterized. Finally necessary and sufficient conditions for a system class to be causal given some dwell-time are stated.},
note = {to appear, open access},
keywords = {controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
In this paper impulse controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. In the case of a system class generated by switching signals with a fixed mode sequence it is shown that either all or almost all systems are impulse-controllable, or that all or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that although all systems are impulse-controllable, the input achieving impulse-free solutions might still depend on the switching times in the future, which causes some causality issues. Therefore, the concept of (quasi-) causal impulse-controllability is introduced and system classes which are (quasi-) causal are characterized. Finally necessary and sufficient conditions for a system class to be causal given some dwell-time are stated. |
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, 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, 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. |
Sutrisno,; Trenn, Stephan Switched linear singular systems in discrete time: Solution theory and observability notions Journal Article In: Systems & Control Letters, vol. 183, no. 105674, pp. 1-11, 2024, (open access). @article{SutrTren24,
title = {Switched linear singular systems in discrete time: Solution theory and observability notions},
author = {Sutrisno and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2023/11/SutrTren24.pdf, Paper},
doi = {10.1016/j.sysconle.2023.105674},
year = {2024},
date = {2024-01-01},
urldate = {2023-11-16},
journal = {Systems & Control Letters},
volume = {183},
number = {105674},
pages = {1-11},
abstract = {We study the solution theory of linear switched singular systems. In a recent paper by Anh et al. (2019), it was highlighted that the common assumption that each mode of the switched system is index-1 is not sufficient to guarantee existence and uniqueness of solutions of the corresponding switched system and the notion of “jointly index-1” was introduced. However, until now it was not clear what conditions are actually required to guarantee existence and uniqueness of solutions if the switching signal is not considered arbitrary. In particular, we study the two relevant situations where the mode sequence is fixed (and the switching times are arbitrary) and where the whole switching signal is fixed. In both cases, we provide conditions in terms of the original system matrices which ensure existence and uniqueness of solutions. We also extend the idea of the one-step map introduced by Anh et al. (2019) to these two cases. It turns out that in the case of a fixed switching signal, the index-1 condition for the individual modes is also not necessary (in addition to not being sufficient). Furthermore, we utilize the established solution theory to provide characterizations of observability and determinability of switched singular systems.},
note = {open access},
keywords = {DAEs, discrete-time, observability, solution-theory, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
We study the solution theory of linear switched singular systems. In a recent paper by Anh et al. (2019), it was highlighted that the common assumption that each mode of the switched system is index-1 is not sufficient to guarantee existence and uniqueness of solutions of the corresponding switched system and the notion of “jointly index-1” was introduced. However, until now it was not clear what conditions are actually required to guarantee existence and uniqueness of solutions if the switching signal is not considered arbitrary. In particular, we study the two relevant situations where the mode sequence is fixed (and the switching times are arbitrary) and where the whole switching signal is fixed. In both cases, we provide conditions in terms of the original system matrices which ensure existence and uniqueness of solutions. We also extend the idea of the one-step map introduced by Anh et al. (2019) to these two cases. It turns out that in the case of a fixed switching signal, the index-1 condition for the individual modes is also not necessary (in addition to not being sufficient). Furthermore, we utilize the established solution theory to provide characterizations of observability and determinability of switched singular systems. |
2023
|
Sutrisno,; Trenn, Stephan Nonlinear switched singular systems in discrete-time: The one-step map and stability under arbitrary switching signals Journal Article In: European Journal of Control, vol. 74, no. 100852, pp. 1-7, 2023, (presented at the 2023 European Control Conference, Bucharest, Rumania; open access). @article{SutrTren23a,
title = {Nonlinear switched singular systems in discrete-time: The one-step map and stability under arbitrary switching signals},
author = {Sutrisno and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/SutrTren23a.pdf, Paper},
doi = {10.1016/j.ejcon.2023.100852},
year = {2023},
date = {2023-11-01},
urldate = {2023-11-01},
journal = {European Journal of Control},
volume = {74},
number = {100852},
pages = {1-7},
abstract = {The solvability of nonlinear nonswitched and switched singular systems in discrete time is studied. We provide necessary and sufficient conditions for solvability. The one-step map that generates equivalent nonlinear (ordinary) systems for solvable nonlinear singular systems under arbitrary switching signals is introduced. Moreover, the stability is studied by utilizing this one-step map. A sufficient condition for stability is provided in terms of (switched) Lyapunov functions.},
note = {presented at the 2023 European Control Conference, Bucharest, Rumania; open access},
keywords = {DAEs, discrete-time, nonlinear, solution-theory, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
The solvability of nonlinear nonswitched and switched singular systems in discrete time is studied. We provide necessary and sufficient conditions for solvability. The one-step map that generates equivalent nonlinear (ordinary) systems for solvable nonlinear singular systems under arbitrary switching signals is introduced. Moreover, the stability is studied by utilizing this one-step map. A sufficient condition for stability is provided in terms of (switched) Lyapunov functions. |
Wijnbergen, Paul; Trenn, Stephan Impulse-free linear quadratic optimal control of switched differential algebraic equations Unpublished 2023, (submitted). @unpublished{WijnTren23pp,
title = {Impulse-free linear quadratic optimal control of switched differential algebraic equations},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2023/11/Preprint-WT231030.pdf, Preprint},
year = {2023},
date = {2023-10-30},
urldate = {2024-06-01},
