1. Feedback Control of Computing Systems M4: Analyzing Composed Systems
Joseph L. Hellerstein
IBM Thomas J Watson Research Center, NY
September 21, 2004

2. Key Results for LTI Systems+
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
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

3. Motivating Example Controller Notes Server Sensor - + The problem
This module addresses the analysis of composed systems: steady state gain, stability, settling time
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

4. M4:
Lecture

5. Outline Canonical feedback loop and its transfer functions
Generalized canonical feedback loop
Complex structures
Reference: “Feedback Control of Computer Systems”, Chapter 4.

6. Block Diagram Basics D(z) Controller Target System R(z) E(z) U(z) +
V(z)
Y(z) +
K(z)
G(z) - +
W(z)
H(z)
Transducer
Permitted operations
Summing signals
Cascading systems
Functional block: component of the system
Arrow: signal
Summation point: addition of signals
Branching point: signal with multiple destinations

7. Canonical Feedback Loop
Noise
Input
Disturbance
Input
D(z)
N(z)
Reference
Input
Measured
Output
Controller
Target System
R(z)
E(z)
U(z) +
V(z)
Y(z) +
G(z)
T(z) +
K(z) + - +
W(z)
H(z)
May also want transfer functions to the control error, which adds another 3.
Many transfer functions are potentially of interest.
For each, we are interested in stability, ssg, settling time.
Transducer
Want to analyze characteristics of the entire system, like its stability, settling time, and accuracy (ability to achieve the reference input).
It’s all done with transfer functions!

8. Canonical Feedback Loop – Reference Input
FR(z)
D(z)=0
N(z)=0
Reference
Input
Measured
Output
Controller
Target System
R(z)
E(z)
U(z) +
V(z)
Y(z) +
T(z) +
K(z)
G(z) - + +
Transducer
W(z)
H(z)
May also want transfer functions to the control error, which adds another 3.
Many transfer functions are potentially of interest.
For each, we are interested in stability, ssg, settling time.
View the dark rectangle as a large transfer function FR(z) with input R(z) and output T(z).
System is stable if largest pole of FR(z) has an absolute value that is less than 1
System is accurate if FR(1)=1
System’s settling time is determined by the largest pole of FR(z)

9. Canonical Feedback Loop Has Many T.F.
Noise
Input
Disturbance
Input
D(z)
N(z)
Reference
Input
Measured
Output
Controller
Target System
R(z)
E(z)
U(z) +
V(z)
Y(z) + +
K(z)
G(z)
T(z) + + -
W(z)
H(z)
A transfer function is specified in terms of its input and output.
May also want transfer functions to the control error, which adds another 3.
Many transfer functions are potentially of interest.
For each, we are interested in stability, ssg, settling time.
Assess
Accuracy
Assess disturbance robustness
Transducer
Assess noise robustness
Transfer function from the reference input to the measured output
Transfer function from the disturbance input to the measured output
Transfer function from the noise input to the measured output

10. Transfer functions in series+
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).
Results Used to Find T.F.
G(z)
W(z)
U(z)
H(z)
Y(z)
G(z)H(z)
is equivalent to
Transfer functions in series
We’ll use these result

11. The only non-zero input is R(z).
Computing FR(z)
The only non-zero input is R(z).
R(z)
E(z)
U(z)
T(z) +
K(z)
G(z) -
Simplified B.D. since D(z)=0=N(z)
W(z)
H(z)
A set of equations relates R(z) to T(z) based on our previous results
This is a transfer function and so can be analyzed just like any other.
W(z) = H(z)T(z) by the definition of a transfer function.
E(z) = R(z)-W(z) since this is an addition of signals.
T(z) = E(z)K(z)G(z) since K(z) and G(z) are in series.
T(z) = (R(z)-H(z)T(z))K(z)G(z) by substitution.

