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.

## 2024 |

Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Journal Article In: IEEE Transactions on Automatic Control, vol. 69, no. 1, 2024, (2023 early access). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, switched-systems @article{HossTren24, 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. |

Sutrisno,; Trenn, Stephan Switched linear singular systems in discrete time: Solution theory and observability notions Journal Article In: Systems & Control Letters, vol. 183, no. 105674, pp. 1-11, 2024, (open access). Abstract | Links | BibTeX | Tags: DAEs, discrete-time, observability, solution-theory, switched-DAEs, switched-systems @article{SutrTren24, We study the solution theory of linear switched singular systems. In a recent paper by Anh et al. (2019), it was highlighted that the common assumption that each mode of the switched system is index-1 is not sufficient to guarantee existence and uniqueness of solutions of the corresponding switched system and the notion of “jointly index-1” was introduced. However, until now it was not clear what conditions are actually required to guarantee existence and uniqueness of solutions if the switching signal is not considered arbitrary. In particular, we study the two relevant situations where the mode sequence is fixed (and the switching times are arbitrary) and where the whole switching signal is fixed. In both cases, we provide conditions in terms of the original system matrices which ensure existence and uniqueness of solutions. We also extend the idea of the one-step map introduced by Anh et al. (2019) to these two cases. It turns out that in the case of a fixed switching signal, the index-1 condition for the individual modes is also not necessary (in addition to not being sufficient). Furthermore, we utilize the established solution theory to provide characterizations of observability and determinability of switched singular systems. |

## 2023 |

Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Output contraction analysis of nonlinear systems Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: nonlinear, stability @unpublished{YinJaya23pp, This paper introduce the notion of output contraction that expands the contraction notion to the time-varying nonlinear systems with output. It pertains to the systems’ property that any pair of outputs from the system converge to each other exponentially. This concept exhibits a more expansive nature when contrasted with another generalized contraction framework known as partial contraction. The first result establishes a connection between the output contraction of a time-varying system and the output exponential stability of its variational system. Subsequently, we derive a sufficient condition for achieving output contraction in time-varying systems by applying the output contraction Lyapunov criterion. Finally, we apply the results to analyze the output exponential stability of nonlinear time-invariant systems. |

Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Journal Article In: Automatica, 2023, (to appear). Abstract | Links | BibTeX | Tags: application, averaging, DAEs, LMIs, switched-DAEs, switched-systems @article{MostTren23pp, 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 Nonlinear Switched Singular Systems in Discrete-time: The One-step Map and Stability Under Arbitrary Switching Signals Journal Article In: European Journal of Control, vol. 74, no. 100852, pp. 1-7, 2023, (presented at the 2023 European Control Conference, Bucharest, Rumania; open access). Abstract | Links | BibTeX | Tags: DAEs, discrete-time, nonlinear, solution-theory, stability, switched-DAEs, switched-systems @article{SutrTren23a, 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 (switched) Lyapunov functions. |

Wijnbergen, Paul; Trenn, Stephan Impulse-free linear quadratic optimal control of switched differential algebraic equations Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: optimal-control, piecewise-smooth-distributions, switched-DAEs, switched-systems @unpublished{WijnTren23pp, In this paper the finite horizon linear quadratic regulator (LQR) problem for switched linear differential algebraic equations is studied. It is shown that for switched DAEs with a switching signal that induces locally finitely many switches the problem can be solved by solving several LQR problems for non-switched DAE recursively. First, it is shown how to solve the non-switched problems for index-1 DAEs followed by an extension of the results to higher index DAEs. The resulting optimal control can be computed based on the solution of a Riccati differential equation expressed in terms of the differential system matrices. The paper concludes with the extension of the results to the LQR problem for general switched DAEs. |

Berger, Thomas; Hackl, Christoph M.; Trenn, Stephan Asymptotic tracking by funnel control with internal models Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: funnel-control, relative-degree, stability @unpublished{BergHack23pp, Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters). |

Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Journal Article In: Automatica, vol. 156, no. 111204, pp. 1-10, 2023, (open access). Abstract | Links | BibTeX | Tags: funnel-control, networks, nonlinear, open-access, relative-degree, synchronization @article{LeeBerg23, When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics. This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes. |

