Modern Control Engineering

616 Chapter 8 / PID Controllers and Modified PID Controllers

+

• 
  

+

• 
  

Ei(s) E(s)

Eo(s)

Z 1 Z^2

C 1 C 2

R 1

R 2

R 4

R 3

R 5

Figure 8–45

Modified PID

controller.

Solution

A–8–3. Consider the electronic circuit involving two operational amplifiers shown in Figure 8–45. This is

a modified PID controller in that the transfer function involves an integrator and a first-order

lag term. Obtain the transfer function of this PID controller.

Solution.Since

and

we have

Also,

Eo(s)

E(s)

=-

R 5

R 4

E(s)

Ei(s)

=-

Z 2

Z 1

=-

AR 2 C 2 s+ 1 BAR 1 C 1 s+ 1 B

C 2 sAR 1 +R 3 +R 1 R 3 C 1 sB

Z 2 =R 2 +

1

C 2 s

Z 1 =

1

1

R 1

+C 1 s

+R 3 =

R 1 +R 3 +R 1 R 3 C 1 s

1 +R 1 C 1 s

=K 0

T 1 +T 2

T 1

c 1 +

1

AT 1 +T 2 Bs

+

T 1 T 2 s

T 1 +T 2

d

=K 0 a 1 +

1

T 1 s

+T 2 s+

T 2

T 1

b

=K 0 a 1 +

1

T 1 s

bA 1 +T 2 sB

=

K 0 A 1 +T 1 sBA 1 +T 2 sB

T 1 s



K

Ka

1

K 0

T 1 s

1 +T 1 s

1

1 +T 2 s

b

U(s)

E(s)

=

K

1 +Ka

1

K 0

T 1 s

1 +T 1 s

1

1 +T 2 s

b

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