Nonlinear DAEs Research Articles

Improved structural methods for nonlinear differential-algebraic equations via combinatorial relaxation

Abstract Differential-algebraic equations (DAEs) are widely used for modelling dynamical systems. In the numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing steps prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, structural methods fail if the DAE has a singular system Jacobian matrix. For such DAEs, methods have been proposed to modify them to other DAEs to which structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. Our methods are implemented as a MATLAB library using MuPAD, and through its application to practical DAEs, we show that our methods can be used as a promising preprocessing of DAEs that the index reduction procedure in MATLAB cannot handle.

An approximation for nonlinear differential-algebraic equations via singular perturbation theory

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 consistency projectors with two existing consistent initialization methods 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, we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the $\mathcal C^1$-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.

A new approach for computing the exact solutions of DAEs in generalized Hessenberg forms

‎In this paper‎, ‎we propose a new method‎, ‎with different approach and economical computing‎, ‎that presents explicit formulas for the exact solutions of a large class DAEs in Hessenberg forms‎. ‎First‎, ‎we illustrate the method for linear time-varying DAEs in Hessenberg forms‎, ‎in order to show the different approach and also we present an algorithm for proposed method that shows the method is as economic as possible‎. ‎Then‎, ‎we describe that the method is efficient for larger classes including special case of non-linear DAEs in Hessenberg forms‎. ‎Some examples are presented to illustrate the proposed method‎.

Internal and External Linearization of Semi-Explicit Differential Algebraic Equations

In this paper, we study two kinds of linearization (internal and external) of nonlinear differential-algebraic equations DAEs of semi-explicit SE form. The difference of external and internal linearization is illustrated by an example of a mechanical system. Moreover, we define different levels of external equivalence for two SE DAEs. The proposed explicitation procedure allows us to treat a given SE DAE as a control system defined up to feedback transformation (a class of control systems). Then sufficient and necessary conditions, expressed via explicitation procedure, are given to describe when a given SE DAE is level-3 externally equivalent to a linear SE DAE of some specific forms. At last, we show by an example that level-2 external linearization of a DAE can be achieved if its explicitation is level-2 input-output linearizable.

Regularization of nonlinear DAEs based on Structural Analysis

Differential algebraic equations (henceforth DAEs), arise from the equation based modelling of physical systems, such as those found in engineering or physics applications, with problems specifically arising from chemical distillation (Washington and Swartz (2011)), electronic circuits (Brenan et al. (1996)) and robotics (Campbell and Griepentrog (1995)). Models are now frequently built interactively using different components from large libraries in environments such as GPROMS, MAPLESIM, SIMULINK and an assortment of tools that use the Modelica language. This way of modelling systems can lead to large scale DAEs (Fritzson (2015)). A common notion is the differentiation index—which is equal to the number of times all or part of the system has to be differentiated in order to solve the problem as an ODE. It is well known that solving a high index (larger than one) DAE directly is numerically difficult, hence modelling tools usually perform some structural analysis to determine the index of the problem. This talk will outline different ways of using this structural analysis in the solution and regularization of DAEs. In particular the Signature Matrix method (Pryce (2001)), Dummy Derivative method (Mattsson and Söderlind (1993)), Structural-Algebraic method (Scholz and Steinbrecher (2013)) and a new approach called Structural Analysis based Dummy Derivatives will be considered.

Towards an AD based diagnosis of singularities of DAEs

AbstractThe detection of singular points of DAEs is a challenging task, which has not been sufficiently considered so far. Combining automatic differentiation (AD) with the projector based analysis of DAEs, it becomes possible to monitor singularities while integrating the DAE. In this brief outline we restrict our considerations to linear DAEs. The corresponding treatment of nonlinear DAEs is based on linearization and can be found in the cited literature. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Projector based integration of DAEs with the Taylor series method using automatic differentiation

Automatic (or Algorithmic) Differentiation (AD) opens new possibilities to analyze and solve DAEs by projector based methods. In this paper, we present a new approach to compute consistent initial values and integrate DAEs up to index two, considering the nonlinear DAE in each time-step as a nonlinear system of equations for Taylor expansions. These systems will be solved by the Newton–Kantorowitsch method, whereas the resulting linear systems are decoupled using the splitting techniques related to the tractability index concept. This approach provides a description of the inherent ODE that allows an application of the classical Taylor series method to the integration of initial value problems. Linear and nonlinear DAEs with index up to two are examined and solved numerically.

On a Newton‐Type Method for Differential‐Algebraic Equations

This paper deals with the approximation of systems of differential‐algebraic equations based on a certain error functional naturally associated with the system. In seeking to minimize the error, by using standard descent schemes, the procedure can never get stuck in local minima but will always and steadily decrease the error until getting to the solution sought. Starting with an initial approximation to the solution, we improve it by adding the solution of some associated linear problems, in such a way that the error is significantly decreased. Some numerical examples are presented to illustrate the main theoretical conclusions. We should mention that we have already explored, in some previous papers (Amat et al., in press, Amat and Pedregal, 2009, and Pedregal, 2010), this point of view for regular problems. However, the main hypotheses in these papers ask for some requirements that essentially rule out the application to singular problems. We are also preparing a much more ambitious perspective for the theoretical analysis of nonlinear DAEs based on this same approach.

Convergence Analysis of Continuous Extension RKM for Index-1 DAE's

For solving nonlinear DAEs of index-1, the convergence results of the continuous extension Runge-Kutta methods are investigated. The theoretical models are very general and even allow including different interpolation formulae along the mesh. The methods are suitable for decoupling the complex, singular systems of instantaneous state.