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, vol. 68, iss. 10, pp. 6409-6416, 2023. Abstract | Links | BibTeX | Tags: LMIs, Lyapunov, nonlinear, stability, switched-systems @article{YinJaya23a, 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. |

Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Journal Article In: Automatica, vol. 156, no. 111208, pp. 1-14, 2023. Abstract | Links | BibTeX | Tags: DAEs, Lyapunov, nonlinear, normal-forms, solution-theory, stability, switched-DAEs, switched-systems @article{ChenTren23, In this paper, we investigate solutions and stability properties of switched nonlinear differential– algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs. |

Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan Stability of switched systems with multiple equilibria: a mixed stable-unstable subsystem case Journal Article In: Systems & Control Letters, vol. 180, no. 105622, pp. 1-9, 2023, (open access). Abstract | Links | BibTeX | Tags: LMIs, Lyapunov, stability, switched-systems @article{YinJaya23b, This paper studies the 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, Linear Matrix Inequality (LMI) conditions are derived that allow for numerical validation of the practical stability of switched affine systems, which include those with all unstable modes. Two examples are provided to verify the theoretical results. |

Sutrisno,; Trenn, Stephan Inhomogeneous Singular Linear Switched Systems in Discrete Time: Solvability, Reachability, and Controllability Characterizations Proceedings Article In: Proc. 62nd IEEE Conf. Decision Control, 2023, (to appear). Abstract | Links | BibTeX | Tags: CDC, controllability, DAEs, discrete-time, solution-theory, switched-DAEs, switched-systems @inproceedings{SutrTren23c, In this paper we study a novel solvability notion for discrete-time singular linear switched systems with inputs. We consider the existence and uniqueness of a solution on arbitrary finite time intervals with arbitrary inputs and arbitrary switching signals, and furthermore, we pay special attention to strict causality, i.e. the current state is only allowed to depend on past values of the state and the input. A necessary and suﬀicient condition for this solvability notion is then established. Furthermore, a surrogate switched system (an ordinary switched system that has equivalent input-output behavior) is derived for any solvable system. By utilizing those surrogate systems, we are able to characterize the reachability and controllability properties of the original singular systems using a geometric approach. |

Chang, Hamin; Trenn, Stephan Design of Q-filter-based Disturbance Observer for Differential Algebraic Equations and a Robust Stability Condition: Zero Relative Degree Case Proceedings Article In: Proc. 62nd IEEE Conf. Decision Control, 2023, (to appear). Abstract | BibTeX | Tags: CDC, DAEs, observer, relative-degree, stability @inproceedings{ChanTren23, While the disturbance observer (DOB)-based controller is widely utilized in various practical applications, there has been a lack of extension of its use to differential algebraic equations (DAEs). In this paper, we introduce several lemmas that establish necessary and/or sufficient conditions for specifying the relative degree of DAEs. Using these lemmas, we also figure out that there is a class of DAEs which can be viewed as linear systems with zero relative degree. For the class of DAEs, we propose a design of Q-filter-based DOB as well as a robust stability condition for systems controlled by the DOB through time domain analysis using singular perturbation theory. The proposed stability condition is verified by an illustrative example. |

Sutrisno,; Yin, Hao; Trenn, Stephan; Jayawardhana, Bayu Nonlinear singular switched systems in discrete-time: solution theory and incremental stability under restricted switching signals Proceedings Article In: Proc. 62nd IEEE Conf. Decision Control, Singapore, 2023, (to appear). Abstract | Links | BibTeX | Tags: CDC, DAEs, discrete-time, nonlinear, solution-theory, stability, switched-DAEs, switched-systems @inproceedings{SutrYin23, In this article the solvability analysis of discrete-time nonlinear singular switched systems with restricted switching signals is studied. We provide necessary and sufficient conditions for the solvability analysis under fixed switching signals and fixed mode sequences. The so-called surrogate systems (ordinary systems that have the equivalent behavior to the original switched systems) are introduced for solvable switched systems. Incremental stability, which ensures that all solution trajectories converge with each other, is then studied by utilizing these surrogate systems. Sufficient (and necessary) conditions are provided for this stability analysis using single and switched Lyapunov function approaches. |

Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin A novel two stages funnel controller limiting the error derivative Journal Article In: Systems & Control Letters, vol. 179, no. 105601, pp. 1-10, 2023, (open access). Abstract | Links | BibTeX | Tags: funnel-control, input-constraints, nonlinear, relative-degree @article{HuTren23, As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme to solve the output-tracking control problem for uncertain nonlinear systems with relative degree one and stable internal dynamics. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel. |

Wijnbergen, Paul; Trenn, Stephan Impulse-controllability of system classes of switched differential algebraic equations Journal Article In: Mathematics of Control, Signals, and Systems, vol. 36, 2023, (open access). Abstract | Links | BibTeX | Tags: controllability, DAEs, piecewise-smooth-distributions, switched-DAEs, switched-systems @article{WijnTren23, 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. |

Hossain, Sumon; Trenn, Stephan Model reduction for switched differential-algebraic equations with known switching signal Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: model-reduction, switched-DAEs @unpublished{HossTrenn23pp, Building on our recently proposed model reduction methods for switched ordinary linear systems we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features which makes a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first apply an exact model reduction which reduces the switched DAE to a switched ordinary systems with jumps and carefully keep track of the impulsive effects. As a second step we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples. |

Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, vol. 154, no. 111065, pp. 1-9, 2023, (open access). Abstract | Links | BibTeX | Tags: controllability, model-reduction, observability, open-access, 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. |

Sutrisno,; Trenn, Stephan Reachability and Controllability Characterizations for Linear Switched Systems in Discrete Time: A Geometric Approach Proceedings Article In: Proc. 2023 European Control Conference (ECC), pp. 2227-2232, Bucharest, Rumania , 2023. Abstract | Links | BibTeX | Tags: controllability, discrete-time, switched-systems @inproceedings{SutrTren23b, 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. |

Lanza, Lukas; Dennstädt, Dario; Worthmann, Karl; Trenn, Stephan; Schaller, Manuel Sampled-data funnel control with zero-order hold Unpublished 2023, (submitted). Abstract | Links | BibTeX | Tags: discrete-time, funnel-control, input-constraints, relative-degree @unpublished{LanzDann23pp, Output reference tracking for nonlinear systems with arbitrary relative degree via sampled-data feedback control with zero-order hold is studied. We propose a novel sample-and-hold feedback controller, which achieves output reference tracking with prescribed transient behaviour of the tracking error. Furthermore, we derive explicit bounds on the maximal required input signal and the global sampling time such that the proposed controller is feasible for all times. The results are illustrated with a numerical example. |

## 2022 |

Chen, Yahao; Trenn, Stephan Impulse-free jump solutions of nonlinear differential-algebraic equations Journal Article In: Nonlinear Analysis: Hybrid Systems, vol. 46, no. 101238, pp. 1-17, 2022. Abstract | Links | BibTeX | Tags: DAEs, nonlinear @article{ChenTren22a, In this paper, we propose a novel notion called impulse-free jump solution for nonlinear differential-algebraic equations (DAEs) of the form E(x)x' = F(x) with inconsistent initial values. The term “impulse-free” means that there are no Dirac impulses caused by jumps from inconsistent initial values, i.e., the directions of jumps stay in ker E(x). We find that the existence and uniqueness of impulse-free jumps are closely related to the notion of geometric index-1 and the involutivity of the distribution defined by ker E(x). Moreover, a singular perturbed system approximation is proposed for nonlinear DAEs; we show that solutions of the perturbed system approximate both impulse-free jump solutions and C1-solutions of nonlinear DAEs. Finally, we show by some examples that our results of impulse-free jumps are useful for the problems like consistent initializations of nonlinear DAEs and transient behavior simulations of electric circuits. |

