ON A REGULARIZATION OF INDEX 2 DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH PROPERLY STATED LEADING TERM

In this article, linear regular index 2 DAEs A(t)[D(t)x(t)]′+B(t)x(t)=q(t) are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.     

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.      

THE EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF DAES

1Intr0ducti0nDifferential-algebraicequations(DAEs)areveryusefu1inwidefields(cf.[1]).Bydifferential-algebraicequations,wemeanthoseequati0nswhosepartsof"derivative"cann0tbeexpressedexplicitly.Forexample,weconsidertheimplicitdifferentialequationwithmappingFsm00thssufficient1y.Itisusuallyreferredt0adifferential-algebraicequation(DAE)whentherank0fD.F(t,x,p)islessthann,wheretheremightbesomepurea1gebraic,whichwecallc0nstraintequations.TheDAEs,inparticular,theexistenceanduniquenessofitssolutions…     

An algorithm for the computation of consistent initial values for differential–algebraic equations

We use boundary value methods to compute consistent initial values for fully implicit nonlinear differential-algebraic equations. The obtained algorithm uses variable order formulae and a deferred correction technique to evaluate the error. A rigorous theory is stated for nonlinear index 1, 2 and 3 DAEs of Hessenberg form. Numerical tests on classical index 1, 2 and 3 DAE problems are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

Constraint preserving integrators for general nonlinear higher index DAEs

Summary. In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) where is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three. Even in this case, the systems were often assumed to have a special structure. Recently a numerical method was proposed that could, in principle, be used to integrate general unstructured higher index solvable DAEs. However, that method did not preserve constraints. This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. Received August 25, 1993 / Revised version received April 7, 1994     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.     

Geometric Properties of Runge-Kutta Discretizations for Nonautonomous Index 2 Differential Algebraic Systems

We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0.     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Element-based index reduction in electrical circuit simulation

The numerical simulation of very large scale integrated circuits is an important tool in the development of new industrial circuits. In the course of the last years, this topic has received increasing attention. Common modeling approaches for circuits lead to differential-algebraic systems (DAEs). In circuit simulation, these DAEs are known to have index 2, given some topological properties of the network. This higher index leads to several undesirable effects in the numerical solution of the DAEs. Recent approaches try to lower the index of DAEs to improve the numerical behaviour. These methods usually involve costly algebraic transformations of the equations. Especially, for large scale circuit equations, these transformations become too costly to be efficient. We will present methods that change the topology of the network itself, while replacing certain elements in oder to obtain a network that leads to a DAE of index 1, while not altering the analytical solution of the DAE. This procedure can be performed prior to the actual numerical simulation. The decreasing of the index usually leads to significantly improved numerical behaviour. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization

This article describes a new algorithm for the computation of consistent initial values for differential-algebraic equations (DAEs). The main idea is to formulate the task as a constrained optimization problem in which, for the differentiated components, the computed consistent values are as close as possible to user-given guesses. The generalization to compute Taylor coefficients results immediately, whereas the amount of consistent coefficients will depend on the size of the derivative array and the index of the DAE. The algorithm can be realized using automatic differentiation (AD) and sequential quadratic programming (SQP). The implementation in Python using AlgoPy and SLSQP has been tested successfully for several higher index problems.     

WAVEFORM RELAXATION METHODS AND ACCURACY INCREASE

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Optimal Control of Differential–Algebraic Equations of Higher Index,Part 2: Necessary Optimality Conditions

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of problems which can be transformed to index one control problems. For these problems we show in the accompanying paper that, if the solutions to the adjoint equations are well–defined, then the first-order approximations to the functionals defining the problem can be expressed in terms of the adjoint variables. In this paper we show that the solutions to the adjoint equations are essentially bounded measurable functions. Then, based on the first order approximations, we derive the necessary optimality conditions for the considered class of control problems. These conditions do not require the transformation of the DAEs to index-one system; however, higher-index DAEs and their associated adjoint equations have to be solved.     

The index of general nonlinear DAEs

Summary. In the last few years there has been considerable research on differential algebraic equations (DAEs) where is identically singular. Much of the mathematical effort has focused on computing a solution that is assumed to exist. More recently there has been some discussion of solvability of DAEs. There has historically been some imprecision in the use of the two key concepts of solvability and index for DAEs. The index is also important in control and systems theory but with different terminology. The consideration of increasingly complex nonlinear DAEs makes a clear and correct development necessary. This paper will try to clarify several points concerning the index. After establishing some new and more precise terminology that we need, some inaccuracies in the literature will be corrected. The two types of indices most frequently used, the differentiation index and the perturbation index, are defined with respect to solutions of unperturbed problems. Examples are given to show that these indices can be very different for the same problem. We define new "maximum indices," which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other. These indices are also related to an index defined in terms of Jacobians. Received November 15, 1993 / Revised version received December 23, 1994     

预测校正公式求解指标2微分代数方程

1．前言微分代数方程（EEES）是经常出现于实际问题中的一类方程．其数值求解已成为常微分方程数值求解领域十分活跃的一个方向．目前微分代数方程求解的数值方法主要是nunge－Kutta型方法及BDF方法．Runge－Kutta型方法在网,问中有详细的介绍．Hairer等人据此编制了软件RADAU,而目前使用最广泛的软件还是PetZold等编制的DASSL．DASSL使用的方法为BDF方法,它在微分代数方程中的应用最早可以追述到Gear的开创性工作问．BDF方法一个很大的优点是刚性稳定．然而对于非刚性的微分代数方程,刚性稳定已不是主要考虑的因素．因此…     

Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1

Summary For the numerical solution of non-stiff semi-explicit differentialalgebraic equations (DAEs) of index 1 half-explicit Runge-Kutta methods (HERK) are considered that combine an explicit Runge-Kutta method for the differential part with a simplified Newton method for the (approximate) solution of the algebraic part of the DAE. Two principles for the choice of the initial guesses and the number of Newton steps at each stage are given that allow to construct HERK of the same order as the underlying explicit Runge-Kutta method. Numerical tests illustrate the efficiency of these methods.     

A Structure Preserving Index Reduction Method For MNA

Transient analysis in industrial chip design leads to very large systems of differential-algebraic equations (DAEs). The numerical solution of these DAEs strongly depends on the so called index of the DAE. In general, the higher the index of the DAE is, the more sensitive the numerical solution will be to errors in the computation. So, it is advisable to use mathematical models with small index or to reduce the index. This paper presents an index reduction method that uses information based on the topology of the circuit. In addition, we show that the presented method retains structural properties of the DAE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Detecting structures in differential algebraic equations: Computational aspects

Differential algebraic equations (DAEs) are often automatically generated, in particular, by coupling different tools. These DAEs are unstructured in the sense that they do not reveal their mathematical structure a priori. In view of a reliable treatment of those DAEs, their mathematical structure should be uncovered and monitored also by computational methods. We discuss several computational aspects of the tractability index concept.     

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.     

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.Dedicated to Germund Dahlquist on the occasion of his 70th birthdayThis author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolivar (CESMa) for permitting her free use of its research facilities.Partial support by the Swedish Research Council for Engineering Sciences TFR under contract no. 222/91-405.