Modern Control Engineering

628 Chapter 8 / PID Controllers and Modified PID Controllers

Figure 8–58

Nyquist plot.

− 1500 − 1000 − 500 0 500 1000 1500

Real Axis

Nyquest Diagram

Imaginary Axis

2500

2000

1500

1000

500

0

− 500

− 1000

−1500

− 2000

−2 500

Figure 8–59

Redrawn Nyquist

plot.

Im

v= 0+ Re

v= 0−

v=−`

v=+`

Using the Nyquist plot obtained here, it is not easy to determine the encirclements of the − 1 +j0

point by the Nyquist locus. Therefore, we need to redraw this Nyquist plot qualitatively to show

the details near the − 1 +j0point. Such a redrawn Nyquist diagram is shown in Figure 8–59.

From this diagram we find that the − 1 +j0point is encircled counterclockwise three times.

Hence,N−3. Since the open-loop transfer function has three poles in the right-half s 1 plane,

we have P3. Then, we have ZN+P0. This means that there are no closed-loop poles in

the right-half s 1 plane. The system is therefore stable.

A–8–8. Show that the I-PD-controlled system shown in Figure 8–60(a) is equivalent to the PID-controlled

system with input filter shown in Figure 8–60(b).

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