abstract = {In this paper the finite horizon linear quadratic regulator (LQR) problem for switched linear differential algebraic equations is studied. It is shown that for switched DAEs with a switching signal that induces locally finitely many switches the problem can be solved by solving several LQR problems for non-switched DAE recursively. First, it is shown how to solve the non-switched problems for index-1 DAEs followed by an extension of the results to higher index DAEs. The resulting optimal control can be computed based on the solution of a Riccati differential equation expressed in terms of the differential system matrices. The paper concludes with the extension of the results to the LQR problem for general switched DAEs.},
note = {submitted},
keywords = {optimal-control, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {unpublished}
}
In this paper the finite horizon linear quadratic regulator (LQR) problem for switched linear differential algebraic equations is studied. It is shown that for switched DAEs with a switching signal that induces locally finitely many switches the problem can be solved by solving several LQR problems for non-switched DAE recursively. First, it is shown how to solve the non-switched problems for index-1 DAEs followed by an extension of the results to higher index DAEs. The resulting optimal control can be computed based on the solution of a Riccati differential equation expressed in terms of the differential system matrices. The paper concludes with the extension of the results to the LQR problem for general switched DAEs. |
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Journal Article In: Automatica, vol. 156, no. 111204, pp. 1-10, 2023, (open access). @article{LeeBerg23,
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/2024/02/LeeBerg23.pdf, Paper
https://arxiv.org/abs/2110.05330, ArXiV},
doi = {10.1016/j.automatica.2023.111204},
year = {2023},
date = {2023-10-01},
urldate = {2023-10-01},
journal = {Automatica},
volume = {156},
number = {111204},
pages = {1-10},
abstract = {When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes.},
note = {open access},
keywords = {funnel-control, networks, nonlinear, relative-degree, synchronization},
pubstate = {published},
tppubtype = {article}
}
When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes. |
Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Journal Article In: Automatica, vol. 156, no. 111208, pp. 1-14, 2023. @article{ChenTren23,
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/2023/06/Preprint-CT230602.pdf, Preprint},
doi = {10.1016/j.automatica.2023.111208},
year = {2023},
date = {2023-10-01},
urldate = {2023-06-02},
journal = {Automatica},
volume = {156},
number = {111208},
pages = {1-14},
abstract = {In this paper, we investigate solutions and stability properties of switched nonlinear differential– algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs.},
keywords = {DAEs, Lyapunov, nonlinear, normal-forms, solution-theory, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
In this paper, we investigate solutions and stability properties of switched nonlinear differential– algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs. |
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin A novel two stages funnel controller limiting the error derivative Journal Article In: Systems & Control Letters, vol. 179, no. 105601, pp. 1-10, 2023, (open access). @article{HuTren23,
title = {A novel two stages funnel controller limiting the error derivative},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/HuTren23.pdf, Paper},
doi = {10.1016/j.sysconle.2023.105601},
year = {2023},
date = {2023-09-01},
urldate = {2023-09-01},
journal = {Systems & Control Letters},
volume = {179},
number = {105601},
pages = {1-10},
abstract = {As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme to solve the output-tracking control problem for uncertain nonlinear systems with relative degree one and stable internal dynamics. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel.},
note = {open access},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme to solve the output-tracking control problem for uncertain nonlinear systems with relative degree one and stable internal dynamics. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel. |
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, 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. |
Sutrisno,; Trenn, Stephan Reachability and Controllability Characterizations for Linear Switched Systems in Discrete Time: A Geometric Approach Proceedings Article In: Proc. 2023 European Control Conference (ECC), pp. 2227-2232, Bucharest, Rumania , 2023. @inproceedings{SutrTren23b,
title = {Reachability and Controllability Characterizations for Linear Switched Systems in Discrete Time: A Geometric Approach},
author = {Sutrisno and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/11/Preprint-ST221125a.pdf, Preprint},
doi = {10.23919/ECC57647.2023.10178124},
year = {2023},
date = {2023-06-13},
urldate = {2023-06-13},
booktitle = {Proc. 2023 European Control Conference (ECC)},
pages = {2227-2232},
address = {Bucharest, Rumania },
abstract = {This article presents the reachability and controllability characterizations for discrete-time linear switched systems under a fixed and known switching signal. A geometric approach is used, and we are able to provide alternative conditions which are more computationally friendly compared to existing results by utilizing the solution formula at switching times. Furthermore, the proposed conditions make it easier to study the dependency of the reachability and controllability on the switching times and the mode sequences; this is a new result currently not investigated in the literature. Some academic examples are provided to illustrate the novel features found in this study.},
keywords = {controllability, discrete-time, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
This article presents the reachability and controllability characterizations for discrete-time linear switched systems under a fixed and known switching signal. A geometric approach is used, and we are able to provide alternative conditions which are more computationally friendly compared to existing results by utilizing the solution formula at switching times. Furthermore, the proposed conditions make it easier to study the dependency of the reachability and controllability on the switching times and the mode sequences; this is a new result currently not investigated in the literature. Some academic examples are provided to illustrate the novel features found in this study. |
2022
|
Chen, Yahao; Trenn, Stephan Impulse-free jump solutions of nonlinear differential-algebraic equations Journal Article In: Nonlinear Analysis: Hybrid Systems, vol. 46, no. 101238, pp. 1-17, 2022, (open access). @article{ChenTren22a,
title = {Impulse-free jump solutions of nonlinear differential-algebraic equations},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/ChenTren22a.pdf, Paper},
doi = {10.1016/j.nahs.2022.101238},
year = {2022},
date = {2022-11-01},
