Analytical derivation of the sinusoidal input describing function for adaptive augmenting control algorithms

The Space Launch System is planned to deliver more payload to orbit than any other space vehicle is currently able to. NASA has developed an Adaptive Augmenting Control (AAC) algorithm that aims to increase robustness and performance of the underlying baseline controller of the launch vehicle. Monte Carlo simulation results have provided an important form of verification already. In addition, flight testing on an F/A-18 research flight vehicle has shown decreased excitation of parasitic dynamics such as bending and slosh modes of the launch vehicle. Moreover, classic linear stability margins have been used to set hard limits to the multiplicative adaptive gain. This paper aims to give deeper insights into AAC’s nonlinear characteristics and their dependency on respective tuning of the adaptive update law from an analytical point of view. It thereby complements existing analysis approaches that are mainly based on heuristic principles, numerical simulations, and flight tests. By applying the well-known concept of describing functions an analytical expression for the Sinusoidal Input Describing Function of the Adaptive Augmenting Control subsystem is derived within this paper. Latter provides a deeper understanding of AAC’s nonlinear transfer characteristics and is used to provide closed-form solutions for possible limit cycle oscillations within the adaptive augmented feedback system and to assess their amplitude and stability depending on AAC’s tuning parameters.