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.

## 2023 |

Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, 2023, (accepted). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, switched-systems @article{HossTren23a, 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. |

Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: DAEs, Lyapunov, nonlinear, normal-forms, solution-theory, stability, switched-DAEs, switched-systems @unpublished{ChenTren23pp, n this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [27, 28], for linear DAEs. Then we show that the common Lyapunov functions stability conditions proposed in our previous work [16] (which differ from the ones in [28]) can be applied to switched nonlinear DAEs with high-index models which are not equivalent to the nonlinear Weierstrass form. Moreover, we generalize the commutativity stability conditions [32] for switched nonlinear ordinary differential equations to the switched nonlinear DAEs case. Finally, some simulation results of switching electrical circuits and numerical examples are given to illustrate the usefulness of the proposed stability conditions. |

Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time Journal Article In: IEEE Transactions on Automatic Control, 2023, (early access). Abstract | Links | BibTeX | Tags: LMIs, Lyapunov, nonlinear, stability, switched-systems @article{YinJaya23, 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. |

## 2022 |

Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Unpublished 2022, (provisionally accepted in IEEE TAC). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, switched-systems @unpublished{HossTren22ppc, 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. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Unpublished 2022, (submitted). Abstract | Links | BibTeX | Tags: application, averaging, DAEs, LMIs, switched-DAEs, switched-systems @unpublished{MostTren22ppb, Linear switched impulsive systems (SIS) are characterized by ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes nontrivial the application of classical averaging techniques. In this paper we consider SIS with pulse width modulation (PWM) and we propose an averaged model whose solution approximates the moving average of the SIS solution with an error which decreases with the multiple of the switching period and by decreasing the PWM period. The averaging result requires milder assumptions on the system matrices with respect to those needed by the previous averaging techniques for SIS. The interest of the proposed model is strengthened by the fact that it reduces to the classical averaged model for PWM systems when there are no jumps in the state. The theoretical results are verified through numerical results obtained by considering a switched capacitor electrical circuit. |

Sutrisno,; Trenn, Stephan 2022, (submitted). Abstract | Links | BibTeX | Tags: controllability, discrete-time, switched-systems @unpublished{SutrTren22ppa, This article presents the reachability and controllability characterizations for discrete-time linear switched systems under a fixed and known switching signal. A geometric approach is used, and we are able to provide alternative conditions which are more computationally friendly compared to existing results by utilizing the solution formula at switching times. Furthermore, the proposed conditions make it easier to study the dependency of the reachability and controllability on the switching times and the mode sequences; this is a new result currently not investigated in the literature. Some academic examples are provided to illustrate the novel features found in this study. |

Sutrisno,; Trenn, Stephan 2022, (submitted). Abstract | Links | BibTeX | Tags: DAEs, discrete-time, nonlinear, solution-theory, stability, switched-DAEs, switched-systems @unpublished{SutrTren22ppb, The solvability of nonlinear nonswitched and switched singular systems in discrete time is studied. We provide necessary and sufficient conditions for solvability. The one-step map that generates equivalent nonlinear (ordinary) systems for solvable nonlinear singular systems under arbitrary switching signals is introduced. Moreover, the stability is studied by utilizing this one-step map. A sufficient condition for stability is provided in terms of Lyapunov functions. |

Hossain, Sumon; Sutrisno,; Trenn, Stephan A time-varying approach for model reduction of singular linear switched systems in discrete time Miscellaneous Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems, 2022. Abstract | Links | BibTeX | Tags: controllability, DAEs, discrete-time, model-reduction, observability, open-access, switched-DAEs, switched-systems @misc{HossSutr22m, We propose a model reduction approach for singular linear switched systems in discrete time with a fixed mode sequence based on a balanced truncation reduction method for linear time-varying discrete-time systems. The key idea is to use the one-step map to find an equivalent time-varying system with an identical input-output behavior, and then adapt available balance truncation methods for (discrete) time-varying systems. The proposed method is illustrated with a low-dimensional academic example. |

Wijnbergen, Paul; Trenn, Stephan Linear quadratic optimal control of switched differential algebraic equations Miscellaneous Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems, 2022. Abstract | Links | BibTeX | Tags: DAEs, open-access, optimal-control, switched-DAEs, switched-systems @misc{WijnTren22mb, In this abstract the finite horizon linear quadratic optimal control problem with constraints on the terminal state for switched differential algebraic equations is considered. Furthermore, we seek for an optimal solution that is impulse-free. In order to solve the problem, a non standard finite horizon problem for non-switched DAEs is considered. Necessary and sufficient conditions on the initial value x0 for solvability of this non standard problem are stated. Based on these results a sequence of subspaces can be defined which lead to necessary and sufficient conditions for solvability of the finite horizon optimal control problem for switched DAEs. |

Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched differential algebraic equations Unpublished 2022, (submitted). Abstract | Links | BibTeX | Tags: controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @unpublished{WijnTren22pp, In this paper impulse controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. In the case of a system class generated by switching signals with a fixed mode sequence it is shown that either all or almost all systems are impulse-controllable, or that all or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that although all systems are impulse-controllable, the input achieving impulse-free solutions might still depend on the switching times in the future, which causes some causality issues. Therefore, the concept of (quasi-) causal impulse-controllability is introduced and system classes which are (quasi-) causal are characterized. Finally necessary and sufficient conditions for a system class to be causal given some dwell-time are stated. |

Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Stability of switched systems with multiple equilibria: a mixed stable-unstable subsystem case Unpublished 2022, (submitted). Abstract | Links | BibTeX | Tags: LMIs, Lyapunov, stability, switched-systems @unpublished{YinJaya22ppb, This paper studies stability of switched systems that are composed of a mixture of stable and unstable modes with multiple equilibria. The main results of this paper include some sufficient conditions concerning set convergence of switched nonlinear systems. We show that under suitable dwell-time and leave-time switching laws, trajectories converge to an initial set and then stay in a convergent set. Based on these conditions, LMI conditions are derived that allow for numerical validation of practical stability of switched affine systems, which include those with all unstable modes. Two examples are provided to verify the theoretical results. |

Hossain, Sumon; Trenn, Stephan A weak Kalman decomposition approach for reduced realizations of switched linear systems Inproceedings In: IFAC-PapersOnLine, pp. 157-162, 2022, (Part of special issue: 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022: Vienna Austria, 27–29 July 2022). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, switched-systems @inproceedings{HossTren22, We propose a novel reduction approach for switched linear systems with a fixed mode sequence based on subspaces related to the (time-varying) reachable and unobservable spaces. These subspaces are defined in such a way that they can be used to construct a weak Kalman decomposition, which is then in turn used to define a reduced switched linear system with an identical input-output behavior. The proposed method is illustrated with a low dimensional academic example. |

Chen, Yahao; Trenn, Stephan Stability analysis of switched nonlinear differential-algebraic equations via nonlinear Weierstrass form Inproceedings In: Proceedings of the 2022 European Control Conference (ECC), pp. 1091-1096, London, 2022. 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. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco An averaged model for switched systems with state jumps applicable for PWM descriptor systems Inproceedings In: Proceedings of the 2022 European Control Conference (ECC), pp. 1085-1090, London, 2022. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched-DAEs, switched-systems @inproceedings{MostTren22b, Switched descriptor systems with pulse width modulation are characterized by modes whose dynamics are described by differential algebraic equations; this type of models can be viewed as switched impulsive systems, i.e. switched systems with ordinary differential equations as modes dynamics and state jumps at the switching time instants. The presence of possible jumps in the state makes the application of the classical averaging technique nontrivial. In this paper we propose an averaged model for switched impulsive systems. The state trajectory of the proposed averaged model is shown to approximate the one of the original system with an error of order of the switching period. The model reduces to the classical averaged model when there are no jumps in the state. The practical interest of the theoretical averaging result is demonstrated through numerical simulations of a switched capacitor electrical circuit. |

Sutrisno,; Trenn, Stephan The one-step function for discrete-time nonlinear switched singular systems Miscellaneous Book of Abstracts - 41th Benelux Meeting on Systems and Control, 2022. Links | BibTeX | Tags: discrete-time, nonlinear, solution-theory, switched-DAEs, switched-systems @misc{SutrTren22m, |

Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time Miscellaneous Book of Abstracts - 41th Benelux Meeting on Systems and Control, 2022. Links | BibTeX | Tags: LMIs, Lyapunov, stability, switched-systems, synchronization @misc{YinJaya22m, |

Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched DAEs Miscellaneous Book of Abstracts - 41th Benelux Meeting on Systems and Control, 2022. Links | BibTeX | Tags: controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @misc{WijnTren22ma, |

Xu, Teke; Water, Alden; Trenn, Stephan Dynamical boundary conditions for the water hammer problem Miscellaneous Book of Abstracts - XVIII International Conference on Hyperbolic Problems: Theory, Numerics, and Applications (HYP 2022), 2022. Links | BibTeX | Tags: PDEs, solution-theory, switched-systems @misc{XuWate22m, |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco A smooth model for periodically switched descriptor systems Journal Article In: Automatica, vol. 136, no. 110082, pp. 1-8, 2022, (open access). Abstract | Links | BibTeX | Tags: averaging, DAEs, open-access, switched-DAEs, switched-systems @article{MostTren22, Switched descriptor systems characterized by a repetitive finite sequence of modes can exhibit state discontinuities at the switching time instants. The amplitudes of these discontinuities depend on the consistency projectors of the modes. A switched ordinary differential equations model whose continuous state evolution approximates the state of the original system is proposed. Sufficient conditions based on linear matrix inequalities on the modes projectors ensure that the approximation error is of linear order of the switching period. The theoretical findings are applied to a switched capacitor circuit and numerical results illustrate the practical usefulness of the proposed model. |

## 2021 |

Hossain, Sumon; Trenn, Stephan Minimality of Linear Switched Systems with known switching signal Inproceedings In: Proceedings in Applied Mathematics and Mechanics, pp. 1-3, 2021, (open access). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, open-access, switched-systems @inproceedings{HossTren21a, Minimal realization is discussed for linear switched systems with a given switching signal. We propose a consecutive forward and backward approach for the time-interval of interest. The forward approach refers to extending the reachable subspace at each switching time by taking into account the nonzero reachable space from the previous mode. Afterwards, the backward approach extends the observable subspace of the current mode by taking observability information from the next mode into account. This results in an overall reduced switched system which is minimal and has the same input-output behavior as original system. Some examples are provided to illustrate the approach. |

