19. | Küsters, Ferdinand; Trenn, Stephan Switch observability for switched linear systems Journal Article Automatica, 87 , pp. 121-127, 2018. @article{KustTren18, title = {Switch observability for switched linear systems}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/10/Preprint-KT170808.pdf, Preprint}, doi = {https://doi.org/10.1016/j.automatica.2017.09.024}, year = {2018}, date = {2018-01-01}, journal = {Automatica}, volume = {87}, pages = {121-127}, abstract = {Mode observability of switched systems requires observability of each individual mode. We consider other concepts of observability that do not have this requirement: Switching time observability and switch observability. The latter notion is based on the assumption that at least one switch occurs. These concepts are analyzed and characterized both for homogeneous and inhomogeneous systems.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Mode observability of switched systems requires observability of each individual mode. We consider other concepts of observability that do not have this requirement: Switching time observability and switch observability. The latter notion is based on the assumption that at least one switch occurs. These concepts are analyzed and characterized both for homogeneous and inhomogeneous systems. |

18. | Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched DAEs: convergence, partial averaging and stability Journal Article Automatica, 82 , pp. 145–157, 2017. @article{MostTren17, title = {Averaging for switched DAEs: convergence, partial averaging and stability}, author = {Elisa Mostacciuolo and Stephan Trenn and Francesco Vasca}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-MTV170407.pdf, Preprint}, doi = {10.1016/j.automatica.2017.04.036}, year = {2017}, date = {2017-08-01}, journal = {Automatica}, volume = {82}, pages = {145--157}, abstract = {Averaging is a useful technique to simplify the analysis of switched systems. In this paper we present averaging results for the class of systems described by switched differential algebraic equations (DAEs). Conditions on the consistency projectors are given which guarantee convergence towards a non-switched averaged system. A consequence of this result is the possibility to stabilize switched DAEs via fast switching. We also study partial averaging in case the consistency projectors do not satisfy the conditions for convergence; the averaged system is then still a switched system, but is simpler than the original. The practical interest of the theoretical averaging results is demonstrated through the analysis of the dynamics of a switched electrical circuit.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Averaging is a useful technique to simplify the analysis of switched systems. In this paper we present averaging results for the class of systems described by switched differential algebraic equations (DAEs). Conditions on the consistency projectors are given which guarantee convergence towards a non-switched averaged system. A consequence of this result is the possibility to stabilize switched DAEs via fast switching. We also study partial averaging in case the consistency projectors do not satisfy the conditions for convergence; the averaged system is then still a switched system, but is simpler than the original. The practical interest of the theoretical averaging results is demonstrated through the analysis of the dynamics of a switched electrical circuit. |

17. | Tanwani, Aneel; Trenn, Stephan Determinability and state estimation for switched differential–algebraic equations Journal Article Automatica, 76 , pp. 17–31, 2017, ISSN: 0005-1098. @article{TanwTren17, title = {Determinability and state estimation for switched differential–algebraic equations}, author = {Aneel Tanwani and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-TT160919.pdf, Preprint}, doi = {10.1016/j.automatica.2016.10.024}, issn = {0005-1098}, year = {2017}, date = {2017-02-01}, journal = {Automatica}, volume = {76}, pages = {17--31}, abstract = {The problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential–algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appropriate notion of observability is presented which accommodates these phenomena. Based on this notion, we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state. In practice, such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential–algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appropriate notion of observability is presented which accommodates these phenomena. Based on this notion, we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state. In practice, such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system. |

16. | Gross, Tjorben B; Trenn, Stephan; Wirsen, Andreas Solvability and stability of a power system DAE model Journal Article Syst. Control Lett., 29 , pp. 12–17, 2016. @article{GrosTren16, title = {Solvability and stability of a power system DAE model}, author = {Tjorben B. Gross and Stephan Trenn and Andreas Wirsen}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-GTW160816.pdf, Preprint}, doi = {10.1016/j.sysconle.2016.08.003}, year = {2016}, date = {2016-11-01}, journal = {Syst. Control Lett.}, volume = {29}, pages = {12--17}, abstract = {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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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. |