12. Computing FD(z)=T(z)/D(z)
The only non-zero input is D(z).
D(z)
N(z)=0
R(z)=0
E(z)
U(z) +
V(z)
Y(z) + +
K(z)
G(z)
T(z) + + -
W(z)
H(z)
W(z) = H(z)T(z) by the definition of a transfer function.
E(z) = -W(z) since this is an addition of signals.
T(z) = E(z)K(z)G(z) + D(z)G(z).
Y(z) = (-H(z)T(z))K(z)G(z) + D(z)G(z) by substitution.

13. Stable if |a|<1, where a is the largest pole of G(z)
Results Used Next
G(z)
Y(z)
U(z)
Stable if |a|<1, where a is the largest pole of G(z)
ssg of G(z) is

14. Properties of Canonical Loop
D(z)
N(z)
Properties of Canonical Loop
R(z)
U(z) +
V(z)
Y(z) +
E(z) +
K(z)
G(z)
T(z) - + +
W(z)
H(z)
Reference to Output
Disturbance to Output
Noise to Output?
As what the T.F.
What can we say about the stability and settling times of these three transfer functions?
They are the same!
When is the system accurate in the sense that T(z)=R(z)?
FR(1)=1
When is the system robust to disturbances and noise?
FD(1)=0= FN(1)

15. Predicting Measured Outputs
D(z)
N(z)
Predicting Measured Outputs
R(z)
E(z)
U(z) +
V(z)
Y(z) + +
K(z)
G(z)
T(z) + + -
W(z)
H(z)
What is T(z) if R(z), D(z), and N(z) are all non-zero?
As what the T.F.

16. Generalized Canonical Feedback Loop
In-1(z)
In+1(z)
I1(z) +
G1(z)
Gn(z)
T(z) + + - +
Hm(z)
H1(z)
What is ?
How can this be simplified in terms of the series of transducers? (Make into a single transducer.)

17. Nested Structures R(z) - T(z) + K1(z) + K2(z) G(z) - Find: H(z) K(z) -
is equivalent to
Approach 1: Solve directly.
Motivate structure: control with different time constants.
Approach 2:
a. Transform into a canonical control loop.
b. Apply result for T.F. of canonical control loop.

18. Summary Control systems have many transfer functions of interest
FR(z) – From reference input to measured output
FD(z) – From distrubance input to measured output
FN(z) – From noise input to measured output
Key properties of system are indicated by the transfer functions
Stability, accuracy, settling time, robustness to disturbance, robustness to noise
Find transfer functions by
Solving a (simple) set of equations
Transforming into a canonical control loop and solving

19. Key Results for LTI Systems+
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
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

20. M4:
Labs

21. Analytic Solution to First Lab Exercise
Block Diagram
ARX Models
Reference
RIS
MaxUsers
Actual
RIS
Control error: e(k)=r*-y(k)
e(k) +
u(k)
y(k)
P controller: u(k)=Ke(k)
Controller
Notes
Server r*
System model: y(k)=(0.43)y(k-1)
+(0.47)u(k-1) -
Assignment
Find FR(z)
Find the steady state error K, K = .1, 1, 3
Find the poles for these same values of K. For each, determine the settling times. How do they compare with simulation results?
Key insights for this system
Accuracy requires a larger K
Stability & short settling times require a smaller K
Overshoot is less (or non-existent for smaller K)
Why are these important properties?

22. Compare with simulations in M1.
Solution
ARX Models
Control error: e(k)=r*-y(k)
P controller: u(k)=Ke(k)
System model: y(k)=(0.43)y(k-1)+(0.47)u(k-1)
e(k) +
Controller
Notes
Server r*
u(k)
y(k) -
Compare with simulations in M1.

23. Homework: Due 9/23/2004 Controller Notes Server Notes Sensor -
D(z)
w(k+1)=0.43w(k)+0.47v(k)
u(k)=u(k-1)+KIe(k)
Controller +
V(z)
Notes
Server
Notes
Sensor - +
Assignment
Find FD(z)
Assess the robustness of this control system to noise.
Use FD(z) to find the KI that produces the smallest settling time. (Hint: plot the largest pole versus KI.
Verify the results of analysis using simulations.