Wijnbergen, Paul; Trenn, Stephan Optimal control of DAEs with unconstrained terminal costs Inproceedings In: Proc. 60th IEEE Conf. Decision and Control (CDC 2021), pp. 5275-5280, 2021. Abstract | Links | BibTeX | Tags: CDC, DAEs, optimal-control, switched-DAEs, switched-systems @inproceedings{WijnTren21b, This paper is concerned with the linear quadratic optimal control problem for impulse controllable differential algebraic equations on a bounded half open interval. Regarding the cost functional, a general positive semi-definite weight matrix is considered in the terminal cost. It is shown that for this problem, there generally does not exist an input that minimizes the cost functional. First it is shown that the problem can be reduced to finding an input to an index-1 DAE that minimizes a different quadratic cost functional. Second, necessary and sufficient conditions in terms of matrix equations are given for the existence of an optimal control. |

Sutrisno,; Trenn, Stephan Observability and Determinability Characterizations for Linear Switched Systems in Discrete Time Inproceedings In: Proc. 60th IEEE Conf. Decision and Control (CDC 2021), pp. 2474-2479, 2021. Abstract | Links | BibTeX | Tags: CDC, discrete-time, observability, switched-systems @inproceedings{SutrTren21b, In this article, we study the observability and determinability for discrete-time linear switched systems. Studies for the observability for this system class are already available in literature, however, we use assume here that the switching signal is known. This leads to less conservative observability conditions (e.g. observability of each individual mode is not necessary for the overall switched system to be observable); in particular, the dependencies of observability on the switching times and the mode sequences are derived; these results are currently not available in the literature on discrete-time switched systems. In addition to observability (which is concerned with recovering the state from the initial time onwards), we also investigate the determinability which is concerned with the ability to reconstruct the state value at the end of the observation interval. We provide several simple examples to illustrate novel features not seen in the continuous time case or for unswitched systems. |

Sutrisno,; Trenn, Stephan Observability of Switched Linear Singular System in Discrete Time: Single Switch Case Inproceedings In: Proc. European Control Conference (ECC21), pp. 267-292, Rotterdam, Netherlands, 2021. Abstract | Links | BibTeX | Tags: discrete-time, observability, switched-systems @inproceedings{SutrTren21a, In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is inde- pendent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability. |

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{HossTren21b, 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. |

Sutrisno,; Trenn, Stephan Observability and Determinability of Discrete Time Switched Linear Singular Systems: Multiple Switches Case Miscellaneous Book of Abstracts - 40th Benelux Workshop on Systems and Control, 2021, (extended abstract). Links | BibTeX | Tags: DAEs, discrete-time, observability, switched-DAEs, switched-systems @misc{SutrTren21m, |

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: LMIs, 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. |

Wijnbergen, Paul; Trenn, Stephan Impulse-free interval-stabilization of switched differential algebraic equations Journal Article In: Systems & Control Letters, vol. 149, pp. 104870.1-10, 2021, (Open Access.). Abstract | Links | BibTeX | Tags: controllability, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems @article{WijnTren21a, In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory. |

## 2020 |

Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan Stability analysis for switched discrete-time linear singular systems Journal Article In: Automatica, vol. 119, no. 109100, 2020. Abstract | Links | BibTeX | Tags: discrete-time, stability, switched-systems @article{AnhLinh20, The stability of arbitrarily switched discrete-time linear singular (SDLS) systems is studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1. We first provide a sufficient stability conditions in terms of Lyapunov functions. Furthermore, we generalize the notion of joint spectral radius of a finite set of matrix pairs, which allows us to fully characterize exponential stability. |

Wijnbergen, Paul; Jeeninga, Mark; Trenn, Stephan On stabilizability of switched differential algebraic equations Inproceedings In: IFAC-PapersOnLine 53-2, pp. 4304-4309, 2020, (Proc. IFAC World Congress 2020, Berlin, Germany. Open access.). Abstract | Links | BibTeX | Tags: DAEs, open-access, stability, switched-DAEs, switched-systems @inproceedings{WijnJeen20, This paper considers stabilizability of switched differential algebraic equations (DAEs). We first introduce the notion of interval stabilizability and show that under a certain uniformity assumption, stabilizability can be concluded from interval stabilizability. A geometric approach is taken to find necessary and sufficient conditions for interval stabilizability. This geometric approach can also be utilized to derive a novel characterization of controllability. |

Hossain, Sumon; Trenn, Stephan A time-varying Gramian based model reduction approach for Linear Switched Systems Inproceedings In: IFAC PapersOnline 53-2, pp. 5629-5634, 2020, (Proc. IFAC World Congress 2020, Berlin, Germany. Open access.). Abstract | Links | BibTeX | Tags: model-reduction, open-access, switched-systems @inproceedings{HossTren20a, We propose a model reduction approach for switched linear system based on a balanced truncation reduction method for linear time-varying systems. The key idea is to approximate the piecewise-constant coefficient matrices with continuous time-varying coefficients and then apply available balance truncation methods for (continuous) time-varying systems. The proposed method is illustrated with a low dimensional academic example. |

