Stability Limits of Uncertain Systems in the Presence of Unmatched Coupled Dynamics and Actuator Dynamics with Adaptive Architectures

Abstract

Uncertain systems with coupled dynamics and actuator dynamics are prevalent in various applications spanning private, commercial, and government sectors. While stabilization and command tracking strategies exist, they often rely on idealized assumptions, such as negligible actuator bandwidth limitations and matched coupled dynamics. In order to mitigate actuator-induced limitations, we first employ standard model reference adaptive control (MRAC) architectures. We then proposed two architectures: modified hedging reference modeled based MRAC and modified expanded reference model based MRAC. In order to deal with coupled dynamics, we used an observer design to estimate the states of unmatched coupled dynamics. In addition, Lyapunov stability analyses are provided to prove closed-loop systems stabilities with the aforementioned MRACs. Specifically, modified hedging- and expanded-based approaches are designed to ensure that the controller does not become destabilized by the actuator’s presence by incorporating the actuator dynamics into the reference model. The minimum required actuator bandwidths are then determined by the quadratic stability condition that is examined via linear matrix inequalities (LMI) analysis, where applicable. Additionally, we investigate the effects of incorporating a low-pass filter to mitigate high-frequency oscillations that can arise from the large MRAC update rates that are required for faster convergence. Furthermore within the modified hedged- and modified expanded reference model-based MRAC schemes, we analyze three approaches to handling ground effect uncertainty directly. Finally, these architectures are applied to the helicopter and quadrotor slung load examples, demonstrating stability limits and revealing the minimum required actuator bandwidths for each approach. For a helicopter, where longitudinal dynamics must be stabilized despite load-induced coupling effects, and for a quadrotor carrying a slung payload example, we assumed similar coupled dynamical effects, where both stabilization and command tracking cases are considered. For the quadrotor, we also consider hovering near the ground as a source of uncertainty and analyze command tracking performance when estimated coupled-load dynamics approach zero. In such cases, commands can be precisely tracked, while otherwise, when the coupled dynamics fail to return zero, the system achieves close proximity to the desired command