Modern Control Engineering

Section 8–7 / Zero-Placement Approach to Improve Response Characteristics 599

we have

(8–6)

The two controllers Gc1andGc2can be determined from Equations (8–5) and (8–6).

EXAMPLE 8–4 Consider the two-degrees-of-freedom control system shown in Figure 8–33. The plant transfer

function is given by

Design controllers and such that the maximum overshoot in the response to the

unit-step reference input be less than 19%, but more than 2%, and the settling time be less than

1 sec. It is desired that the steady-state errors in following the ramp reference input and acceler-

ation reference input be zero. The response to the unit-step disturbance input should have a small

amplitude and settle to zero quickly.

To design suitable controllers and first note that

To simplify the notation, let us define

Gc=Gc1+Gc2

Then

=

10

s(s+1)+10Gc

Y(s)

D(s)

=

Gp

1 +Gp Gc

=

10

s(s+1)

1 +

10

s(s+1)

Gc

Y(s)

D(s)

=

Gp

1 +GpAGc1+Gc2B

Gc1(s) Gc2(s),

Gc1(s) Gc2(s)

Gp(s)=

10

s(s+1)

Gp(s)

=

AKa-a 1 Bs+AKb-a 0 B+AKg-a 2 Bs^2

Ks

1

A(s)

Gc2= c

as+b+gs^2

s

-

a 1 s+a 0 +a 2 s^2

Ks

d

1

A(s)

Gc 1 (s) Gp(s)

R(s) U(s) Y(s)

D(s)

Gc 2 (s)

+

+

+

• 
  

+

• 
  

Figure 8–33
Two-degrees-
of-freedom control
system.