As mentioned in Section 11.4, the continuous-time Nyquist criterion can be extended to a...

As mentioned in Section 11.4, the continuous-time Nyquist criterion can be extended to allow for poles of G(s)H(s) on the jω -axis. In this problem, we will illustrate the general technique for doing this by means of several examples. Con-sider a continuous-time feedback system with

When G(s)H(s) has a pole at s = 0, we modify the contour of Figure 11.19 by avoiding the origin. To do this, we indent the contour by adding a semicircle of infinitesimal radius ∈ into the right-half plane. [See Figure P11.44(a).] Thus, only a small part of the right-half plane is not enclosed by the modified contour, and its area goes to zero as we let ∈ → 0. Consequently, as M → ∞the contour will enclose the entire right-half plane. As in the text, G(s)H(s) is a constant (in this case zero) along the circle of infinite radius. Thus, to plot G(s)H(s) along the contour, we need only plot it for the portion of the contour consisting of the fro-axis and the infinitesimal circle.

(a) Show that

where is the point where the infinitesimal semicircle meets the jω-axis just below the origin and is the corresponding point just above the origin.

(b) Use the result of part (a) together with eq. (P11.44-1) to verify that Figure P11.44(b) is an accurate sketch of G(s)H(s) along the portions of the contour

(d) Using the Nyquist plot of Figure P11.44(c), find the range of values of K for which the closed-loop feedback system is stable. (Note: As presented in the text, the continuous-time Nyquist criterion states that, for closed-loop sys-tem stability, the net number of clockwise encirclements of the point —1/K must equal minus the net number of right-half plane poles of G(s)H(s). In the present example, note that the pole of G(s)H(s) at s = 0 is outside the modified contour. Consequently, it is not included in counting the poles of 0(s)H(s) in the right-half plane [i.e., only poles of G(s)H(s) strictly inside the right-half plane are counted in applying the Nyquist criterion]. Thus, in this case, since G(s)H(s) has no poles strictly inside the right-half plane, we must have no encirclements of the point s = —11K for closed-loop system stability.)

In each case, use the Nyquist criterion to determine the range of values of K (if any such range exists) for which the closed-loop system is stable. Also, use another method (root locus or direct calculation of the closed-loop poles as a function of K) to provide a partial check of the correctness of your Nyquist plot. [Note: In sketching the Nyquist plots, you may find it useful to sketch the Bode plots of G(s)H(s) first. It may also be helpful to determine the values of w for which G(jω)H (jω) is real.]

In these cases there are two poles on the imaginary axis; accordingly, you will need to modify the contour of Figure 11.19 to avoid each of them. Use infinitesimal semicircles, as in Figure P11.44(a).