urldate = {2022-11-01},
journal = {Nonlinear Analysis: Hybrid Systems},
volume = {46},
number = {101238},
pages = {1-17},
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 = {open access},
keywords = {DAEs, nonlinear},
pubstate = {published},
tppubtype = {article}
}
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; Sutrisno,; Trenn, Stephan A time-varying approach for model reduction of singular linear switched systems in discrete time Miscellaneous Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems, 2022. @misc{HossSutr22m,
title = {A time-varying approach for model reduction of singular linear switched systems in discrete time},
author = {Sumon Hossain and Sutrisno and Stephan Trenn},
url = {https://epub.uni-bayreuth.de/id/eprint/6809/, Book of Extended Abstracts
https://stephantrenn.net/wp-content/uploads/2023/01/HossSutr22m.pdf, Extended Abtract},
year = {2022},
date = {2022-09-12},
urldate = {2023-01-23},
abstract = {We propose a model reduction approach for singular linear switched systems in discrete time with a fixed mode sequence based on a balanced truncation reduction method for linear time-varying discrete-time systems. The key idea is to use the one-step map to find an equivalent time-varying system with an identical input-output behavior, and then adapt available balance truncation methods for (discrete) time-varying systems. The proposed method is illustrated with a low-dimensional academic example.},
howpublished = {Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems},
keywords = {controllability, DAEs, discrete-time, model-reduction, observability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
We propose a model reduction approach for singular linear switched systems in discrete time with a fixed mode sequence based on a balanced truncation reduction method for linear time-varying discrete-time systems. The key idea is to use the one-step map to find an equivalent time-varying system with an identical input-output behavior, and then adapt available balance truncation methods for (discrete) time-varying systems. The proposed method is illustrated with a low-dimensional academic example. |
Wijnbergen, Paul; Trenn, Stephan Linear quadratic optimal control of switched differential algebraic equations Miscellaneous Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems, 2022. @misc{WijnTren22mb,
title = {Linear quadratic optimal control of switched differential algebraic equations},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://epub.uni-bayreuth.de/id/eprint/6809/, Book of Extended Abstracts
https://stephantrenn.net/wp-content/uploads/2023/01/WijnTren22mb.pdf, Extended Abstract},
year = {2022},
date = {2022-09-12},
urldate = {2022-09-12},
abstract = {In this abstract the finite horizon linear quadratic optimal control problem with constraints on the terminal state for switched differential algebraic equations is considered. Furthermore, we seek for an optimal solution that is impulse-free. In order to solve the problem, a non standard finite horizon problem for non-switched DAEs is considered. Necessary and sufficient conditions on the initial value x0 for solvability of this non standard problem are stated. Based on these results a sequence of subspaces can be defined which lead to necessary and sufficient conditions for solvability of the finite horizon optimal control problem for switched DAEs.},
howpublished = {Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems},
keywords = {DAEs, optimal-control, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
In this abstract the finite horizon linear quadratic optimal control problem with constraints on the terminal state for switched differential algebraic equations is considered. Furthermore, we seek for an optimal solution that is impulse-free. In order to solve the problem, a non standard finite horizon problem for non-switched DAEs is considered. Necessary and sufficient conditions on the initial value x0 for solvability of this non standard problem are stated. Based on these results a sequence of subspaces can be defined which lead to necessary and sufficient conditions for solvability of the finite horizon optimal control problem for switched DAEs. |
Hossain, Sumon; Trenn, Stephan A weak Kalman decomposition approach for reduced realizations of switched linear systems Proceedings Article In: IFAC-PapersOnLine, pp. 157-162, 2022, (Part of special issue: 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022: Vienna Austria, 27–29 July 2022). @inproceedings{HossTren22,
title = {A weak Kalman decomposition approach for reduced realizations of switched linear systems},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/06/Preprint-HT220613.pdf, Preprint},
doi = {10.1016/j.ifacol.2022.09.088},
year = {2022},
date = {2022-07-27},
urldate = {2022-07-27},
booktitle = {IFAC-PapersOnLine},
volume = {55},
number = {20},
pages = {157-162},
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 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 = {Part of special issue: 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022: Vienna Austria, 27–29 July 2022},
keywords = {controllability, model-reduction, observability, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
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 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. |
Chen, Yahao; Trenn, Stephan Stability analysis of switched nonlinear differential-algebraic equations via nonlinear Weierstrass form Proceedings Article In: Proceedings of the 2022 European Control Conference (ECC), pp. 1091-1096, London, 2022. @inproceedings{ChenTren22b,
title = {Stability analysis of switched nonlinear differential-algebraic equations via nonlinear Weierstrass form},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-CT220329.pdf, Preprint},
doi = {10.23919/ECC55457.2022.9838148},
year = {2022},
date = {2022-07-12},
urldate = {2022-07-12},
booktitle = {Proceedings of the 2022 European Control Conference (ECC)},
pages = {1091-1096},
address = {London},
abstract = {In this paper, we propose some sufficient conditions for checking the asymptotic stability of switched nonlinear differential-algebraic equations (DAEs) under arbitrary switch- ing signal. We assume that each model of a given switched DAE is externally equivalent to a nonlinear Weierstrass form. With the help of this form, we can define nonlinear consistency projectors and jump-flow solutions for switched nonlinear DAEs. Then we use a different approach from the paper [12] to study the stability of switched DAEs via a novel notion called the jump-flow explicitation, which attaches a nonlinear control system to a given nonlinear DAE and can be used to simplify the common Lyapunov function conditions for both the flow and the jump dynamics of switched nonlinear DAEs. At last, a numerical example is given to illustrate how to check the stability of a switched nonlinear DAE by constructing a common Lyapunov function.
