Convex Bound of Nonlinear Dynamical Errors for Stochastic Optimal Control

TL;DR

The paper tackles the degradation of linear covariance controllers in nonlinear dynamics by formulating a convex upper bound on the nonlinear remainder from Taylor linearization, using higher-order state-transition tensors up to order $m^*$. This bound is embedded as a convex objective in a covariance-steering framework, enabling efficient optimization of controller gains without re-linearization. The main contributions are a tractable convex surrogate for nonlinear dynamics, a nonlinear minimization formulation that balances positional and velocity nonlinearities, and a halo-orbit stationkeeping case in the CR3BP showing improved alignment with nonlinear simulations and tighter quantile predictions. The results indicate that minimizing nonlinear exposure yields more reliable Gaussian-based predictions in nonlinear regimes, enhancing mission assurance in astrodynamics while preserving computational efficiency of convex optimization.

Abstract

Applying linear controllers to nonlinear systems requires the dynamical linearization about a reference. In highly nonlinear environments such as cislunar space, the region of validity for these linearizations varies widely and can negatively affect controller performance if not carefully formulated. This paper presents a formulation that minimizes the nonlinear errors experienced by linear covariance controllers. The formulation involves upper-bounding the remainder term from the linearization process using higher-order terms in a Taylor series expansion, and resolving it into a convex function. This can serve as a cost function for controller gain optimization, and its convex nature allows for efficient solutions through convex optimization. This formulation is then demonstrated and compared with the current methods within a halo orbit stationkeeping scenario. The results show that the formulation proposed in this paper maintains the Gaussianity of the distribution in nonlinear simulations more effectively, thereby allowing the linear covariance controller to perform more as intended in nonlinear environments.

Figures (5)

• Figure 1: Diagram of Nonlinearity Effects in Linear Covariance Steering.
• Figure 2: Southern L$_2$ halo family in barycenter-centered synodic frame. Examined orbit in yellow.
• Figure 3: Comparison of the final linear covariance between the minimum covariance and minimum nonlinearity approaches. Origin normalize to halo orbit.
• Figure 4: Monte Carlo ($n_{\text{samples}}=1,000$): Distribution at the terminal time comparing the minimum covariance and minimum nonlinearity approaches. Origin normalize to halo orbit.
• Figure 5: Monte Carlo ($n_{\text{samples}}=1,000$): Comparison of the upper-bound quantile function $\text{Quant}_{\text{ub}}(\cdot)$ for state deviations predicted by the linear covariance with the actual quantile function $\text{Quant}(\cdot)$ obtained from Monte Carlo simulations.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3