Modern Control Engineering

Based on the Bode diagram of Figure 8–15, we choose the value of b.The term

(bs+1)needs to give the phase margin of at least 50°. By simple MATLAB trials, we

findb=0.25to give the phase margin of at least 50° and gain margin of ±qdB. There-

fore, by choosing b=0.25,we have

and the open-loop transfer function of the designed system becomes

MATLAB Program 8–4 produces the Bode diagram of the open-loop transfer function.

The resulting Bode diagram is shown in Figure 8–16. From it we see that the static ve-

locity error constant is 4 sec–1, the phase margin is 55°, and the gain margin is ±qdB.

=

5 s^2 + 21 s+ 4

s^3 +s

Open-loop transfer function=

4 ( 5 s+ 1 )(0.25s+ 1 )

s

1

s^2 + 1

Gc(s)=

4 ( 5 s+ 1 )(0.25s+ 1 )

s

580 Chapter 8 / PID Controllers and Modified PID Controllers

MATLAB Program 8–4

num = [5 21 4];

den = [1 0 1 0];

w = logspace(-2,2,100);

bode(num,den,w)

title('Bode Diagram of 4(5s+1)(0.25s+1)/[s(s^2+1)]')

Frequency (rad/sec)

Bode Diagram of 4(5s + 1)(0.25s + 1)/[s(s^2 + 1)]

− 200

− 100

− 50

0

50

− 150

100

− 50

Phase (deg); Magnitude (dB)

0

100

50

10 –2 10 –1 100 101 102

Figure 8–16

Bode diagram of

4(5s+1)(0.25s+1)/

CsAs^2 +1BD.

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