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. |

Hossain, Sumon; Trenn, Stephan A weak Kalman decomposition approach for reduced realizations of switched linear systems Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin Funnel control for relative degree one nonlinear systems with input saturation Proceedings Article In: Proceedings of the 2022 European Control Conference (ECC), pp. 227-232, London, 2022. Abstract | Links | BibTeX | Tags: funnel-control, input-constraints, nonlinear, relative-degree @inproceedings{HuTren22, The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly. |

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, |

Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Journal Article In: Automatica, vol. 141, no. 110276, pp. 13, 2022, (open access). Abstract | Links | BibTeX | Tags: funnel-control, nonlinear, synchronization @article{LeeTren22, In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying (undirected) graph structure. The proposed coupling law is motivated by the so-called funnel control method studied in adaptive control under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by a high-gain coupling; consequently we call our novel synchronization method ‘(node-wise) funnel coupling.’ By adjusting the conventional proof technique in the funnel control study, we are even able to obtain asymptotic synchronization with the same funnel coupling law. Moreover, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, is analyzed in this paper. In particular, we introduce a single scalar dynamics called ‘emergent dynamics’ which describes the emergent synchronized behavior of the multi-agent system under funnel coupling. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. We illustrate this idea via the example of a distributed median solver based on funnel coupling. |

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, |

Berger, Thomas; Ilchmann, Achim; Trenn, Stephan Quasi feedback forms for differential-algebraic systems Journal Article In: IMA Journal of Mathematical Control and Information, vol. 39, iss. 2, pp. 533-563, 2022, (open access, published online October 2021). Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms, open-access @article{BergIlch22, We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example state space transformations, invertible transformations from the left, and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold. |

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{MostTren22a, 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 Proceedings Article 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. |

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

Wijnbergen, Paul; Trenn, Stephan Optimal control of DAEs with unconstrained terminal costs Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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, |

Chen, Yahao; Trenn, Stephan; Respondek, Witold Normal forms and internal regularization of nonlinear differential-algebraic control systems Journal Article In: International Journal of Robust and Nonlinear Control, vol. 2021, no. 31, pp. 6562-6584, 2021, (open access). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, normal-forms, open-access, solution-theory @article{ChenTren21d, In this paper, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, i.e., when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs. |

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

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

Chen, Yahao; Trenn, Stephan An approximation for nonlinear differential-algebraic equations via singular perturbation theory Proceedings Article In: Proceedings of 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS21), IFAC-PapersOnLine, pp. 187-192, Brussels, Belgium, 2021, (open access). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, normal-forms, open-access, solution-theory @inproceedings{ChenTren21c, In this paper, we study the jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector introduced in Liberzon and Trenn (2009) for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods (one is from the paper Liberzon and Trenn (2012) and the other is given by a MATLAB function) to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter and we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the C1-solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs. |

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

Chen, Yahao; Trenn, Stephan The differentiation index of nonlinear differential-algebraic equations versus the relative degree of nonlinear control systems Proceedings Article In: PAMM · Proc. Appl. Math. Mech. 2020, pp. e202000162, Wiley-VCH GmbH, 2021, (Open Access.). Abstract | Links | BibTeX | Tags: DAEs, nonlinear, normal-forms, relative-degree @inproceedings{ChenTren21a, It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differen- tiation index for nonlinear differential-algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems. |

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 |

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

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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Utility of edge-wise funnel coupling for asymptotically solving distributed consensus optimization Proceedings Article In: Proc. European Control Conference (ECC 2020), pp. 911-916, Saint Petersburg, Russia, 2020. Abstract | Links | BibTeX | Tags: funnel-control, networks, nonlinear, synchronization @inproceedings{LeeBerg20, A new approach to distributed consensus optimization is studied in this paper. The cost function to be minimized is a sum of local cost functions which are not necessarily convex as long as their sum is convex. This benefit is obtained from a recent observation that, with a large gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields. However, design of the large coupling gain requires global information such as network structure and individual agent dynamics. In this paper, we employ a nonlinear time-varying coupling of diffusive type, which we call `edge-wise funnel coupling.' This idea is borrowed from adaptive control, which enables decentralized design of distributed optimizers without knowledge of global information. Remarkably, without a common internal model, each agent achieves asymptotic consensus to the optimal solution of the global cost. We illustrate this result by a network that asymptotically finds the least-squares solution of a linear equation in a distributed manner. |

