Subject: Process Control Please provide full answers (b, c)

1. In a metal production process, a step change from 0.7MPa to 0.95MPa in actual pressure results in the measured recorder response shown in the sketch below with the vertical axis simply in units of millimetres of deflection; the sketch has not been drawn to scale. (a) (50\%) Assuming second-order dynamics, calculate all important parameters and write an approximate system transfer function in the form \[ \frac{\boldsymbol{R}}{\boldsymbol{P}}=G(s)=\frac{K}{\tau^{2} s^{2}+2 \zeta \varangle+1} \] where R is the Laplace transform of the deflection response signal in mm units and P is the Laplace transform on the input (step) pressure deviation signal in MPa units. (b) (25%) If the deflection signal is represented more accurately with a 'dead time' in its measurement of 3s vis--vis the second-order model signal described above, i.e. as r(t)=rso(t)U(t3) and a proportional controller with gain Kc is used, determine the amplitude ratio and phase angle of the open-loop transfer function if there is unity feedback. (c) (25\%) Determine the maximum allowable gain Kc for borderline stability in conjunction with this 'dead time', and what would the period of response be under these conditions