1. 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

2. 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:

3. ODE model to State space model

4. ODE model to State space model
m<n

5. ODE model to State space model

6. Transfer Function

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

8. Input Output System H(s) 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

9. Example: controller Proportional controller: C(s) = KP =const
E(s)
U(s)
Proportional controller: C(s) = KP =const
Integral controller: C(s) = KI/s
Derivative controller: C(s) = KDs
PI controller: C(s) = KP + KI/s
PD controller: C(s) = KP + KDs
PID controller: C(s) = KP + KI/s + KDs

10. Negative Feedback Control System
From lecture 1:
Negative Feedback Control System + + +
CONTROLLED
DEVICE
CONTROLLER -
FEEDBACK
ELEMENT

12. 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

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

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

15. + G1 + G2 + -

16. >> 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

17. >> 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)

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

19. >> 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)

20. 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 block (1/G1)
If arrow direction remains, no change in block
For example:
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

21. 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
against, against x G1 G2 x G1 G2 y y
against
along  z G3 z G3
1/G2 x G1 G2 x G1 G3
1/G3 G2 y y  z G3 z

24. No pure series/parallel/feedback Needs to move a block, but which one?
Find TF from U to Y: I2 I1 - Vc 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.

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

26. - U + y - U + y U y

27. 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.

28. + U + + Y + - - + U + + Y + - - + U + Y + -

29. fig_03_18b

30. 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

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

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

33. fig_03_19
Finally, add the two components to get the overall Y