1. 3. Systems and Transfer function
Discrete-time system revision
Discrete-time system
A/D and D/A converters
Sampling frequency and sampling theorem
Nyquist frequency
Aliasings
Z-transform & inverse Z-transform
The output of a D/A converter

2. 3.1 Zero-order-hold (ZOH)
A Zero-order hold in a system
x(t)
x(kT)
h(t)
Zero-order
Hold
Sampler

3. 3.1 Zero-order-hold (ZOH)
How does a signal change its form in a discrete-time system?
The input signal x(t) is sampled at discrete instants and the sampled signal is passed through the zero-order-hold (ZOH). The ZOH circuit smoothes the sampled signal to produce the signal h(t), which is a constant from the last sampled value until the next sample is available. That is

4. 3.1 Zero-order-hold (ZOH)
Transfer function of Zero-order-hold
The figure below shows a combination of a sampler and a zero-order hold.
x(t)
x(kT)
h(t)
Zero-order
Hold
Sampler

5. 3.1 Zero-order-hold (ZOH)
Assume that the signal x(t) is zero for t<0, then the output h(t)is related to x(t) as follows:
h(t) t

6. 3.1 Zero-order-hold (ZOH)As
The Laplace transform of the above equation becomes

7. 3.1 Zero-order-hold (ZOH)As
Therefore
Finally, we obtain the transfer function of a ZOH as

8. 3.1 Zero-order-hold (ZOH)
There are also first-order-hold and high-order-hold although they are not used in control system.

9. 3.1 Zero-order-hold (ZOH)
A zero-order-hold creates one sampling interval delay in input signal.

10. 3.1 Zero-order-hold (ZOH)
First-order-hold

11. 3.1 Zero-order-hold (ZOH)
First-order-hold and high-order-hold does not bring us much advantages except in some special cases. Therefore, in a control system, usually a ZOH is employed. The device to implement a ZOH is a D/A converter.
If not told, always suppose there is a ZOH in a digital control system.

12. 3.2 Plants with ZOH
Given a discrete-time system, the transfer function of a combination of a ZOH and the plant can be written as GHP(z) in Z-domain. HP, here, means the ZOH and the Plant.
ZOH
GP(s)
GHP(z)

13. 3.2 Plants with ZOH
The continuous time transfer function GHP(s)=G0(s)GP(s)
The discrete time transfer function

14. 3.2 Plants with ZOH Example 1: Given a ZOH and a plant
Determine their Z-domain transfer function.

15. 3.2 Plants with ZOH Example 2: Given a ZOH and a plant
Determine their z-domain transfer function.

16. 3.2 Plants with ZOH Answer: Exercise 1: Given a ZOH and a plant
Determine their z-domain transfer function.
Answer:

17. Assignment 1
You are required to implement a digital PID controller which will enable a control object with a transfer function of
where K=0.2, n=10 rad/s, and =0.3.
to track a) a unit step signal, and b) a unit ramp signal.
1) Simulate this control object and find the responses using Matlab or other packages/computer languages.

18. Assignment 1
2) Choose a suitable sample period for a control loop for G(s) and explain your choice.
3)* Derive the discrete-time system transfer function GHP(z) from G(s).
4) Design a digital PID controller for the discrete-time system, and optimize its parameters with respect to the performance criterion below using steepest descent minimization process .
5) Simulate the resulting closed-loop system and find the responses. Swapping the input signals a) and b), discuss the resulting responses.

19. 3.3 Represent a system in difference equation
For we have
Let A=1-e-T and B=e-T, then the transfer function can be rewritten as

20. 3.3 Represent a system in difference equation
Simulate the above system
1) Parameters and input: A=1-e-T, B=e-T , x(k)=1
2) initial condition: x(k-1)=0, y(k)=y(k-1)=0, k=0
3) Simulation
While k<100 do
y(k)=Ax(k-1)+By(k-1); Calculate output
x(k-1)=x(k); y(k-1)=y(k); x(k)=1; k=k+1; Update data
print k, x(k), y(k); Display step, input & output
End

21. 3.3 Represent a system in difference equation
Let T=1, we have A= and B=0.3679
For a unit step input, the response is
y(k)=0.6321x(k-1) y(k-1) k=
x(k)
y(k)

22. 3.3 Represent a system in difference equation

23. Assignment 1
1)* Simulate this control object and find the responses using Matlab or other packages/computer languages.
Hints: Method 1

24. Assignment 1
Hints: Method 2

25. 3.4 System stability We can rewrite the difference equation as
If A=1 and =0.9, for an impulse input we have k
x(k)
y(k) …
It decreases exponentially, a stable system.

