Modern Control System Theory and Design, 2nd Edition

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 ...