Consider the feedback system of Figure P11.39 with (a) Plot the root locus for...

Consider the feedback system of Figure P11.39 with

(a) Plot the root locus for K > 0.

(b) Plot the root locus for K < 0. (Note: Be careful with this root locus. By applying the angle criterion on the real axis, you will find that as K is decreased from zero, the closed loop approaches z = +∞ along the positive real axis and then returns along the negative real axis from z = —∞. Check that this is in fact the case by explicitly solving for the closed-loop pole as a function of K. At what value of K is the pole at | z | = ∞?)

(c) Find the full range of values of K for which the closed-loop system is stable.

(d) The phenomenon observed in part(b) is a direct consequence of the fact that in this example the numerator and denominator of G(z)H(z) have the same degree. When this occurs in a discrete-time feedback system, it means that there is a delay-free loop in the system. That is, the output at a given point in time is being fed back into the system and in turn affects its own value at the same point in time. To see that this is the case in the system we are considering here, write the difference equation relating y[n] and e[n]. Then write e[n] in terms of the input and output for the feedback system. Contrast this result with that of the feedback system with

The primary consequence of having delay-free loops is that such feed-back systems cannot be implemented in the form depicted. For example, for the system of eq. (P11.39-1), we cannot first calculate e[n] and then y[n], because e[n] depends on y[n]. Note that we can perform this type of calculation for the system of eq. (P11.39-2), since e[n] depends on y[n — 1]

(e) Show that the feedback system of eq. (P11.39-1) represents a causal system, except for the value of K for which the closed-loop pole is at | z | = ∞.