@DSPinfinity Variable and long delays are notoriously hard for PID. So are processes with undamped oscillatory poles (especially bad in this case with load disturbances, less a problem with setpoint changes.) When you say
dynamics explicitly are you asking for a time domain case, such as this system with dead time: \$\frac{\text{d}}{\text{d}t}y=-\frac12 y+\frac12u(t-6)\$? Or what, exactly?