15. | Küsters, Ferdinand; Trenn, Stephan Duality of switched DAEs Journal Article Math. Control Signals Syst., 28 (3), pp. 25, 2016. @article{KustTren16a, title = {Duality of switched DAEs}, author = {Ferdinand Küsters and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KT160627.pdf, Preprint}, doi = {10.1007/s00498-016-0177-2}, year = {2016}, date = {2016-07-01}, journal = {Math. Control Signals Syst.}, volume = {28}, number = {3}, pages = {25}, abstract = {We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs. |

14. | Küsters, Ferdinand; Ruppert, Markus G -M; Trenn, Stephan Controllability of switched differential-algebraic equations Journal Article Syst. Control Lett., 78 (0), pp. 32 - 39, 2015, ISSN: 0167-6911. @article{KustRupp15, title = {Controllability of switched differential-algebraic equations}, author = {Ferdinand Küsters and Markus G.-M. Ruppert and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-KRT150122.pdf, Preprint}, doi = {10.1016/j.sysconle.2015.01.011}, issn = {0167-6911}, year = {2015}, date = {2015-01-01}, journal = {Syst. Control Lett.}, volume = {78}, number = {0}, pages = {32 - 39}, abstract = {We study controllability of switched differential–algebraic equations. We are able to establish a controllability characterization where we assume that the switching signal is known. The characterization takes into account possible jumps induced by the switches. It turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study controllability of switched differential–algebraic equations. We are able to establish a controllability characterization where we assume that the switching signal is known. The characterization takes into account possible jumps induced by the switches. It turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times. |

13. | Berger, Thomas; Trenn, Stephan Kalman controllability decompositions for differential-algebraic systems Journal Article Syst. Control Lett., 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 = {}, 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. |

12. | Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article IEEE Trans. Autom. Control, 58 (12), pp. 3126–3141, 2013. @article{LibeTren13b, title = {The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130702.pdf, Preprint}, doi = {10.1109/TAC.2013.2277631}, year = {2013}, date = {2013-08-16}, journal = {IEEE Trans. Autom. Control}, volume = {58}, number = {12}, pages = {3126--3141}, abstract = {The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions. |

11. | Hackl, Christoph M; Hopfe, Norman; Ilchmann, Achim; Mueller, Markus; Trenn, Stephan Funnel control for systems with relative degree two Journal Article SIAM J. Control Optim., 51 (2), pp. 965–995, 2013. @article{HackHopf13, title = {Funnel control for systems with relative degree two}, author = {Christoph M. Hackl and Norman Hopfe and Achim Ilchmann and Markus Mueller and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/HackHopf13.pdf, Paper}, doi = {10.1137/100799903 }, year = {2013}, date = {2013-03-19}, journal = {SIAM J. Control Optim.}, volume = {51}, number = {2}, pages = {965--995}, abstract = {Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. |

10. | Berger, Thomas; Trenn, Stephan Addition to ``The quasi-Kronecker form for matrix pencils'' Journal Article SIAM J. Matrix Anal. & Appl., 34 (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 = {}, 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. |

9. | Berger, Thomas; Trenn, Stephan The quasi-Kronecker form for matrix pencils Journal Article SIAM J. Matrix Anal. & Appl., 33 (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 = {}, 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. |

8. | Liberzon, Daniel; Trenn, Stephan Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability Journal Article Automatica, 48 (5), pp. 954–963, 2012. @article{LibeTren12, title = {Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability}, author = {Daniel Liberzon and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT111011.pdf, Preprint}, doi = {10.1016/j.automatica.2012.02.041}, year = {2012}, date = {2012-05-01}, journal = {Automatica}, volume = {48}, number = {5}, pages = {954--963}, abstract = {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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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. |

