Modern Control Engineering

Section 8–7 / Zero-Placement Approach to Improve Response Characteristics 607

tor of the closed-loop transfer function Y(s)/R(s).] To eliminate the offset in the response to the

step reference input and eliminate the steady-state errors in following the ramp reference input

and acceleration reference input, the numerator of Y(s)/R(s)must be equal to the last three

terms of the denominator, as mentioned earlier. That is,

or

Thus, is a PID controller. Since is given as

we obtain

Thus, is a derivative controller. A block diagram of the designed system is shown in

Figure 8–37.

The closed-loop transfer function Y(s)/R(s)now becomes

Y(s)

R(s)

=

20.4s^2 +122.44s+259.68

s^3 +20.4s^2 +122.44s+259.68

Gc2(s)

=-0.1s

= a1.94s+12.244+

25.968

s

b- a2.04s+12.244+

25.968

s

b

Gc2(s)=Gc(s)-Gc1(s)

Gc(s)=Gc1(s)+Gc2(s)=

1.94s^2 +12.244s+25.968

s

Gc1(s) Gc(s)

Gc1(s)=2.04s+12.244+

25.968

s

10sGc1(s)=20.4s^2 +122.44s+259.68

R(s) Y(s)

D(s)

Gc 2 (s)

Gc 1 (s)

0.1s

+

+

+

+

+ 10

s(s+1)

25.968

s

2.04s+12.244+

Figure 8–37
Block diagram of the
designed system.