1. Background Knowledge Expected
Elementary functions
Complex numbers
Common test input signals: impulse, step, ramp, acceleration, sinusoidal, exponential
Differential equations
Laplace transform
Forward transform and its properties
Inverse transform and partial fraction expansion
Initial value theorem and final value theorem
Use of Laplace xform to solve diff eq
Matlab

2. System models Time domain models
High order ordinary differential equation model
Contains only input variables, output variables, their derivatives, and constant parameters
Proper: highest output derivative order is greatest
Highest order derivative of output = system order

3. System models Time domain models
State space model: state equation + output equation
State equation: a set of 1st order diff eq on state variables
Output equation: output as function of state and input
Linear systems:

4. ODE model to State space model

5. Transfer Function

6. State space model to TF
A, B, C, D are matrices

7. Input Output System Y(s) = H(s)X(s)
x(t)
Output
y(t)
H(s)
Y(s) = H(s)X(s)
If the input x(t)=δ (t), the output is called the impulse response.
If the input x(t)=u(t), the output is called the step response.
If the input x(t)=Asin(wt), and H(s) is stable, output steady state is A|H(jw)|sin(wt+H(jw))
Poles: values of s at which TF  infinity
Zeros: values of s at which TF = 0

8. 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

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

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

11. + G1 + G2 + -

12. >> 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

13. >> 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)

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

15. >> 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)

16. 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

17. 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

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

21. - U + y - U + y U y

22. 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.

23. + U + + Y + - - + U + + Y + - - + U + Y + -

24. fig_03_18b

25. 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

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

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

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