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

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, |

Hu, Jiaming; Trenn, Stephan Sliding mode observer based hysteresis compensation control for piezoelectric stacks Miscellaneous Book of Abstracts - 39th Benelux Meeting on Systems and Control, 2020. Links | BibTeX | Tags: application, nonlinear @misc{HuTren20m, |

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, |

Boon, Marko; van den Bosch, Mark; Breeuwsma, Paul; Bucchianico, Alessandro Di; Emampour, Mona; van Ginkel, Bart; Hepkema, Tjebbe; Holzinger, Philipp; Timmerman, Rik; Trenn, Stephan Event Driven Model with an Objective to Control Traffic Lights in the Netherlands Miscellaneous Scientific Proceedings 157th European Study Group with Industry (SWI 2020), 2020. Abstract | Links | BibTeX | Tags: application @misc{BoonBosc20m, The study group participants of SWI 2020 with regard to the challenge proposed by the company Sweco were tasked to initiate a discrete-event dynamic model into Smart Traffic. Smart Traffic is cloud driven software developed by Sweco, implementing real-time predictive traf- fic signal control. Currently, the microscopic traffic simulator SUMO is being used within Smart Traffic to predict the traffic pattern for the short-term future, with the purpose of optimising traffic signal settings. However, in practice, microscopic traffic simulators appear to be too slow and hence infeasible considering its application. We employ discrete-event simulations as a tool to predict the future traffic state both efficiently and effectively, even though those type of simulations are usually employed in a different context. We were in particular advised to focus on devising an event-driven model for a single, general intersection. This enabled us to create a thorough mathematical basic model. We are able to study various performance characteristics of the traffic light, such as the total delay or to the total squared delay. Accompanied with the mathematical basic model, we deliver in correspondence a fully functioning program written in Python. Our article includes a detailed yet relatively simple example based on this program. This example additionally demonstrates the difference in an optimal outcome when using the total delay or the total squared delay. Ultimately, we note that our model is easily extendable and several feasible extensions are proposed in this article. |

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. |

Trenn, Stephan The Laplace transform and inconsistent initial values Miscellaneous Extended Abstract, 2020, (accepted for cancelled MTNS 20/21, presented at MTNS 2022). Abstract | Links | BibTeX | Tags: DAEs, piecewise-smooth-distributions, solution-theory, switched-DAEs @misc{Tren20m, Switches in electrical circuits may lead to Dirac impulses in the solution; a real word example utilizing this effect is the spark plug. Treating these Dirac impulses in a mathematically rigorous way is surprisingly challenging. This is in particular true for arguments made in the frequency domain in connection with the Laplace transform. A survey will be given on how inconsistent initials values have been treated in the past and how these approaches can be justified in view of the now available solution theory based on piecewise-smooth distributions. |

## 2019 |

Lee, Jin Gyu; Trenn, Stephan Asymptotic tracking via funnel control Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 4228-4233, Nice, France, 2019. Abstract | Links | BibTeX | Tags: CDC, funnel-control @inproceedings{LeeTren19, Funnel control is a powerful and simple method to solve the output tracking problem without the need of a good system model, without identification and without knowledge how the reference signal is produced, but transient behavior as well as arbitrary good accuracy can be guaranteed. Until recently, it was believed that the price to pay for these very nice properties is that only practical tracking and not asymptotic tracking can be achieved. Surprisingly, this is not true! We will prove that funnel control – without any further assumptions – can achieve asymptotic tracking. |

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

Anh, Pham Ky; Linh, Pham Thi; Thuan, Do Duc; Trenn, Stephan The one-step-map for switched singular systems in discrete-time Proceedings Article 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. |

Trenn, Stephan Asymptotic tracking with funnel control Proceedings Article In: PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2019, (online). Abstract | Links | BibTeX | Tags: funnel-control, stability @inproceedings{Tren19, Funnel control is a strikingly simple control technique to ensure model free practical tracking for quite general nonlinear systems. It has its origin in the adaptive control theory, in particular, it is based on the principle of high gain feedback control. The key idea of funnel control is to chose the feedback gain large when the tracking error approaches the prespecified error tolerance (the funnel boundary). It was long believed that it is a theoretical limitation of funnel control not being able to achieve asymptotic tracking, however, in this contribution it will be shown that this is not the case. |

Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks Journal Article In: Automatica, vol. 101, pp. 280-289, 2019. Abstract | Links | BibTeX | Tags: DAEs, networks, observability @article{PatiTesi19, A problem of characterizing conditions under which a topological change in a network of differential algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogenous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network. |

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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Trenn, Stephan Edge-wise funnel synchronization Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 821 - 822, WILEY-VCH Verlag, 2017, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: funnel-control, networks, nonlinear, synchronization @inproceedings{Tren17, Recently, it was suggested in [Shim & Trenn 2015] to use the idea of funnel control in the context of synchronization of multi-agent systems. In that approach each agent is able to measure the difference of its own state and the average state of its neighbours and this synchronization error is used in a typical funnel gain feedback law, see e.g. [Ilchmann & Ryan 2008]. Instead of considering one error signal for each node of the coupling graph (corresponding to an agent) it is also possible to consider one error signal for each edge of the graph. In contrast to the node-wise approach this edgewise funnel synchronization approach results (at least in simulations) in a predictable consensus trajectory. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Switch observability for homogeneous switched DAEs Proceedings Article 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. |

Küsters, Ferdinand; Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks with applications to power grids Proceedings Article In: Proc. 20th IFAC World Congress 2017, pp. 7333 - 7338, Toulouse, France, 2017, ISSN: 2405-8963. Abstract | Links | BibTeX | Tags: application, DAEs, networks, observability @inproceedings{KustPati17a, The ability to detect topology variations in dynamical networks defined by differential algebraic equations (DAEs) is considered. We characterize the existence of initial states, for which topological changes are indiscernible. A key feature of our characterization is the ability to verify indiscernibility just in terms of the nominal topology. We apply the results to a power grid model and also discuss the relationship to recent mode-detection results for switched DAEs. |

Kall, Jochen; Kausar, Rukhsana; Trenn, Stephan Modeling water hammers via PDEs and switched DAEs with numerical justification Proceedings Article 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 Proceedings Article 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. |

Berger, Thomas; Reis, Timo; Trenn, Stephan Observability of linear differential-algebraic systems: A survey Book Section In: Ilchmann, Achim; Reis, Timo (Ed.): Surveys in Differential-Algebraic Equations IV, pp. 161–219, Springer-Verlag, Berlin-Heidelberg, 2017. Abstract | Links | BibTeX | Tags: DAEs, observability, survey @incollection{BergReis17, We investigate different concepts related to observability of linear constant coefficient differential-algebraic equations. Regularity, which, loosely speaking, guarantees existence and uniqueness of solutions for any inhomogeneity, is not required in this article. Concepts like impulse observability, observability at infinity, behavioral observability, strong and complete observability are described and defined in the time-domain. Special emphasis is placed on a normal form under output injection, state space and output space transformation. This normal form together with duality is exploited to derive Hautus type criteria for observability. We also discuss geometric criteria, Kalman decompositions and detectability. Some new results on stabilization by output injection are proved. |

## 2016 |

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

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

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 Proceedings Article 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. |

Küsters, Ferdinand; Trenn, Stephan; Wirsen, Andreas Observer design based on constant-input observability for DAEs Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 813–814, WILEY-VCH Verlag, 2016, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: DAEs, observability, observer @inproceedings{KustTren16b, For differential-algebraic equations (DAEs) an observability notion is considered which assumes the input to be unknown and constant. Based on this, an observer design is proposed. |

