| 73. | Chang, Hamin; Song, Donghyeon; Trenn, Stephan; Shim, Hyungbo Disturbance observer with switched output redefinition for robust stabilization of non-minimum phase linear systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). @inproceedings{ChanSong24,Q-filter-based disturbance observer (DOB) has emerged as a powerful robust control technique renowned for its effectiveness in mitigating disturbances and addressing plant uncertainties. Despite its advantage, a key limitation of the Q-filter-based DOB lies in its requirement for plants to be of minimum phase. In this paper, we introduce an approach allowing the utilization of the Q-filter-based DOB as a stabilizing controller for non-minimum phase linear systems based on switched output redefinition of the systems. By redefining the output of systems to be controlled periodically, the approach stabilizes unstable internal dynamics of the systems as well as the original output. The proposed method is verified by an illustrative example. |
| 72. | Pour, Atiyeh Karimi; Trenn, Stephan Funnel control for impulsive switched systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). @inproceedings{KariTren24,Impulsive switched systems encompass various modes, each exhibiting distinct behaviours. Typically, a switching sequence orchestrates transitions between these modes, where state jumps may occur, potentially undermining output tracking performance or system stability. This work introduces a funnel controller tailored for relative degree one nonlinear impulsive switched systems. Notably, this controller operates solely based on system output without necessitating knowledge of system dynamics. Unlike classical funnel controllers with fixed boundaries, the proposed method dynamically adjusts the funnel boundary for each approaching jump, aiming to preserve adherence to the original boundary. No precise knowledge of jump instances or maps is required; approximate jump intervals and an upper bound for maximum jump height suffice. Theoretical analysis establishes that the error remains within the funnel, facilitating successful reference signal tracking. Performance validation is demonstrated via numerical simulation. |
| 71. | Chen, Yahao; Trenn, Stephan Solution concepts for linear piecewise affine differential-algebraic equations Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). @inproceedings{ChenTren24,In this paper, we introduce a definition of solu- tions for linear piecewise affine differential-algebraic equations (PWA-DAEs). Firstly, to address the conflict between projector-based jump rule and active regions, we propose a concept called state-dependent jump path. Unlike the conventional perspective that treats jumps as discrete-time dynamics, we interpret them as continuous dynamics, parameterized by a virtual time-variable. Secondly, by adapting the hybrid time-domain solution theory for continuous-discrete hybrid systems, we define the concept of jump-flow solutions for PWA-DAEs with the help of Filippov solutions for differential inclusions. Subsequently, we study various boundary behaviors of jump-flow solutions. Finally, we apply the proposed solution concepts in simulating a state-dependent switching circuit. |
| 70. | Berger, Thomas; Hackl, Christoph M.; Trenn, Stephan Asymptotic tracking by funnel control with internal models Proceedings Article In: Proc. European Control Conference (ECC24), Stockholm, Sweden, 2024, (to appear). @inproceedings{BergHack24,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). |
| 69. | 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, pp. 8489-8494, IEEE, 2023. @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. |
| 68. | 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, pp. 5869-5874, IEEE, Singapore, 2023. @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. |
| 67. | 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, pp. 914-919, IEEE, Singapore, 2023. @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. |
| 66. | 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. @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. |
| 65. | 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). @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. |
| 64. | 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. @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. |
| 63. | 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. @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. |
| 62. | 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. @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. |
| 61. | 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). @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. |
| 60. | 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. @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. |
| 59. | 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. @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. |
| 58. | Sutrisno,; Trenn, Stephan Observability of Singular Linear Switched Systems in Discrete Time: Single Switch Case Proceedings Article In: Proc. European Control Conference (ECC21), pp. 267-292, Rotterdam, Netherlands, 2021. @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. |
| 57. | 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. @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. |
| 56. | 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). @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. |
| 55. | 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). @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. |
| 54. | 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.). @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. |
| 53. | 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.). @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. |
| 52. | 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.). @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. |
| 51. | 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.). @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. |
| 50. | 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. @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. |
| 49. | 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. @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. |
| 48. | 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. @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. |
| 47. | 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. @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. |
| 46. | 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. @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. |
| 45. | Trenn, Stephan Asymptotic tracking with funnel control Proceedings Article In: PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2019, (online). @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. |
| 44. | 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)). @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. |
