Consider the chamber temperature control system of Problem 8.4-3. Suppose that a variable gain \(K\) is added to the plant-. The pulse transfer function for this system is given in Problem 8.4-3.

(a) Plot the root locus for this system, and find the value of \(K>0\) for which the system is stable.

(b) Find the time constant for the system with \(K=1\).

(c) Assuming the resulting system to be stable, find \(\mathrm{K}\) such that the steady-state error in the uncompensated system is 0.05 , or 5 percent. Is the resulting system stable? [See part (a).]

(d) Design a phase-lag controller such that the characteristic-equation root is approximately the same for the gain \(\mathrm{K}\) found in part (c). Hint: Choose \(z_{p}=0.99999\).

(e) By computer, verify the design in part (d) by finding the characteristic-equation roots.

Problem 8.4-3