Modeling and Adjoints for Continuous Systems

Abstract

Approximate solution of the differential equations of state of continuous systems by various numerical integration schemes is standard practice in trajectory optimization and control work, and the resulting truncation error incurred represents the main error in many applications. Suppression of the deleterious effects of this particular type of error is of increasing interest as double precision arithmetic becomes routinely available for round-off error reduction, and especially so when high overall accuracy is needed.

In the situation met in numerical optimization of trajectories, what is important is accuracy of partial derivatives in a special sense: compatibility of the function and its derivatives. That is, if the partial derivatives of the terminal state with respect to trajectory parameters are accurate representations of the partials of the terminal state as calculated through the integration model, this is enough for purposes of avoiding adverse effects on convergence of successive approximation iterations. From previous numerical experiments, this is known to be of particular importance when conjugate direction methods are used, as these happen to require unusually accurate first partials in order to actually realize the quadratic convergence theoretically attainable.

When the mathematical model of the system consists of differential equations, it is theoretically correct to use differential equations for the adjoint (influence function) variables as well. The actual numerical solution of both state and adjoint variables, however, is obtained by using finite-difference equation approximations. The existence of truncation error in the state is inevitable but, by suitable construction of the adjoint difference equation, it is possible to obtain the compatibility just discussed.

The procedure is to recognize that (for numerical purposes) the system model consists of difference equations, and to directly derive compatible adjoint difference equations. The adjoint variables will then be correct influence functions for the numerically computed state variables, rather than approximate influence functions for the theoretically continuous state variables. The construction of the adjoint system for difference equation models is straightforward. If the typical “integration routine” involves, say, fourth differences, the adjoint system will also involve fourth differences. The intervals and coefficients for the adjoint system, however, will differ from those of the dynamic system. Thus, the compatible adjoint system will use the same order difference equations in a different (although quite connected) “integration routine.”

A considerable indirect benefit arising from the use of a compatible adjoint is that very large integration steps can be tolerated without incurring errors of the type affecting convergence. Thus a minimum may be found, with a modest expenditure of computer time, which provides a good starting point for optimization with the small integration steps usually appropriate.

The complete paper appears in the April 1969 issue of the Journal of Optimization Theory and Applications.