},
keywords = {DAEs, nonlinear, solution-theory, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
In this paper, we propose some sufficient conditions for checking the asymptotic stability of switched nonlinear differential-algebraic equations (DAEs) under arbitrary switch- ing signal. We assume that each model of a given switched DAE is externally equivalent to a nonlinear Weierstrass form. With the help of this form, we can define nonlinear consistency projectors and jump-flow solutions for switched nonlinear DAEs. Then we use a different approach from the paper [12] to study the stability of switched DAEs via a novel notion called the jump-flow explicitation, which attaches a nonlinear control system to a given nonlinear DAE and can be used to simplify the common Lyapunov function conditions for both the flow and the jump dynamics of switched nonlinear DAEs. At last, a numerical example is given to illustrate how to check the stability of a switched nonlinear DAE by constructing a common Lyapunov function.
|
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, 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. |
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin Funnel control for relative degree one nonlinear systems with input saturation Proceedings Article In: Proceedings of the 2022 European Control Conference (ECC), pp. 227-232, London, 2022. @inproceedings{HuTren22,
title = {Funnel control for relative degree one nonlinear systems with input saturation},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-HTZ220329.pdf, Preprint},
doi = {10.23919/ECC55457.2022.9837979},
year = {2022},
date = {2022-07-12},
urldate = {2022-07-12},
booktitle = {Proceedings of the 2022 European Control Conference (ECC)},
pages = {227-232},
address = {London},
abstract = {The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly. |
Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched DAEs Miscellaneous Book of Abstracts - 41th Benelux Meeting on Systems and Control, 2022. @misc{WijnTren22ma,
title = {Impulse-controllability of system classes of switched DAEs},
author = {Paul Wijnbergen and Stephan Trenn},
editor = {Alain Vande Wouwer and Michel Kinnaert and Emanuele Garone and Laurent Dewasme and Guilherme A. Pimentel},
url = {https://stephantrenn.net/wp-content/uploads/2022/08/WijnTren22ma.pdf, Abstract
https://www.beneluxmeeting.nl/2022/uploads/images/2022/boa_BeneluxMeeting2022_Web_betaV12_withChairs.pdf, Book of Abstracts},
year = {2022},
date = {2022-07-05},
urldate = {2022-07-05},
howpublished = {Book of Abstracts - 41th Benelux Meeting on Systems and Control},
keywords = {controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
|
Berger, Thomas; Ilchmann, Achim; Trenn, Stephan Quasi feedback forms for differential-algebraic systems Journal Article In: IMA Journal of Mathematical Control and Information, vol. 39, iss. 2, pp. 533-563, 2022, (open access, published online October 2021). @article{BergIlch22,
title = {Quasi feedback forms for differential-algebraic systems},
author = {Thomas Berger and Achim Ilchmann and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2023/01/BergIlch22.pdf, Paper
https://arxiv.org/abs/2102.12713, arXiv:2102.12713},
doi = {10.1093/imamci/dnab030},
year = {2022},
date = {2022-06-01},
urldate = {2022-06-01},
journal = {IMA Journal of Mathematical Control and Information},
volume = {39},
issue = {2},
pages = {533-563},
abstract = {We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example state space transformations, invertible transformations from the left, and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold.},
note = {open access, published online October 2021},
keywords = {controllability, DAEs, normal-forms},
pubstate = {published},
tppubtype = {article}
}
We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example state space transformations, invertible transformations from the left, and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold. |
2021
|
Hossain, Sumon; Trenn, Stephan Minimality of Linear Switched Systems with known switching signal Proceedings Article In: Proceedings in Applied Mathematics and Mechanics, pp. 1-3, 2021, (open access). @inproceedings{HossTren21a,
title = {Minimality of Linear Switched Systems with known switching signal},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/08/HossTren21a.pdf, Paper},
doi = {10.1002/pamm.202100067},
year = {2021},
date = {2021-12-14},
urldate = {2021-12-14},
booktitle = {Proceedings in Applied Mathematics and Mechanics},
volume = {21},
number = {e202100067},
pages = {1-3},
abstract = {Minimal realization is discussed for linear switched systems with a given switching signal. We propose a consecutive forward and backward approach for the time-interval of interest. The forward approach refers to extending the reachable subspace at each switching time by taking into account the nonzero reachable space from the previous mode. Afterwards, the backward approach extends the observable subspace of the current mode by taking observability information from the next mode into account. This results in an overall reduced switched system which is minimal and has the same input-output behavior as original system. Some examples are provided to illustrate the approach.},
note = {open access},
keywords = {controllability, model-reduction, observability, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