26. 3.4 System stability If K=1 and =1.2, we have k 0 1 2 3 4 ...
x(k)
y(k) …
It increases exponentially, an unstable system.

27. 3.4 System stability If K=1 and = -0.8, we have k 0 1 2 3 4 ...
x(k)
y(k) …
It decays exponentially, and alternates in sign, a gradual stable system.

28. 3.4 System stability
It is clear that the value of  determines the system stability. Why  is so important?
First, let A=1, we have
From the transfer function, we can see that z= is a pole of the system. The pole of the system will determine the nature of the response.

29. 3.4 System stability
For continuous system, we have stable, critical stable and unstable areas in s domain.
Stable area
Unstable area
Critical stable area

30. 3.4 System stability
What is the stable area, critical stable area and unstable area for a discrete system in Z domain ?
Stable area: unit circle
Critical stable: on the unit circle
Unstable area: outside of the unit circle

31. 3.4 System stability As For the critical stable area in s domain s=j,
As  is from 0 to , then the angle will be greater than 2. That is the critical area forms a unit circle in Z domain.

32. 3.4 System stability
If we choose a point from the stable area at S domain, eg s=- a + j, we have
Let eg s=-  + j
The stable area in Z domain is within a unit circle around the origin.

33. 3.4 System stability
Exercise 2: Prove that the unstable area in Z domain is the area outside the unit circle.
Hint: Follow the above procedures.

34. 3.4 System stability
Z domain responses 1

35. 3.5 Closed-loop transfer function
Computer controlled system
Gc(z)
ZOH
GP(s)
R(z)
E(z)
M(z)
GHP(z)
Computer system
C(z)
Plant

36. 3.5 Closed-loop transfer function
Let’s find out the closed-loop transfer function

37. 3.5 Closed-loop transfer function
C(z): output; E(z): error
R(z): input; M(z): controller output
GC(z): controller
GP(z)/G(z): plant transfer function
GHP(z): transfer function of plant + ZOH
T(z): closed-loop transfer function
GC(z)GHP(z): open-loop transfer function
1+ GC(z)GHP(z)=0: characteristic equation

38. 3.6 System block diagram C(z) + G(s) R(s) - H(s) C(s) C(z) + G(s) R(s)

39. 3.6 System block diagram The difference between G(z)H(z) and GH(z)
G(z)H(z)=Z[G(s)]Z[H(s)]
GH(z)=Z[G(s)H(s)]
Usually, G(z)H(z)  GH(z)
G(z)H(z) means they are connected through a sampler. Whereas GH(z) they are connected directly.

40. 3.6 System block diagram
Example: Find the closed-loop transfer function for the system below.
Solution: The open-loop is G1(z)G2H(z).
The forward path is G1(z)G2(z).
G1(s)
H(s) - +
R(s)
C(z)
G2(s)

41. 3.6 System block diagram
G1(s)
H(s) - +
R(s)
C(z)
G2(s)

42. 3.6 System block diagram
*Exercise 3: Find the output for the closed-loop system below.
G(s)
H(s) - +
R(s)
C(s)
C(z)

43. 3.6 System block diagram
*Exercise 4: Find the output for the closed-loop system below.
G1(s)
H(s) - +
R(s)
C(z)
G2(s)

44. Reading
Study book
Module 3: Systems and transfer functions (Please try the problems on page )
Textbook
Chapter 3 : Z-plane analysis of discrete-time control system (pages & ).

45. Tutorial Exercise 1: Given a ZOH and a plant
Determine their z-domain transfer function.

46. Tutorial
You are required to implement a digital PID controller which will enable a control object with a transfer function of
where K=0.2, n=10 rad/s, and =0.3.
to track a) a unit step signal, and b) a unit ramp signal.
1) Simulate this control object and find the responses using Matlab or other packages/computer languages.

47. Tutorial
2) Choose a suitable sample period for a control loop for G(s) and explain your choice.
3) Derive the discrete-time system transfer function GHP(z) from G(s).
4) Design a digital PID controller for the discrete-time system, and optimize its parameters with respect to the performance criterion below using steepest descent minimization process .
5) Simulate the resulting closed-loop system and find the responses. Swapping the input signals a) and b), discuss the resulting responses.

48. Tutorial
2) Choose a suitable sample period for a control loop for G(s) and explain your choice.
Sampling theorem
Input signal
Bandwidth of a system
Bold plots
Applying sampling theorem
Sampling frequency