1. Feedback Control of Computing Systems M3: Systems and Transfer Functions
Joseph L. Hellerstein
IBM Thomas J Watson Research Center, NY
September 21, 2004

2. This module focuses on systems: components that transform signals
Motivating Example
This module focuses on systems: components that transform signals
Controller
Notes
Server
Sensor - +
The problem
Want to find y(k) in terms of KI so can design control system that is stable, accurate, settles quickly, and has small overshoot.
But this is difficult to do with ARX models.
The Solution
Use a different representation

3. M3:
Lecture

4. Agenda Motivation and definition Examples of transfer functions
Interpretations of transfer functions
Transfer function for systems in series
Steady state gain
Poles
Settling times
Reference: “Feedback Control of Computer Systems”, Chapter 3.

5. Motivation and Definition
ARX model relates u(k) to y(k)
Transfer function expresses this relationship in the z domain
y(k) = (a)y(k-1)
+(b)u(k-1)
u(k)
y(k)
G(z)
Y(z)
U(z)
A transfer function is specified in terms of its input and output.
Intuition for G(z): Effect of “kicking the system”

6. Constant Transfer Function3
y(k) = au(k)
u(k)
y(k)
U(z)
Y(z) a 3

7. Time-Delay Transfer Function
z-1
y(k) = u(k-1)
u(k)
y(k)
U(z)
Y(z)
z-1
z-1

8. Combining Simple Transfer Functions
3z-1
y(k) = au(k-1)
u(k)
y(k)
U(z)
Y(z)
az-1
3z-1

9. Additional Terms in T.F. U(z) Y(z) y(k) = au(k-1)+bu(k-2) y(k) u(k)
3z-1+2z-2
U(z)
Y(z)
y(k) = au(k-1)+bu(k-2)
u(k)
y(k)
U(z)
Y(z)
az-1+bz-2

10. Geometric Sum of T.F. U(z) Y(z) y(k) = u(k)+au(k-1)+a2u(k-2)+… y(k)
z/(z-0.5)
U(z)
Y(z)
y(k) = u(k)+au(k-1)+a2u(k-2)+…
u(k)
y(k)
U(z)
Y(z)
1 + az-1 + a2z-2 + …
z/(z-a)
Y(z)
U(z)

11. T.F. Interpretation 1 U(z) Y(z) z/(z-0.5)
Signal generated by an impulse input
Example:
G(z) is the same as the y(k) signal

12. T.F. Interpretation 2 U(z) Y(z)
Sum of effects at different time delays 1
0.5z-1
0.25z-2
0.125z-3
z/(z-0.5)
Y(z)
U(z)
Example:

13. Complicated Transfer Functions
Decompose into a sum of geometrics
Y(z)
U(z)
Partial fraction expansion allows rational polynomials to be decomposed into a sum of geometrics
Poles of the original polynomial are the poles of the geometrics

14. Converting ARX Models into T.F.
Approach
Replace u(k+n) by znU(z) and replace y(k+n) by znY(z)
Express Y(z) as a function of U(z)
Divide both sides by U(z)
Example:

15. Converting T.F. into ARX Model
Approach
Express transfer function as Y(z)=G(z)U(z)
Multiply both sides by the denominator of G(z)
Convert Y(z) and U(z) into the time domain
Example 1:
Example 2:

16. Transfer Functions In Series?
u(k)
G(z)
W(z)
U(z)
H(z)
Y(z)
G(z)H(z)
T.F. provide an easy way to analyze the behavior of complex structures.

17. M3:
Labs Part 1

18. Lab 1: T.F. for Notes Server + Notes Sensor
u(k)
w(k)
y(k)
Find the transfer function H(z) from u(k) to y(k)
1. Notes Server T.F.
Compare simulation of
Server + Sensor
H(z)
2. Notes Sensor T.F.
3. Compute the product

19. M3:
Lecture Continued

20. Poles of a Transfer Function
Poles: Values of z for which the denominator is 0.
Example:
Poles: 0.3, 0.2
Poles
Determine stability
Major effect on settling time, overshoot
Larger |a|
Slower convergence
|a|>1
Does not converge
a<0
Oscillates

21. Almost All You’ll Ever Need to Know About Poles
Explain output signals shown in terms of a unit step input

22. Steady State Gain (ssg) of a Transfer Function
from final value thm
Definition and result: Steady state divided by steady state input.
ssg of G(z) is
U(z)
Y(z)
G(z)
Example:

23. Settling Time (ks) of a System
from geometric
Definition and result: Time until an input signal is within 2% of its steady state value
G(z)
Y(z)
z/(z-1)
(unit step)
Examples:
do more time steps

24. Summary of Key Results Interpretation of transfer functions
Signal created by an impulse (impulse response)
Sum of time delayed effects
Encoding of a ARX model
Transfer functions in series
End-to-end T.F. is the product of the individual T.F.
Steady state gain
Magnitude of effect on the output for a unit change in input
Settling time
Time until steady state is reached after applying a unit input

25. Key Results for LTI Systems (LTI = Linear Time Invariant)+
A(z)
C(z)
B(z)
Adding signals:
If {a(k)} and {b(k)} are signals, then {c(k)=a(k)+b(k)} has Z-Transform A(z)+B(z).
Key Results for LTI Systems (LTI = Linear Time Invariant)
G(z)
Y(z)
U(z)
G(z)
W(z)
U(z)
H(z)
Y(z)
G(z)H(z)
is equivalent to
Transfer functions in series
Stable if |a|<1, where a is the largest pole of G(z)
ssg of G(z) is

26. M3:
Labs Part 2

27. Why? Causality Constraint
The degree of z in the numerator of a transfer function cannot exceed
that of the denominator
Why?
Hint: Find the ARX model for a T.F. that violates the causality constraint.

28. Settling Time Questions
Notes
Server
Notes
Sensor
u(k)
w(k)
y(k)
1. How do the sum of the settling times of the Notes Server and the Notes Sensor
compare to the settling time of these two systems in series? Why?
2. How does the settling time of the Notes Sensor change if the coefficient of w(k) is changed to 0.9? What if the coefficient of y(k) is changed to 0.4?
3. How does the settling time of the combined (series) system change if the coefficient of w(k) in the Notes Sensor changes to 0.7?