L1 adaptive prescribed-time control

Abstract

This paper develops an $L_1$ adaptive prescribed-time control (PTC) to address external perturbations and uncertainties, where a low-pass filter with time-varying cutoff frequency is proposed. A generalized time-varying scaling function is introduced to achieve prescribed-time control (PTC) and practical prescribed-time control (PPTC). The proposed control strategy innovatively incorporates the design philosophy of $L_1$ adaptation into PTC to achieve smooth convergence and effective management of external perturbations and uncertainties. The $L_1$ adaptive control framework ensures the capability to achieve PTC by elaborately setting the time-varying cutoff frequency. The proposed method also achieves the uniform distribution of the control energy over the time interval. It reduces the sensitivity of the control energy subject to abnormal initial states or time-varying functions that tend to infinity. Finally, we present numerical simulations and comparisons with existing methods to demonstrate effectiveness and superiority of the proposed $L_1$ adaptive PTC.