Higher Index Differential Algebraic Equations Research Articles

Jacobian reuse in explicit integrators for higher index DAEs

Systems F( y′, y, t) = 0 with F y′ identically singular are known as differential algebraic equations (DAEs) and occur in a variety of applications. The index v is one measure of numerical difficulty. Most numerical methods for DAEs either require special structure or low index. Two alternative approaches have been proposed for numerically integrating more general higher index DAEs. This paper examines some of the mathematical issues involved in the efficient implementation of the “explicit integration” method. It is first shown that the reuse of Jacobians can lead to the integration of discontinuous vector fields. It is then proven that these discontinuous fields can be successfully integrated. Computational examples back up the theory. A comparison to a standard integrator on an index three control problem illustrates that while the explicit approach can be somewhat more expensive computationally, it can be easier to apply, and does not suffer from order reduction in the higher index variables.

Regularization of Higher-Index Differential-Algebraic Equations with Rank-Deficient Constraints

In this paper we present several regularizations for higher-index differential-algebraic equations with rank-deficient or singular constraints. These types of problems arise, for example, in the solution of constrained mechanical systems, when a mechanism's trajectory passes through or near a kinematic singularity. We derive a class of regularizations for these problems which is based on minimization of the norm of the constraints. The new regularizations are analogous to trust-region methods of numerical optimization. We give convergence results for the regularizations and present some numerical experiments which illustrate their effectiveness.

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

In this paper we generalize the class of explicit partitioned Runge-Kutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in ${\Bbb R}^m$ to systems on manifolds embedded in ${\Bbb R}^m$ . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods.

Sequential Regularization Methods for Nonlinear Higher-Index DAEs

Sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints. This paper extends an earlier work [SIAM J. Numer. Anal., 33 (1996), pp. 1921--1940] to nonlinear problems and to differential algebraic equations (DAEs) with an index higher than 2. Rather than having one winning method, this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances. We propose sequential regularization methods for index-2 and index-3 DAEs, both with and without constraint singularities. In the case of no constraint singularity we prove convergence results. Numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods. The examples include constrained multibody systems.

Sequential Regularization Methods for Higher Index DAEs with Constraint Singularities: The Linear Index-2 Case

Standard stabilization techniques for higher index differential-algebraic equations (DAEs) often involve elimination of the algebraic solution components. This may not work well if there are singularity points where the constraint Jacobian matrix becomes rank deficient. This paper proposes instead a sequential regularization method (SRM)---a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components. For linear index-2 DAEs we consider both initial value problems (IVPs) and boundary value problems (BVPs). The convergence of the SRM is described and proved in detail. We believe that this is the first convergence proof for any method for DAEs with this type of constraint singularities. Moreover, the regularization parameter in our method is not necessarily very small, so the SRM is an important improvement over usual regularization methods. Various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out.

Generalized Inverses of Differential-Algebraic Operators

In the theoretical treatment of linear differential-algebraic equations one must deal with inconsistent initial conditions, inconsistent inhomogeneities, and undetermined solution components. Often their occurrence is excluded by assumptions to allow a theory along the lines of differential equations. This paper aims at a theory that generalizes the well-known least squares solution of linear algebraic equations to linear differential-algebraic equations and that fixes a unique solution even when the initial conditions or the inhomogeneities are inconsistent or when undetermined solution components are present. For that a higher index differential-algebraic equation satisfying some mild assumptions is replaced by a so-called strangeness-free differential-algebraic equation with the same solution set. The new equation is transformed into an operator equation and finally generalized inverses are developed for the underlying differential-algebraic operator.

Computational Optimal Control.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

Constraint preserving integrators for general nonlinear higher index DAEs

In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) $f(x', x, t) = 0$ where $f_{x'}$ 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.

High-Index Differential Algebraic Equations

ABSTRACT In the last few years there has been considerable research on differential algebraic equations (DAEs) f(x1, x, t) = 0, where fx 1 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, because of technical difficulties and because many basic applications including constrained mechanical systems have this index. This paper discusses several situations where DAEs of index higher than three occur naturally. It will also discuss the relationship between certain concepts in nonlinear control theory such as relative degree and zero dynamics, the index, and constrained mechanical systems.

Progress on a general numerical method for nonlinear higher index DAEs II

A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays. This paper discusses progress on the implementation of this method, resolves some of the issues involved, and lists some remaining problems. Computational experience on two prescribed path control problems is presented showing that the approach should prove practical for many applications.

Stabilization of DAEs and invariant manifolds

Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.

Least squares completions for nonlinear differential algebraic equations

A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods. The theoretical foundations, generality, and limitations of this approach remain to be determined. This paper carefully examines several key aspects of this approach. The emphasis is on general results rather than specific results based on the structure of various applications.

Projected collocation for higher-order higher-index differential-algebraic equations

Higher-order, higher-index Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs) are considered. These types of systems arise in a variety of applications, including multibody systems. We extend a class of recently introduced projected implicit Runge-Kutta methods and define a new class of projected piecewise polynomial collocation methods for the solution of these problems. Stabilizing reformulations are considered as well, and a new projected invariant method is proposed. The higher-order ODE part of the DAE is collocated directly by a piecewise polynomial. A projection modification helps restore all the properties of stability and superconvergence which a corresponding collocation method for an ODE possesses. Higher-order collocation at Radau points is recommended for initial-value problems. The projection methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations, like collocation at Gauss points. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation and loss of accuracy; the new methods overcome these difficulties. For higher-index problems we consider reformulation methods of stabilizing index reduction. We propose new methods of projected invariants to handle particularly tough higher-index problems. The advantages offered by these methods are demonstrated numerically.

Modelling for dynamic simulation of chemical processes: the index problem

A set of ordinary differential equations (ODEs) and algebraic equations (AEs) can usually be employed to represent a class of models for dynamic simulation of processes. Together the set of ODEs and AEs is generally known as a differential-algebraic equation (DAE) system. It is customary to classify a DAE system in terms of its index, this being a non-negative integer. The index of a DAE system indicates the degree of difficulty associated with the numerical solution of the system. In the communication all DAE systems with index greater than unity are referred to as higher-index DAE systems. The objective of the work is to show through analysis of the model equations, and specific examples, how some of the causes of higher-index DAE systems can be identified and remedied

On the Asymptotic Representation of a Regularization Approach to Nonlinear Semiexplicit Higher-Index Differential-Algebraic Equations

On the Asymptotic Representation of a Regularization Approach to Nonlinear Semiexplicit Higher-Index Differential-Algebraic Equations

On a geometrical interpretation of differential-algebraic equations

The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE, and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lower-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.

Higher-index differential-algebraic equations: analysis and numerical treatment

Higher-index differential-algebraic equations: analysis and numerical treatment

Accurate solution of differential-algebraic optimization problems

Accurate solution of differential-algebraic optimization problems