One issue that must always be taken into account by the system designer is the possible...

One issue that must always be taken into account by the system designer is the possible effect of unmodeled aspects of the system one is attempting to stabilize or modify through feedback. In this problem, we provide an illustration of why this is the case. Consider a continuous-time feedback system, and suppose that

(a) Use root-locus techniques to show that the closed-loop system will be stable if K is chosen large enough.

(b) Suppose that the system we are trying to stabilize by feedback actually has a system function

The added factor can be thought of as representing a first-order system in cascade with the system of eq. (P11.37-1). Note that the time constant of the added first order system is extremely small and thus will appear to have a step response that is almost instantaneous. For this reason, one often neglects such factors in order to obtain simpler and more tractable models that capture all of the important characteristics of the system. However, one must still keep these neglected dynamics in mind in obtaining a useful feedback design. To see why this is the case, show that if G(s) is given by eq. (P11.37-2) and H(s) is as in eq. (P11.37-3), then the closed-loop system will be unstable if K is chosen zoo large. Hint: See Problem 11.34.

(c) Use root-locus techniques to show that if

then the feedback system will be stable for all values of K sufficiently large if H(s) is given by eq. (P11.37-1)