Wijnbergen, Paul; Trenn, Stephan Impulse controllability of switched differential-algebraic equations Inproceedings In: Proc. European Control Conference (ECC 2020), pp. 1561-1566, Saint Petersburg, Russia, 2020. Abstract | Links | BibTeX | Tags: controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{WijnTren20, This paper addresses impulse controllability of switched DAEs on a finite interval. First we present a forward approach where we define certain subspaces forward in time. These subpsaces are then used to provide a sufficient condition for impulse controllability. In order to obtain a full characterization we present afterwards a backward approach, where a sequence of subspaces is defined backwards in time. With the help of the last element of this backward sequence, we are able to fully characterize impulse controllability. All results are geometric results and thus independent of a coordinate system. |

Hossain, Sumon; Trenn, Stephan Model reduction of switched systems in time-varying approach Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. Links | BibTeX | Tags: model-reduction, switched-systems @misc{HossTren20m, |

Wijnbergen, Paul; Trenn, Stephan A forward approach to controllability of switched DAEs Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. Links | BibTeX | Tags: controllability, switched-DAEs, switched-systems @misc{WijnTren20m, |

Iervolino, Raffaele; Vasca, Francesco; Trenn, Stephan Discontinuous Lyapunov functions for discontinous piecewise-affine systems Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21). Abstract | Links | BibTeX | Tags: LMIs, Lyapunov, stability, switched-systems @misc{IervTren20m, 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. We first introduce the feasible Filippov solution concept 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 highlighted 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. |

## 2019 |

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. |

Tanwani, Aneel; Trenn, Stephan Detectability and observer design for switched differential algebraic equations Journal Article In: Automatica, vol. 99, pp. 289-300, 2019. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{TanwTren19, This paper studies detectability for switched linear differential–algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component of the system is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption. |

## 2018 |

Gross, Tjorben B.; Trenn, Stephan; Wirsen, Andreas Switch induced instabilities for stable power system DAE models Inproceedings In: IFAC-PapersOnLine, pp. 127-132, 2018, (Proc. IFAC Conf. Analysis Design Hybrid Systems (ADHS 2018)). Abstract | Links | BibTeX | Tags: application, stability, switched-DAEs, switched-systems @inproceedings{GrosTren18, It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenoma can indeed occur for a linearized DAE model of power grids. By making certain topological assumptions on the power grid, we can ensure stability under arbitrary switching. |

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. |

Küsters, Ferdinand; Trenn, Stephan Switch observability for switched linear systems Journal Article In: Automatica, vol. 87, pp. 121-127, 2018. Abstract | Links | BibTeX | Tags: observability, switched-systems @article{KustTren18, 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. |

## 2017 |

Iervolino, Raffaele; Trenn, Stephan; Vasca, Francesco Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions Inproceedings In: Proc. 56th IEEE Conf. Decis. Control, pp. 5894 - 5899, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: CDC, LMIs, stability, switched-systems @inproceedings{IervTren17, State-dependent switched systems characterized by piecewise affine (PWA) dynamics in a polyhedral partition of the state space are considered. Sufficient conditions on the vectors fields such that the solution crosses the common boundaries of the polyhedra are expressed in terms of quadratic inequalities constrained to the polyhedra intersections. A piecewise quadratic (PWQ) function, not necessarily continuous, is proposed as a candidate Lyapunov function (LF). The sign conditions and the negative jumps at the boundaries are expressed in terms of linear matrix inequalities (LMIs) via cone-copositivity. A sufficient condition for the asymptotic stability of the PWA system is then obtained by finding a PWQ-LF through the solution of a set LMIs. Numerical results with a conewise linear system and an opinion dynamics model show the effectiveness of the proposed approach. |

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. |

Küsters, Ferdinand; Patil, Deepak; Trenn, Stephan Switch observability for a class of inhomogeneous switched DAEs Inproceedings In: Proc. 56th IEEE Conf. Decis. Control, pp. 3175 - 3180, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, switched-DAEs, switched-systems @inproceedings{KustPati17b, Necessary and sufficient conditions for switching time and switch observability of a class of inhomogeneous switched differential algebraic equations (DAEs) are obtained. A characterization of initial states and inputs for which switched DAEs are switch unobservable is also provided by using the zeros of an augmented system obtained by combining the output of two modes suitably. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch-observer for switched linear systems Inproceedings In: Proc. 56th IEEE Conf. Decis. Control, pp. 1749 - 1754, Melbourne, Australia, 2017. Abstract | Links | BibTeX | Tags: CDC, observability, observer, switched-systems @inproceedings{KustTren17b, To determine the switching signal and the state of a switched linear system, one usually requires mode observability. This requires that all individual modes are observable and that the modes are distinguishable. In theory, it allows to determine the active mode in an arbitrarily short time. If one enlarges the observation to an interval that contains a switch, both assumptions (observability of each mode and clearly distinct dynamics) can be relaxed. In [Küsters and Trenn 2017] this concept, called switch observability, was formalized. It is of particular interest for fault identification. Based on switch observability, we propose an observer. This observer combines the information obtained before and after a switching instant to determine both the state and the switching signal. It is analyzed and illustrated in an example. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched DAEs: convergence, partial averaging and stability Journal Article In: Automatica, vol. 82, pp. 145–157, 2017. Abstract | Links | BibTeX | Tags: averaging, DAEs, stability, switched-DAEs, switched-systems @article{MostTren17, 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. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch observability for homogeneous switched DAEs Inproceedings In: Proc. 20th IFAC World Congress 2017, pp. 9355 - 9360, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{KustTren17a, We introduce the notions of switching time observability and switch observability for homogeneous switched differential-algebraic equations (DAEs). In contrast to mode detection, they do not require observability of the individual modes and are thus more suitable for fault detection and identification. Based on results in (Küsters and Trenn, 2017) for switched ordinary differential equations (ODEs), we characterize these notions for homogeneous switched DAEs and propose an observer for switch observable systems. |