| 43. | 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). @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. |
| 42. | 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. @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. |
| 41. | 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. @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. |
| 40. | 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. @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. |
| 39. | 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. @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. |
| 38. | Trenn, Stephan Edge-wise funnel synchronization Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 821 - 822, WILEY-VCH Verlag, 2017, ISSN: 1617-7061. @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. |
| 37. | 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. @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. |
| 36. | 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. @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. |
| 35. | 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. @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. |
| 34. | 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. @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. |
| 33. | 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. @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. |
| 32. | 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. @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. |
| 31. | 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. @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. |
| 30. | 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. @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. |
| 29. | 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. @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. |
| 28. | 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. @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 |
| 27. | 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. @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. |
| 26. | 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. @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. |
| 25. | 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. @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. |
| 24. | 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. @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. |
| 23. | 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. @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. |
| 22. | 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. @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. |
| 21. | 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. @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. |
| 20. | 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. @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. |
| 19. | 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. @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. |
| 18. | 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. @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. |
| 17. | 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. @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. |
| 16. | 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. @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. |
| 15. | 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. @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. |
| 14. | 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. @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. |
| 13. | 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. @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. |
| 12. | 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. @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. |
| 11. | 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. @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. |
| 10. | 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. @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. |
| 9. | 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. @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. |
| 8. | 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. @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. |
| 7. | 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. @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. |
| 6. | Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller Proceedings Article In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. @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. |
| 5. | 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. @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. |
| 4. | 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. @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. |
| 3. | 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. @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. |
| 2. | French, Mark; Trenn, Stephan lp gain bounds for switched adaptive controllers Proceedings Article In: Proc. 44th IEEE Conf. Decis. Control and European Control Conf. (ECC), pp. 2865–2870, 2005. @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. |
| 1. | 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. @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. |
Conferences
| 73. | Disturbance observer with switched output redefinition for robust stabilization of non-minimum phase linear systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). |
| 72. | Funnel control for impulsive switched systems Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). |
| 71. | Solution concepts for linear piecewise affine differential-algebraic equations Proceedings Article In: Proc. 63rd IEEE Conf. Decision Control (CDC 2024), IEEE Milan, Italy, 2024, (to appear). |
| 70. | Asymptotic tracking by funnel control with internal models Proceedings Article In: Proc. European Control Conference (ECC24), Stockholm, Sweden, 2024, (to appear). |
| 69. | 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, pp. 8489-8494, IEEE, 2023. |
| 68. | Inhomogeneous singular linear switched systems in discrete time: Solvability, reachability, and controllability Characterizations Proceedings Article In: Proc. 62nd IEEE Conf. Decision Control, pp. 5869-5874, IEEE, Singapore, 2023. |
| 67. | 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, pp. 914-919, IEEE, Singapore, 2023. |
| 66. | 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. |
| 65. | 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). |
| 64. | 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. |
| 63. | 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. |
| 62. | 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. |
