Advanced control engineering (S.R. Burns, 2001)

Transformation

Equation

Block diagram

Equivalent block diagram

1. Combining

X

Y

X

Y

blocks in

Y =(G1G2)X

G1

G2

G1G2

cascade

2. Combining

blocks in

X

Y

+

+

parallel; or

Y = G1X –G2X

G1–G2

eliminating a

X

G1

+

Y

forward loop

+

–

3. Removing a

G2

X

.

G1

+

Y

block from

Y = G

X +G X

G2

G2

a forward

1

– 2

+

–

path

4. Eliminating

Y = G (X + G Y )

X +

Y

X

G1

Y

a feedback

1

– 2

G1

.

1

+

loop

–

– G1G2

+

5. Removing a

X

1 +

Y

block from

Y = G1(X

+

G

2

G1G2

a feedback

– G2Y)

G2

–

loop

+

6. Rearranging

W

+

+

W

+

+

+

summing

+

+

+

Z

+

+

Z = W – X –Y

Y

Z

points

X

–

–

–

–

Y

X

7. Moving a

X

+

Z

X +

G

Z

summing

G

+

Z = GX +Y

+

point ahead

–

–

1

of a block

–

Y

Y

G

8. Moving a

X +

Z

X

G

+

summing

G

+

Z

point

+

+

Z = G(X –Y)

–

beyond

Y

–

Y

G

a block

9. Moving a

X

Y

X

G

take-off

Y = GX

G

Y

point ahead

Y

of a block

Y

G

10. Moving a

X

Y

X

G

. Y

G

take-off

Y = GX

point beyond

X

X

1

a block

G

1

H3

1

G1

G4

R(s)

+ –

+

C(s)

G1G2

G3G4

–

–

H1

H2

H3

G1G4

R(s)

+ –

G1G2

G3G4

C(s)

1 + G1G2H1

1 + G3G4H2

#

$

# #

$

#

#

R1(s) +

+

C(s)

G1

G2

–

–

H2

H1

+

+

R2(s)

R1(s) +

G1G2

C(s)

Ra

La

ia(t)

Tm

θ(t) ω(t)

$

$

$

$

$

$

#

$ #

Tm(t)

(Nm)_

Increasing ea(t)

ω(t) (rad/s)

Ef(s)

1

If(s)

Tm(s)

Lfs + Rf

Kr

$

Tm(t)

Tm(t)

Increasing ef(t)

(Nm)

(Nm)

Kf

Rf

ef(t) (V)

ω(t) (rad/s)

(a)

(b)

Pe = 0

Ps

Pe = 0

Q2

Q1

Xo, xo

P2 V2

A

A P1 V1

m

(2)

(1)

Qleak