1. Block Diagrams A line is a signal A block is a gain A circle is a sum
Due to h.f. noise, use proper blocks: num deg ≤ den deg
Try to use just horizontal or vertical lines
Use additional “ ” to help
e.g. x y G
y = Gx + x s Σ + -
s = x + z - y y z Σ + x s + + z - y

2. Block Diagram Algebra Series: Parallel: x y G1 G2 x y G1 G2 G1 + x y x
G1 + G2 + G2

3. Feedback:
Proof: + e x x y G1 y  - b G2

4. + G1 + G2 + -

5. >> s=tf('s')
Transfer function: s
>> G1=(s+1)/(s+2)
s + 1
-----
s + 2
>> G2=5/(s+5) 5
s + 5
>> G=G1*G2
Transfer function:
5 s + 5
s^2 + 7 s + 10
>> H=G1+G2
s^ s + 15
>> HF=feedback(G1, G2)
s^2 + 6 s + 5
s^ s + 15

6. >> delay1=tf(1,1,'inputdelay',0.05)
Transfer function:
exp(-0.05*s) * 1
>> H2=HF*delay1
s^2 + 6 s + 5
exp(-0.05*s) *
s^ s + 15
>> stepresp=H2*1/s
exp(-0.05*s) *
s^ s^ s
>> step(H2)

7. Quarter car suspension
Series
R(s) + y -
R(s) + y
Feedback -
R(s) y

8. >> b=sym('b');
>> m=sym('m');
>> k=sym('k');
>> s=sym('s');
>> G1=b*s+k
G1 =
b*s+k
>> G2=1/m*1/s*1/s
G2 =
1/m/s^2
>> G=G1*G2
G =
(b*s+k)/m/s^2
>> Gcl=G/(1+G)
Gcl =
(b*s+k)/m/s^2/(1+(b*s+k)/m/s^2)
>> simplify(Gcl)
ans =
(b*s+k)/(m*s^2+b*s+k)

9. Move a block (G1) across a into all touching lines:
pick-up point
summation
Move a block (G1) across a into all touching lines:
If arrow direction changes, invert
If arrow direction remains, no change in block
e.g.
along arrow
no change
along arrow x y x y G1 G2 G1 G2
no change z G3 G1
along arrow
along arrow z G3

10. x G1 G2 x G1 G2 y y  z G3 z G3
1/G2 x G1 G2 x G1 G3
1/G3 G2 y y  z G3 z

13. I2 I1 - Vc U + y + - I2 - Vc U + y + - - U + y + -

14. - U + y - U + y U y

15. No pure series/parallel/feedback Needs to move a block, but which one?
Find TF from U to Y: + U + + Y + - -
No pure series/parallel/feedback
Needs to move a block, but which one?
Key: move one block to create pure series or parallel or feedback!
So move either left or right.

16. + U + + Y + - - + U + + Y + - - + U + Y + -

17. fig_03_16
fig_03_16 is wrong
Should be H1(s)G2(s)

18. fig_03_17
fig_03_17 is wrong

19. fig_03_18b

20. fig_03_19 Can use superposition: First set D=0, find Y due to R
Then set R=0, find Y due to D
Finally, add the two component to get the overall Y

21. fig_03_20
First set D=0, find Y due to R

22. Then set R=0, find Y due to D
fig_03_21 G2

23. fig_03_19
Finally, add the two component to get the overall Y