Below you find an interactive list of all my publications, which can be filtered by keywords, year, publication type and coauthors. There are also static lists of my books/book-chapters as well as journal-, conference-, and submitted publications.

## 2022 |

Chen, Yahao; Trenn, Stephan Stability analysis of switched nonlinear differential-algebraic equations via nonlinear Weierstrass form Inproceedings In: Proceeding of European Control Conference (ECC22), 2022, (to appear). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, solution-theory, stability, switched-DAEs, switched-systems @inproceedings{ChenTren22b, In this paper, we propose some sufficient conditions for checking the asymptotic stability of switched nonlinear differential-algebraic equations (DAEs) under arbitrary switch- ing signal. We assume that each model of a given switched DAE is externally equivalent to a nonlinear Weierstrass form. With the help of this form, we can define nonlinear consistency projectors and jump-flow solutions for switched nonlinear DAEs. Then we use a different approach from the paper [12] to study the stability of switched DAEs via a novel notion called the jump-flow explicitation, which attaches a nonlinear control system to a given nonlinear DAE and can be used to simplify the common Lyapunov function conditions for both the flow and the jump dynamics of switched nonlinear DAEs. At last, a numerical example is given to illustrate how to check the stability of a switched nonlinear DAE by constructing a common Lyapunov function. |

Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Unpublished 2022, (submitted). Abstract | Links | BibTeX | Tags: DAEs, Lyapunov, nonlinear, normal-forms, solution-theory, stability, switched-DAEs, switched-systems @unpublished{ChenTren22ppb, In this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs).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., [22, 23], for linear DAEs. Then we show that the common Lyapunov functions stability conditions proposed in our previous work [12] (which differ from the ones in [23]) 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 [27] for switched nonlinear ordinary differential equations to the 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. |

## 2021 |

Trenn, Stephan Distributional restriction impossible to define Journal Article In: Examples and Counterexamples, vol. 1, no. 100023, pp. 1-4, 2021, (open access). Abstract | Links | BibTeX | Tags: open-access, piecewise-smooth-distributions, solution-theory @article{Tren21, A counterexample is presented showing that it is not possible to define a restriction for distributions. |

Hossain, Sumon; Trenn, Stephan Minimal realization for linear switched systems with a single switch Inproceedings In: Proc. European Control Conference (ECC21), pp. 1168-1173, Rotterdam, Netherlands, 2021. Abstract | Links | BibTeX | Tags: controllability, normal-forms, observability, solution-theory, switched-systems @inproceedings{HossTren21, 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). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, normal-forms, open-access, solution-theory @article{ChenTren21d, 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 On geometric and differentiation index of nonlinear differential-algebraic equations Inproceedings In: IFAC-PapersOnLine (Proceedings of the MTNS 2020/21), pp. 186-191, IFAC Elsevier, 2021, (open access). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, solution-theory @inproceedings{ChenTren21b, We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices. |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions Journal Article In: IEEE Transactions on Automatic Control, vol. 66, no. 4, pp. 1513-1528, 2021. Abstract | Links | BibTeX | Tags: nonlinear, open-access, solution-theory, stability, switched-systems @article{IervTren21, Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In the first part of the paper the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result. |

Chen, Yahao; Trenn, Stephan An approximation for nonlinear differential-algebraic equations via singular perturbation theory Inproceedings In: Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC Brussels, Belgium, 2021, (open access). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, normal-forms, open-access, solution-theory @inproceedings{ChenTren21c, 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. |

Trenn, Stephan; Unger, Benjamin Unimodular transformations for DAE initial trajectory problems Inproceedings In: PAMM · Proc. Appl. Math. Mech., pp. e202000322, Wiley-VCH GmbH, 2021, (Open Access.). Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory @inproceedings{TrenUnge20, We consider linear time-invariant differential-algebraic equations (DAEs). For high-index DAEs, it is often the first step to perform an index reduction, which can be realized with a unimodular matrix. In this contribution, we illustrate the effect of unimodular transformations on initial trajectory problems associated with DAEs. |

