Impact of delayed acceleration feedback on the classical car-following model

Delayed acceleration feedback (DAF) is known to have a positive impact on the stability properties of dynamical models in several applications. Motivated by this, we study the impact of DAF on the classical car-following model (CCFM). First, we show that DAF shrinks the locally stable region. We then show that the resulting model, similar to the CCFM, loses local stability via a Hopf bifurcation. However, DAF decreases the amplitude and increases the frequency of the resulting limit cycles. Additionally, we show that DAF is detrimental to the CCFM due to $(i)$ destruction of the non-oscillatory property of its solutions, $(ii)$ increased risk of string instability and $(iii)$ reduced resilience towards parametric uncertainty. Thus, we report a relevant application wherein DAF degrades the performance across several metrics of practical interest. Thus, our work motivates a systematic study of controllers that can guarantee performance enhancement. We complement our analytical results with stability charts and a bifurcation diagram.