43. | Lanza, Lukas; Dennstädt, Dario; Worthmann, Karl; Schmitz, Philipp; Sen, Gökcen Devlet; Trenn, Stephan; Schaller, Manuel Sampled-data funnel control and its use for safe continual learning Journal Article In: Systems & Control Letters, vol. 192, no. 105892, pp. 1-12, 2024, (open access). @article{LanzDann24, We propose a novel sampled-data output-feedback controller for nonlinear systems of arbitrary relative degree that ensures reference tracking within prescribed error bounds. We provide explicit bounds on the maximum input signal and the required uniform sampling time. A key strength of this approach is its capability to serve as a safety filter for various learning-based controller designs, enabling the use of learning techniques in safety- critical applications. We illustrate its versatility by integrating it with two different controllers: a reinforcement learning controller and a non-parametric predictive controller based on Willems et al.’s fundamental lemma. Numerical simulations illustrate effectiveness of the combined controller design. |

42. | 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, iss. 2, pp. 351–380, 2024, (to appear, open access). @article{WijnTren24a, 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. |

41. | Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco Averaging for switched impulsive systems with pulse width modulation Journal Article In: Automatica, vol. 160, no. 111447, pp. 1-12, 2024, (open access). @article{MostTren24, 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. |

40. | 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). @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. |

39. | 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, pp. 535-542, 2024. @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. |

38. | 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). @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. |

37. | 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. @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. |

36. | 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. @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. |

35. | 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). @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. |

34. | 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). @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. |

33. | 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). @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. |

32. | 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). @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. |

31. | 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, (open access). @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. |

30. | 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). @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. |

29. | 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). @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. |

28. | 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). @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. |

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

26. | 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). @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. |

25. | 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. @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. |

24. | 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.). @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. |

23. | 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). @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. |

22. | 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. @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. |

21. | Patil, Deepak; Tesi, Pietro; Trenn, Stephan Indiscernible topological variations in DAE networks Journal Article In: Automatica, vol. 101, pp. 280-289, 2019. @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. |

20. | Tanwani, Aneel; Trenn, Stephan Detectability and observer design for switched differential algebraic equations Journal Article In: Automatica, vol. 99, pp. 289-300, 2019. @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. |

19. | Küsters, Ferdinand; Trenn, Stephan Switch observability for switched linear systems Journal Article In: Automatica, vol. 87, pp. 121-127, 2018. @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. |

18. | 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. @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. |

17. | 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. @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. |

16. | 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. @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. |

15. | Küsters, Ferdinand; Trenn, Stephan Duality of switched DAEs Journal Article In: Math. Control Signals Syst., vol. 28, no. 3, pp. 25, 2016. @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. |

14. | 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. @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. |

13. | 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. @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. |

12. | 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. @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. |

11. | 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. @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. |

10. | 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. @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. |

9. | 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. @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. |

8. | 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. @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. |

7. | 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). @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. |

6. | Trenn, Stephan Regularity of distributional differential algebraic equations Journal Article In: Math. Control Signals Syst., vol. 21, no. 3, pp. 229–264, 2009. @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. |

5. | Trenn, Stephan A normal form for pure differential algebraic systems Journal Article In: Linear Algebra Appl., vol. 430, no. 4, pp. 1070 – 1084, 2009. @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. |

4. | 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. @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. |

3. | 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. @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. |

2. | 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. @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. |

1. | 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. @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*. |

# Journals

43. | Sampled-data funnel control and its use for safe continual learning Journal Article In: Systems & Control Letters, vol. 192, no. 105892, pp. 1-12, 2024, (open access). |

42. | Impulse-controllability of system classes of switched differential algebraic equations Journal Article In: Mathematics of Control, Signals, and Systems, vol. 36, iss. 2, pp. 351–380, 2024, (to appear, open access). |

41. | Averaging for switched impulsive systems with pulse width modulation Journal Article In: Automatica, vol. 160, no. 111447, pp. 1-12, 2024, (open access). |

40. | 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). |

39. | Midpoint based balanced truncation for switched linear systems with known switching signal Journal Article In: IEEE Transactions on Automatic Control, vol. 69, no. 1, pp. 535-542, 2024. |

38. | 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). |

37. | 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. |

36. | On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Journal Article In: Automatica, vol. 156, no. 111208, pp. 1-14, 2023. |

35. | 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). |

34. | 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). |

33. | 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). |

32. | Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, vol. 154, no. 111065, pp. 1-9, 2023, (open access). |

31. | Impulse-free jump solutions of nonlinear differential-algebraic equations Journal Article In: Nonlinear Analysis: Hybrid Systems, vol. 46, no. 101238, pp. 1-17, 2022, (open access). |

30. | 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). |

29. | 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). |

28. | A smooth model for periodically switched descriptor systems Journal Article In: Automatica, vol. 136, no. 110082, pp. 1-8, 2022, (open access). |

27. | Distributional restriction impossible to define Journal Article In: Examples and Counterexamples, vol. 1, no. 100023, pp. 1-4, 2021, (open access). |

26. | 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). |

25. | 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. |

24. | Impulse-free interval-stabilization of switched differential algebraic equations Journal Article In: Systems & Control Letters, vol. 149, pp. 104870.1-10, 2021, (Open Access.). |

23. | 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). |

22. | Stability analysis for switched discrete-time linear singular systems Journal Article In: Automatica, vol. 119, no. 109100, 2020. |

21. | Indiscernible topological variations in DAE networks Journal Article In: Automatica, vol. 101, pp. 280-289, 2019. |

20. | Detectability and observer design for switched differential algebraic equations Journal Article In: Automatica, vol. 99, pp. 289-300, 2019. |

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

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

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

16. | Solvability and stability of a power system DAE model Journal Article In: Syst. Control Lett., vol. 97, pp. 12–17, 2016. |

15. | Duality of switched DAEs Journal Article In: Math. Control Signals Syst., vol. 28, no. 3, pp. 25, 2016. |

14. | Controllability of switched differential-algebraic equations Journal Article In: Syst. Control Lett., vol. 78, no. 0, pp. 32 - 39, 2015, ISSN: 0167-6911. |

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

12. | 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. |

11. | Funnel control for systems with relative degree two Journal Article In: SIAM J. Control Optim., vol. 51, no. 2, pp. 965–995, 2013. |

10. | 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. |

9. | The quasi-Kronecker form for matrix pencils Journal Article In: SIAM J. Matrix Anal. & Appl., vol. 33, no. 2, pp. 336–368, 2012. |

8. | Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability Journal Article In: Automatica, vol. 48, no. 5, pp. 954–963, 2012. |

7. | 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). |

6. | Regularity of distributional differential algebraic equations Journal Article In: Math. Control Signals Syst., vol. 21, no. 3, pp. 229–264, 2009. |

5. | A normal form for pure differential algebraic systems Journal Article In: Linear Algebra Appl., vol. 430, no. 4, pp. 1070 – 1084, 2009. |

4. | 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. |

3. | Pneumatic cylinders: modelling and feedback force-control Journal Article In: Int. J. Control, vol. 79, no. 6, pp. 650–661, 2006. |

2. | Tracking control: performance funnels and prescribed transient behaviour Journal Article In: Syst. Control Lett., vol. 54, no. 7, pp. 655–670, 2005. |

1. | Input constrained funnel control with applications to chemical reactor models Journal Article In: Syst. Control Lett., vol. 53, no. 5, pp. 361–375, 2004. |