Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Unpublished 2023, (submitted). @unpublished{ChenTren23pp,
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/2023/03/Preprint-CT230302.pdf, Preprint},
year = {2023},
date = {2023-03-02},
urldate = {2022-05-05},
abstract = {n this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion
of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and
uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [27, 28], for linear DAEs. Then we show
that the common Lyapunov functions stability conditions proposed in our previous work [16] (which differ from the ones in
[28]) can be applied to switched nonlinear DAEs with high-index models which are not equivalent to the nonlinear Weierstrass
form. Moreover, we generalize the commutativity stability conditions [32] for switched nonlinear ordinary differential equations
to the switched nonlinear DAEs case. Finally, some simulation results of switching electrical circuits and numerical examples
are given to illustrate the usefulness of the proposed stability conditions.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
n this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion
of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and
uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [27, 28], for linear DAEs. Then we show
that the common Lyapunov functions stability conditions proposed in our previous work [16] (which differ from the ones in
[28]) can be applied to switched nonlinear DAEs with high-index models which are not equivalent to the nonlinear Weierstrass
form. Moreover, we generalize the commutativity stability conditions [32] for switched nonlinear ordinary differential equations
to the switched nonlinear DAEs case. Finally, some simulation results of switching electrical circuits and numerical examples
are given to illustrate the usefulness of the proposed stability conditions. |
Lanza, Lukas; Dennstädt, Dario; Worthmann, Karl; Trenn, Stephan; Schaller, Manuel Sampled-data funnel control with zero-order hold Unpublished 2023, (submitted). @unpublished{LanzDann23pp,
title = {Sampled-data funnel control with zero-order hold},
author = {Lukas Lanza and Dario Dennstädt and Karl Worthmann and Stephan Trenn and Manuel Schaller},
url = {https://stephantrenn.net/wp-content/uploads/2023/03/Preprint-LDWTS230301.pdf, Preprint
https://arxiv.org/abs/2303.00523, arXiv},
year = {2023},
date = {2023-03-01},
abstract = {Output reference tracking for nonlinear systems with arbitrary relative degree via sampled-data feedback control with zero-order hold is studied. We propose a novel sample-and-hold feedback controller, which achieves output reference tracking with prescribed transient behaviour of the tracking error. Furthermore, we derive explicit bounds on the maximal required input signal and the global sampling time such that the proposed controller is feasible for all times. The results are illustrated with a numerical example.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Output reference tracking for nonlinear systems with arbitrary relative degree via sampled-data feedback control with zero-order hold is studied. We propose a novel sample-and-hold feedback controller, which achieves output reference tracking with prescribed transient behaviour of the tracking error. Furthermore, we derive explicit bounds on the maximal required input signal and the global sampling time such that the proposed controller is feasible for all times. The results are illustrated with a numerical example. |
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin A novel two stages funnel controller limiting the error derivative Unpublished 2023, (submitted). @unpublished{HuTren23pp,
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/2023/02/Preprint-HTZ230227.pdf, Preprint},
year = {2023},
date = {2023-02-27},
urldate = {2022-11-30},
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 in this paper. 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 = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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 in this paper. 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. |
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Unpublished 2023, (submitted). @unpublished{LeeBerg23pp,
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/2023/01/Preprint-LBTS230116.pdf, Preprint
https://arxiv.org/abs/2110.05330, ArXiV},
year = {2023},
date = {2023-01-16},
urldate = {2022-06-29},
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 = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Unpublished 2022, (provisionally accepted in IEEE TAC). @unpublished{HossTren22ppc,
title = {Midpoint based balanced truncation for switched linear systems with known switching signal},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/12/Preprint-HT221222.pdf, Preprint},
year = {2022},
date = {2022-12-21},
urldate = {2022-12-21},
abstract = {We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size.},
note = {provisionally accepted in IEEE TAC},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size. |
Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Unpublished 2022, (submitted). @unpublished{MostTren22ppb,
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/2022/11/Preprint-MTV221128.pdf, Preprint},
year = {2022},
date = {2022-11-28},
urldate = {2022-11-28},
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 = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
Sutrisno,; Trenn, Stephan Reachability and Controllability Characterizations for Linear Switched Systems in Discrete Time: A Geometric Approach Unpublished 2022, (submitted). @unpublished{SutrTren22ppa,
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},
year = {2022},
date = {2022-11-25},
urldate = {2022-11-25},
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.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
Sutrisno,; Trenn, Stephan Nonlinear Switched Singular Systems in Discrete-time: The One-step Map and Stability Under Arbitrary Switching Signals Unpublished 2022, (submitted). @unpublished{SutrTren22ppb,
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/2022/11/Preprint-ST221125b.pdf, Preprint},
year = {2022},
date = {2022-11-25},
urldate = {2022-11-25},
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 Lyapunov functions.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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 Lyapunov functions. |
Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched differential algebraic equations Unpublished 2022, (submitted). @unpublished{WijnTren22pp,
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},
year = {2022},
date = {2022-08-06},
urldate = {2022-08-06},
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 = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
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. |
Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Stability of switched systems with multiple equilibria: a mixed stable-unstable subsystem case Unpublished 2022, (submitted). @unpublished{YinJaya22ppb,
title = {Stability of switched systems with multiple equilibria: a mixed stable-unstable subsystem case},
author = {Hao Yin and Bayu Jayawardhana and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2022/08/Preprint-YJT220730.pdf, Preprint},
year = {2022},
date = {2022-07-30},
urldate = {2022-07-30},
abstract = {This paper studies stability of switched systems that are composed of a mixture of stable and unstable modes with multiple equilibria. The main results of this paper include some sufficient conditions concerning set convergence of switched nonlinear systems. We show that under suitable dwell-time and leave-time switching laws, trajectories converge to an initial set and then stay in a convergent set. Based on these conditions, LMI conditions are derived that allow for numerical validation of practical stability of switched affine systems, which include those with all unstable modes. Two examples are provided to verify the theoretical results.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
This paper studies stability of switched systems that are composed of a mixture of stable and unstable modes with multiple equilibria. The main results of this paper include some sufficient conditions concerning set convergence of switched nonlinear systems. We show that under suitable dwell-time and leave-time switching laws, trajectories converge to an initial set and then stay in a convergent set. Based on these conditions, LMI conditions are derived that allow for numerical validation of practical stability of switched affine systems, which include those with all unstable modes. Two examples are provided to verify the theoretical results. |
