A cascade control configuration is sometimes used to separate different objectives of feedback control. For instance, first an inner loop can be used to stabilize the process, and then an outer loop can be used to control the output to track set-point changes and reject disturbances. An example of a cascade control system is shown in Fig. E. The process transfer function is unstable.

(a) Determine the maximum range of K_inner values for which the inner loop will be stable.

(b) Now assume K_c2 = 6 and G_c1 is a PI controller with gain K_c1 and ?_I. Find values of K_c1 and ?_I such that the closed-loop poles of the transfer function from Y_SP, to Y(s) are at s = ?0.5 ? O.2/.

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