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Lecture 2 - what defines a dynamic model? - 1 Tuesday, August 2, 11
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Lecture 2- what defines a dynamic model? -

1

Tuesday, August 2, 11

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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

3

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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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<2>

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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

18

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?

Will it rain enough this season?Tuesday, August 2, 11

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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

50

100

150

200

250

300

350

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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!


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