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. |

Tanwani, Aneel; Trenn, Stephan Observer design for detectable switched differential-algebraic equations Inproceedings In: Proc. 20th IFAC World Congress 2017, pp. 2953 - 2958, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, stability, switched-DAEs, switched-systems @inproceedings{TanwTren17b, This paper studies detectability for switched linear differential-algebraic equations (DAEs) and its application in synthesis of observers. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observability, we propose the notion of interval-wise detectability: If the output of the system is constrained to be identically zero over an interval, then the norm of the corresponding state trajectories scales down by a certain factor over that interval. Conditions are provided under which the interval-wise detectability leads to asymptotic stability of zero-output-constrained state trajectories. An application is demonstrated in designing state estimators. Decomposing the state into observable and unobservable components, we show that if the observable component in the estimator is reset appropriately and persistently, then the estimation error converges to zero asymptotically under the interval-wise detectability assumption. |

Tanwani, Aneel; Trenn, Stephan Determinability and state estimation for switched differential–algebraic equations Journal Article In: Automatica, vol. 76, pp. 17–31, 2017, ISSN: 0005-1098. Abstract | Links | BibTeX | Tags: DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{TanwTren17, 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. |

## 2016 |

Küsters, Ferdinand; Trenn, Stephan Duality of switched DAEs Journal Article In: Math. Control Signals Syst., vol. 28, no. 3, pp. 25, 2016. Abstract | Links | BibTeX | Tags: controllability, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{KustTren16a, 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. |

Trenn, Stephan Stabilization of switched DAEs via fast switching Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 827–828, WILEY-VCH Verlag, 2016, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: averaging, DAEs, stability, switched-DAEs, switched-systems @inproceedings{Tren16, Switched differential algebraic equations (switched DAEs) can model dynamical systems with state constraints together with sudden structural changes (switches). These switches may lead to induced jumps and can destabilize the system even in the case that each mode is stable. However, the opposite effect is also possible; in particular, the question of finding a stabilizing switching signal is of interest. Two approaches are presented how to stabilize a switched DAE via fast switching. |

## 2015 |

Küsters, Ferdinand; Trenn, Stephan Duality of switched ODEs with jumps Inproceedings In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 4879–4884, 2015. Abstract | Links | BibTeX | Tags: CDC, controllability, observability, switched-systems @inproceedings{KustTren15b, Duality between controllability/reachability and determinability/observability of switched systems with jumps is proven. The duality result is based on the recent characterization of controllability for switched differential-algebraic equations (DAEs) which share many properties with switched ordinary differential equations (ODEs) with jumps. Here we view the switching signal as given and fixed, which makes the overall switched system time-varying, in particular controllability and reachability do not coincide anymore. |

Trenn, Stephan Distributional averaging of switched DAEs with two modes Inproceedings In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 3616–3620, 2015. Abstract | Links | BibTeX | Tags: averaging, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{Tren15, The averaging technique is a powerful tool for the analysis and control of switched systems. Recently, classical averaging results were generalized to the class of switched differential algebraic equations (switched DAEs). These results did not consider the possible Dirac impulses in the solutions of switched DAEs and it was believed that the presence of Dirac impulses does not prevent convergence towards an average model and can therefore be neglected. It turns out that the first claim (convergence) is indeed true, but nevertheless the Dirac impulses cannot be neglected, they play an important role for the resulting limit. This note first shows with a simple example how the presence of Dirac impulses effects the convergence towards an averaged model and then a formal proof of convergence in the distributional sense for switched DAEs with two modes is given. |

