In Example 16.1, the ultimate gain for the primary controller was found to be 43.3 when Kc2 = 5.

(a) Derive the closed-loop transfer functions for Y1/D1 and Y1/D2 as a function of Kc1 and Kc2.

(b) Examine the effect of Kc2 on the critical gain of Kc1 by varying Kc2 from 1 to 20, For what values of Kc2 do the benefits of cascade control seem to be less important? Is there a stability limit on K?

(c) Integral action was not included in either primary or secondary loops. First set Kc2 = 5, τ/1 = ∞ and τ/2 = 5 min. Find the ultimate controller gain using the Routh array. Then repeat the stability calculation for τ/1 = 5 min and τ/2 = ∞ and compare the two results. Is offset for Y1 eliminated in both cases for step changes in D1 or D2?