Modern Control Engineering

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

The frequency vaat the intersection of the initial –40-dBdecade segment (or its

extension) with the 0-dB line gives the square root of Kanumerically. This can be seen

from the following:

which yields

Plotting Bode Diagrams with MATLAB. The command bodecomputes magni-

tudes and phase angles of the frequency response of continuous-time, linear, time-

invariant systems.

When the command bode(without left-hand arguments) is entered in the

computer,MATLABproduces a Bode plot on the screen. Most commonly used bode

commands are

bode(num,den)

bode(num,den,w)

bode(A,B,C,D)

bode(A,B,C,D,w)

bode(A,B,C,D,iu,w)

bode(sys)

When invoked with left-hand arguments, such as

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

bodereturns the frequency response of the system in matrices mag,phase, and w.No

plot is drawn on the screen. The matrices magandphasecontain magnitudes and phase

angles of the frequency response of the system, evaluated at user-specified frequency

points. The phase angle is returned in degrees. The magnitude can be converted to deci-

bels with the statement

magdB = 20*log10(mag)

Other Bode commands with left-hand arguments are

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

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

[mag,phase,w] = bode(A,B,C,D)

[mag,phase,w] = bode(A,B.C,D,w)

[mag,phase,w] = bode(A,B,C,D,iu,w)

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

To specify the frequency range, use the command logspace(d1,d2)orlogspace

(d1,d2,n). logspace(d1,d2)generates a vector of 50 points logarithmically equally spaced

between decades 10 d1and 10 d2. (50 points include both endpoints. There are 48 points

between the endpoints.) To generate 50 points between 0.1 radsec and 100 radsec,

enter the command

w = logspace(-1,2)

va= 1 Ka

20 log 2

Ka

AjvaB

22 =^20 log^1 =^0

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