2023
|
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Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Unpublished 2023, (submitted). @unpublished{ChenTren23pp,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 = {DAEs, Lyapunov, nonlinear, normal-forms, solution-theory, stability, switched-DAEs, switched-systems},
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tppubtype = {unpublished}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. |
2022
|
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, open-access},
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 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; Respondek, Witold Normal forms and internal regularization of nonlinear differential-algebraic control systems Journal Article In: International Journal of Robust and Nonlinear Control, vol. 2021, no. 31, pp. 6562-6584, 2021, (open access). @article{ChenTren21d,
title = {Normal forms and internal regularization of nonlinear differential-algebraic control systems},
author = {Yahao Chen and Stephan Trenn and Witold Respondek},
url = {https://stephantrenn.net/wp-content/uploads/2021/06/ChenTren21d.pdf, Paper},
doi = {10.1002/rnc.5623},
year = {2021},
date = {2021-04-13},
urldate = {2021-04-13},
journal = {International Journal of Robust and Nonlinear Control},
volume = {2021},
number = {31},
pages = {6562-6584},
abstract = {In this paper, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, i.e., when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs.},
note = {open access},
keywords = {DAEs, nonlinear, normal-forms, open-access, solution-theory},
pubstate = {published},
tppubtype = {article}
}
In this paper, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, i.e., when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs. |
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, open-access, 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. |
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. |
2014
|
Berger, Thomas; Trenn, Stephan Kalman controllability decompositions for differential-algebraic systems Journal Article In: Syst. Control Lett., vol. 71, pp. 54–61, 2014, ISSN: 0167-6911. @article{BergTren14,
title = {Kalman controllability decompositions for differential-algebraic systems},
author = {Thomas Berger and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-BT140603.pdf, Preprint},
doi = {10.1016/j.sysconle.2014.06.004},
issn = {0167-6911},
year = {2014},
date = {2014-01-01},
journal = {Syst. Control Lett.},
volume = {71},
pages = {54--61},
abstract = {We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition (KCD). The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part. This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems.},
keywords = {controllability, DAEs, normal-forms},
pubstate = {published},
tppubtype = {article}
}
We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition (KCD). The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part. This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems. |
2013
|
Berger, Thomas; Trenn, Stephan Addition to ``The quasi-Kronecker form for matrix pencils'' Journal Article In: SIAM J. Matrix Anal. & Appl., vol. 34, no. 1, pp. 94–101, 2013. @article{BergTren13,
title = {Addition to ``The quasi-Kronecker form for matrix pencils''},
author = {Thomas Berger and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/BergTren13.pdf, Paper},
doi = {10.1137/120883244},
year = {2013},
date = {2013-02-11},
journal = {SIAM J. Matrix Anal. & Appl.},
volume = {34},
number = {1},
pages = {94--101},
abstract = {We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences.},
keywords = {DAEs, normal-forms, solution-theory},
pubstate = {published},
tppubtype = {article}
}
We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences. |
2012
|
Berger, Thomas; Trenn, Stephan The quasi-Kronecker form for matrix pencils Journal Article In: SIAM J. Matrix Anal. & Appl., vol. 33, no. 2, pp. 336–368, 2012. @article{BergTren12,
title = {The quasi-Kronecker form for matrix pencils},
author = {Thomas Berger and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/BergTren12.pdf, Paper},
doi = {10.1137/110826278},
year = {2012},
date = {2012-05-03},
journal = {SIAM J. Matrix Anal. & Appl.},
volume = {33},
number = {2},
pages = {336--368},
abstract = {We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit.},
keywords = {DAEs, normal-forms, solution-theory},
pubstate = {published},
tppubtype = {article}
}
We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit. |
Berger, Thomas; Ilchmann, Achim; Trenn, Stephan The quasi-Weierstraß form for regular matrix pencils Journal Article In: Linear Algebra Appl., vol. 436, no. 10, pp. 4052–4069, 2012, (published online February 2010). @article{BergIlch12a,
title = {The quasi-Weierstraß form for regular matrix pencils},
author = {Thomas Berger and Achim Ilchmann and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-BIT091017.pdf, Preprint
http://dx.doi.org/10.1016/S0024-3795(11)00688-4, Corrections (see Paragraph 6 of Note to Editors)},
doi = {10.1016/j.laa.2009.12.036},
year = {2012},
date = {2012-01-01},
journal = {Linear Algebra Appl.},
volume = {436},
number = {10},
pages = {4052--4069},
abstract = {Regular linear matrix pencils A- E d in K^{n x n}[d], where K=Q, R or C, and the associated differential algebraic equation (DAE) E x' = A x are studied. The Wong sequences of subspaces are investigate and invoked to decompose the K^n into V* + W*, where any bases of the linear spaces V* and W* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V* and ``pure'' inconsistent initial values W* - {0}. Furthermore, V* and W* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A- E d lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E x' = A x.},
note = {published online February 2010},
keywords = {DAEs, normal-forms, solution-theory},
pubstate = {published},
tppubtype = {article}
}
Regular linear matrix pencils A- E d in K^{n x n}[d], where K=Q, R or C, and the associated differential algebraic equation (DAE) E x' = A x are studied. The Wong sequences of subspaces are investigate and invoked to decompose the K^n into V* + W*, where any bases of the linear spaces V* and W* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V* and ``pure'' inconsistent initial values W* - {0}. Furthermore, V* and W* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A- E d lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E x' = A x. |
2009
|
Trenn, Stephan A normal form for pure differential algebraic systems Journal Article In: Linear Algebra Appl., vol. 430, no. 4, pp. 1070 – 1084, 2009. @article{Tren09a,
title = {A normal form for pure differential algebraic systems},
author = {Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre081215.pdf, Preprint},
doi = {10.1016/j.laa.2008.10.004},
year = {2009},
date = {2009-01-01},
journal = {Linear Algebra Appl.},
volume = {430},
number = {4},
pages = {1070 -- 1084},
abstract = {In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability.},
keywords = {controllability, DAEs, normal-forms, observability, relative-degree},
pubstate = {published},
tppubtype = {article}
}
In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability. |
