Introduction to Engineering Systems, ESD.00 System Dynamics · Modes of Behavior Ref: Figure 4-1, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world,

Introduction to Engineering Systems, ESD.00

System Dynamics

Lecture 3

Dr. Afreen Siddiqi

From Last Time: Systems ThinkingFrom Last Time: Systems Thinking• “we can’t do just one thing” – things are

interconnected and our actions have numerous effects that we often do not anticipate or realize.

• Many times our policies and efforts aimed towards some objective fail to produce the desired outcomes, rather we often make matters worsematters worse

Ref: Figure 1-4, J. Sterman, Business Dynamics: Systems

• Systems Thinking involves holistic Thinking and Modeling for a complex world, McGraw Hill, 2000

consideration of our actions

Decisions

Environment

Goals

Image by MIT OpenCourseWare.

Dynamic ComplexityDynamic Complexity• Dynamic (changing over time) • Governed by feedback erned by f (actions feedback on themselves) eedback on themselvGov eedback (actions f es) • Nonlinear (effect is rarely proportional to cause, and what happens locally often doesn’t

apply in distant regions) • History‐dependent (taking one road often precludes taking others and determines your

destination, you can’t unscramble an egg) • Adaptive (the capabilities and decision rules of agents in complex systems change over

time) • Counterintuitive (cause and effect are distant in time and space) • Policy resistant (many seemingly obvious solutions to problems fail or actually worsen the

situation) • Ch i d b d ff ( h l i f diff f hCharacterized by trade‐offs (the long run is often different from the shhort‐run response,

due to time delays. High leverage policies often cause worse‐before‐better behavior while low leverage policies often generate transitory improvement before the problem grows worse.

Modes of BehaviorModes of Behavior

Ref: Figure 4-1, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world, McGraw Hill, 2000

Exponential Growth Goal Seeking

Time

Time

Time

Time

Time

Time

S-shaped Growth

Overshoot and CollapseGrowth with OvershootOscillation


e

Exponential GrowthExponential Growth• Arises from positive (self‐reinforcing)

feedback.feedback. • In pure exponential growth the state of

the system doubles in a fixed period of time. • Same amount of time to grow from

1 to 2, and from 1 billion to 2billion!

• Self‐reinforcing feedback can be a declining loop as well (e.g. stock prices)

• Common example: compound interestCommon example: compound interest, population growth


Time

Net increase

rate

State of the

systemR

+

+

State of the system


Exponential Growth: ExamplesExponential Growth: Examples

4004 8008

8060

8088

286

386

486

PentiumK5

K6

K6-IIIK7

P4

PIIPIII

AtomBarton

K8

K10

Itanium 2Core 2 DuoCell

Core 2 Quad

G80POWER6

RV770

Quad-Core Itanium TukwilaGT200

Dual-Core Itanium 2

Itanium 2 with 9 MB cache

Curve shows ‘Moore’s Law’;Transistor count doublingevery two years

CPU Transistor Counts 1971-2008 & Moore’s Law

2,300

10,000

100,000

1000,000

10,000,000

100,000,000

1,000,000,0002,000,000,000

1971 1980 1990 2000 2008

Date of Introduction

Tra

nsi

sto

r co

un

t

Ref: wikipedia Image by MIT OpenCourseWare.

Some Positive Feedbacksunderlying Moore’s Lawunderlying Moore s Law

Ref: Exhibit 4-7 J Sterman Instructor’s Manual 7

Ref: Exhibit 4 7, J. Sterman, Instructor s Manual, Business Dynamics: Systems Thinking and Modeling for a complex world, McGraw Hill, 2000

Applicationsfor chips

Demand forIntel chips

RevenueR&D budget, Investment

in manufactutringcapability

Design and fabricationcapability

Transistors per chips

New uses,New needs

Price premiumfor performance

Chip performancerelative to competitors

Computer performance

Differentiationadvantage

Price premium

Design bootstrap

Design toolcapability

Design andfabricationexperience Learning

by doing

R5

R4

R1

R2

R3

+

+

+

+

+

+

+

++

+

+

+

+

+

+


a e s cou e ac e o

Goal SeekingGoal Seeking• Negative loops seek balance, and

equilibrium, and try to bring the systemequilibrium, and try to bring the systemto a desired state (goal).

• Positive loops reinforce change, while negative loops counteract change oreg oop c a g disturbances.

• Negative loops have a process to comppare desired state to current state and take corrective action.

• Pure exponential decay is characterized byy its half life – the time it takes for half the remaining gap to be eliminated.

