Stephan Trenn

University of Groningen | RUG · Jan C. Willems Center for Systems and Control

This survey aims at giving a comprehensive overview of the solution theory of linear differential algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial...

In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form Ex = Ax + Bu where E is, in general, a singular matrix and u is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can a...

The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switch...

We study switched nonlinear differential algebraic equations (DAEs) with re-spect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for switched linear DAEs to establish a solution framework for switched nonlinear DAEs. In par-ticular, we allow induced jumps in the soluti...

This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron (MLP). For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on th...

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and anothe...

We propose a novel learning-based tracking controller for nonlinear systems of arbitrary relative degree. Here, we use sample-and-hold input signals and derive a bound on the required sampling frequency. While the controller guarantees tracking within prescribed, possibly time-varying bounds on the error signal, system data is collected at runtime...

This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and noncontracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global exponential stability of its variational system. Based on this equivalence property, sufficient conditions for...

In this paper, impulse-controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. For a system class generated by switching signals with a...

We propose a novel learning-based tracking controller for nonlinear systems of arbitrary relative degree. Here, we use sample-and-hold input signals and derive a bound on the required sampling frequency. While the controller guarantees tracking within prescribed, possibly time-varying bounds on the error signal, system data is collected at runtime...

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 th...

In this paper, we propose a novel notion called impulse-free jump solution for nonlinear differential–algebraic equations (DAEs) of the form E(x)ẋ=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 the jumps stay in kerE(x)....

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 syst...

In this paper impulse controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. In the case of a system class generated by switching signa...

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-call...

In this paper, we study solutions and stability for switched nonlinear differential-algebraic equations (DAEs). A novel notion of solutions, called the impulse-free (jump-flow) solution, is proposed and a geometric characterization for its existence and uniqueness is given as a nonlinear version of the impulse-free condition used in, e.g., [22, 23]...

In this paper, we propose a novel notion called impulse-free jump solution for nonlinear differential-algebraic equations (DAEs) of the form E(x)ẋ = 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)....

In this paper, we propose some sufficient conditions for checking the asymptotic stability of switched nonlinear differential-algebraic equations (DAEs) under arbitrary switching 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 con...

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 th...

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. Afterw...

A counterexample is presented showing that it is not possible to define a restriction for distributions.

In a recent work by three of the authors, in order to enforce synchronization for scalar heterogeneous multi-agent systems with some useful characteristics, a node-wise funnel coupling law was proposed. The emergent dynamics, to which each of the agents synchronizes, was characterized and it was studied how networks can be synthesized which exhibit...

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...

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 d...

In this article, 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 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...

In this paper, we study 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 for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consiste...

In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-f...

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...

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.

It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differentiation 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...

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...

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 graph structure. Meanwhile, we consider undirected graphs and scalar input affine s...

A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations 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 electri...

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 condition in terms of Lyapunov functions. Furthermore, we generalize the notion of joint spectral radius of a finite set of ma...

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 Z...

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 beh...

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 interva...

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 syste...

Switched system can be considered as a special class of piecewise continuous linear time varying systems. In this contribution , we propose a model reduction approach for piecewise continuous linear time varying switched systems. The technique is mainly based on balancing based model order reduction (MOR) method for linear time varying systems. We...

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 a...

This paper studies detectability for switched linear differential-algebraic equations (DAEs) and its application to the synthesis of observers, which generate asymptotically converging state estimates. Equating detectability to asymptotic stability of zero-output-constrained state trajectories, and building on our work on interval-wise observabilit...

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations arise naturally, if a feedback controller is applied to a descriptor system, since 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 withi...

A problem of characterizing conditions under which a topological change in a network of differential–algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topologic...

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 tha...

Mode observability of switched systems requires observability of each individual mode. We consider other concepts of observability that do not have this requirement: Switching time observability and switch observability. The latter notion is based on the assumption that at least one switch occurs. These concepts are analyzed and characterized both...

It is well known that for switched systems the overall dynamics can be unstable despite stability of all individual modes. We show that this phenomenon 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 for the linearized DAE mode...

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. [I...

Averaging is a useful technique to simplify the analysis of switched systems. In this paper we present averaging results for the class of systems described by switched differential algebraic equations (DAEs). Conditions on the consistency projectors are given which guarantee convergence towards a non-switched averaged system. A consequence of this...

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)...

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 th...

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...

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...

We investigate different concepts related to observability of linear constant coefficient differential-algebraic equations. Regularity, which, loosely speaking, guarantees existence and uniqueness of solutions for any inhomogeneity, is not required in this article. Concepts like impulse observability, observability at infinity, behavioral observabi...

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 modele...

The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential–algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condi...

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...

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. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential-algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appr...

We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability charac...

Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. The main difference in these approaches is how the switching signal is viewed: Is it a fixed and known function of time, is it an unknown external signal, is it the res...

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. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Wein...

We study controllability of switched differential–algebraic equations. We are able to establish a controllability characterization where we assume that the switching signal is known. The characterization takes into account possible jumps induced by the switches. It turns out that controllability not only depends on the actual switching sequence but...