Modern Control Engineering

630 Chapter 8 / PID Controllers and Modified PID Controllers

A–8–10. In some cases it is desirable to provide an input filter as shown in Figure 8–61(a). Notice that the

input filter is outside the loop. Therefore, it does not affect the stability of the closed-

loop portion of the system. An advantage of having the input filter is that the zeros of the closed-loop

transfer function can be modified (canceled or replaced by other zeros) so that the closed-

loop response is acceptable.

Show that the configuration in Figure 8–61(a) can be modified to that shown in Figure 8–61(b),

where The compensation structure shown in Figure 8–61(b) is some-

times called command compensation.

Solution.For the system of Figure 8–61(a), we have

(8–15)

For the system of Figure 8–61(b), we have

Thus

or

(8–16)

By substituting into Equation (8–16), we obtain

=Gf(s)

Gc(s)Gp(s)

1 +Gc(s)Gp(s)

C(s)

R(s)

=

CGf(s)Gc(s)-Gc(s)+Gc(s)DGp(s)

1 +Gc(s)Gp(s)

Gd(s)=CGf(s)- 1 DGc(s)

C(s)

R(s)

=

CGd(s)+Gc(s)DGp(s)

1 +Gc(s)Gp(s)

C(s)=Gp(s)EGd(s)R(s)+Gc(s)CR(s)-C(s)DF

C(s)=Gp(s)U(s)

E(s)=R(s)-C(s)

U(s)=Gd(s)R(s)+Gc(s)E(s)

C(s)

R(s)

=Gf(s)

Gc(s)Gp(s)

1 +Gc(s)Gp(s)

Gd(s)=CGf(s)- 1 DGc(s).

Gf(s)

(a)

(b)

Gc(s)

R(s) C(s)

Gf(s) Gp(s)

Gc(s)

R(s) E(s) C(s)

Gd(s)

Gp(s)

U(s)

+

+– +

Figure 8–61 +

(a) Block diagram of

control system with

input filter;

(b) modified block

diagram.

Openmirrors.com