How come there is no difference between velocity and position?

I mean:

position PID:  u(t) = K_p * e(t) +K_i * T_s * [e(0) + e(1) + ... e(t)] +u(0)

velocity PID:   u(t) = u(t-1) + K_p * (e(t) - e(t-1)) + K_i *T_s * e(t) 

Learn some ethics first. There is no reason to downvote.

I think Pavel is correct in questioning that "theoretically there is no difference between them". 

My understanding is that there is a difference. The positional form has the P, I & D terms all in the fwd path; whereas the velocity form has only the I term in the fwd path and the P & D terms in the bwd path. This results in a different transfer function for the reference input to the plant output and consequently different behaviour. The result is that for to step changes in reference signal, the so-called "kick" (i.e. a sharp spike) in the plant output is avoided.

See  "Discrete-Time Control Systems"  by Katsuhiko Ogata.

.... actually, I think the "kick" is usually most visible in the the plant input (i.e. the controller output, with possible saturation problems) not in the plant output. 

The equations you wrote are actually for PI controllers, since there is no D term.

The theoretical proof would simply be:

position PI:  u(t) = K_p * e(t) +K_i * T_s * [e(0) + e(1) + ... + e(t-1) + e(t)] +u(0)

so,

u(t-1) = K_p * e(t-1) +K_i * T_s * [e(0) + e(1) + ... + e(t-1)] +u(0)

and,

u(t) - u(t-1) = K_p * (e(t) - e(t-1)) + K_i * T_s * e(t)

therefore the following equation would be equivalent,

velocity PI:   u(t) = u(t-1) + K_p * (e(t) - e(t-1)) + K_i *T_s * e(t)

These equations can easily be expanded to also include the D term.

With how many bits is each digital signal represented by your DSP? Because the different ways of adding (mainly for the integral term) might cause a rounding error build up.

Hi

Position form of PID gives output as the control variable. Now if we arrange the control system in a way that the control signal is driven directly by an integrator, e.g., a motor. It gives the velocity of the control variable. The control variable is then obtained by integrating its velocity. The PID controlled system of this type is called a velocity form. Velocity form PID has many practical uses, like wind-up protection and bumpless parameter changes, are easy to implement using the velocity form.

See a long discussion and block diagram from "Control System Design " by K J Astrom 2002. 

I have implemented PI controller with feedforward for gimbal tracking. It is very accurate once it locks the target. But initially it is having some jump. How to avoid it. Will velocity pid feedforward controller help ? any suugestions.