Efficient Gradient-Based Optimization of Differential-Algebraic Equation Systems

Abstract

Differential algebraic equations (DAEs) arise in a variety of practical engineering systems. This work presents an open-source framework to numerically obtain the solution to DAEs, and further obtain the gradient of functions of interest with respect to design variables. A time-domain forward solver based on the stable backward difference formula (BDF) is implemented. The gradients are efficiently obtained using the adjoint method. Prior work has shown that the computation of six partial derivatives are sufficient to obtain the solution of both the forward and adjoint problems. In this work, the framework is implemented using the computational system design language (CSDL). CSDL enables the user to write high-level python code and automates the computation of the required derivatives. Thus, the framework saves time for users who are prototyping DAE formulations and solving complex engineering problems. The framework is validated against three different benchmark problems: spring-mass-damper system, pendulum system, and RLC circuit. The gradient from the adjoint solver is compared to that calculated using the finite difference method. It is found that the error of the both the forward and adjoint solutions is negligible. The implementation of benchmark problems also serves as examples of how users can formulate their own DAE systems in the open-source framework.