Integration Error Regularization in Direct Optimal Control using Embedded Runge Kutta Methods

In order to solve continuous-time optimal control problems, direct methods transcribe the infinite-dimensional problem to a nonlinear program (NLP) using numerical integration methods. In cases where the integration error can be manipulated by the chosen control trajectory, the transcription might produce spurious local NLP solutions as a by-product. While often this issue can be addressed by increasing the accuracy of the integration method, this is not always computationally acceptable, e.g., in the case of embedded optimization. Therefore, alternatively, we propose to estimate the integration error using established embedded Runge-Kutta methods and to regularize this estimate in the NLP cost function, using generalized norms. While this regularization is effective at eliminating spurious solutions, it inherently comes with a loss of optimality of valid solutions. The regularization can be tuned to minimize this loss, using a single parameter that can be intuitively interpreted as the maximum allowable estimated local integration error. In a numerical example based on a system with stiff dynamics, we show how this methodology enables the use of a computationally cheap explicit integration method, achieving a speedup of a factor of 3 compared to an otherwise more suitable implicit method, with a loss of optimality of only 3\%.