How to implement PID controller in Closed-loop system Identification?

I want to identify the performance of a closed-loop feedback control system in MatLab. So, in the end I want to have poles, zeros, bode etc. of the closed loop system:

   yset ---(err)---[Gc]---(u)---[Gp]---y
         |___________yfeedb____________|

What I have:

    PID settings
    Known structure Gp (1st/2nd order)
    Closed-loop data (yset, yfeedb, y) (y == yfeedb)

What I did:

    Set up Gp with unknown parameters Kp,tau,td
    Write Gp to state-space form

What I want:

    Use lsim to simulate y with certain parameters for Gp.
    Obtain the right parameters for Gp by minimization ysim-y.
    Finally: have the TF of the whole CL system.

Problem:

    How to implement the PID controller?

I used the velocity form of PI in order to calculate u. With u and Gp in steady-state form, I am able to calculate y. However, how can I implement a PID controller in it?

    Is there a velocity form for PID?
    Or can I make a overall TF (OTF) with feedback(Gp*Gc,1) and use OTF and error to calculate y??