In this paper, we improve the dynamics of an autonomous car-following system with two classical compensation methods: proportional-differential (PD) and velocity feedback (VF) control methods. Based on an optimal velocity model, the system transfer function was derived and control block diagram was given with an open-loop transfer function. Three theorems were proposed and proven with the use of small gain theorem. The analytical results show that two compensation factors play important roles in stabilizing the car-following system. In the simulation, three dynamical models were derived in reverse from the closed-loop transfer function and the numerical results show that, with increasing factors, the stability of carfollowing was enhanced and traffic jams were eliminated. The analytical results are supported by the numerical results.

In this paper, we discretize continuous car-following model into a difference equation. With the use of discrete control theory, the discrete car-following model was linearized and the impulse transfer function was obtained. Based on the definition of traffic jam, one Lemma was proposed to judge system stability and two theorems were given to judge the non-jam conditions for controlled and uncontrolled systems. Moreover, the feedback gain was designed for a controlled system. Numerical simulations were conducted to show the validity of the discrete car-following model and control scheme. The results are in agreement with the theoretical results.