Author: stephan

@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)

https://github.com/Mrc-Smn/QWF_lean, Lean formalization of the main result (Theorem 2.6)},

doi = {10.1016/j.laa.2009.12.036},

year  = {2012},

date = {2012-01-01},

urldate = {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 = {},

pubstate = {published},

tppubtype = {article}

}

@article{KariTren25,

title = {Funnel-based output tracking control for nonlinear impulsive switched systems},

author = {Atiyeh Karimi-Pour and Stephan Trenn and Moosa Ayati and Mohammad Reza Zakerzadeh},

url = {https://stephantrenn.net/wp-content/uploads/2025/12/KariTren25.pdf, Paper},

doi = {10.1016/j.nahs.2025.101661},

year  = {2025},

date = {2025-12-01},

urldate = {2025-12-01},

journal = {Nonlinear Analysis: Hybrid Systems},

volume = {60},

number = {101661},

pages = {1-25},

abstract = {We study output tracking for nonlinear impulsive switched systems with global relative degree one under prescribed performance requirements. Classical funnel control is not directly applicable in this setting, since output jumps can cause violations of funnel constraints. To address this, we design an adjusted funnel boundary that contracts prior to jumps and expands afterward, computed offline based on the stability of the internal dynamics and bounded jump maps. We also derive sufficient conditions ensuring bounded control input. To obtain tighter bounds, practical ISS is employed in place of BIBO stability, yielding smaller input requirements. Additional refinements include asymmetric jump bounds, level-set adjustments, and real-time funnel adaptation, which further improve performance. Numerical examples confirm stability and practical tracking under disturbance impulses and switching.},

note = {open access},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@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/2025/07/HossTren25.pdf, Paper

https://doi.org/10.5281/zenodo.8133788, Matlab implementation},

doi = {10.52825/dae-p.v3i.688},

year  = {2025},

date = {2025-02-04},

urldate = {2025-02-04},

journal = {DAE Panel},

volume = {3},

number = {688},

pages = {1-26},

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 = {open access},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@article{YinJaya23a,

title = {On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time},

author = {Hao Yin and Bayu Jayawardhana and Stephan Trenn},

url = {https://stephantrenn.net/wp-content/uploads/2022/04/Preprint-YJT221110.pdf, Preprint},

doi = {10.1109/TAC.2023.3237492},

year  = {2023},

date = {2023-10-01},

urldate = {2023-01-16},

journal = {IEEE Transactions on Automatic Control},

volume = {68},

issue = {10},

pages = {6409-6416},

abstract = {This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and non- contracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global ex- ponential stability of its variational system. Based on this equiva- lence property, sufficient conditions for a mode-dependent average dwell/leave-time based switching law to be contractive are estab- lished. Correspondingly, LMI conditions are derived that allow for numerical validation of contraction property of nonlinear switched systems, which include those with all non-contracting modes.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}