Minimal realization is discussed for linear switched systems with a given switching signal. We propose a consecutive forward and backward approach for the time-interval of interest. The forward approach refers to extending the reachable subspace at each switching time by taking into account the nonzero reachable space from the previous mode. Afterwards, the backward approach extends the observable subspace of the current mode by taking observability information from the next mode into account. This results in an overall reduced switched system which is minimal and has the same input-output behavior as original system. Some examples are provided to illustrate the approach. |
Wijnbergen, Paul; Trenn, Stephan Optimal control of DAEs with unconstrained terminal costs Proceedings Article In: Proc. 60th IEEE Conf. Decision and Control (CDC 2021), pp. 5275-5280, 2021. @inproceedings{WijnTren21b,
title = {Optimal control of DAEs with unconstrained terminal costs},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/09/Preprint-WT210927.pdf, Preprint},
doi = {10.1109/CDC45484.2021.9682950},
year = {2021},
date = {2021-09-27},
urldate = {2021-09-27},
booktitle = {Proc. 60th IEEE Conf. Decision and Control (CDC 2021)},
pages = {5275-5280},
abstract = {This paper is concerned with the linear quadratic optimal control problem for impulse controllable differential algebraic equations on a bounded half open interval. Regarding the cost functional, a general positive semi-definite weight matrix is considered in the terminal cost. It is shown that for this problem, there generally does not exist an input that minimizes the cost functional. First it is shown that the problem can be reduced to finding an input to an index-1 DAE that minimizes a different quadratic cost functional. Second, necessary and sufficient conditions in terms of matrix equations are given for the existence of an optimal control.},
keywords = {DAEs, optimal-control, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
This paper is concerned with the linear quadratic optimal control problem for impulse controllable differential algebraic equations on a bounded half open interval. Regarding the cost functional, a general positive semi-definite weight matrix is considered in the terminal cost. It is shown that for this problem, there generally does not exist an input that minimizes the cost functional. First it is shown that the problem can be reduced to finding an input to an index-1 DAE that minimizes a different quadratic cost functional. Second, necessary and sufficient conditions in terms of matrix equations are given for the existence of an optimal control. |
Hossain, Sumon; Trenn, Stephan Minimal realization for linear switched systems with a single switch Proceedings Article In: Proc. European Control Conference (ECC21), pp. 1168-1173, Rotterdam, Netherlands, 2021. @inproceedings{HossTren21b,
title = {Minimal realization for linear switched systems with a single switch},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/04/Preprint-HT210406.pdf, Preprint},
doi = {10.23919/ECC54610.2021.9654948},
year = {2021},
date = {2021-06-29},
urldate = {2021-06-29},
booktitle = {Proc. European Control Conference (ECC21)},
pages = {1168-1173},
address = {Rotterdam, Netherlands},
abstract = {We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple examples to illustrate the approach.},
keywords = {controllability, normal-forms, observability, solution-theory, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
We discuss the problem of minimal realization for linear switched systems with a given switching signal and present some preliminary results for the single switch case. The key idea is to extend the reachable subspace of the second mode to include nonzero initial values (resulting from the first mode) and also extend the observable subspace of the first mode by taking information from the second mode into account. We provide some simple examples to illustrate the approach. |
Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential-algebraic equations Proceedings Article In: IFAC-PapersOnLine (Proceedings of the MTNS 2020/21), pp. 186-191, IFAC Elsevier, 2021, (open access). @inproceedings{ChenTren21b,
title = {On geometric and differentiation index of nonlinear differential-algebraic equations},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/ChenTren21b.pdf, Paper},
doi = {10.1016/j.ifacol.2021.06.075},
year = {2021},
date = {2021-04-06},
urldate = {2021-04-06},
booktitle = {IFAC-PapersOnLine (Proceedings of the MTNS 2020/21)},
volume = {54},
number = {9},
pages = {186-191},
publisher = {Elsevier},
organization = {IFAC},
abstract = {We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices.},
note = {open access},
keywords = {DAEs, nonlinear, solution-theory},
pubstate = {published},
tppubtype = {inproceedings}
}
We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices. |
Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions Journal Article In: IEEE Transactions on Automatic Control, vol. 66, no. 4, pp. 1513-1528, 2021. @article{IervTren21,
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/2020/02/Preprint-ITV200204.pdf, Preprint},
doi = {10.1109/TAC.2020.2996597},
year = {2021},
date = {2021-04-01},
urldate = {2021-04-01},
journal = {IEEE Transactions on Automatic Control},
volume = {66},
number = {4},
pages = {1513-1528},
