1. Review last lectures

2. centrifugal (fly-ball) governor
1788 Picture shows an operation principle of the fly-ball (centrifugal) speed governor developed by James Watt.

3. simple feedback system
heat transfer Qout
desired temperature _
room temperature
Qin S
thermostat
switch
air con
office room +

4. closed-loop (feedback) system
error or actuating signal
input or reference
disturbance
summing junction or comparator
plant
output or controlled variable
control signal + S
input filter (transducer)
controller
actuator
process _
sensor or output transducer
sensor noise

5. closed-loop system advantages disadvantages high accuracy
not sensitive on disturbance
controllable transient response
controllable steady state error
more complex
more expensive
possibility of instability
recalibration needed
need for output measurement

6. open-loop system input or reference disturbance plant
output or controlled variable
control signal
input filter (transducer)
controller
actuator
process

7. open-loop system advantages disadvantages simple construction
ease of maintenance
less expensive
no stability problem
no need for output measurement
disturbances cause errors
changes in calibration cause errors
output may differ from what is desired
recalibration needed

8. Example 1: Liquid Level System
(input flow)
Goal: Design the input valve control to maintain a constant height regardless of the setting of the output valve
Input valve control
float
This can be applied to computer systems as a fluid model of queues: inflow is in work/sec; volume represents work in system; R relates to service rate.
(resistance)
(height)
(output flow)
(volume)
Output
valve

9. Example 2: Admission Control
Goal: Design the controller to maintain a constant queue length regardless of the workload
Users
Administrator
Controller
RPCs
Sensor
Server
Reference
value
Queue
Length
Tuning
control
Log
Jlh: May want a slide that explains how this controller works?

10. Why Control Theory Systematic approach to analysis and design
Transient response
Consider sampling times, control frequency
Taxonomy of basic controls
Select controller based on desired characteristics
Predict system response to some input
Speed of response (e.g., adjust to workload changes)
Oscillations (variability)
Approaches to assessing stability and limit cycles
Jlh: Some edits on this slide

12. Controller Design Methodology
Start
System Modeling
Controller Design
Block diagram construction
Controller Evaluation
Transfer function formulation and validation
Objective achieved? Y
Stop
Model Ok? N Y N

13. Control System Goals Regulation Tracking Optimization
thermostat, target service levels
Tracking
robot movement, adjust TCP window to network bandwidth
Optimization
best mix of chemicals, minimize response times

14. Approaches to System Modelling
First Principles
Based on known laws
Physics, Queuing theory
Difficult to do for complex systems
Experimental (System ID)
Statistical/data-driven models
Requires data
Is there a good “training set”?

23. Laplace transforms
The Laplace transform of a signal f(t) is defined as
The Laplace transform is an integral transform that changes a function of t to a function of a complex variable s = s + jw
The inverse Laplace transform changes the function of s back to a function of t

24. Laplace transforms of basic functions
f(t)
F(s)
Unit impulse
d (t)
Unit step
u (t)
Exponential
e−at
Sine wave
sin(t)
Cosine wave
cos(t)
Polynomial tn
e−at x(t)
Note:
f(t) = 0, t < 0

25. Example
f (t)
Laplace transform:
signal 1 t w w s s

26. Properties of Laplace transforms
Linear operator: if
and
then
for any two signals f1(t) and f2(t) and any two constants a1 and a2
Time delay:

27. Properties of Laplace transforms cont.
Laplace transforms of derivatives: if
then

28. Properties of Laplace transforms cont.
Laplace transform of integrals:
The Laplace transform changes differential equations in t into arithmetic equations in s

29. Laplace Transform Properties

30. Using Laplace transforms to solve ODEs
The Laplace transform can be used to solve differential equations
Method:
Transform the differential equation into the ‘Laplace domain’ (equation in t → equation in s)
Rearrange to get the solution
Transform the solution back from the Laplace domain to the time domain (signal in s → signal in t)
Usually the Laplace transform (step 1) and the inverse transform (step 3) are done using a Table of Laplace transforms

31. Example m = 1 kg k = 2 N/m b = 3 Ns/m
Use Laplace transforms to find the unforced response of a spring-mass-damper with initial conditions x
m = 1 kg
k = 2 N/m
b = 3 Ns/m k m f b x
Equation of motion m f
Free body diagram

32. Example – solution
Take Laplace transform of both sides of equation of motion:
Equation of motion in Laplace domain is

33. The system can be in motion if
Rearrange:
External force
Initial conditions
The system can be in motion if
An external force is applied
The initial conditions are not an equilibrium state (not zero)
Apply initial conditions:

34. Partial fraction expansion:
Use tables to find inverse Laplace transform
System response (in time domain) is
x(t) x0 t

35. Partial fraction expansion
A partial fraction expansion can be used to find the inverse transform of
This can be expanded as
Note that So

36. Example - Partial fraction expansion
Find the partial fraction expansion of
This can be expanded as So

37. Other examples 1. 2. 3.

38. Insights from Laplace Transforms
What the Laplace Transform says about f(t)
Value of f(0)
Initial value theorem
Does f(t) converge to a finite value?
Poles of F(s)
Does f(t) oscillate?
Value of f(t) at steady state (if it converges)
Limiting value of F(s) as s ---> 0

42. Transfer Function Definition
H(s) = Y(s) / X(s)
Relates the output of a linear system (or component) to its input
Describes how a linear system responds to an impulse
All linear operations allowed
Scaling, addition, multiplication
X(s)
H(s)
Y(s)

43. Block Diagrams
Pictorially expresses flows and relationships between elements in system
Blocks may recursively be systems
Rules
Cascaded (non-loading) elements: convolution
Summation and difference elements
Can simplify

48. Block Diagram of System
Disturbance
Reference Value + S
Controller S
Plant –
Transducer

49. Combining Blocks
Reference Value + S
Combined Block –
Transducer

50. Key Transfer Functions
Reference + S
Controller
Plant –
Jlh: Can we make the diagram bigger and the title smaller?
Transducer

58. Class Home page: http://saba.kntu.ac.ir/eecd/People/aliyari/