Lecture 2
- what defines a dynamic model? -
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Office hours
Tu 11:00am-1:00pm
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Review
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<1>
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Democratizing technology
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Democratization processes
1.Tech devices
2.Information
3.Finance/marketing
Changes our range of choices (globalization)
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Image from blog.vortx.com
Democratization of tech devices
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Computerization, miniaturization
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Telecommunication
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Digitization
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Compression
5.9 billions
peoples
use a
cellphone!
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Democratization of information
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Satellite dishes
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TV
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Internet
26 million
members
in North
and Latin
America,
the United
Kingdom
and
Ireland
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Democratization of e-trade/marketing
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Automated loan kiosks
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e-trade
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e-commerce
World’s
largest
online
retailer,
$48.07
billion
revenue
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Analog v. digital
Source: McKinsey Global Institute Report. Big data: The
next frontier for innovation, competition, and productivity.
2011
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Democratization
BIG
DATA
tech devices
information
finance/marketing
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Claim:
Democratization of technology
is the biggest innovation
process of modern society
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<2>
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What this course is about...
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Introduction to dynamic systems theory
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Model + analyze + design of feedback systems
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Basic principles of feedback
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Understand how to use these principles to develop
innovative technologies
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Analysis + Design of Systems
Black box methodologies
Model-based methodologies
Model-Based methodologies
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Use mathematical methods of addressing problems
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Analysis + design based on models
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A prediction of how the system will behave
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Feedback can lead to counter-intuitive behavior
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Help sort out what is going on
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Today
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What are models?
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Define concepts of state, dynamics, inputs and
outputs
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Overview dynamic modeling techniques:
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differential equations
-
difference equations
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World data
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The world
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Research
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Research expenditure
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Research employees
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Research papers
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Research growth
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Population
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Population 1960
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Population 2050?
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Predictions by the U.N.
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Population
Dynamic population model
exponential growth
2-point limit cycles
logistic growth to
a carrying capacity
4–point limit cycles
stable equilibrium dynamics
chaotic dynamics
Time
Time
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Courtesy of Michael Bonsall
What are models?
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A simplified, quantified representation of a
system or process used to answer questions
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How?
via mathematical analysis and simulation
What for?
to assist calculations and predictions
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Models serve as a means of understanding the
mechanism of a process, predicting
relationships and outcomes, and inferring the
existence and role of [information in a system]
Jeff G. Bohn, Thinking Systematically About Policy,
IEEE Technology and Society Magazine. winter 2000/2001
What are dynamic models?
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Dynamical v. statistical modeling
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Statistical modeling focuses on how certain
variable correlate with other variables
➡
What influences what?
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Dynamic modeling focuses on the structure, not
statistical technique.
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Tries to answer the “why” question by describing
the structure of the system
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“Causation across time” occurs because a
variable's derivative has been affected
instantaneously
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courtesy from cortneybrown.com
Democritus (“Father of modern science”)
“I would rather discover one causal
relation than be the King of Persia”
Works
Ethics
...
Mathematics
...
Literature
...
The weather:
what causes precipitation?
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How much will it rain in the morning?
Will it rain in the next 5-10 days?
Different questions
→ different models!
What are the conditions that will cause
it to rain in the morning?
Models don’t have to be perfect
→ feedback provides robustness
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The model you use depends on
the questions you want to
answer
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Terminology
State captures effects of the past
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independent quantities that determine future evolution
Inputs describe external excitation
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extrinsic to the system dynamics
Dynamics describe state evolution
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update rule for system state
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function of current state + inputs
Outputs describe measured quantities
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function of state + inputs (not independent variables)
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often subset of the state
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Modeling Properties
Choice of state is not unique
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many choices of variables can act as the state
Choice of inputs and outputs depend on point of view
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inputs: factors that are external to the model you
are building
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outputs: what variables can you measure:
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what you can sense
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what parts of the component model interact
with other component models
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Types of models
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Ordinary differential equations
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Difference equations
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Discrete event
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Partial differential equations
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Hybrid models
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Cellular automata
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Ordinary differential equations
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Second order model
Example #1: Spring Mass System
u(t)
Applications
! Flexible structures (many apps)
Questions we want to answer
! Suspension systems (eg, “Alice”)
! Molecular and quantum dynamics
q2
q1
m2
m1
k1
k2
How much do masses move
Questions we want to answer
as a frequency of the
! How much do masses move as a
force?
functionforcing
of the forcing
frequency?
•
k3
! What happens if I change the values
• masses?
of the
What happens if I change
fly into
the air if of
I take
thatmasses
! Will Alice
the
values
the
speed bump at 25 mph?
b
Modeling
assumptions
• Will
it fly into the air if I
! Mass, spring, and damper constants
take a speed bump at 30
are fixed and known
satisfy Hooke’s law
! Springskm/h
! Damper is (linear) viscous force,
proportional to velocity
8 Oct 07
R. M. Murray, Caltech CDS
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Second order model
q2
Example #1: Spring Mass System
m1 q̈1 = k2 (q2 q1 ) k1 q1
u(t)
Applications
m2 q̈2 = k3 (u q2 ) k2 (q2
! Flexible structures (many apps)
! Suspension systems (eg, “Alice”)
! Molecular and quantum dynamics
q1
m2
m1
k1
k2
k3
b
Questions we want to answer
! How much do masses move as a
function of the forcing frequency?
! What happens if I change the values
of the masses?
! Will Alice fly into the air if I take that
speed bump at 25 mph?
Modeling assumptions
! Mass, spring, and damper constants
are fixed and known
! Springs satisfy Hooke’s law
! Damper is (linear) viscous force,
proportional to velocity
8 Oct 07
R. M. Murray, Caltech CDS
48
4
q1 )
bq̇2
Ordinary
differential
difference equations
Summary: system modeling
Model = state + inputs + outputs + dynamics
dx
= f (x, u)
dt
y = h(x)
xk+1 = f (xk , uk )
yk+1 = h(xk+1 )
Choice of model depends on questions you want answer!
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