2025
|
Hossain, Sumon; Trenn, Stephan Model reduction for switched differential-algebraic equations with known switching signal Journal Article In: DAE Panel, 2025, (to appear). @article{HossTren25,
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/2025/01/Preprint-HT250114.pdf, Preprint
https://doi.org/10.5281/zenodo.8133788, Matlab implementation},
year = {2025},
date = {2025-01-14},
urldate = {2025-01-14},
journal = {DAE Panel},
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 that make 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 to 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 applying an exact model reduction which reduces the switched DAE to a switched ordinary system with jumps and which carefully keeps 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 = {to appear},
keywords = {model-reduction, switched-DAEs},
pubstate = {published},
tppubtype = {article}
}
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 that make 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 to 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 applying an exact model reduction which reduces the switched DAE to a switched ordinary system with jumps and which carefully keeps 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. |
2024
|
Trenn, Stephan; Sutrisno,; Thuan, Do Duc; Ha, Phi Model reduction of singular switched systems in discrete time Unpublished 2024, (submitted). @unpublished{TrenSutr24pp,
title = {Model reduction of singular switched systems in discrete time},
author = {Stephan Trenn and Sutrisno and Do Duc Thuan and Phi Ha},
url = {https://stephantrenn.net/wp-content/uploads/2025/01/Preprint-TSTP241108.pdf, Preprint},
year = {2024},
date = {2024-11-08},
urldate = {2024-11-08},
abstract = {Based on our recently established solution characterization of switched singular descriptor systems in discrete time, we propose a time-varying balanced truncation method. For that we consider the switched system on a finite time interval and define corresponding time-varying reachability and observability Gramians. We then show that these capture essential quantitative information about reachable and observable state directions. Based on these Gramians we formulate a time-varying balanced truncation method resulting in a fully-time varying linear system with possible varying state dimensions. We illustrate this method with a small dynamic Leontief model, where we can reduce the size to one third without altering the input-output behavior significantly. We also show that the method is suitable for a medium size random descriptor system (100 x100) resulting in a time-varying system of less then a tenth of the size where the outputs of the original and reduced system are indistinguishable.},
note = {submitted},
keywords = {DAEs, discrete-time, model-reduction, switched-DAEs, switched-systems},
pubstate = {published},
tppubtype = {unpublished}
}
Based on our recently established solution characterization of switched singular descriptor systems in discrete time, we propose a time-varying balanced truncation method. For that we consider the switched system on a finite time interval and define corresponding time-varying reachability and observability Gramians. We then show that these capture essential quantitative information about reachable and observable state directions. Based on these Gramians we formulate a time-varying balanced truncation method resulting in a fully-time varying linear system with possible varying state dimensions. We illustrate this method with a small dynamic Leontief model, where we can reduce the size to one third without altering the input-output behavior significantly. We also show that the method is suitable for a medium size random descriptor system (100 x100) resulting in a time-varying system of less then a tenth of the size where the outputs of the original and reduced system are indistinguishable. |
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. |
2023
|
Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, vol. 154, no. 111065, pp. 1-9, 2023, (open access). @article{HossTren23a,
title = {Reduced realization for switched linear systems with known mode sequence},
author = {Sumon Hossain and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2024/02/HossTren23a.pdf, Paper
https://doi.org/10.5281/zenodo.6410136, Matlab sources},
doi = {10.1016/j.automatica.2023.111065},
year = {2023},
date = {2023-08-01},
urldate = {2023-08-01},
journal = {Automatica},
volume = {154},
number = {111065},
pages = {1-9},
abstract = {We consider switched linear systems with mode-dependent state-dimensions and/or state jumps and propose a method to obtain a switched system of reduced size with identical input-output behavior. Our approach is based in considering time-dependent reachability and unobservability spaces as well as suitable extended reachability and restricted unobservability spaces together with the notion of a weak Kalman decomposition. A key feature of our approach is that only the mode sequence of the switching signal needs to be known and not the exact switching times. However, the size of a minimal realization will in general depend on the mode durations, hence it cannot be expected that our method always leads to minimal realization. Nevertheless, we show that our method is optimal in the sense that a repeated application doesn’t lead to a further reduction and we also highlight a practically relevant special case, where minimality is achieved.},
note = {open access},
keywords = {controllability, model-reduction, observability, switched-systems},
pubstate = {published},
tppubtype = {article}
}
We consider switched linear systems with mode-dependent state-dimensions and/or state jumps and propose a method to obtain a switched system of reduced size with identical input-output behavior. Our approach is based in considering time-dependent reachability and unobservability spaces as well as suitable extended reachability and restricted unobservability spaces together with the notion of a weak Kalman decomposition. A key feature of our approach is that only the mode sequence of the switching signal needs to be known and not the exact switching times. However, the size of a minimal realization will in general depend on the mode durations, hence it cannot be expected that our method always leads to minimal realization. Nevertheless, we show that our method is optimal in the sense that a repeated application doesn’t lead to a further reduction and we also highlight a practically relevant special case, where minimality is achieved. |
2022
|
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. |
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. |
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. |
2020
|
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. |
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}
}
|