7. | Berger, Thomas; Ilchmann, Achim; Trenn, Stephan The quasi-Weierstraß form for regular matrix pencils Journal Article Linear Algebra Appl., 436 (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 = {}, 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. |

6. | Trenn, Stephan Regularity of distributional differential algebraic equations Journal Article Math. Control Signals Syst., 21 (3), pp. 229–264, 2009. @article{Tren09b, title = {Regularity of distributional differential algebraic equations}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre090806.pdf, Preprint}, doi = {10.1007/s00498-009-0045-4}, year = {2009}, date = {2009-12-01}, journal = {Math. Control Signals Syst.}, volume = {21}, number = {3}, pages = {229--264}, abstract = {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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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. |

5. | Trenn, Stephan A normal form for pure differential algebraic systems Journal Article Linear Algebra Appl., 430 (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 = {}, 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. |

4. | Trenn, Stephan Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units Journal Article IEEE Transactions on Neural Networks, 19 (5), pp. 836–844, 2008, ISSN: 1045-9227. @article{Tren08a, title = {Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units}, author = {Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre071012.pdf, Preprint}, doi = {10.1109/TNN.2007.912306}, issn = {1045-9227}, year = {2008}, date = {2008-05-01}, journal = {IEEE Transactions on Neural Networks}, volume = {19}, number = {5}, pages = {836--844}, abstract = {This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function. |

3. | Ilchmann, Achim; Sawodny, Oliver; Trenn, Stephan Pneumatic cylinders: modelling and feedback force-control Journal Article Int. J. Control, 79 (6), pp. 650–661, 2006. @article{IlchSawo06, title = {Pneumatic cylinders: modelling and feedback force-control}, author = {Achim Ilchmann and Oliver Sawodny and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IST050502.pdf, Preprint}, doi = {10.1080/00207170600645875}, year = {2006}, date = {2006-01-01}, journal = {Int. J. Control}, volume = {79}, number = {6}, pages = {650--661}, abstract = {In this paper, we model, analyse, and control an experimental set-up of a servo pneumatic cylinder. The dynamic behaviour of pneumatic actuator systems is dominant by non-linear functions. First, a mathematical model for the pneumatic system is derived. Secondly, we investigate the mathematical properties of this model and show boundedness and positiveness of certain variables. Thirdly, we prove that a proportional output feedback controller with saturation achieves practical tracking a wide class of reference trajectories. We verify the theoretical results and the effectiveness of the control by experiments.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we model, analyse, and control an experimental set-up of a servo pneumatic cylinder. The dynamic behaviour of pneumatic actuator systems is dominant by non-linear functions. First, a mathematical model for the pneumatic system is derived. Secondly, we investigate the mathematical properties of this model and show boundedness and positiveness of certain variables. Thirdly, we prove that a proportional output feedback controller with saturation achieves practical tracking a wide class of reference trajectories. We verify the theoretical results and the effectiveness of the control by experiments. |

2. | Ilchmann, Achim; Ryan, Eugene P; Trenn, Stephan Tracking control: Performance funnels and prescribed transient behaviour Journal Article Syst. Control Lett., 54 (7), pp. 655–670, 2005. @article{IlchRyan05, title = {Tracking control: Performance funnels and prescribed transient behaviour}, author = {Achim Ilchmann and Eugene P. Ryan and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IRT041112.pdf, Preprint}, doi = {10.1016/j.sysconle.2004.11.005}, year = {2005}, date = {2005-01-01}, journal = {Syst. Control Lett.}, volume = {54}, number = {7}, pages = {655--670}, publisher = {Elsevier}, abstract = {Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given lambda >0 (arbitrarily small), determine a feedback strategy which ensures that, for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than lambda (that is, ||e(t)|| < lambda for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t)=K_F(t,e(t)), where K_F is any continuous function exhibiting two specific properties, the first of which ensures that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary. |

