Stabilization of numerical solutions of boundary value problems exploiting invariants

Abstract

A b s t r a c t Solving boundary value problems (BVPs) numerically is an important task when dealing with problems of optimal control. In this paper the numerical solution of BVPs for differential algebraic equations (DAEs) is discussed. The method of choice is multiple shooting. Optimal control problems are higher index DAEs in the case of singular controls or state constraints. The common procedure of solving higher index DAEs is to reduce the index by differentiating the algebraic equations until index 1 DAEs or ordinary differential equations (ODEs) are obtained which can be treated directly. Unfortunately, the numerical solution of the index reduced problems often suffers from instabilities introducing a drift from the original algebraic conditions. We interprete the higher index constraints as invariants of the ODE and exploit these invariants in order to improve accuracy, stability and efficiency by a new projection technique. Obviously, conservation properties e.g. for energy or momentum can be used in this sense, but the symplectic structure of Pontryagin's Maximum principle allows for deriving further invariants. Solving the shooting equations by Newton's method requires the computation of sensitivity matrices. This is performed by solving the initial value problems for the variational ODEs together with their invariants or by differentiation of the discretization scheme. The techniques are demonstrated on the example of a flight path optimization problem. I n t r o d u c t i o nIn this paper we discuss the numerical stabilization of higher index differential-algebraic boundary value pro-Roughly spoken, the index is defined as the number of times the algebraic equations (lb) must be differentiated with respect to the independent variable in order to obtain an ODE for the algebraic variable y, (index reduction)." Higher index DAE BVPs arise in the field of trajectory optimization if there are singular controls, i.e. if the second derivative of the Hamiltonian is singular or in the case of state constraints. However, the index of a DAE is closely related to the numerical difficulties encountered in its solution.So far, the discussion of BVPs is mostly restricted to the case of ordinary differential equations (ODEs). In [3] the numerical solution of BVPs has been extended to index 1 DAEs. Since higher index DAEs cannot be treated directly by discretization methods, one has to reduce the index by differentiation of the algebraic equtaions. This leads to numerical instabilities in the sense, that the numerical solution of the index reduced system does not fulfill the original higher index equations. Furthermore, due to error propagation this instability often leads to a significant drift from these equations. In Sec. 4 we propose a projection technique to avoid this drift and to improve the stability of the numerical solution of the initial value problems. In Sec. 3 we show how the index reduced equations can be interpreted as ODEs with invariants. We give additional invariants due to the symplectic structur...

Numerical Methods in Multibody Dynamics

“…We briefly describe the main ideas of this approach and refer the reader to [EMS93] for details. Index reduction by differentiating the constraints transforms these into an index-l problem, for which we just described an adequate formulation for applying shooting techniques.…”