| 61. | Minimality of Linear Switched Systems with known switching signal Proceedings Article In: Proceedings in Applied Mathematics and Mechanics, pp. 1-3, 2021, (open access). |
| 60. | Optimal control of DAEs with unconstrained terminal costs Proceedings Article In: Proc. 60th IEEE Conf. Decision and Control (CDC 2021), pp. 5275-5280, 2021. |
| 59. | 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. |
| 58. | Observability of Singular Linear Switched Systems in Discrete Time: Single Switch Case Proceedings Article In: Proc. European Control Conference (ECC21), pp. 267-292, Rotterdam, Netherlands, 2021. |
| 57. | Minimal realization for linear switched systems with a single switch Proceedings Article In: Proc. European Control Conference (ECC21), pp. 1168-1173, Rotterdam, Netherlands, 2021. |
| 56. | 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). |
| 55. | 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). |
| 54. | Unimodular transformations for DAE initial trajectory problems Proceedings Article In: PAMM · Proc. Appl. Math. Mech., pp. e202000322, Wiley-VCH GmbH, 2021, (Open Access.). |
| 53. | 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.). |
| 52. | 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.). |
| 51. | 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.). |
| 50. | Impulse controllability of switched differential-algebraic equations Proceedings Article In: Proc. European Control Conference (ECC 2020), pp. 1561-1566, Saint Petersburg, Russia, 2020. |
| 49. | 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. |
| 48. | Asymptotic tracking via funnel control Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 4228-4233, Nice, France, 2019. |
| 47. | Delay regularity of differential-algebraic equations Proceedings Article In: Proc. 58th IEEE Conf. Decision Control (CDC) 2019, pp. 989-994, Nice, France, 2019. |
| 46. | 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. |
| 45. | Asymptotic tracking with funnel control Proceedings Article In: PAMM - Proc. Appl. Math. Mech., WILEY-VCH Verlag, 2019, (online). |
| 44. | 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)). |
| 43. | 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). |
| 42. | 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. |
| 41. | Impulses in structured nonlinear switched DAEs Proceedings Article In: Proc. 56th IEEE Conf. Decis. Control, pp. 3181 - 3186, Melbourne, Australia, 2017. |
| 40. | Switch observability for a class of inhomogeneous switched DAEs Proceedings Article In: Proc. 56th IEEE Conf. Decis. Control, pp. 3175 - 3180, Melbourne, Australia, 2017. |
| 39. | Switch-observer for switched linear systems Proceedings Article In: Proc. 56th IEEE Conf. Decis. Control, pp. 1749 - 1754, Melbourne, Australia, 2017. |
| 38. | Edge-wise funnel synchronization Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 821 - 822, WILEY-VCH Verlag, 2017, ISSN: 1617-7061. |
| 37. | Switch observability for homogeneous switched DAEs Proceedings Article In: Proc. 20th IFAC World Congress 2017, pp. 9355 - 9360, Toulouse, France, 2017, ISSN: 2405-8963. |
| 36. | 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. |
| 35. | 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. |
| 34. | 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. |
| 33. | Differential-algebraic inclusions with maximal monotone operators Proceedings Article In: Proc. 55th IEEE Conf. Decis. Control, Las Vegas, USA, pp. 610–615, 2016. |
| 32. | 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. |
| 31. | 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. |
| 30. | Duality of switched ODEs with jumps Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 4879–4884, 2015. |
| 29. | Distributional averaging of switched DAEs with two modes Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 3616–3620, 2015. |
| 28. | On detectability of switched linear differential-algebraic equations Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2957–2962, 2015. |
| 27. | Averaging for non-homogeneous switched DAEs Proceedings Article In: Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, pp. 2951–2956, 2015. |
| 26. | 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. |
| 25. | Controllability characterization of switched DAEs Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 643–644, WILEY-VCH Verlag, 2015, ISSN: 1617-7061. |
| 24. | Partial averaging for switched DAEs with two modes Proceedings Article In: Proc. 2015 European Control Conf. (ECC), Linz, Austria, pp. 2896–2901, 2015. |
| 23. | 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. |
| 22. | Nondecreasing Lyapunov functions Proceedings Article In: Proc. 21st Int. Symposium Math. Theory Networks Systems (MTNS), pp. 1038–1043, 2014. |
| 21. | Controllability of switched DAEs: the single switch case Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 15–18, Wiley-VCH Verlag GmbH, 2014. |
| 20. | 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. |
| 19. | 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. |
| 18. | An averaging result for switched DAEs with multiple modes Proceedings Article In: Proc. 52nd IEEE Conf. Decis. Control, Florence, Italy, pp. 1378 - 1383, 2013. |
| 17. | Averaging for switched DAEs Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 489–490, WILEY-VCH Verlag, 2013, ISSN: 1617-7061. |
| 16. | On averaging for switched linear differential algebraic equations Proceedings Article In: Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 2163 – 2168, 2013. |
| 15. | 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. |
| 14. | 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. |
| 13. | 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. |
| 12. | 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. |
| 11. | 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. |
| 10. | 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. |
| 9. | 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. |
| 8. | 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. |
| 7. | On observability of switched differential-algebraic equations Proceedings Article In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5656–5661, 2010. |
| 6. | The bang-bang funnel controller Proceedings Article In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. |
| 5. | 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. |
| 4. | Distributional solution theory for linear DAEs Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 10077–10080, WILEY-VCH Verlag, 2008, ISSN: 1617--7061. |
| 3. | Analogue Implementation of the funnel controller Proceedings Article In: PAMM - Proc. Appl. Math. Mech., pp. 823–824, WILEY-VCH Verlag, 2006, ISSN: 1617-7061. |
| 2. | lp gain bounds for switched adaptive controllers Proceedings Article In: Proc. 44th IEEE Conf. Decis. Control and European Control Conf. (ECC), pp. 2865–2870, 2005. |
| 1. | Adaptive tracking within prescribed funnels Proceedings Article In: Proc. 2004 IEEE Int. Conf. Control Appl., pp. 1032–1036, 2004. |