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Lecture 2- what defines a dynamic model? -
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Office hours
Mon: 8:00a.m. - 9:00a.m.
Tue: 5:00p.m. - 6:00p.m.
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Review
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<1>
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it takes 2% of Americans to feed us all,
and 5% to make everything we need
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it takes 2% of Americans to feed us all,
and 5% to make everything we needeverything else will be service and
information technology, ...
T. J. Rodgers
founder and CEO ofCypress Semiconductor
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A knowledge-based economy
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Where do I fit into the globalcompetition + opportunities?
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El sueño Colombiano
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Local challenges?
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?Technology = Part of thesolution
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Technology is anything invented after you were born
Alan Kay
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Technology is anything thatdoesn't work yet
Danny Hillis
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BIG
DATA
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Class Project:develop a dynamic model that
resembles real-world data
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About this course
• Feedback principles for any educated engineers’ background
• Significantly broader than the traditionalintroductory course
• Challenges in an Information-Rich world
• Dynamic systems concepts and tools fundamentalto a broader range of non-traditional audiences
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About this course
• Feedback principles for any educated engineers’ background
• Significantly broader than the traditionalintroductory course
• Challenges in an Information-Rich world
• Dynamic systems concepts and tools fundamentalto a broader range of non-traditional audiences
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Feedback
Sensing + Computation + Actuation
Principles Design of Dynamics + Robustness to Uncertainties
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Analysis + Design of Systems
Black box methodologies
Model-based methodologies
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Model-Based methodolo ies
• Use mathematical methods of addressing problems
• Analysis + design based on models
• A prediction of how the system will behave
• Feedback can lead to counter-intuitive behavior
• Help sort out what is going on
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Toda
• What are models?
•
Define concepts of state, dynamics, inputs andoutputs
• Overview dynamic modeling techniques:
- differential equations
- difference equations
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Visualizing data
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The World
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Research
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Research ex enditure
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Research em lo ees
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Research a ers
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Research rowth
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Population
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Population 1960
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Population 2050
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Population 2300
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Courtesy of Michael Bonsall
exponential growth
Time
logistic growth toa carrying capacity
stable equilibrium dynamics
2-point limit cycles
4–point limit cycles
chaotic dynamics
Time
P o p u l a t i o n
Modelin o ulation d namics
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What are models?
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A simplified, quantified representation of asystem 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, predictingrelationships 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
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What are dynamic models?
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D i l t ti ti l d li
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• Statistical modeling focuses on how certainvariable correlate with other variables
➡ shows influence
• Dynamic modeling focuses on the structure
D namical v. statistical modelin
courtesy from cortneybrown.com39
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D i l t ti ti l d li
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• Statistical modeling focuses on how certainvariable correlate with other variables
➡ shows influence
• Dynamic modeling focuses on the structure
D namical v. statistical modelin
independentvariables
dependent
variables
courtesy from cortneybrown.com
s t a t i s t i c a l m o d e l i n g
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D i l t ti ti l d li
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• Statistical modeling focuses on how certainvariable correlate with other variables
➡ shows influence
• Dynamic modeling focuses on the structure
D namical v. statistical modelin
independentvariables
dependent
variables
courtesy from cortneybrown.com
s t a t i s t i c a l m o d e l i n g
d y n a mi c m
o d e l i n g
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Wh t i d i d li ?
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What is d namic modelin ?
• Think dynamically, not just what influences what
• Not a statistical technique
• Tries to answer the “why” question by describingthe structure of the system
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e.g.
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How much will it rain in the morning?
How much will it rain tomorrow?
Will it rain in the next 5-10 days?
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How much will it rain in the morning?
How much will it rain tomorrow?
Will it rain in the next 5-10 days?
Will it rain enough this season?
Different questions→ different models!
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How much will it rain in the morning?
How much will it rain tomorrow?
Will it rain in the next 5-10 days?
Will it rain enough this season?
Different questions→ different models!
