Modern Control Engineering

Section 8–7 / Zero-Placement Approach to Improve Response Characteristics 609

Assume also that the controller is of PID type. That is,

The controller is of P or PD type. [If involves integral control action, then this will

introduce a ramp component in the input signal, which is not desirable. Therefore, should

not include the integral control action.] Thus, we assume that

where may be zero.

Let us design controllers and such that the responses to the step-disturbance

input and the step-reference input are of “desirable characteristics” in the sense that

1. The response to the step-disturbance input will have a small peak and eventually approach
   zero. (That is, there will be no steady-state error.)


2. The response to the step reference input will exhibit less than 25%overshoot with a settling
   time less than 2 sec. The steady-state errors to the ramp reference input and acceleration ref-
   erence input should be zero.
   The design of this two-degrees-of-freedom control system may be carried out by following the
   steps 1 and 2 below.


3. Determine so that the response to the step-disturbance input is of desirable characteristics.


4. Design so that the responses to the reference inputs are of desirable characteristics
   without changing the response to the step disturbance considered in step 1.

Design of First, note that we assumed the noise input N(s)to be zero. To obtain the re-

sponse to the step-disturbance input, we assume that the reference input is zero. Then the block

diagram which relates Y(s)andD(s)can be drawn as shown in Figure 8–40. The transfer func-

tionY(s)/D(s)is given by

Y(s)

D(s)

=

Gp

1 +Gc1 Gp

Gc1(s):

Gc2(s)

Gc1(s)

Gc1(s) Gc2(s)

Tˆd

Gc2(s)=KˆpA 1 +Tˆd sB

Gc2(s)

Gc2(s) Gc2(s)

Gc1(s)=Kpa 1 +

1

Ti s

+Td sb

Gc1(s)

Gc 1 (s) Gp(s)

Gc 2 (s)

Y(s)

N(s)

U(s)

D(s)

R(s)

B(s)

+– ++ ++

++

Figure 8–39
Two-degrees-of-
freedom control
system.

D(s) Y(s)

+

• Gp(s)

Figure 8–40 Gc 1 (s)
Control system.