  In this paper, we study the internal stability and string stability of a
vehicle platoon under the constant time headway spacing (CTHS) policy and the
multiple-predecessor-following (MPF) vehicle-to-vehicle information flow
topology. More specifically, we depart from the conventional
Proportional-Integral-Derivative (PID) controller design for such systems and
we propose the design of an observer-based controller. For designing our
observer-based controller, we first design a distributed observer, with which
each follower estimates their position, speed and acceleration error with
respect to the leader. The observer is designed by means of constructing an
observer matrix whose parameters should be determined. Next, we simplify the
design of the matrix of the observer in such a way that the design boils down
to choosing a single scalar value; this design further simplifies the structure
of the controller, whose simplicity enables the derivation of string stability
conditions by means of a frequency response method. Subsequently, the string
stability conditions for a given time headway, are transformed to conditions
for the controller parameters. We obtain controller parameters that satisfy the
stability conditions by designing a novel heuristic search algorithm.
Furthermore, we extend the search algorithm by incorporating a bisection-like
algorithm, which allows to obtain (within some deviation tolerance) the minimum
available value of the time headway. Finally, we provide insights about how to
finalize the observer-based controller parameters from above algorithms to
avoid the peaking phenomenon. The performance of the proposed observer-based
controller is demonstrated via illustrative examples. Additionally, a
comparison with a widely-used PID controller for MPF topology shows that our
proposed observer-based controller has better convergence performance.
