### Differential-Algebraic Equations: A Projector Based Analysis (Differential-Algebraic Equations Forum)

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The main result we prove is that AA improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step.

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In addition to improving the convergence rate, our results indicate that AA increases the radius of convergence. Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations.

Results of several numerical tests for fluid applications are given which illustrate the theory. The Stokes problem is motivated by computation of flow in the Geosciences. Teubner Verlagsgesellschaft, Leipzig, With German, French and Russian summaries. Horn and Charles R. Hinze , R. Pinnau , M. Ulbrich , and S. MR [HV95] J. Heinrich and C. Vionnet , The penalty method for the Navier-Stokes equations , Arch. Methods Engrg. Hairer and G. Wanner , Solving ordinary differential equations.

II , 2nd ed. Stiff and differential-algebraic problems. Analysis and numerical solution. MR [LO93] Ch. Lubich and A. Ostermann , Runge-Kutta methods for parabolic equations and convolution quadrature , Math. Chan, W. Cook, E. Hairer, J. Hastad, A. Iserles, H. Langtangen, C. Lions, C.

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Lubich, A. We show that the augmented Wong sequences can be exploited for a transformation of the system into a Kalman controllability decomposition KCD. The KCD decouples the system into a completely controllable part, an uncontrollable part given by an ordinary differential equation and an inconsistent part, which is behaviorally controllable but contains no completely controllable part.

This decomposition improves a known KCD from a behavioral point of view. We conclude the paper with some features of the KCD in the case of regular systems. We study linear differential-algebraic control systems and investigate decompositions with respect to controllability properties.

## Numerical Solution of Differential-Algebraic Equations

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. 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.

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. 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. 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. 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.

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. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels.

We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions.

The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. 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. 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. 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.

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We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. In the presence of sudden structural changes e. A special feature of switched DAEs is the presence of induced jumps or even Dirac impulses in the solution. This chapter studies stability of switched DAEs taking into account the presence of these jumps and impulses.

### Table of Contents

For a rigorous mathematical treatment it is first necessary to introduce a suitable solution space - the space of piecewise-smooth distributions. Within this distributional solution space the notion of stability encompasses impulse-freeness which is studied first. Afterwards stability under arbitrary and slow switching is investigated.

A generalization to switched DAEs of a classical result concerning stability and commutativity is presented as well as a converse Lyapunov theorem. The theoretical results are illustrated with intuitive examples. Differential algebraic equations DAEs are used to model dynamical systems with constraints given by algebraic equations. The system is assumed to have strict relative degree two with weakly stable zero dynamics.

The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller i is applicable to relative degree one systems, ii allows for input constraints provided a feasibility condition formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state holds, iii is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective.

Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. Paper doi In our recent paper [T. Berger and S. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences. We refine a result concerning singular matrix pencils and the Wong sequences.

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 values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks. This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations DAEs.

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. We present a new framework to describe and study switched behaviors. 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. For linear switched differential algebraic equations DAEs we consider the problem of characterizing the maximal exponential growth rate of solutions.

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. We study observability of switched differential-algebraic equations DAEs for arbitrary switching. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form.

The results are illustrated with an example given by a simple electrical circuit. We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.

We study switched nonlinear differential algebraic equations DAEs with respect to existence and nature of solutions as well as stability. 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. For switched ordinary differential equations ODEs it is well known that exponential stability under arbitrary switching yields the existence of a common Lyapunov function.

We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive.

We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. So the latter can be viewed as a generalized Jordan form. Preprint Corrections see Paragraph 6 of Note to Editors doi The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments.

However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the consistency projectors. In order to capture this impulsive behavior the space of piecewise-smooth distributions is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs including the uniqueness of the jumps induced by the switches.

With the help of the consistency projector a condition is formulated whether a switch or fault can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role. In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation switched DAE , i.

This result is generalized to linear switched differential algebraic equations DAEs. As in the ODE case we are also able to construct a common quadratic Lyapunov function. 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.

## Differential-Algebraic Equations: A Projector Based Analysis | fekypymo.ga

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. 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 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. We investigate observability of switched differential algebraic equations.

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. 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.

Preprint Preprint long version doi The solutions x and the inhomogeneities f are assumed to be distributions generalized functions. As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well.

Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses.

A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs i. Sufficient and necessary conditions for the regularity of distributional DAEs are given. 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. This paper studies linear switched differential algebraic equations DAEs , i. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability. If the matrix E is not invertible, these equations contain differential as well as algebraic equations. A main goal of this dissertation is the consideration of certain distributions or generalized functions as solutions and studying time-varying DAEs, whose coefficient matrices have jumps.

Therefore, a suitable solution space is derived. This solution space allows to study the important class of switched DAEs. The space of piecewise-smooth distributions is introduced as the solution space. For this space of distributions, it is possible to define a multiplication, hence DAEs can be studied whose coefficient matrices have also distributional entries. For distributional DAEs, existence and uniqueness of solutions are studied, therefore, the concept of regularity for distributional DAEs is introduced.

Necessary and sufficient conditions for existence and uniqueness of solutions are derived. Sufficient conditions are given which ensure that all solutions of a switched DAE are impulse free. Furthermore, it is studied which conditions ensure that arbitrary switching between stable subsystems yield a stable overall system.

Finally, controllability and observability for distributional DAEs are studied. Download Book Cover Publication-Website. For a given approximation order, explicit formulas for the necessary number of hidden units and its distributions to the hidden layers of the MLP are derived. These formulas depend only on the number of input variables and on the desired approximation order. The concept of approximation order encompasses Kolmogorov-Gabor polynomials or discrete Volterra series, which are widely used in static and dynamic models of nonlinear systems. The results are obtained by considering structural properties of the Taylor polynomials of the function in question and of the MLP function.

This paper considers the approximation of sufficiently smooth multivariable functions with a multilayer perceptron MLP. 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.

A solution theory for switched linear differential—algebraic equations DAEs is developed. 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.