Frequency Response Characteristics
Non-unity Feedback Systems
Example M9.1
Consider a non-unity feedback system with forward-path transfer function
and feedback-path transfer function
Resonance peak, resonance frequency, and bandwidth of the closed-loop frequency response may be obtained by generating Bode plot of the closed-loop transfer function.
s = tf('s');
G = 10/(s*(s+5));
H = 1/(0.1*s+1);
M = feedback(G,H);
w = logspace(-1,1);
bode(M,w);
grid;
This MATLAB program produces Bode plots for the closed-loop system shown in Fig.M9.1. Right-clicking away from a curve brings up a menu. From this menu, select Characteristics --> Peak Response. When selected, a dot appears on the curve at the appropriate point Let your mouse rest on the point to read the parameters of the resonance peak. Now point the mouse at any point on the magnitude curve and left-click. The arrow curser changes to four-pointed arrow. Hold down the left mouse button and drag the four-pointed arrow along the magnitude curve. By trial-and-error, locate the point where magnitude is - 3dB. The frequency at this point is the bandwidth.
Fig. M9.1
From Fig.M9.1, we obtain the following:
Mr = 0.311dB,
= 1.53 rad/sec., and
= 3.38 rad/sec.
The gain margin and phase margin can be obtained from the Bode plot of G(s)H(s). The Bode plot and the performance measures generated by the command margin(G*H) are shown in Fig. M9.2.
Fig. M9.2