Tanwani, Aneel; Trenn, Stephan On detectability of switched linear differential-algebraic equations Inproceedings In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2957–2962, 2015. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, stability, switched-DAEs, switched-systems @inproceedings{TanwTren15, This paper addresses the notion of detectability for continuous-time switched systems comprising linear differential-algebraic equations (DAEs). It relates to studying asymptotic stability of the set of state trajectories corresponding to zero input and zero output, with a fixed switching signal. Due to the nature of solutions of switched DAEs, the problem reduces to analyzing stability of the trajectories emanating from a non-vanishing unobservable subspace, for which we first derive a geometric expression. The stability of state trajectories starting from that subspace can then be checked in two possible ways. In the first case, detectability of switched DAE is shown to be equivalent to the asymptotic stability of a reduced order discrete-time switched system. In the second approach, the solutions from a non-vanishing unobservable subspace are mapped to the solutions of a reduced order continuous system with time-varying switching ordinary differential equations (ODEs). As a special case of the later approach, the reduced order switched system is time-invariant if the unobservable subspace is invariant for all subsystems |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for non-homogeneous switched DAEs Inproceedings In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2951–2956, 2015. Abstract | Links | BibTeX | Tags: application, averaging, CDC, DAEs, switched-DAEs, switched-systems @inproceedings{MostTren15b, Averaging is widely used for approximating the dynamics of switched systems. The validity of an averaged model typically depends on the switching frequency and on some technicalities regarding the switched system structure. For homogeneous linear switched differential algebraic equations it is known that an averaged model can be obtained. In this paper an averaging result for non-homogeneous switched systems is presented. A switched electrical circuit illustrates the practical interest of the result. |

Küsters, Ferdinand; Trenn, Stephan Controllability characterization of switched DAEs Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 643–644, WILEY-VCH Verlag, 2015, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: controllability, DAEs, switched-DAEs, switched-systems @inproceedings{KustTren15a, We study controllability of switched differential algebraic equations (switched DAEs) with fixed switching signal. Based on a behavioral definition of controllability we are able to establish a controllability characterization that takes into account possible jumps and impulses induced by the switches. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Partial averaging for switched DAEs with two modes Inproceedings In: Proc. 2015 European Control Conf. (ECC), Linz, Austria, pp. 2896–2901, 2015. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched-DAEs, switched-systems @inproceedings{MostTren15a, In this paper an averaging result for switched systems whose modes are represented by means of differential algebraic equations (DAEs) is presented. Homogeneous switched DAEs with periodic switchings between two modes are considered. It is proved that a (switched) averaged system can be defined also in the presence of state jumps whose amplitude does not decrease with the increasing of the switching frequency. A switched capacitor electrical circuit is considered as an illustrative example. |

Petreczky, Mihály; Tanwani, Aneel; Trenn, Stephan Observability of switched linear systems Incollection In: Djemai, Mohamed; Defoort, Michael (Ed.): Hybrid Dynamical Systems, vol. 457, pp. 205–240, 2015. Abstract | Links | BibTeX | Tags: observability, switched-DAEs, switched-systems @incollection{PetrTanw15, Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. We will present these different approaches, with a focus on the recently introduced notion of large-time observability in the context of switched linear systems and its geometric characterization. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times, and it is shown that the proposed conditions have a denseness property with respect to switching times. To relieve the computation burden, some relaxed conditions that do not rely on the switching times are given; this allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential-algebraic systems is discussed. Detailed examples are included to illustrate the geometric conditions and to better understand the theoretical development. |

Küsters, Ferdinand; Ruppert, Markus G. -M.; Trenn, Stephan Controllability of switched differential-algebraic equations Journal Article In: Syst. Control Lett., vol. 78, no. 0, pp. 32 - 39, 2015, ISSN: 0167-6911. Abstract | Links | BibTeX | Tags: controllability, DAEs, switched-DAEs, switched-systems @article{KustRupp15, 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. |

## 2014 |

Defoort, Michael; Djemai, Mohamed; Trenn, Stephan Nondecreasing Lyapunov functions Inproceedings In: Proc. 21st Int. Symposium Math. Theory Networks Systems (MTNS), pp. 1038–1043, 2014. Abstract | Links | BibTeX | Tags: Lyapunov, nonlinear, stability, switched-systems @inproceedings{DefoDjem14, We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switched systems for which no piecewise-quadratic (classical) Lyapunov function exists. |

Ruppert, Markus G. -M.; Trenn, Stephan Controllability of switched DAEs: the single switch case Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 15–18, Wiley-VCH Verlag GmbH, 2014. Abstract | Links | BibTeX | Tags: controllability, switched-DAEs, switched-systems @inproceedings{RuppTren14, We study controllability of switched DAEs and formulate a definition of controllability in the behavioral sense. In order to characterize controllability for switched DAEs we first present new characterizations of controllability of non-switched DAEs based on the Wong-sequences. Afterwards a first result concerning the single-switch case is presented. |

## 2013 |

Tanwani, Aneel; Trenn, Stephan An observer for switched differential-algebraic equations based on geometric characterization of observability Inproceedings In: Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 5981–5986, 2013. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, observer, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TanwTren13, Based on our previous work dealing with geometric characterization of observability for switched differential-algebraic equations (switched DAEs), we propose an observer design for switched DAEs that generates an asymptotically convergent state estimate. Without assuming the observability of individual modes, the central idea in constructing the observer is to filter out the maximal information from the output of each of the active subsystems and combine it with the previously extracted information to obtain a good estimate of the state after a certain time has passed. In general, observability only holds when impulses in the output are taken into account, hence our observer incorporates the knowledge of impulses in the output. This is a distinguishing feature of our observer design compared to observers for switched ordinary differential equations. |

