In many cases, a single traditional PID controller is adequate to track an object and match both position and velocity of the object.
With the error term of the PID in terms of position, the integral term ("I" of the PID) and the derivative term (the "D" term in the PID) each help compensate for the velocity of the object.
No matter what control system you use, if the object makes a sudden unexpected movement, the tracking will lag behind -- that's part of the nature of our causal universe.
(Please tell us if there are any, even completely infeasible, ways of eliminating all lag from an object tracker).
There are various feedforward approaches that can reduce (but never eliminate) the lag in a PID controller.
In your case, rather than the (traditional, error-amplifying) approach that makes D proportional to the derivative of the position error,
you may have less lag by directly measuring the speed of the object and the speed of your tracker and setting D proportional to their difference.
(Or perhaps proportional to the speed of the object alone).
If you want the tracker to arrive at the velocity of the object at the same instant that it arrive at the position of the object,
then you want a control system "without overshoot".
There are lots of applications that require such a control system,
and PID control systems can be tuned such that they have no overshoot after a step impulse.
(You probably already know that most PID control systems are tuned using a different critera that allows some overshoot and oscillation in order to reduce the lag in settling time; people using such systems use the term "overdamped" to describe settings that have no overshoot).