Ref: Figure 4-4, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world McGraw Hill 2000Thinking and Modeling for a complex world, McGraw Hill, 2000

State ofthe system

Discrepancy

Correctiveaction

B

Goal (desired stateof system)

Time

Goal

-

+

+

+

State ofthe system


‐

OscillationOscillation• This is the third fundamental mode of

behavior.vior.beha

• It is caused by goal‐seeking behavior, but results from constant ‘over‐shoots’but results from constant over shoots and ‘under‐shoots’

•• The over shoots and under shoots result The over‐shoots and under‐shoots result due to time delays‐ the corrective action continues to execute even when system reaches desired state giving rise to thereaches desired state giving rise to the oscillations.


State ofthe system

Discrepancy

Correctiveaction

B

Goal (desired stateof system)

+

+

+

-

Administrative and decisionmaking delays

Measurement, reportingand perception delays

Action delays

Time

State ofthe system

Goal

Del

ay

Delay

Delay


•

Interpreting BehaviorInterpreting Behavior • Connection between structure and behavior helps in generating hypotheses

• If exponential growth is observed ‐> some reinforcing feedback loop is dominant over the time horizon of behavior

• If oscillations are observed, think of time delays and goal‐seeking behavior.

• Past data shows historical behavior, the future maybe different. Dormantshows historical behavior future maybe different. Dormant Past data , theunderlying structures may emerge in the future and change the ‘mode’

• It is useful to think what future ‘modesmodes’ can be how to plan and manage them It is useful to think what future can be, how to plan and manage them

• Exponential growth gets limited by negative loops kicking in/becoming dominant l tlater on

10

Limits of Causal Loop DiagramsLimits of Causal Loop Diagrams• Causal loop diagrams (CLDs) help

– in capturing mental models, and

– showing interdependencies and

– feedback processes.

• CLDs cannot – capture accumulations (stocks) and flowscapture accumulations (stocks) and flows – help in determining detailed dynamics

Stocks, Flows and Feedback are central concepts in System Dynamics

StockssStock• Stocks are accumulations, aggregations,

summations over time

• Stocks characterize/describe the state of the system

• Stocks change with inflows and outflows

• Stocks provide memory and give inertia by accumulating past inflows; they are the sources of delays.

• Stocks, by accumulating flows, decouple the inflows and outflows of a system and cause variations such as oscillations over time. variations such as oscillations over time.

Stock

Flow

Valve Valve (flow regulator)

Source or Sink

OutflowInflow

Stock



Mathematics of Stocksthema of StocksMa tics • Stock and flow diagramming were based

on a hydraulic metaphor

• Stocks integrate their flows:

• The net flow is rate of change of stock:


Inflow

Outflow

Stock

OutflowInflow

Stock



Stocks and Flows ExamplesStocks and Flows Examples

Ref: Table 6-1, J. Sterman, Business Dynamics: Systems Snapshot Test: Thinking and Modeling for a complex world, McGraw Hill, 2000

Freeze the system in time – things that are measurable in the snapshot are stocksthe snapshot are stocks.

Field Stocks Flows

Mathematics, Physics and Engineering Integrals, States, Statevariables, Stocks

Buffers, Inventories

Levels

Stocks, Balance sheet items

Reactants and reaction productsChemistry

Manufacturing

Economics

Accounting

Derivatives, Rates ofchange, Flows

Reaction Rates

Throughput

Rates

Flows, Cash flow or Incomestatement items


ExampleExample


15

20

10

0

Net

flo

w(u

nits/

seco

nd)


ExampleExample


16

100

50

00 5 10 15 20

Flow

s (u

nits/

tim

e)

Out flowIn flow


Flow RatesFlow Rates• Model systems as networks of stocks

and flows linked by information and flows linked by informationfeedbacks from the stocks to therates.

• Rates can be influenced by stocks, other constants (variables that change very slowly) and exogenous variablesvery slowly) and exogenous variables (variables outside the scope of the model).

• Stocks only change via inflows and outflows.

17

Net rate of change

Stock

Net rate of change

Stock

Exogenous variable

Constant



Auxiliary VariablesAuxiliary Variables

Ref: Figure 8-?, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world, McGraw Hill, 2000

• Auxiliary variables are neither stocks nor flows, but intermediate conceppts for clarityy

• Add enough structure tomake polarities clear

18

Net birth rate

??

+

Food

Population


Aggregation and Boundaries ation and Boundaries ‐ IIAggreg

• Identifyy main stocks in the syystemand then flows that alter them.