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. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco An averaging result for switched DAEs with multiple modes Inproceedings In: Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 1378 - 1383, 2013. Abstract | Links | BibTeX | Tags: averaging, CDC, DAEs, switched-DAEs, switched-systems @inproceedings{IannPedi13b, The major motivation of the averaging technique for switched systems is the construction of a smooth average system whose state trajectory approximates in some sense the state trajectory of the switched system. Averaging of dynamic systems represented by switched ordinary differential equations (ODEs) has been widely analyzed in the literature. The averaging approach can be useful also for the analysis of switched differential algebraic equations (DAEs). Indeed by analyzing the evolution of the switched DAEs state it is possible to conjecture the existence of an average model. However a trivial generalization of the ODE case is not possible due to the presence of state jumps. In this paper we discuss the averaging approach for switched DAEs and an approximation result is derived for homogenous switched linear DAE with periodic switching signals commuting among several modes. This approximation result extends a recent averaging result for switched DAEs with only two modes. Numerical simulations confirm the validity of the averaging approach for switched DAEs. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco Averaging for switched DAEs Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 489–490, WILEY-VCH Verlag, 2013, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched-DAEs, switched-systems @inproceedings{IannPedi13c, Switched differential-algebraic equations (switched DAEs) E_sigma(t) x'(t) = A_sigma(t) x(t) are suitable for modeling many practical systems, e.g. electrical circuits. When the switching is periodic and of high frequency, the question arises whether the solutions of switched DAEs can be approximated by an average non-switching system. It is well known that for a quite general class of switched ordinary differential equations (ODEs) this is the case. For switched DAEs, due the presence of the so-called consistency projectors, it is possible that the limit of trajectories for faster and faster switching does not exist. Under certain assumptions on the consistency projectors a result concerning the averaging for switched DAEs is presented. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco On averaging for switched linear differential algebraic equations Inproceedings In: Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 2163 – 2168, 2013. Abstract | Links | BibTeX | Tags: averaging, DAEs, switched-DAEs, switched-systems @inproceedings{IannPedi13a, Averaging is an effective technique which allows the analysis and control design of nonsmooth switched systems through the use of corresponding simpler smooth averaged systems. Approximation results and stability analysis have been presented in the literature for dynamic systems described by switched ordinary differential equations. In this paper the averaging technique is shown to be useful also for the analysis of switched systems whose modes are represented by means of differential algebraic equations (DAEs). An approximation result is derived for a simple but representative homogenous switched DAE with periodic switching signals and two modes. Simulations based on a simple electric circuit model illustrate the theoretical result. |

Trenn, Stephan Stability of switched DAEs Incollection In: Daafouz, Jamal; Tarbouriech, Sophie; Sigalotti, Mario (Ed.): Hybrid Systems with Constraints, pp. 57–83, London, 2013. Abstract | Links | BibTeX | Tags: DAEs, stability, switched-DAEs, switched-systems @incollection{Tren13b, Differential algebraic equations (DAEs) are used to model dynamical systems with constraints given by algebraic equations. In the presence of sudden structural changes (e.g. switching or faults) this leads to a switched DAE. A special feature of switched DAEs is the presence of induced jumps or even Dirac impulses in the solution. This chapter studies stability of switched DAEs taking into account the presence of these jumps and impulses. For a rigorous mathematical treatment it is first necessary to introduce a suitable solution space - the space of piecewise-smooth distributions. Within this distributional solution space the notion of stability encompasses impulse-freeness which is studied first. Afterwards stability under arbitrary and slow switching is investigated. A generalization to switched DAEs of a classical result concerning stability and commutativity is presented as well as a converse Lyapunov theorem. The theoretical results are illustrated with intuitive examples. |

## 2012 |

Trenn, Stephan; Willems, Jan C. Switched behaviors with impulses - a unifying framework Inproceedings In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 3203-3208, 2012, ISSN: 0743-1546. Abstract | Links | BibTeX | Tags: CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TrenWill12, We present a new framework to describe and study switched behaviors. We allow for jumps and impulses in the trajectories induced either implicitly by the dynamics after the switch or explicitly by “impacts”. With some examples from electrical circuit we motivate that the dynamical equations before and after the switch already uniquely define the “dynamics” at the switch, i.e. jumps and impulses. On the other hand, we also allow for external impacts resulting in jumps and impulses not induced by the internal dynamics. As a first theoretical result in this new framework we present a characterization for autonomy of a switched behavior. |

Trenn, Stephan; Wirth, Fabian Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms Inproceedings In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 2666–2671, 2012, ISSN: 0191-2216. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{TrenWirt12b, For linear switched differential algebraic equations (DAEs) we consider the problem of characterizing the maximal exponential growth rate of solutions. It is shown that a finite exponential growth rate exists if and only if the set of consistency projectors associated to the family of DAEs is product bounded. This result may be used to derive a converse Lyapunov theorem for switched DAEs. Under the assumption of irreducibility we show that a construction reminiscent of the construction of Barabanov norms is feasible as well. |

