Stability of the collision-proof and related braking laws

This paper discusses three braking control laws: one is the collision-proof law put forth by Schubring, the other two are modifications of this law. The collision-proof law requires that the kinetic energy of a following vehicle never exceeds that of a leading vehicle by more than the amount which can be dissipated in the distance between them. The other two laws assume, in addition, that there is a finite system time delay, and that there is a time delay which is partially compensated for by projecting velocity measurements ahead in time. Stability of control laws is an important criterion for their evaluation. Local stability, the subject of this paper, is the stability of a given vehicle with respect to the position of the lead vehicle in the platoon. It is found that the collision-proof control law is locally stable with respect to small velocity perturbations. The following vehicle approaches its stable platoon position exponentially, the time constant being (µg/u)^-1, where µ is the road surface-tire coefficient of friction, g is the acceleration of gravity (=9.807 m/s²), and u is the platoon velocity before disturbance. With the incorporation of a system time delay, Δ, the local stability is dependent upon the parameter C = µgΔ/u. If C<e^-1, the platoon is locally stable, whereas if C>π/2 (=1.571) it is unstable. When e^-1<C<π/2, the platoon oscillates, but the oscillations are damped.