## 2020 |

Borsche, Raul; Kocoglu, Damla; Trenn, Stephan A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs Journal Article In: Mathematics of Control, Signals, and Systems (MCSS), vol. 32, pp. 455-487, 2020, (Open Access). Abstract | Links | BibTeX | Tags: DAEs, delay, networks, open-access, PDEs, piecewise-smooth-distributions, solution-theory, switched-DAEs @article{BorsKoco20, A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions. |

Chen, Yahao; Trenn, Stephan On geometric and differentiation index of nonlinear differential algebraic equations Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. Links | BibTeX | Tags: DAEs, nonlinear, solution-theory @misc{ChenTren20m, |

Trenn, Stephan The Laplace transform and inconsistent initial values Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs @misc{Tren20m, Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions. |

## 2019 |

Trenn, Stephan; Unger, Benjamin Delay regularity of differential-algebraic equations Inproceedings In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 989-994, Nice, France, 2019. Abstract | Links | BibTeX | Tags: CDC, DAEs, delay, solution-theory @inproceedings{TrenUnge19, We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations can arise if a feedback controller is applied to a descriptor system and the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise-smooth distributions and an algorithm to determine whether a DDAE is delay-regular. |

Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan The one-step-map for switched singular systems in discrete-time Inproceedings In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 605-610, Nice, France, 2019. Abstract | Links | BibTeX | Tags: CDC, discrete-time, solution-theory, switched-systems @inproceedings{AnhLinh19, We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-step- map which can be used to provide explicit solution formulas for general switching signals. |

## 2018 |

Kausar, Rukhsana; Trenn, Stephan Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs Inproceedings In: Klingenberg, Christian; Westdickenberg, Michael (Ed.): Theory, Numerics and Applications of Hyperbolic Problems II, pp. 123-135, Springer, Cham, 2018, ISBN: 978-3-319-91548-7, (Presented at XVI International Conference on Hyperbolic Problems (HYP2016), Aachen). Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, PDEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @inproceedings{KausTren18, In water distribution network instantaneous changes in valve and pump settings introduce jumps and sometimes impulses. In particular, a particular impulsive phenomenon which occurs due to sudden closing of valve is the so called water hammer. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). We observed that under some suitable assumptions the PDEs usually used to describe water flows can be simplified to differential algebraic equations (DAEs). The idea is to model water hammer phenomenon in the switched DAEs framework due to its special feature of studying such impulsive effects. To compare these two modeling techniques, a system of hyperbolic PDE model and the switched DAE model for a simple set up consisting of two reservoirs, six pipes and three valve is presented. The aim of this contribution is to present results of both models as motivation for the claim that a switched DAE modeling framework is suitable for describing a water hammer. |

## 2017 |

Kausar, Rukhsana; Trenn, Stephan Impulses in structured nonlinear switched DAEs Inproceedings In: Proc. 56th IEEE Conf. Decis. Control, pp. 3181 - 3186, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: application, CDC, DAEs, nonlinear, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @inproceedings{KausTren17b, Switched nonlinear differential algebraic equations (DAEs) occur in mathematical modeling of sudden transients in various physical phenomenons. Hence, it is important to investigate them with respect to the nature of their solutions. The few existing solvability results for switched nonlinear DAEs exclude Dirac impulses by definition; however, in many cases this is too restrictive. For example, in water distribution networks the water hammer effect can only be studied when allowing Dirac impulses in a nonlinear switched DAE description. We investigate existence and uniqueness of solutions with impulses for a general class of nonlinear switched DAEs, where we exploit a certain sparse structure of the nonlinearity. |

Kall, Jochen; Kausar, Rukhsana; Trenn, Stephan Modeling water hammers via PDEs and switched DAEs with numerical justification Inproceedings In: Proc. 20th IFAC World Congress 2017, pp. 5349 - 5354, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: application, DAEs, nonlinear, PDEs, solution-theory, switched-DAEs, switched-systems @inproceedings{KallKaus17, In water distribution networks instantaneous changes in valve and pump settings may introduces jumps and peaks in the pressure. In particular, a well known phenomenon in response to the sudden closing of a valve is the so called water hammer, which (if not taken into account properly) may destroy parts of the water network. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). After discussing this PDE model we propose a simplified model using switched differential-algebraic equations (DAEs). Switched DAEs are known to be able to produce infinite peaks in response to sudden structural changes. These peaks (in the mathematical form of Dirac impulses) can easily be predicted and may allow for a simpler analysis of complex water networks in the future. As a first step toward that goal, we verify the novel modeling approach by comparing these two modeling techniques numerically for a simple set up consisting of two reservoirs, a pipe and a valve. |

