1. Brief Review of Control Theory
E0397 Lecture.
Some PPT slides based on Columbia Course by authors of the book
Feedback Control of Computing Systems
Hellerstein, Diao, Parekh, Tilbury

2. A Feedback Control System
Error
Data
Reference input: objective
Control input: manipulated to affect output
Disturbance input: other factors that affect the target system
Transduced output: result of manipulation
Components
Target system: what is controlled
Controller: exercises control
Transducer: translates measured outputs
Given target system, transducer, Control theory finds controller that adjusts control input to achieve given measured output in the presence of disturbances.

3. Control System Examples
AC temperature control
Controlled Variable: temperature
Control variables: fan speed, time for which compressor is on?
Car cruise control
Controlled variable: Car speed
Control variable: Angle of pushing of accelerator and brake

4. A Feedback Control System
Road surface,
gradient
Random variation
Embedded
System
that does
cruise control
Desired
Speed
Distance
Travelled,
Time taken
Car
Error
Pedal
Angle
Setting
Speed
calculator
Speed
Actuator:
The entity that implements the control
Sensor: the entity that measures the output
Odometer,
Car clock
Mechanical
Device that
Changes pedal
angle
Feedback!

5. A virtual machine as a FBCS
Other Virtual Machines
Doing some
interfering work
Random variation
Controller:
Process in
Domain 0 that
periodically
changes
Cap value of
Guest
Domain
Desired
Response
time
Response
Time (of
Applications)
Xen VM
(Guest Domain)
With applications
running
Error
CPU
Cap
Value
E.g.
Smoothing
“filter”
Smoothed
Response
time
Actuator:
The entity that implements the control
Sensor: the entity that measures the output
E.g. a “proxy” that
Can measure
Time taken
Xen hypervisor
scheduler
Feedback!

6. A Feedback Control System
Error
Data
Reference input: objective
Control input: manipulated to affect output
Disturbance input: other factors that affect the target system
Transduced output: result of manipulation
Components
Target system: what is controlled
Controller: exercises control
Transducer: translates measured outputs
Given target system, transducer, Control theory finds controller that adjusts control input to achieve given measured output in the presence of disturbances.

7. Control System Goals Reference Tracking Disturbance Rejection
Ensure that measured output “follows” (tracks) a target desired level
Disturbance Rejection
Maintain measured output at a given stable level even in presence of “disturbances”
Optimization
No reference input may be given. Maintain measured output and control input at “optimal” levels
Minimize petrol consumption, maximize speed (not available in reality!!)
Minimize power consumed by CPU, maximize performance

8. Control System: Basic Working
u(k)
x(k)
e(k)
y(k)
“Control Loop” executed every T time units.
y(k): Value of measure at time instant k
u(k): Value of control input at time instant k
r(k): Value of “reference” at time instant k
e(k): Error between reference and measured value
Next value of control input, i.e. u(k+1) to be computed based on feedback: e(k)

9. Determining Change in Control Input
Change in u should depend on relationship between y and u
Relationship should be known
If known, then why feedback? Why not:
y = f(u)
(Car speed as function of accelerator pedal angle)
 u = f-1(y). For a desired reference, just calculate the control input required by using inverse. This is called feed forward
Feed-forward

10. Feedforward (model based)
Problems:
Need accurate model
If model wrong, control input can be totally wrong
Imagine wrong model b/w accel. pedal and car speed
Does not take into account time-dependent behaviour
E.g. how long will system require to attain desired value
If car was at speed S1, we set cruise control to speed S2, if we set pedal angle to f-1(S2), will it immediately go to speed S2?
Cannot take care of “disturbances”
E.g. if environmental conditions change, model may not be applicable.
What if car was climbing a slope? (And angle was measured on flat ground?)
Feedback addresses many of these problems.

11. Feedback Control
Rather than use only some offline model, also take into account the immediate measured effect of the value of your control variable
If car at S1, want to go to S2, S1 < S2 (positive error), pedal must be pushed further down.
We still need some idea of the relationship between “input” and “output”
Pedal should be pushed? Or released? If pushed – by what angle?
(Intuitively, for larger S2 - S1, angle of pushing in must be larger)
Another e.g. if e(k) = r(k) – y(k) is positive
Should u(k) increase or decrease?
If u is CPU frequency, y is response time. If e(k) is positive, u should?
Increase/Decrease by how much?
If increase/decrease too much:
Measured output will keep on oscillating above and below reference: Unstable behavior

12. …Feedback control
Since we want control to work in a timely manner (not only in “steady-state”), relationship should be time dependent
New value of y: y(k+1) will generally depend on y(k) and u(k)
Speed of car at this instant, will depend on what speed was at the beginning of previous interval, and the pedal angle in the previous interval
Similarly, queuing delay at this intv’l will depend on queuing delay of previous instant, and CPU frequency setting of previous interval

13. System Modeling Need a model to capture this relationship over time
Can be first order (depending on 1 previous value), or higher order (more than 1 previous values).
First Order Linear model: y(k+1) = a y(k) + b u(k)
Can be done by “first principles” or “analytical modeling”
E.g. If y is response time of queueing system, u is service time, it may be possible to find an equation to relate y(k+1) to y(k) and u(k)

14. System Modeling If “first principles” difficult: empirical model
Run controlled experiments on target system, record values of y(k), u(k), run regression models to find a and b.
How to do this correctly is a field called “system identification”
Main issues:
Relationship may not be actually be linear

15. Non-linear relationshipsm l {
Number in System
Response Time
Utilization
E.g. number in queueing system vs buffer size
Relationship can be “linearized” in certain regions. Centre of such a region: Operating Point
Operating point:
Operating range
Values of N,K for which model applies.
Offset value
Difference from operating point
Explain relationship between NIS and K using the Markov chain.
Plot does not include time since assuming steady state
No time index since in steady state
Linear Relationship is made between y and u defined as “offsets”

16. Control “Laws”
Once system model is made, need to design “control laws”
Control Laws relate error to new value of control input:
From e(k) determine value of u(k)
Controllers should have good SASO properties:
STABILITY - ACCURACY - SETTLING TIME - OVERSHOOT
Deeply mathematical theory for deriving these laws
Transfer functions, Poles
Estimating SASO properties of control laws
Using this understanding to design good control laws which have good SASO properties
Cannot go into details

17. Properties of Control Systems
Stability
Accuracy
Short settling
Small overshoot
Unstable System

18. Types of Controllers (Control Laws)
Proportional Control Law
u(k) = Kpe(k) (Kp is a constant, called controller gain)
Can be made stable, short settling time, less overshoot – but may be inaccurate
Integral Control Law
u(k) = u(k-1) + KI e(k) (KI is controller gain)
=u(k-2) + KI e(k-1) + KI e(k)
= u(0) +…+ KI * [ e(1) + e(2) + … +e(k) ]
Keeps reacting to previous errors also
Can show that this law will lead to zero error
Has longer settling time

19. Conclusion Padala paper uses integral control law in most places.
It uses empirical methods for generating system model
Slides gave just enough required background.
Control theory is a huge field with lots of books (mainly used in EE, MECH, CHEM)
Read books/papers/if further interested