Modern Control Engineering

476 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

MATLAB Approach to Get Resonant Peak, Resonant Frequency, and Band-

width. The resonant peak is the value of the maximum magnitude (in decibels) of the

closed-loop frequency response. The resonant frequency is the frequency that yields the

maximum magnitude. MATLAB commands to be used for obtaining the resonant peak

and resonant frequency are as follows:

[mag,phase,w] = bode(num,den,w); or [mag,phase,w] = bode(sys,w);

[Mp,k] = max(mag);

resonant_peak = 20*log10(Mp);

resonant_frequency = w(k)

The bandwidth can be obtained by entering the following lines in the program:

n = 1;

while 20*log10(mag(n)) > = -3; n = n + 1;

end

bandwidth = w(n)

For a detailed MATLAB program, see Example 7–23.

EXAMPLE 7–23 Consider the system shown in Figure 7–78. Using MATLAB, obtain a Bode diagram for the closed-

loop transfer function. Obtain also the resonant peak, resonant frequency, and bandwidth.

MATLAB Program 7–12 produces a Bode diagram for the closed-loop system as well as the

resonant peak, resonant frequency, and bandwidth. The resulting Bode diagram is shown in

MATLAB Program 7–12

nump = [1];

denp = [0.5 1.5 1 0];

sysp = tf(nump,denp);

sys = feedback(sysp,1);

w = logspace(-1,1);

bode(sys,w)

[mag,phase,w] = bode(sys,w);

[Mp,k] = max(mag);

resonant_peak = 20*log10(Mp)

resonant_peak =

5.2388

resonant_frequency = w(k)

resonant_frequency =

0.7906

n = 1;

while 20*log(mag(n))> = -3; n = n + 1;

end

bandwidth = w(n)

bandwidth =

1.2649

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