Hossain, Sumon; Trenn, Stephan Model reduction for switched differential-algebraic equations with known switching signal Unpublished 2023, (submitted). @unpublished{HossTrenn23pp,
title = {Model reduction for switched differential-algebraic equations with known switching signal},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2023/09/Preprint-HT230816.pdf, Preprint
https://doi.org/10.5281/zenodo.8133789, Matlab implementation},
year = {2023},
date = {2023-08-16},
abstract = {Building on our recently proposed model reduction methods for switched ordinary linear systems we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features which makes a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first apply an exact model reduction which reduces the switched DAE to a switched ordinary systems with jumps and carefully keep track of the impulsive effects. As a second step we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples.},
note = {submitted},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Building on our recently proposed model reduction methods for switched ordinary linear systems we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features which makes a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first apply an exact model reduction which reduces the switched DAE to a switched ordinary systems with jumps and carefully keep track of the impulsive effects. As a second step we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples. |
Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Unpublished 2023, (submitted). @unpublished{MostTren23pp,
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/2023/07/Preprint-MTV230714.pdf, Preprint},
year = {2023},
date = {2023-07-14},
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. |
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. |
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. |
