Brief Review of Control Theory

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
Distance Travelled, Time taken
Desired Speed
Car
Pedal Angle Setting
Error
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
Response Time (of Applications)
Desired Response time
Xen VM (Guest Domain) With applications running
CPU Cap Value
Error
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
• 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
r(k)
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(k1) 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 lt 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(k1) 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 intvl 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(k1) 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(k1) 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 relationships
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

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
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