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.},
keywords = {LMIs, Lyapunov, nonlinear, solution-theory, stability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
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. |
Chen, Yahao; Trenn, Stephan An approximation for nonlinear differential-algebraic equations via singular perturbation theory Proceedings Article In: Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC-PapersOnLine, pp. 187-192, Brussels, Belgium, 2021, (open access). @inproceedings{ChenTren21c,
title = {An approximation for nonlinear differential-algebraic equations via singular perturbation theory},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/ChenTren21c.pdf, Paper
},
doi = {10.1016/j.ifacol.2021.08.496},
year = {2021},
date = {2021-03-26},
urldate = {2021-03-26},
booktitle = {Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC-PapersOnLine},
volume = {54},
number = {5},
pages = {187-192},
address = {Brussels, Belgium},
abstract = {In this paper, we study the jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector introduced in Liberzon and Trenn (2009) for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods (one is from the paper Liberzon and Trenn (2012) and the other is given by a MATLAB function) to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter and we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the C1-solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs.},
note = {open access},
keywords = {DAEs, nonlinear, normal-forms, solution-theory},
pubstate = {published},
tppubtype = {inproceedings}
}
In this paper, we study the jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector introduced in Liberzon and Trenn (2009) for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods (one is from the paper Liberzon and Trenn (2012) and the other is given by a MATLAB function) to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter and we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the C1-solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs. |
Trenn, Stephan; Unger, Benjamin Unimodular transformations for DAE initial trajectory problems Proceedings Article In: PAMM · Proc. Appl. Math. Mech., pp. e202000322, Wiley-VCH GmbH, 2021, (Open Access.). @inproceedings{TrenUnge20,
title = {Unimodular transformations for DAE initial trajectory problems},
author = {Stephan Trenn and Benjamin Unger},
url = {https://stephantrenn.net/wp-content/uploads/2021/01/pamm.202000322.pdf, Paper},
doi = {10.1002/pamm.202000322},
year = {2021},
date = {2021-01-26},
booktitle = {PAMM · Proc. Appl. Math. Mech.},
volume = {20},
number = {1},
pages = {e202000322},
publisher = {Wiley-VCH GmbH},
abstract = {We consider linear time-invariant differential-algebraic equations (DAEs). For high-index DAEs, it is often the first step to perform an index reduction, which can be realized with a unimodular matrix. In this contribution, we illustrate the effect of unimodular transformations on initial trajectory problems associated with DAEs.},
note = {Open Access.},
keywords = {DAEs, piecewise-smooth-distributions, solution-theory},
pubstate = {published},
tppubtype = {inproceedings}
}
We consider linear time-invariant differential-algebraic equations (DAEs). For high-index DAEs, it is often the first step to perform an index reduction, which can be realized with a unimodular matrix. In this contribution, we illustrate the effect of unimodular transformations on initial trajectory problems associated with DAEs. |
Chen, Yahao; Trenn, Stephan The differentiation index of nonlinear differential-algebraic equations versus the relative degree of nonlinear control systems Proceedings Article In: PAMM · Proc. Appl. Math. Mech. 2020, pp. e202000162, Wiley-VCH GmbH, 2021, (Open Access.). @inproceedings{ChenTren21a,
title = {The differentiation index of nonlinear differential-algebraic equations versus the relative degree of nonlinear control systems},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/01/pamm.202000162.pdf, Paper},
doi = {10.1002/pamm.202000162},
year = {2021},
date = {2021-01-25},
booktitle = {PAMM · Proc. Appl. Math. Mech. 2020},
volume = {20},
number = {1},
pages = {e202000162},
publisher = {Wiley-VCH GmbH},
abstract = {It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differen- tiation index for nonlinear differential-algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems.},
note = {Open Access.},
keywords = {DAEs, nonlinear, normal-forms, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differen- tiation index for nonlinear differential-algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems. |
Wijnbergen, Paul; Trenn, Stephan Impulse-free interval-stabilization of switched differential algebraic equations Journal Article In: Systems & Control Letters, vol. 149, pp. 104870.1-10, 2021, (Open Access.). @article{WijnTren21a,
title = {Impulse-free interval-stabilization of switched differential algebraic equations},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/01/24-SCL149-104870.pdf, Paper},
doi = {10.1016/j.sysconle.2020.104870},
year = {2021},
date = {2021-01-23},
urldate = {2021-01-23},
journal = {Systems & Control Letters},
volume = {149},
pages = {104870.1-10},
abstract = {In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory.
},
note = {Open Access.},
keywords = {controllability, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory.