1. | Ilchmann, Achim; Trenn, Stephan Input constrained funnel control with applications to chemical reactor models Journal Article Syst. Control Lett., 53 (5), pp. 361–375, 2004. @article{IlchTren04, title = {Input constrained funnel control with applications to chemical reactor models}, author = {Achim Ilchmann and Stephan Trenn}, url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-IT040715.pdf, Preprint}, doi = {10.1016/j.sysconle.2004.05.014}, year = {2004}, date = {2004-01-01}, journal = {Syst. Control Lett.}, volume = {53}, number = {5}, pages = {361--375}, publisher = {Elsevier}, abstract = {Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Error feedback control is considered for a class of exothermic chemical reactor models. The control objective is that the temperature T evolves within a prespecified performance envelope or ``funnel'' around the set point temperature T*. A simple error feedback control with input constraints of the form u(t)=sat(-k(t)[T(t)-T*] + u*), u* an offset, is introduced which achieves the objective in the presence of disturbances corrupting the measurement. The gain k(t) is a function of the error e(t)=T(t)-T* and its distance to the funnel boundary. The input constraints have to satisfy certain feasibility assumptions in terms of the model data and the operating point T*. |

# Journals

19. | Switch observability for switched linear systems Journal Article Automatica, 87 , pp. 121-127, 2018. |

18. | Averaging for switched DAEs: convergence, partial averaging and stability Journal Article Automatica, 82 , pp. 145–157, 2017. |

17. | Determinability and state estimation for switched differential–algebraic equations Journal Article Automatica, 76 , pp. 17–31, 2017, ISSN: 0005-1098. |

16. | Solvability and stability of a power system DAE model Journal Article Syst. Control Lett., 29 , pp. 12–17, 2016. |

15. | Duality of switched DAEs Journal Article Math. Control Signals Syst., 28 (3), pp. 25, 2016. |

14. | Controllability of switched differential-algebraic equations Journal Article Syst. Control Lett., 78 (0), pp. 32 - 39, 2015, ISSN: 0167-6911. |

13. | Kalman controllability decompositions for differential-algebraic systems Journal Article Syst. Control Lett., 71 , pp. 54–61, 2014, ISSN: 0167-6911. |

12. | The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article IEEE Trans. Autom. Control, 58 (12), pp. 3126–3141, 2013. |

11. | Funnel control for systems with relative degree two Journal Article SIAM J. Control Optim., 51 (2), pp. 965–995, 2013. |

10. | Addition to ``The quasi-Kronecker form for matrix pencils'' Journal Article SIAM J. Matrix Anal. & Appl., 34 (1), pp. 94–101, 2013. |

9. | The quasi-Kronecker form for matrix pencils Journal Article SIAM J. Matrix Anal. & Appl., 33 (2), pp. 336–368, 2012. |

8. | Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability Journal Article Automatica, 48 (5), pp. 954–963, 2012. |

7. | The quasi-Weierstraß form for regular matrix pencils Journal Article Linear Algebra Appl., 436 (10), pp. 4052–4069, 2012, (published online February 2010). |

6. | Regularity of distributional differential algebraic equations Journal Article Math. Control Signals Syst., 21 (3), pp. 229–264, 2009. |

5. | A normal form for pure differential algebraic systems Journal Article Linear Algebra Appl., 430 (4), pp. 1070 – 1084, 2009. |

4. | Multilayer Perceptrons: Approximation Order and Necessary Number of Hidden Units Journal Article IEEE Transactions on Neural Networks, 19 (5), pp. 836–844, 2008, ISSN: 1045-9227. |

3. | Pneumatic cylinders: modelling and feedback force-control Journal Article Int. J. Control, 79 (6), pp. 650–661, 2006. |

2. | Tracking control: Performance funnels and prescribed transient behaviour Journal Article Syst. Control Lett., 54 (7), pp. 655–670, 2005. |

1. | Input constrained funnel control with applications to chemical reactor models Journal Article Syst. Control Lett., 53 (5), pp. 361–375, 2004. |