Projects, publications, slides and other research related news
The details of the counterexample I used in my PhD-thesis to proof that a distributional restriction is not possible in general is now accepted for publication as a paper in the journal Examples and Counterexamples. The reviewing process took quite some time, but in the end it was accepted without any changes.
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. |
The following two papers have been accepted for (online) presentation at the 60th IEEE CDC:
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. |
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. |
Our manuscript
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. |
Our paper
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. |
Furthermore, Yahao and I have finished a follow-up paper discussing the generalization of the consistency-projector to the nonlinear case:
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. |
In the last few days we have prepared the final version of some accepted conference papers, I have therefore updated the preprints of the following papers:
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. |
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. |
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. |
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. |
Our paper
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. |
Our following papers were accepted for publication in Automatica and IEEE Transactions on Automatic Control, I have updated the preprint and will add bibliographical data as soon as possible.
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. |
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. |
[Update] For the Automatica paper there is a 50-day free download available.
All our submissions to ECC’20 and IFAC World Congress 2020 were accepted for publication. I have updated the preprints to the final version and as soon as the bibliographical data is available I will also update those.
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. |
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. |
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. |
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. |
My short paper about asymptotic funnel control (see below) was one of the most downloaded papers of the PAMM in 2019!
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. |
I have uploaded the final version of my three accepted CDC papers
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. |
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. |
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. |