Abstract
Asa result of the time lag in transmission, the data obtained by the sensor is delayed and does not reflect the state at the current moment. The effects of input delay are often overlooked in active inference (AIF), which may lead to significant deviations in state estimation and increased prediction errors, particularly when the system is subjected to a sudden external stimulus. In this paper, a theoretical framework of delayed feedback active inference (DAIF) is proposed to enhance the applicability of AIF to real systems. The probability model of DAIF is defined by incorporating a control distribution into that of AIF. The free energy of DAIF is defined as the sum of the quadratic state, sense, and control prediction error. A predicted state derived from previous states is defined and introduced as the expectation of the prior distribution of the real-time state. A proportional-integral (PI)-like control based on the predicted state is taken to be the expectation of DAIF preference control, whose gain coefficient is inversely proportional to the measurement accuracy variance. To adaptively compensate for external disturbances, a second-order inverse variance accuracy replaces the fixed sensory accuracy of preference control. The simulation results of the trajectory tracking control of a quadrotor unmanned aerial vehicle (UAV) show that DAIF performs better than AIF in state estimation and disturbance resistance.