## 2015 |

Küsters, Ferdinand; Trenn, Stephan Duality of switched ODEs with jumps Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Shim, Hyungbo; Trenn, Stephan A preliminary result on synchronization of heterogeneous agents via funnel control Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2229–2234, 2015. Abstract | Links | BibTeX | Tags: CDC, funnel-control, networks, nonlinear, stability, synchronization @inproceedings{ShimTren15, We propose a new approach to achieve practical synchronization for heterogeneous agents. Our approach is based on the observation that a sufficiently large (but constant) gain for diffusive coupling leads to practical synchronization. In the classical setup of high-gain adaptive control, the funnel controller gained popularity in the last decade, because it is very simple and only structural knowledge of the underlying dynamical system is needed. We illustrate with simulations that “funnel synchronization” may be a promising approach to achieve practical synchronization of heterogeneous agents without the need to know the individual dynamics and the algebraic connectivity of the network (i.e., the second smallest eigenvalue of the Laplacian matrix). For a special case we provide a proof, but the proof for the general case is ongoing research. |

Küsters, Ferdinand; Trenn, Stephan Controllability characterization of switched DAEs Proceedings Article 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 Proceedings Article 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. |

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. |

Petreczky, Mihály; Tanwani, Aneel; Trenn, Stephan Observability of switched linear systems Book Section 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. |

## 2014 |

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

Defoort, Michael; Djemai, Mohamed; Trenn, Stephan Nondecreasing Lyapunov functions Proceedings Article 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 Proceedings Article 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. |

Berger, Thomas; Trenn, Stephan Kalman controllability decompositions for differential-algebraic systems Journal Article In: Syst. Control Lett., vol. 71, pp. 54–61, 2014, ISSN: 0167-6911. Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms @article{BergTren14, We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition (KCD). The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part. This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems. |

## 2013 |

Tanwani, Aneel; Trenn, Stephan An observer for switched differential-algebraic equations based on geometric characterization of observability Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article In: IEEE Trans. Autom. Control, vol. 58, no. 12, pp. 3126–3141, 2013. Abstract | Links | BibTeX | Tags: funnel-control, input-constraints, nonlinear, relative-degree @article{LibeTren13b, 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. |

Iannelli, Luigi; Pedicini, Carmen; Trenn, Stephan; Vasca, Francesco On averaging for switched linear differential algebraic equations Proceedings Article 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. |

Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller: time delays and case study Proceedings Article In: Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 1669–1674, 2013. Abstract | Links | BibTeX | Tags: application, funnel-control, input-constraints, nonlinear, relative-degree @inproceedings{LibeTren13a, We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered. |

Trenn, Stephan Stability of switched DAEs Book Section 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. |

Hackl, Christoph M.; Hopfe, Norman; Ilchmann, Achim; Mueller, Markus; Trenn, Stephan Funnel control for systems with relative degree two Journal Article In: SIAM J. Control Optim., vol. 51, no. 2, pp. 965–995, 2013. Abstract | Links | BibTeX | Tags: application, funnel-control, input-constraints, nonlinear, relative-degree @article{HackHopf13, 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. |

Berger, Thomas; Trenn, Stephan Addition to ``The quasi-Kronecker form for matrix pencils'' Journal Article In: SIAM J. Matrix Anal. & Appl., vol. 34, no. 1, pp. 94–101, 2013. Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergTren13, We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336--368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences. |

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

## 2012 |

Trenn, Stephan; Willems, Jan C. Switched behaviors with impulses - a unifying framework Proceedings Article 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 Proceedings Article 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 Proceedings Article 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. |

Berger, Thomas; Trenn, Stephan The quasi-Kronecker form for matrix pencils Journal Article In: SIAM J. Matrix Anal. & Appl., vol. 33, no. 2, pp. 336–368, 2012. Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergTren12, We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit. |

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

Trenn, Stephan; Wirth, Fabian A converse Lyapunov theorem for switched DAEs Proceedings Article 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. |

Hackl, Christoph M.; Trenn, Stephan The bang-bang funnel controller: An experimental verification Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 735–736, GAMM Annual Meeting 2012, Darmstadt Wiley-VCH Verlag GmbH, Weinheim, 2012. Abstract | Links | BibTeX | Tags: application, funnel-control, input-constraints, nonlinear, relative-degree @inproceedings{HackTren12, We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive. |

