Nonlinear effects of actuator rate and acceleration limits on closed-loop systems: a describing function approach

Actuator nonlinearities can significantly affect control systems, leading to performance degradation and even loss of stability. Physical constraints such as rate and acceleration limits are particularly detrimental in applications where rapid actuation is required, yet their combined effects remain largely unexplored. This paper  investigates the nonlinear dynamic behaviour induced by rate and acceleration limits in closed-loop systems, focusing on their steady-state response to sinusoidal excitation.  The saturation regimes associated with these nonlinearities are fully characterised, and their analytical boundaries are represented in a two-dimensional parameter space defined by normalised rate and acceleration limits. Sinusoidal describing  functions are derived for each regime, providing a unified frequency-domain representation of the actuator dynamics. These formulations are employed to analyse the impact of actuator nonlinearities on closed-loop dynamics, including the onset of
nonlinear behaviour, phase lag and gain reduction. Analytical conditions for the occurrence of jump resonance are derived, along with the lowest frequency where multiple steady-state solutions appear, leading to potential abrupt changes in system response. The applicability of the proposed framework is demonstrated through both an illustrative first-order system and a realistic high-order aeroservoelastic model for gust load alleviation, where the interaction between actuator nonlinearities and closed-loop dynamics is shown to produce multiple jump resonance scenarios and isolated nonlinear response branches. The results highlight the critical role of actuator rate and acceleration limits in high-bandwidth control applications and provide practical insights for frequency-domain stability assessment and preliminary feedback control system design.