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 induced by these actuator limits are fully characterized, and their analytical boundaries are represented in a two-dimensional parameter space defined by normalized rate and acceleration limits. Sinusoidal describing functions are derived for each regime, capturing the actuator dynamics in the frequency domain. These are used to analyse the effects of actuator nonlinearities on closed-loop performance, including the onset of nonlinear behaviour, phase lag and gain reduction. The conditions for the presence of jump resonance are analytically derived, along with the lowest frequency where multiple solutions appear, leading to potential abrupt changes in system response. The findings highlight how these nonlinear phenomena can severely degrade the performance and stability of closed loops, underlining the importance of accounting for actuator rate and acceleration limits in the analysis and design of feedback control systems.