1. A process has the transfer function K exp(-Ts) G(s) (1,5 +1) where K, T and Tp have nominal values of 1.0, 2.0 minutes and 5.0 minutes, respectively, and the control loop has unity feedback. (a) (20%) Develop the controller transfer function and the settings to be used in this control loop using the Controller Synthesis Formula with t. = 5 minutes in the closed loop transfer function X(S) exp(-T's) R(S) T.5+1 Note that in generating the expression for Go(s), you may use the Pad approximant ( T 2 exp(-Ts) = T + 2 1-as S (b) (40%) For a long dead time and tight' control, derive the de facto type (or mode) of controller to be used, along with expressions for its key properties, such as gain and appropriate time constant(s). (c) (40%) Suppose that the controller settings found in (a) are employed and that the process gain, K, changes unexpectedly from 1.0 to 1.0 + a without change in the synthesised controller settings. For what values of a will the closed-loop system be stable