Tanwani, Aneel; Trenn, Stephan Observability of switched differential-algebraic equations for general switching signals Inproceedings In: Proc. 51st IEEE Conf. Decis. Control, Maui, USA, pp. 2648–2653, 2012. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, switched-DAEs, switched-systems @inproceedings{TanwTren12, We study observability of switched differential-algebraic equations (DAEs) for arbitrary switching. We present a characterization of observability and a related property called determinability. These characterizations utilize the results for the single-switch case recently obtained by the authors. Furthermore, we study observability conditions when only the mode sequence of the switching signal (and not the switching times) are known. This leads to necessary and sufficient conditions for observability and determinability. We illustrate the results with illustrative examples. |

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; Wirth, Fabian A converse Lyapunov theorem for switched DAEs Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 789–792, WILEY-VCH Verlag, 2012, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{TrenWirt12a, For switched ordinary differential equations (ODEs) it is well known that exponential stability under arbitrary switching yields the existence of a common Lyapunov function. The result is known as a “converse Lyapunov Theorem”. In this note we will present a converse Lyapunov theorem for switched differential algebraic equations (DAEs) as well as the construction of a Barabanov norm for irreducible switched DAEs. |

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. |

## 2011 |

Liberzon, Daniel; Trenn, Stephan; Wirth, Fabian Commutativity and asymptotic stability for linear switched DAEs Inproceedings In: Proc. 50th IEEE Conf. Decis. Control and European Control Conf. ECC 2011, Orlando, USA, pp. 417–422, 2011. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{LibeTren11, For linear switched ordinary differential equations with asymptotically stable constituent systems, it is well known that commutativity of the coefficient matrices implies asymptotic stability of the switched system under arbitrary switching. This result is generalized to linear switched differential algebraic equations (DAEs). Although the solutions of a switched DAE can exhibit jumps it turns out that it suffices to check commutativity of the “flow” matrices. As in the ODE case we are also able to construct a common quadratic Lyapunov function. |

## 2010 |

Domínguez-García, Alejandro D.; Trenn, Stephan Detection of impulsive effects in switched DAEs with applications to power electronics reliability analysis Inproceedings In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5662–5667, 2010. Abstract | Links | BibTeX | Tags: application, CDC, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{DomiTren10, This paper presents an analytical framework for detecting the presence of jumps and impulses in the solutions of switched differential algebraic equations (switched DAEs). The framework can be applied in the early design stage of fault-tolerant power electronics systems to identify design flaws that could jeopardize its reliability. The system is described by a switched differential algebraic equation, accounting for both fault-free system configurations and the configurations that arise after component faults, where each configuration p is defined by a pair of matrices (Ep;Ap). For each configuration p, the so called consistency projector is obtained from the pair (Ep;Ap). Based on the consistency projectors of all possible configurations, conditions for impulse-free and jump-free solutions of the switched DAE are established. A case-study of a dual redundant buck converter is presented to illustrate the framework. |

Tanwani, Aneel; Trenn, Stephan On observability of switched differential-algebraic equations Inproceedings In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5656–5661, 2010. Abstract | Links | BibTeX | Tags: CDC, DAEs, observability, piecewise-smooth-distributions, switched-DAEs, switched-systems @inproceedings{TanwTren10, We investigate observability of switched differential algebraic equations. The article primarily focuses on a class of switched systems comprising of two modes and a switching signal with a single switching instant. We provide a necessary and sufficient condition under which it is possible to recover the value of state trajectory (globally in time) with the help of switching phenomenon, even though the constituent subsystems may not be observable. In case the switched system is not globally observable, we discuss the concept of forward observability which deals with the recovery of state trajectory after the switching. A necessary and sufficient condition that characterizes forward observability is presented. |

Liberzon, Daniel; Trenn, Stephan The Bang-Bang Funnel Controller (long version) Miscellaneous Extended Conference Manuscript, 2010, (long version of corresponding CDC paper). Abstract | Links | BibTeX | Tags: CDC, funnel-control, input-constraints, switched-systems @misc{LibeTren10m, A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. |

## 2009 |

Liberzon, Daniel; Trenn, Stephan On stability of linear switched differential algebraic equations Inproceedings In: Proc. Joint 48th IEEE Conf. Decis. Control and 28th Chinese Control Conf., pp. 2156–2161, 2009. Abstract | Links | BibTeX | Tags: CDC, DAEs, Lyapunov, stability, switched-DAEs, switched-systems @inproceedings{LibeTren09, This paper studies linear switched differential algebraic equations (DAEs), i.e., systems defined by a finite family of linear DAE subsystems and a switching signal that governs the switching between them. We show by examples that switching between stable subsystems may lead to instability, and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms compared to those observed in switched systems described by ordinary differential equations (ODEs). We prove two sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The first result states that a common Lyapunov function guarantees stability under arbitrary switching when an additional condition involving consistency projectors holds (this extra condition is not needed when there are no jumps, as in the case of switched ODEs). The second result shows that stability is preserved under switching with sufficiently large dwell time. |

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 |

## 2005 |

French, Mark; Trenn, Stephan l In: Proc. 44th IEEE Conf. Decis. Control and European Control Conf. (ECC), pp. 2865–2870, 2005. Abstract | Links | BibTeX | Tags: CDC, stability, switched-systems @inproceedings{FrenTren05, A class of discrete plants controlled by a switching adaptive strategy is considered, and l^p bounds, 1 ≤ p ≤ ∞, are obtained for the closed loop gain relating input and output disturbances to internal signals. |