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

TL;DR

The paper addresses spurious minima in direct optimal control caused by discretization error during transcription. It introduces an embedded Runge-Kutta–based error estimator and a norm-based regularization term ϕ(w;E_max,p,q) in the NLP objective, controlled by a single tuning parameter e_max that caps the estimated local error. This approach provides numerical robustness and enables cheaper explicit integration schemes at a modest, tunable loss of optimality, demonstrated to yield substantial online-like speedups. A Hang Glider problem with stiff dynamics shows that a ~3x speedup is achievable while maintaining trajectory accuracy comparable to high-accuracy references. The work offers a practical route to balance accuracy and efficiency in direct OCP solvers under tight computation budgets.

Abstract

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\%.

Figures (8)

• Figure 1: Spurious optimal state trajectory of the discretized problem \ref{['eq:simpleExample_OG']} using a 4-stage Legendre collocation integration scheme, and with initialization $u_{0,b} = 10$.
• Figure 2: NLP cost function over the feasible set for different integration methods
• Figure 3: Left: Butcher tableau of an embedded integrator pair. Middle: Heun-Euler of Order 2(1). Right: extension of an existing IRK method with an embedded integrator.
• Figure 4: Minimal example, cont.: Endpoint value, integration error and objective over the feasible set. The difference between the endpoints of the embedded and original integrator is used to regularize the objective.
• Figure 5: Solution of the discretized hang glider problem using a Heun-Euler integration scheme, unregularized $e_\mathrm{rel,max} = 10^{2}$ (blue) and regularized $e_\mathrm{rel,max} = 10^{-2}$ (orange). To avoid a large integration error, a less aggressive control strategy is optimal.