Incorrect integrator result with saturation

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Hello,
I am implementing a PID loop where the input signal will be an acceleration signal that will be fed into a double integrator plant. As I have a limitation on my plant's ability to accelerate, I have placed a saturation on the PID block.
When I plot out acceleration signal and velocity signal after all this, I see something weird: when acceleration signal gets saturated, velocity signal does not change at all... which makes no sense as I expect to see output signal from integrator to keep increasing if what goes into it is held at constant.
What could possibly be causing this problem?

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