### 6.12. RELATIONSHIP BETWEEN CLOSED-LOOP FREQUENCY RESPONSE AND THE TIME-DOMAIN RESPONSE

Section 6.10 has illustrated how the closed-loop frequency-domain response may be obtained from the open-loop transfer function. The next logical question to ask is how to determine the relationship between *M*_{p} and the peak overshoot one obtains in the time domain. In Chapter 4, we defined the time at which the peak overshoot occurs as *t*_{p} in terms of *ζ* and *ω*_{n} [see Eq. (4.29)]. For example, does an *M*_{p} of 1.3 mean a 30% transient overshoot in the time domain?

**Table 6.20. MATLAB Program to Obtain the Nichols Chart for the System whose Open-Loop Transfer Function is shown in Eq. (6.121)**

num = [780 1344.8273]
den = [1 30.5747 217.2414 114.9425 0]
a = −[.25:.25:1 2 5 10:10:170 179.99];
b = [−24 − 18 − 12 − 9 − 7 − 5: − 1 −.5:.25:.5 1:5 7 9 12];
[x,y] = nichgrid([−360 0 − 24 36],a,b,3);
[mag, ph] = bode(num,den,logspace(-1,2));
plot(ph,20 _{*} log10(mag));
W = [5 1 2 5 7 9 12];
[mag, ph] = bode(num,den,w);
plot(ph,20 _{*} log10(mag),‘ _{*} g’)
title(‘Nichols Frequency Response Plot’) |

**Figure 6.52** Nichols chart with superimposed.

This problem has been analyzed for the general, unity-feedback system of Figure 6.45 [22 ...