Consistent initialization for DAEs in Hessenberg form

For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.

Computation of consistent initial values for properly stated index 3 DAEs

The computation of consistent initial values is one of the basic problems when solving initial or boundary value problems of DAEs. For a given DAE it is, in fact, not obvious how to formulate the initial conditions that lead to a uniquely solvable IVP. The existing algorithms for the solution of this problem are either designed for fixed index, or they require a special structure of the DAE or they need more than the given data (e.g. additional differentiations). In this paper, combining the results concerning the solvability of DAEs with properly stated leading terms with an appropriate method for the approximation of the derivative, we propose an algorithm that provides the necessary data to formulate the initial conditions and which works at least for nonlinear DAEs up to index 3. Illustrative examples are given.

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index. In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case. We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls. Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.

Automated behavioral modeling and analytical model-order reduction by application of symbolic circuit analysis for multi-physical systems

The aim of symbolic analysis that has its origin in the design of analog circuits is the extraction of dominant system behavior by automated derivation of approximated symbolic formulas. Since exact symbolic analysis will yield exceptionally complex expressions even for rather small systems a class of symbolic approximation techniques have been developed that allow a reduction of the complexity of symbolic equations and their later solution by means of mixed symbolic and numerical strategies. Hence, it becomes possible to reduce the underlying nonlinear differential-algebraic systems of equations (DAE systems) of component-based networks and systems to a behavioral description of a predefined accuracy. It is a major advantage of the approach that the model simplification is performed by an automatic error control and that the simplified model is physically interpretable again. The contribution will give an overview of the symbolic tool Analog Insydes algorithms for extraction of dominant behavior of linear systems, e.g. formulas for poles and zeros as well as algorithms for generating behavioral models from nonlinear DAEs. Moreover, the underlying methodology has been extended to the application of analysis and modeling of gas-pipeline nets and mixed electrical and mechanical systems. For the latter a library was developed in cooperation with the Fraunhofer IIS/EAS for symbolic models of micro-mechanical elements that can be connected to networks, even together with electrical components.

Applying fixed point homotopy to nonlinear DAEs deriving from switching circuits

The time-domain analysis of switching circuits is a time consuming process as the change of switch state produces sharp discontinuities in switch variables. In this paper, a method for fast time-domain analysis of switching circuits is described. The proposed method is based on piecewise temporal transient analysis windows joined by DC analysis at the switching instants. The DC analysis is carried out by means of fixed point homotopy to join operating points between consecutive time windows. The proposed method guarantees accurate results reducing the number of iterations needed to simulate the circuit.

Logarithmic norms and nonlinear DAE stability

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certain unbounded operators, i.e. partial differential algebraic equations. Perturbation bounds are obtained from differential inequalities for any given norm by using the relation between Dini derivatives and semi-inner products. Simple discretizations are also considered.

Efficient Simulation of Batch Distillation Processes by Using Orthogonal Collocation

Orthogonal collocation on finite elements is applied to discretize the DAE system for the simulation of multiple-fraction batch distillation processes. A detailed dynamic tray-by-tray model is used to describe batch columns more accurately which, however, leads to a set of model equations composed of nonlinear DAEs with a fairly high dimension. In addition, batch distillation operation usually takes a long period of time and therefore it costs large computational expense to simulate such processes. The use of orthogonal collocation is demonstrated to obtain a stable and highly accurate algebraic representation of the differential equations so as to improve the computational efficiency significantly. Because of the orthogonality of the polynomials introduced to approximate the state variables within a time interval, large integration steps can be taken with the collocation approximation without reducing the computational accuracy. Through simulation of two real batch distillation processes it is found that with this discretization approach 50% CPU time can be saved in comparison to the implicit Euler method normally used.

Mixed symbolic–numerical computations with general DAEs II: An applications case study

A variety of theorems and properties of nonlinear DAEs were discussed in part I. This paper illustrates many of these ideas within the context of analyzing a specific nonlinear system that exhibits a variety of interesting features.

Topological index‐calculation of DAEs in circuit simulation

AbstractIn electric circuit simulation we are confronted with highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. The classic and the charge‐oriented modified analysis are shown to lead to the same DAE‐index if the circuit models satisfy some natural assumptions.We present a topological criteria for calculating the index. This makes it possible to determine the index also for high‐dimensional circuit equation systems.

Recent Results in Solving Index-2 Differential-Algebraic Equations in Circuit Simulation

In electric circuit simulation the charge-oriented modified nodal analysis may lead to highly nonlinear DAEs with low smoothness properties. They may have index 2 but they do not belong to the class of Hessenberg form systems that are well understood. In the present paper, on the background of a detailed analysis of the resulting structure, it is shown that charge-oriented modified nodal analysis yields the same index as does the classical modified nodal analysis. Moreover, for index 2 DAEs in the charge-oriented case, a further careful analysis with respect to solvability, linearization, and numerical integration is given.

Chaotic waveform relaxation methods for dynamical systems

For solving the large dynamical systems of initial value problems that can be described by a system of mixed implicit differential equations, the waveform relaxation (WR) methods are proposed and investigated by some mathematicians. In this paper, in order to compute the WR methods on parallel processors and to avoid the synchronization costs, the chaotic waveform relaxation (WR) methods are presented and their convergence is investigated. As for the special cases, the nonlinear ODEs, linear ODEs and semi-explicit DAEs are discussed, respectively.