|
2020
|
Borsche, Raul; Kocoglu, Damla; Trenn, Stephan A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Journal Article In: Mathematics of Control, Signals, and Systems (MCSS), vol. 32, pp. 455-487, 2020, (Open Access). @article{BorsKoco20,
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/2020/11/23-MCSS2020.pdf, Paper},
doi = {10.1007/s00498-020-00267-7},
year = {2020},
date = {2020-11-18},
urldate = {2020-11-18},
journal = {Mathematics of Control, Signals, and Systems (MCSS)},
volume = {32},
pages = {455-487},
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.},
note = {Open Access},
keywords = {DAEs, delay, networks, PDEs, piecewise-smooth-distributions, solution-theory, switched-DAEs},
pubstate = {published},
tppubtype = {article}
}
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. |
Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan Stability analysis for switched discrete-time linear singular systems Journal Article In: Automatica, vol. 119, no. 109100, 2020. @article{AnhLinh20,
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/2020/02/Preprint-ALTT200515.pdf, Preprint},
doi = {10.1016/j.automatica.2020.109100},
year = {2020},
date = {2020-09-01},
urldate = {2020-09-01},
journal = {Automatica},
volume = {119},
number = {109100},
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. We first provide a sufficient stability conditions in terms of Lyapunov functions. Furthermore, we generalize the notion of joint spectral radius of a finite set of matrix pairs, which allows us to fully characterize exponential stability.},
keywords = {discrete-time, stability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
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. We first provide a sufficient stability conditions in terms of Lyapunov functions. Furthermore, we generalize the notion of joint spectral radius of a finite set of matrix pairs, which allows us to fully characterize exponential stability. |
Wijnbergen, Paul; Jeeninga, Mark; Trenn, Stephan On stabilizability of switched differential algebraic equations Proceedings Article In: IFAC-PapersOnLine 53-2, pp. 4304-4309, 2020, (Proc. IFAC World Congress 2020, Berlin, Germany. Open access.). @inproceedings{WijnJeen20,
title = {On stabilizability of switched differential algebraic equations},
author = {Paul Wijnbergen and Mark Jeeninga and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/06/WijnJeen20.pdf, Paper},
doi = {10.1016/j.ifacol.2020.12.2580},
year = {2020},
date = {2020-07-06},
booktitle = {IFAC-PapersOnLine 53-2},
pages = {4304-4309},
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. This geometric approach can also be utilized to derive a novel characterization of controllability.},
note = {Proc. IFAC World Congress 2020, Berlin, Germany. Open access.},
keywords = {DAEs, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
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. This geometric approach can also be utilized to derive a novel characterization of controllability. |
Hossain, Sumon; Trenn, Stephan A time-varying Gramian based model reduction approach for Linear Switched Systems Proceedings Article In: IFAC PapersOnline 53-2, pp. 5629-5634, 2020, (Proc. IFAC World Congress 2020, Berlin, Germany. Open access.). @inproceedings{HossTren20a,
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/2021/06/HossTren20a.pdf, Paper (open access)},
doi = {10.1016/j.ifacol.2020.12.1580},
year = {2020},
date = {2020-07-05},
urldate = {2020-07-05},
booktitle = {IFAC PapersOnline 53-2},
pages = {5629-5634},
abstract = {We propose a model reduction approach for switched linear system based on a balanced truncation reduction method for linear time-varying systems. The key idea is to approximate the piecewise-constant coefficient matrices with continuous time-varying coefficients and then apply available balance truncation methods for (continuous) time-varying systems. The proposed method is illustrated with a low dimensional academic example.},
note = {Proc. IFAC World Congress 2020, Berlin, Germany. Open access.},
keywords = {model-reduction, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
We propose a model reduction approach for switched linear system based on a balanced truncation reduction method for linear time-varying systems. The key idea is to approximate the piecewise-constant coefficient matrices with continuous time-varying coefficients and then apply available balance truncation methods for (continuous) time-varying systems. The proposed method is illustrated with a low dimensional academic example. |
Wijnbergen, Paul; Trenn, Stephan Impulse controllability of switched differential-algebraic equations Proceedings Article In: Proc. European Control Conference (ECC 2020), pp. 1561-1566, Saint Petersburg, Russia, 2020. @inproceedings{WijnTren20,
title = {Impulse controllability of switched differential-algebraic equations},
author = {Paul Wijnbergen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2020/02/Preprint-WT200204.pdf, Preprint},
doi = {10.23919/ECC51009.2020.9143713},
year = {2020},
date = {2020-05-15},
booktitle = {Proc. European Control Conference (ECC 2020)},
pages = {1561-1566},
address = {Saint Petersburg, Russia},
abstract = {This paper addresses impulse controllability of switched DAEs on a finite interval. First we present a forward approach where we define certain subspaces forward in time. These subpsaces are then 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.},
keywords = {controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
This paper addresses impulse controllability of switched DAEs on a finite interval. First we present a forward approach where we define certain subspaces forward in time. These subpsaces are then 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. |
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Proceedings Article In: Proc. European Control Conference (ECC 2020), pp. 911-916, Saint Petersburg, Russia, 2020. @inproceedings{LeeBerg20,
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/2020/02/Preprint-LBTS200204.pdf, Preprint},
doi = {10.23919/ECC51009.2020.9143983},
year = {2020},
date = {2020-05-14},
booktitle = {Proc. European Control Conference (ECC 2020)},
pages = {911-916},
address = {Saint Petersburg, Russia},
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.},
keywords = {funnel-control, networks, nonlinear, synchronization},
pubstate = {published},
tppubtype = {inproceedings}
}
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. |
Wijnbergen, Paul; Trenn, Stephan A forward approach to controllability of switched DAEs Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. @misc{WijnTren20m,
title = {A forward approach to controllability of switched DAEs},
author = {Paul Wijnbergen and Stephan Trenn},
editor = {Raffaella Carloni and Bayu Jayawardhana and Erjen Lefeber},
url = {https://www.beneluxmeeting.nl/2020/uploads/papers/boa.pdf, Book of Abstracts
https://stephantrenn.net/wp-content/uploads/2021/03/WijnTren20.pdf, Extended Abstract},
year = {2020},
date = {2020-03-12},
howpublished = {Book of Abstracts - 39th Benelux Meeting on Systems and Control},
keywords = {controllability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
|
Hossain, Sumon; Trenn, Stephan Model reduction of switched systems in time-varying approach Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. @misc{HossTren20m,