## 2016 |

Camlibel, Kanat; Iannelli, Luigi; Tanwani, Aneel; Trenn, Stephan Differential-algebraic inclusions with maximal monotone operators Inproceedings In: Proc. 55th IEEE Conf. Decis. Control, Las Vegas, USA, pp. 610–615, 2016. Abstract | Links | BibTeX | Tags: CDC, DAEs, nonlinear, solution-theory @inproceedings{CamlIann16, The term differential-algebraic inclusions (DAIs) not only describes the dynamical relations using set-valued mappings, but also includes the static algebraic inclusions, and this paper considers the problem of existence of solutions for a class of such dynamical systems described by the inclusion ddt Px in -M(x) for a symmetric positive semi-definite matrix P in R^(n x n), and a maximal monotone operator M:R^n => R^n. The existence of solutions is proved using the tools from the theory of maximal monotone operators. The class of solutions that we study in the paper have the property that, instead of the whole state, only Px is absolutely continuous and unique. This framework, in particular, is useful for studying passive differential-algebraic equations (DAEs) coupled with maximal monotone relations. Certain class of irregular DAEs are also covered within the proposed general framework. Applications from electrical circuits are included to provide a practical motivation. |

Gross, Tjorben B.; Trenn, Stephan; Wirsen, Andreas Solvability and stability of a power system DAE model Journal Article In: Syst. Control Lett., vol. 97, pp. 12–17, 2016. Abstract | Links | BibTeX | Tags: application, DAEs, Lyapunov, networks, solution-theory, stability @article{GrosTren16, The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization for the linearized power system DAE solely in terms of the network topology. As an extension to previous result we allow for higher order generator dynamics. Furthermore, we show that any solvable power system DAE is automatically of index one, which means that it is also numerically well posed. Finally, we show that any solvable power system DAE is stable but not asymptotically stable. |

## 2014 |

Gross, Tjorben B.; Trenn, Stephan; Wirsen, Andreas Topological solvability and index characterizations for a common DAE power system model Inproceedings In: Proc. 2014 IEEE Conf. Control Applications (CCA), pp. 9–14, IEEE 2014. Abstract | Links | BibTeX | Tags: application, DAEs, networks, nonlinear, solution-theory @inproceedings{GrosTren14, For the widely-used power system model consisting of the generator swing equations and the power flow equations resulting in a system of differential algebraic equations (DAEs), we introduce a sufficient and necessary solvability condition for the linearized model. This condition is based on the topological structure of the power system. Furthermore we show sufficient conditions for the linearized DAE-system and a nonlinear version of the model to have differentiation index equal to one. |

## 2013 |

Costantini, Giuliano; Trenn, Stephan; Vasca, Francesco Regularity and passivity for jump rules in linear switched systems Inproceedings In: Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 4030–4035, 2013, ISSN: 0191-2216. Abstract | Links | BibTeX | Tags: CDC, DAEs, solution-theory, switched-DAEs, switched-systems @inproceedings{CostTren13, A wide class of linear switched systems (LSS) can be represented by a sequence of modes each one described by a set of differential algebraic equations (DAEs). LSS can exhibit discontinuities in the state evolution, also called jumps, when the state at the end of a mode is not consistent with the DAEs of the successive mode. Then the problem of defining a proper state jump rule arises when an inconsistent initial condition is given. Regularity and passivity conditions provide two conceptually different jump maps respectively. In this paper, after proving some preliminary result on the jump analysis within the regularity framework, it is shown the equivalence of regularity-based and passivity-based jump rules. A switched capacitor electrical circuit is used to numerically confirm the theoretical result. |

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. Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergTren13, 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. |

Trenn, Stephan Solution concepts for linear DAEs: a survey Incollection In: Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations I, pp. 137–172, Springer-Verlag, Berlin-Heidelberg, 2013. Abstract | Links | BibTeX | Tags: DAEs, solution-theory, survey @incollection{Tren13a, This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks. |

