We found the err feedback has a very large value at the beginning which charged the integrator a lot. We may want to investigate why it has this large value. A simple method to skip this is to add a limit to filter this large value.
We found the anti-windup is working too slow. We'll try to change the anti-windup gain.
How to calculate the back-calculation gain Kb. The inverse of this gain is the time constant of the anti-windup loop [1].
Reference:
[1] K. Åström, T. Hägglund, Advanced PID Control, ISA, Research Triangle Park, NC, August 2005.