Models don’t have to be perfect→ feedback provides robustness
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The model you use depends on
the questions you want toanswer
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Modelin terminolo
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Modelin terminolo
State captures effects of the past
• independent quantities that determine future evolution
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Modelin terminolo
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Modelin terminolo
State captures effects of the past
• independent quantities that determine future evolution
Inputs describe external excitation
• extrinsic to the system dynamics
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Modelin terminolo
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Modelin terminolo
State captures effects of the past
• independent quantities that determine future evolution
Inputs describe external excitation
• extrinsic to the system dynamics
Dynamics describe state evolution
• update rule for system state
• function of current state + inputs
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Modelin terminolo
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Modelin terminolo
State captures effects of the past
• independent quantities that determine future evolution
Inputs describe external excitation
• extrinsic to the system dynamics
Dynamics describe state evolution
• update rule for system state
• function of current state + inputs
Outputs describe measured quantities
• function of state + inputs (not independent variables)
• often subset of the state
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Modelin Pro erties
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Modelin Pro erties
Choice of state is not unique
• many choices of variables can act as the state
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Modelin Pro erties
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Modelin Pro erties
Choice of state is not unique
• many choices of variables can act as the state
Choice of inputs and outputs depend on point of view
• inputs: factors that are external to the model youare building
• outputs: what variables can you measure:
- what you can sense
- what parts of the component model interact
with other component models
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T es of models
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T es of models
• Ordinary differential equations
• Difference equations
• Discrete event
• Partial differential equations
• Hybrid models
• Cellular automata
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Ordinary differential equations
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Second order model
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b
k 3
m1m2
q1
u(t)
q2
k 2k 1
Second order model
Questions we want to answer
• How much do masses moveas a frequency of theforcing force?
• What happens if I changethe values of the masses
• Will it fly into the air if Itake a speed bump at 30
km/h
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Second order model
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m1q̈1 = k2(q2 q1) k1q1
m2q̈2 = k3(u q2) k2(q2 q1) bq̇2
b
k 3
m1m2
q1
u(t)
q2
k 2k 1
Second order model
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Ordinary differential difference equations
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Questions we want to answer:
• Given the current population of rabbits and foxes, what will it benext year
• If we hunt down lots of foxes in a given year what will the effecton the rabbit and fox population be?
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H k : number of rabbits in period k
Lk : number of foxes in period k
uk : amount of rabbit foodyk : number of rabbits and foxes
state
inputs + outputs
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H k : number of rabbits in period k
Lk : number of foxes in period k
uk : amount of rabbit foodyk : number of rabbits and foxes
state
inputs + outputs
H k+1 = H k + br(u)H k aLkH k
Lk+1 = Lkdf Lk + aLkH k,dynamics
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H k : number of rabbits in period k
Lk : number of foxes in period k
uk : amount of rabbit foodyk : number of rabbits and foxes
state
inputs + outputs
H k+1 = H k + br(u)H k aLkH k
Lk+1 = Lkdf Lk + aLkH k,dynamics
br(u) : rabbit birth rate (per year,
depends no food supply)
df : fox death rate (per year)
a : interaction term
parameters
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Po ulation d namics
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Courtesy of Michael Bonsall
exponential growth
Time
logistic growth toa carrying capacity
stable equilibrium dynamics
2-point limit cycles
4–point limit cycles
chaotic dynamics
Time
P o p u l a t i o n
Po ulation d namics
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Po ulation d namics
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Courtesy of Michael Bonsall
exponential growth
Time
logistic growth toa carrying capacity
stable equilibrium dynamics
2-point limit cycles
4–point limit cycles
chaotic dynamics
Time
Po ulation d namics
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Matlab simulation (see handout)
! Discrete time model, “simulated”through repeated addition
Comparison with data
1850 1870 1890 1910 19300
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100
150
200
250
300
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Summar : s stem modelin
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Su a s ste ode
Model = state + inputs + outputs + dynamics
Choice of model depends on questions you want answer!