## 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. Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergTren12, 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. |

Liberzon, Daniel; Trenn, Stephan Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability Journal Article In: Automatica, vol. 48, no. 5, pp. 954–963, 2012. Abstract | Links | BibTeX | Tags: DAEs, nonlinear, solution-theory, stability, switched-DAEs, switched-systems @article{LibeTren12, We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov’s direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively. |

Trenn, Stephan Switched differential algebraic equations Incollection In: Vasca, Francesco; Iannelli, Luigi (Ed.): Dynamics and Control of Switched Electronic Systems - Advanced Perspectives for Modeling, Simulation and Control of Power Converters, pp. 189–216, Springer, London, 2012. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @incollection{Tren12, In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form $Ex'=Ax+Bu$ where $E$ is, in general, a singular matrix and $u$ is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments. However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the consistency projectors. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. In order to capture this impulsive behavior the space of piecewise-smooth distributions is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs (including the uniqueness of the jumps induced by the switches). With the help of the consistency projector a condition is formulated whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role. |

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). Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergIlch12a, 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 Regularity of distributional differential algebraic equations Journal Article In: Math. Control Signals Syst., vol. 21, no. 3, pp. 229–264, 2009. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory @article{Tren09b, Time-varying differential algebraic equations (DAEs) of the form E x' = A x + f are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given. |

Trenn, Stephan Impulse free solutions for switched differential algebraic equations Miscellaneous Preprint, 2009, (After the initial submission, I decided not to revise this manuscript and instead included most of the content in the paper "Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability" (joint work with Daniel Liberzon), which appeared 2012 in Automatica.). Abstract | Links | BibTeX | Tags: piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @misc{Tren09m, Linear switched differential algebraic equations (switched DAEs) are studied. First, a suitable solution space is introduced, the space of so called piecewise-smooth distributions. Secondly, sufficient conditions are given which ensure that all solutions of the switched DAE are impulse and/or jump free. These conditions are easy to check and are expressed directly in the systems original data. As an example a simple electrical circuit with a switch is analyzed. |

Trenn, Stephan Distributional differential algebraic equations PhD Thesis Institut für Mathematik, Technische Universität Ilmenau, 2009. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs, switched-systems @phdthesis{Tren09d, Linear implicit differential equations of the form Ex'=Ax+f are studied. If the matrix E is not invertible, these equations contain differential as well as algebraic equations. Hence Ex'=Ax+f is called differential algebraic equation (DAE). A main goal of this dissertation is the consideration of certain distributions (or generalized functions) as solutions and studying time-varying DAEs, whose coefficient matrices have jumps. Therefore, a suitable solution space is derived. This solution space allows to study the important class of switched DAEs. The space of piecewise-smooth distributions is introduced as the solution space. For this space of distributions, it is possible to define a multiplication, hence DAEs can be studied whose coefficient matrices have also distributional entries. A distributional DAE is an equation of the form Ex'=Ax+f where the matrices E and A contain piecewise-smooth distributions as entries and the solutions x as well as the inhomogeneities f are also piecewise-smooth distributions. For distributional DAEs, existence and uniqueness of solutions are studied, therefore, the concept of regularity for distributional DAEs is introduced. Necessary and sufficient conditions for existence and uniqueness of solutions are derived. As special cases, the equations x'=Ax+f (distributional ODEs) and Nx'=x+f (pure distributional DAE) are studied and explicit solution formulae are given. Switched DAEs are distributional DAEs with piecewise constant coefficient matrices. Sufficient conditions are given which ensure that all solutions of a switched DAE are impulse free. Furthermore, it is studied which conditions ensure that arbitrary switching between stable subsystems yield a stable overall system. Finally, controllability and observability for distributional DAEs are studied. For this, it is accounted for the fact that input signals can contain impulses, hence an ``instantaneous'' control is theoretically possible. For a DAE of the form Nx'=x+bu |

## 2008 |

Trenn, Stephan Distributional solution theory for linear DAEs Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 10077–10080, WILEY-VCH Verlag, 2008, ISSN: 1617--7061. Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory @inproceedings{Tren08b, A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. |