National Instruments 370753C-01

Chapter 5 Classical Feedback Analysis

By combining the information from the two plots, reflecting them across

the real axis to account for the negative frequency response and augmenting

them with a line closing the contour in the clockwise direction, you obtain

the sketch of the encirclement pattern shown in Figure 5-9. In this figure,

Nyquist contour is formed by drawing the system’s Nyquist plot for all

positive frequencies, reflecting it about the real axis to show plot for

negative frequencies, and completing the closed contour in a clockwise

direction.

Figure 5-9. Nyquist Contour Formed by Drawing the System’s Nyquist Plot

for All Positive Frequencies

Because your open-loop system has no unstable (right-half plane) poles,

the number of unstable closed-loop poles for a given gain K will be equal

to the number of times the contour encircles the (–1/K, 0) point.

Referring back to the Nyquist plots, you see that for K> 168

(or –1/K> –0.006), you have two encirclements of the (–1/K, 0) point,

and thus two unstable closed-loop poles. For gain values less than 167,

the closed-loop system is stable. You can verify this with a small

experiment, using a value of 169 for K:

sysu = 169*sys;

sysucl = feedback(sysu);

poles(sysucl)

ans (a column vector) =

Imaginary

Real