Berger, Thomas; Ilchmann, Achim; Trenn, Stephan The quasi-Weierstraß form for regular matrix pencils Journal Article In: Linear Algebra Appl., vol. 436, no. 10, pp. 4052–4069, 2012, (published online February 2010). Abstract | Links | BibTeX | Tags: DAEs, normal-forms, solution-theory @article{BergIlch12a, Regular linear matrix pencils A- E d in K^{n x n}[d], where K=Q, R or C, and the associated differential algebraic equation (DAE) E x' = A x are studied. The Wong sequences of subspaces are investigate and invoked to decompose the K^n into V* + W*, where any bases of the linear spaces V* and W* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V* and ``pure'' inconsistent initial values W* - {0}. Furthermore, V* and W* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A- E d lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E x' = A x. |

Trenn, Stephan Switched differential algebraic equations Book Section 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article 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 Proceedings Article In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. Abstract | Links | BibTeX | Tags: CDC, funnel-control, input-constraints, nonlinear, relative-degree @inproceedings{LibeTren10, 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. |

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 |

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

Liberzon, Daniel; Trenn, Stephan On stability of linear switched differential algebraic equations Proceedings Article 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 A normal form for pure differential algebraic systems Journal Article In: Linear Algebra Appl., vol. 430, no. 4, pp. 1070 – 1084, 2009. Abstract | Links | BibTeX | Tags: controllability, DAEs, normal-forms, observability, relative-degree @article{Tren09a, 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. |

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

## 2008 |

Trenn, Stephan Multilayer perceptrons: approximation order and necessary number of hidden units Journal Article In: IEEE Transactions on Neural Networks, vol. 19, no. 5, pp. 836–844, 2008, ISSN: 1045-9227. Abstract | Links | BibTeX | Tags: neural-networks @article{Tren08a, 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. |

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

## 2006 |

Mandaloju, Nagendra P.; Trenn, Stephan Analogue Implementation of the funnel controller Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 823–824, WILEY-VCH Verlag, 2006, ISSN: 1617-7061. Abstract | Links | BibTeX | Tags: application, funnel-control, nonlinear @inproceedings{MandTren06, In many tracking control problems, pre-specified bounds for the evolution of the tracking error should be met. The ‘funnel controller’ addresses this requirement and guarantees transient performance for a fairly large class of systems. In addition, only structural assumptions on the underlying system are made; the exact knowledge of the system parameters is not required. This is in contrast to most classical controllers where only asymptotic behaviour can be guaranteed and the system parameters must be known or estimated. Until now, the funnel controller was only studied theoretically. We will present the results of an analogue implementation of the funnel controller. The results show that the funnel controller works well in reality, i.e. it guarantees the pre-specified error bounds. The implementation is an analogue circuit composed of standard devices and is therefore suitable for a broad range of applications. |

Ilchmann, Achim; Sawodny, Oliver; Trenn, Stephan Pneumatic cylinders: modelling and feedback force-control Journal Article In: Int. J. Control, vol. 79, no. 6, pp. 650–661, 2006. Abstract | Links | BibTeX | Tags: application, input-constraints @article{IlchSawo06, 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. |

## 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. |

Ilchmann, Achim; Ryan, Eugene P.; Trenn, Stephan Tracking control: performance funnels and prescribed transient behaviour Journal Article In: Syst. Control Lett., vol. 54, no. 7, pp. 655–670, 2005. Abstract | Links | BibTeX | Tags: funnel-control @article{IlchRyan05, 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. |

## 2004 |

Ilchmann, Achim; Ryan, Eugene P.; Trenn, Stephan Adaptive tracking within prescribed funnels Proceedings Article In: Proc. 2004 IEEE Int. Conf. Control Appl., pp. 1032–1036, 2004. Abstract | Links | BibTeX | Tags: funnel-control, nonlinear, stability @inproceedings{IlchRyan04b, Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given lambda > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e 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 feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary. |

Ilchmann, Achim; Trenn, Stephan Input constrained funnel control with applications to chemical reactor models Journal Article In: Syst. Control Lett., vol. 53, no. 5, pp. 361–375, 2004. Abstract | Links | BibTeX | Tags: application, funnel-control, input-constraints @article{IlchTren04, 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*. |