Our paper
A novel two stages funnel controller limiting the error derivative Journal Article In: Systems & Control Letters, vol. 179, no. 105601, pp. 1-10, 2023, (open access). |
Switched nonlinear DAE paper accepted
Our paper
Chen, Yahao; Trenn, Stephan On impulse-free solutions and stability of switched nonlinear differential-algebraic equations Journal Article In: Automatica, vol. 156, no. 111208, pp. 1-14, 2023. @article{ChenTren23,In this paper, we investigate solutions and stability properties of switched nonlinear differential– algebraic equations (DAEs). We introduce a novel concept of solutions, called impulse-free (jump-flow) solutions, and provide a geometric characterization that establishes their existence and uniqueness. This characterization builds upon the impulse-free condition utilized in previous works such as Liberzon and Trenn (2009, 2012), which focused on linear DAEs. However, our formulation extends this condition to nonlinear DAEs. Subsequently, we demonstrate that the stability conditions based on common Lyapunov functions, previously proposed in our work (Chen and Trenn, 2022) (distinct from those in Liberzon and Trenn (2012)), can be effectively applied to switched nonlinear DAEs with high-index models. It is important to note that these models do not conform to the nonlinear Weierstrass form. Additionally, we extend the commutativity stability conditions presented in Mancilla-Aguilar (2000) from switched nonlinear ordinary differential equations to the case of switched nonlinear DAEs. To illustrate the efficacy of the proposed stability conditions, we present simulation results involving switching electrical circuits and provide numerical examples. These examples serve to demonstrate the practical utility of the developed stability criteria in analyzing and understanding the behavior of switched nonlinear DAEs. |
Switched model reduction paper accepted
Our paper
Hossain, Sumon; Trenn, Stephan Midpoint based balanced truncation for switched linear systems with known switching signal Journal Article In: IEEE Transactions on Automatic Control, vol. 69, no. 1, pp. 535-542, 2024. @article{HossTren24,We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size. |
Top Downloaded Paper
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. |
Reduced realization paper accepted in Automatica
Our manuscript
Hossain, Sumon; Trenn, Stephan Reduced realization for switched linear systems with known mode sequence Journal Article In: Automatica, vol. 154, no. 111065, pp. 1-9, 2023, (open access). @article{HossTren23a,We consider switched linear systems with mode-dependent state-dimensions and/or state jumps and propose a method to obtain a switched system of reduced size with identical input-output behavior. Our approach is based in considering time-dependent reachability and unobservability spaces as well as suitable extended reachability and restricted unobservability spaces together with the notion of a weak Kalman decomposition. A key feature of our approach is that only the mode sequence of the switching signal needs to be known and not the exact switching times. However, the size of a minimal realization will in general depend on the mode durations, hence it cannot be expected that our method always leads to minimal realization. Nevertheless, we show that our method is optimal in the sense that a repeated application doesn’t lead to a further reduction and we also highlight a practically relevant special case, where minimality is achieved. |
Contraction paper available online
My first collabaration with Bayu and his PhD-student Hao has been accepted for publication and is now availble as early access:
Yin, Hao; Jayawardhana, Bayu; Trenn, Stephan On contraction analysis of switched systems with mixed contracting-noncontracting modes via mode-dependent average dwell time Journal Article In: IEEE Transactions on Automatic Control, vol. 68, iss. 10, pp. 6409-6416, 2023. @article{YinJaya23a,This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and non- contracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global ex- ponential stability of its variational system. Based on this equiva- lence property, sufficient conditions for a mode-dependent average dwell/leave-time based switching law to be contractive are estab- lished. Correspondingly, LMI conditions are derived that allow for numerical validation of contraction property of nonlinear switched systems, which include those with all non-contracting modes. |
Nonlinear Jumps for DAEs paper online
The paper
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. |
Three papers accepted for ECC
The following three papers have been accepted for presentation at the 2022 European Control Conference (ECC22) in London, UK:
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. |
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. |
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. |
Funnel synchronisation paper accepted
Finally our paper
Lee, Jin Gyu; Trenn, Stephan; Shim, Hyungbo Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling Journal Article In: Automatica, vol. 141, no. 110276, pp. 13, 2022, (open access). @article{LeeTren22,In this paper, we introduce a nonlinear time-varying coupling law, which can be designed in a fully decentralized manner and achieves approximate synchronization with arbitrary precision, under only mild assumptions on the individual vector fields and the underlying (undirected) graph structure. The proposed coupling law is motivated by the so-called funnel control method studied in adaptive control under the observation that arbitrary precision synchronization can be achieved for heterogeneous multi-agent systems by a high-gain coupling; consequently we call our novel synchronization method ‘(node-wise) funnel coupling.’ By adjusting the conventional proof technique in the funnel control study, we are even able to obtain asymptotic synchronization with the same funnel coupling law. Moreover, the emergent collective behavior that arises for a heterogeneous multi-agent system when enforcing arbitrary precision synchronization by the proposed funnel coupling law, is analyzed in this paper. In particular, we introduce a single scalar dynamics called ‘emergent dynamics’ which describes the emergent synchronized behavior of the multi-agent system under funnel coupling. Characterization of the emergent dynamics is important because, for instance, one can design the emergent dynamics first such that the solution trajectory behaves as desired, and then, provide a design guideline to each agent so that the constructed vector fields yield the desired emergent dynamics. We illustrate this idea via the example of a distributed median solver based on funnel coupling. |
Smooth approximation of switched DAEs paper accepted
Our paper
Mostacciuolo, Elisa; Trenn, Stephan; Vasca, Francesco A smooth model for periodically switched descriptor systems Journal Article In: Automatica, vol. 136, no. 110082, pp. 1-8, 2022, (open access). @article{MostTren22a,Switched descriptor systems characterized by a repetitive finite sequence of modes can exhibit state discontinuities at the switching time instants. The amplitudes of these discontinuities depend on the consistency projectors of the modes. A switched ordinary differential equations model whose continuous state evolution approximates the state of the original system is proposed. Sufficient conditions based on linear matrix inequalities on the modes projectors ensure that the approximation error is of linear order of the switching period. The theoretical findings are applied to a switched capacitor circuit and numerical results illustrate the practical usefulness of the proposed model. |