This paper studies the internal stability and string stability of a vehicle platooning of constant time headway spacing policy with a varying-speed leader using a multiple-predecessor-following strategy via vehicle-to-vehicle communication. Unlike the common case in which the leaders speed is constant and different kinds of Proportional-Integral-Derivative controllers are implemented, in this case, the fact that the leader has a time-varying speed necessitates the design of an observer. First, in order to estimate its position, speed and acceleration error with respect to the leader, each follower designs an observer. The observer is designed by means of constructing an observer matrix whose parameters should be determined. We simplifies the design of the matrix of the observer in such a way that the design boils down to choosing a scalar value. The resulting observer turns out to have a third order integrator dynamics, which provides an advantage of simplifying the controller structure and, hence, derive conditions for string stability using a frequency response method. A new heuristic searching algorithm is developed to deduce the controller parameter conditions, given a fixed time headway, for string stability. Additionally, a bisection-like algorithm is incorporated into the above algorithm to obtain the minimum (with some deviation tolerance) available value of the time headway by fixing one controller parameter. The effectiveness of the internal and string stabilities of the proposed observer-based controller is demonstrated via comparison examples.
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