title = {Model reduction of switched systems in time-varying approach},
author = {Sumon Hossain and Stephan Trenn},
editor = {Raffaella Carloni and Bayu Jayawardhana and Erjen Lefeber},
url = {https://www.beneluxmeeting.nl/2020/uploads/papers/boa.pdf, Book of Abstracts
https://stephantrenn.net/wp-content/uploads/2021/03/HossTren20.pdf, Extended Abstract},
year = {2020},
date = {2020-03-12},
howpublished = {Book of Abstracts - 39th Benelux Meeting on Systems and Control},
keywords = {model-reduction, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
|
Trenn, Stephan The Laplace transform and inconsistent initial values Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21, presented at MTNS 2022). @misc{Tren20m,
title = {The Laplace transform and inconsistent initial values},
author = {Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-Tre200122.pdf, Extended Abstract},
year = {2020},
date = {2020-01-22},
urldate = {2020-01-22},
abstract = {Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions.},
howpublished = {Extended Abstract},
note = {accepted for cancelled MTNS 20/21, presented at MTNS 2022},
keywords = {DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs},
pubstate = {published},
tppubtype = {misc}
}
Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions. |
Iervolino, Raffaele; Vasca, Francesco; Trenn, Stephan Discontinuous Lyapunov functions for discontinous piecewise-affine systems Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). @misc{IervTren20m,
title = {Discontinuous Lyapunov functions for discontinous piecewise-affine systems},
author = {Raffaele Iervolino and Francesco Vasca and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2020/01/Preprint-ITV200122.pdf, Extended Abstract},
year = {2020},
date = {2020-01-22},
urldate = {2020-01-22},
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. We first introduce the feasible Filippov solution concept 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 highlighted 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. },
howpublished = {Extended Abstract},
note = {accepted for cancelled MTNS 20/21},
keywords = {LMIs, Lyapunov, stability, switched-systems},
pubstate = {published},
tppubtype = {misc}
}
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. We first introduce the feasible Filippov solution concept 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 highlighted 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. |
2019
|
Lee, Jin Gyu; Trenn, Stephan Asymptotic tracking via funnel control Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 4228-4233, Nice, France, 2019. @inproceedings{LeeTren19,
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},
doi = {10.1109/CDC40024.2019.9030274},
year = {2019},
date = {2019-12-13},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
pages = {4228-4233},
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.},
keywords = {funnel-control},
pubstate = {published},
tppubtype = {inproceedings}
}
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. |
Trenn, Stephan; Unger, Benjamin Delay regularity of differential-algebraic equations Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 989-994, Nice, France, 2019. @inproceedings{TrenUnge19,
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},
doi = {10.1109/CDC40024.2019.9030146},
year = {2019},
date = {2019-12-12},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
pages = {989-994},
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.},
keywords = {DAEs, delay, solution-theory},
pubstate = {published},
tppubtype = {inproceedings}
}
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. |
Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan The one-step-map for switched singular systems in discrete-time Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 605-610, Nice, France, 2019. @inproceedings{AnhLinh19,
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},
doi = {10.1109/CDC40024.2019.9030154},
year = {2019},
date = {2019-12-11},
urldate = {2019-12-11},
booktitle = {Proc. 58th IEEE Conf. Decision Control (CDC) 2019},
pages = {605-610},
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.},
keywords = {discrete-time, solution-theory, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
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. |
Trenn, Stephan Asymptotic tracking with funnel control Proceedings Article In: PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2019, (online). @inproceedings{Tren19,
title = {Asymptotic tracking with funnel control},
author = {Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2019/11/45-PAMM19-201900071.pdf, Paper},
doi = {10.1002/pamm.201900071},
year = {2019},
date = {2019-09-09},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
journal = {PAMM - Proc. Appl. Math. Mech.},
publisher = {WILEY-VCH Verlag},
abstract = {Funnel control is a strikingly simple control technique to ensure model free practical tracking for quite general nonlinear systems. It has its origin in the adaptive control theory, in particular, it is based on the principle of high gain feedback control. The key idea of funnel control is to chose the feedback gain large when the tracking error approaches the prespecified error tolerance (the funnel boundary). It was long believed that it is a theoretical limitation of funnel control not being able to achieve asymptotic tracking, however, in this contribution it will be shown that this is not the case.},
note = {online},
keywords = {funnel-control, stability},
pubstate = {published},
tppubtype = {inproceedings}
}
Funnel control is a strikingly simple control technique to ensure model free practical tracking for quite general nonlinear systems. It has its origin in the adaptive control theory, in particular, it is based on the principle of high gain feedback control. The key idea of funnel control is to chose the feedback gain large when the tracking error approaches the prespecified error tolerance (the funnel boundary). It was long believed that it is a theoretical limitation of funnel control not being able to achieve asymptotic tracking, however, in this contribution it will be shown that this is not the case. |
Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks Journal Article In: Automatica, vol. 101, pp. 280-289, 2019. @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 = {DAEs, networks, observability},
pubstate = {published},
tppubtype = {article}
}
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. |
Tanwani, Aneel; Trenn, Stephan Detectability and observer design for switched differential algebraic equations Journal Article In: Automatica, vol. 99, pp. 289-300, 2019. @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 = {DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {article}
}
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. |
2018
|
Gross, Tjorben B.; Trenn, Stephan; Wirsen, Andreas Switch induced instabilities for stable power system DAE models Proceedings Article In: IFAC-PapersOnLine, pp. 127-132, 2018, (Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)). @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 = {application, stability, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {inproceedings}
}
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. |
