ABSTRACT
The Nyquist criterion is a frequency-based method used to determine the stability or relative stability of feedback control systems [1–4]. Let GH(s) be the transfer function of the open-loop plant and sensor in a single-loop negative-feedback continuous-time system. A contour that includes the entire righthalf of the s-plane is first mapped into the GH(s-plane by using either s-plane vectors or Bode diagrams of magnitude and phase versus frequency ω. This polar plot mapping is often considered to be the most difficult step in the procedure. The number of closed-loop poles in the righthalf s-plane Z is calculated as the sum of P and N, where P is the number of open-loop poles in the righthalf s-plane and N is the number of encirclements of the −1 + j0 point in the GH(s)-plane. The closed-loop system is stable if and only if Z = 0. It is important to specify N as positive if the encirclements are in the same direction as the contour in the s-plane and negative if in the opposite direction. Different ranges of gain K can yield different numbers of encirclements and, hence, different stability results. Moreover, the application of the Nyquist criterion for the stability of a closed-loop system can be presented succinctly in a table for different ranges of K.
Example 11.1
Consider a negative unity feedback system having G(s) = K/(s + 1)3. For the section of the s-plane contour along the positive jω-axis shown in Figure 11.1a, the corresponding G(s)-plane contour is the polar plot of G(jω), as ω varies from 0 to +∞, shown in Figure 11.1b, for K = 10. The frequency at which the phase of G(jω) is −180° is designated as the phase crossover frequency ωpc, which for this example is √ 3 = 1.732 rad / s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315218403/e3ae489a-b4d0-4be4-acb7-275a744c8397/content/eq535.tif"/> . Observe that ωpc could be obtained alternatively by setting the imaginary part of G(jω) equal to 0. Next, the infinite arc that encloses the entire righthalf s-plane maps into a point at the origin in the G(s)-plane, and the remaining section (the negative jω-axis) is obtained as the mirror reflection of Figure 11.1b about the real axis, as shown in Figure 11.1c. Finally, the G(s)-plane plot for K < 0 is the mirror reflection of Figure 11.1c about the jω-axis, as shown in Figure 11.1d. Table 11.1 shows stability results for different ranges of K.
The three open-loop poles are located at s = −1. Therefore, none of the open-loop poles are inside the righthalf s-plane contour and, thus P = 0. The real part of G ( j ω pc ) = G ( j √ 3 ) = − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315218403/e3ae489a-b4d0-4be4-acb7-275a744c8397/content/eq536.tif"/> when K = 8, which is the upper limit of the range shown in Row 1 of the table. In words, the G(s)-plane contour for positive 11-2 Kin Figure 11.1c passes through the −1 + j0 point for K = 8; the −1 + j0 point is not encircled (N = 0) for K < 8; and it is encircled twice (N = 2) for K > 8. These results are shown in Rows 1 and 2 of the table. For the negative K case, shown in Figure 11.1d, the real part of G(j0) = −1 when K = −1. Similarly, the results in Rows 3 and 4 are obtained for the ranges of negative K. In brief, the Nyquist criterion has shown (Rows 1 and 3) that the closed-loop system is stable for −1 < K < 8.
Nyquist criterion application to Example 11.1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315218403/e3ae489a-b4d0-4be4-acb7-275a744c8397/content/fig11_1.tif"/> Nyquist Criterion Results for Example 11.1Range of K
P
N
Z
Stability
0 < K < 8
0
0
0
Yes
8 < K < ∞
0
2
2
No
–1 < K < 0
0
0
0
Yes
–∞ < K < −1
0
1
1
No
Example 11.2
Let a negative unity feedback system have a plant transfer function G(s) = K/[s(s + 2)2]. The s-plane contour along the positive jω-axis from Example 11.1 is modified by inserting an infinitesimal arc of radius ε that excludes the open-loop pole at the origin from inside the contour, as shown in Figure 11.2a. The G(s)-plane contour corresponding to this arc is an infinite arc extending from the positive real axis clockwise through the fourth quadrant and connecting with the polar plot of G(jω), as ω varies from ε to +∞. The complete G(s)-plane contour is shown in Figure 11.2b for K = 20, and ωpc is equal to 2 rad/s. Table 11.2 shows stability results for different ranges of K.
The open-loop poles are located at s = –2, –2, and the origin. Therefore, again no open-loop poles are inside the righthalf s-plane contour and, thus P = 0. The real part of G(jωpc) = G(j1) = −1 when K = 16, which is the upper limit of the range, shown in Row 1 of the table. In words, the G(s)-plane contour for positive K in Figure 11.2b passes through the −1 + j0 point for K = 16; the −1 + j0 point is not encircled 11-3(N = 0) for K < 16; and it is encircled twice (N = 2) for K > 16. These results are shown in Rows 1 and 2 of the table. For the negative K case, there is one encirclement of −1 + j0 for all K< 0 and consequently Z = P + N = 0 + 1 = 1, as shown in Row 3. In summary, the Nyquist criterion has shown (Row 1) that the closed-loop system is stable for 0 < K < 16.
Nyquist criterion application to Example 11.2. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315218403/e3ae489a-b4d0-4be4-acb7-275a744c8397/content/fig11_2.tif"/> Nyquist Criterion Results for Example 11.2Range of K
P
N
Z
Stability
0 < K < 16
0
0
0
Yes
16 < K < ∞
0
2
2
No
–∞ < K < 0
0
1
1
No
Example 11.3
To illustrate the Nyquist criterion for a system having an unstable open-loop plant, consider the negative unity feedback system with G(s) = K(s + 1)/[s(s – 1)]. The s-plane contour along the positive jω-axis is again modified with an infinitesimal arc of radius ε. The infinite arc extends from the negative real axis clockwise through the second quadrant and connects with the polar plot of G(jω), as ω varies from ε to +∞. The complete G(s)-plane contour is shown in Figure 11.3 for K = 2, and ωpc is equal to 1 rad/s. Table 11.3 shows stability results for different ranges of K.
11-4The open-loop poles are located at s = +1 and the origin. Therefore, one open-loop pole (s = +1) lies inside the righthalf s-plane contour and, thus P = 1. The real part of G(jωpc) = G(j1) = −1 when K = 1, which is the upper limit of the range, shown in Row 1 of the table. In words, the G(s)-plane contour for positive Kin Figure 11.3 passes through the −1 + j0 point for K = 1; the −1 + j0 point is encircled once in the same direction as the s-plane contour (N = +1) for K < 1; and it is encircled once in the opposite direction as the s-plane contour (N = −1) for K > 1. These results are shown in Rows 1 and 2 of the table. For the negative K case, there are no encirclements of −1 + j0 for all K < 0 and consequently Z = P + N = 1 + 0 = 1, as shown in Row 3. In summary, the Nyquist criterion has shown (Row 2) that the closed-loop system is stable for 1 < K < ∞.
Nyquist criterion application to Example 11.3. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315218403/e3ae489a-b4d0-4be4-acb7-275a744c8397/content/fig11_3.tif"/> Nyquist Criterion Results for Example 11.3Range of K
P
N
Z
Stability
0 < K < 1
1
+ 1
2
No
1 < K < ∞
1
–1
0
Yes
–∞ < K < 0
1
0
1
No