• Choose a level of aggregation and b d i f th tboundaries for the system

• Aggregation is number of internal stocks chosen

• Boundaries show how far upstream and downstream (of the flflow)) the system iis modeledh d l d

19

Aggregation and Boundaries ation and Boundaries ‐ IIIIAggreg• One can ‘challenge the clouds’, i.e.

make previous sources or sinks explicit make previous sources or sinks explicit. • We can disaggregate our stocks further

to capture additional dynamics. • Stocks with short ‘residence time’

relative to the modeled time horizon can be lumped togethercan be lumped together

• Level of aggregation depends on purpose of model

• It is better to start simple and then add details.

20

From Structure to BehaviorFrom Structure to Behavior• The underlying structure of the

system defines the time‐based behavior.

• Consider the simplest case: the state of the system is affected by its rate of change.

Ref: Figure 8-1, & 8-2 J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex worldSystems Thinking and Modeling for a complex world, McGraw Hill, 2000

McGraw

Net rate of change

Stock

Net increase

rate

State of the

systemR

+

+

Images by MIT OpenCourseWare.

Population GrowthPopulation Growth

•

•


Note: units of ‘b’, fractional growth rate are 1/time

Consider the population Model:

The mathematical representation of this structure is:

Net birth rate = fractional birth rate * population

Net birth rate

R+

+

Fractional netbirth rate

Population


Phase‐Plots for Exponential GrowthPhase Plots for Exponential Growth• Phase plot is a graph of system state

vs. rate of change of state vs. rate of change of state

• Phase plot of a first‐order, linear positive feedback system is a straightpositive feedback system is a straightline

• If the state of the system is zero If the state of the system is zero, the• the rate of change is also zero

• The origin however is an unstable equilibrium. Ref: Figure 8-3, J. Sterman, Business Dynamics: Systems Thinkingg and Modelingg for a compplex world,, McGraw Hill,, 2000

23

dS/dt = Net Inflow Rate = gS

State of the system (units)

Net

inflow

rat

e (u

nits/

tim

e)

00

Unstable equilibrium

g

1


24

Time Plots Time Plots • Fractional growth rate g =

0.7%/time unit 0.7%/time unit

• Initial state state = 1.

• State doubles every 100 time units

• Every time state of the systemdoubles, so too does the absoluterate of increase

Ref: Figure 8-4, J. Sterman, Business Dynamics: Systems Thinkingg and Modelingg for a compplex world,, McGraw Hill, 2000,

128 256 512 1024

10

8

6

4

2

00

t = 1000

t = 900

t = 800

t = 700

Net

inflow

(units/

tim

e)

State of system (units)

1024

512

256

128

0

10.24

5.12

2.56

1.28

00 200 400 600 800 1000

Time

Net in

flow

(units/tim

e)

Sta

te o

f th

e sy

stem

(units)

Behavior

Net inflow(right scale)

State of the system(left scale)

Images by MIT OpenCourseWare.

Rule of 70Rule of 70

• Exponential growth is one of the most powerful processes. • The rate of increase grows as the state of the system grows. • It has the remarkable property that the state of the system doubles in fixed

period of time. • If the doubling time is say 100 time units, it will take 100 units to go from 2

to 4, and another 100 units to go from 1000 to 2000 and so on. • To find doubling time:

Negative Feedback andExponential DecayExponential Decay

• First‐order linear neggative feedback systems generate exponential decay

• The net outflow is proportional to theThe net outflow is proportional to thesize of the stock

•• The solution is given by: S(t) = S e ‐dt The solution is given by: S(t) = So e dt

Net Inflow = -Net Outflow = -d*S • Examples: d: fractional decay rate [1/time]d: fractional decay rate [1/time]

Ref: Figure 8-6, J. Sterman, Business Dynamics: Systems Reciprocal of d is average lifetime Thinking and Modeling for a complex world, McGraw Hill, 2000 units in stock.

Net outflow rate

B

Fractional decayrate d

++

S state of thesystem


Phase Plot for Exponential DecayPlot for Exponential yPhase Deca

• IIn thhe phhase‐pllot, thhe net rate off change is a straight line with negative slope

• The origin is a stable equilibrium, a minor perturbation in state S increases ththe ddecay rate tto bring systtem bb k t ack tot b i zero – deviations from the equilibriumare self‐correcting

• The goal in exponential decay is implicit and equal to zero Ref: Figure 8-7, J. Sterman, Business Dynamics: Systems

Thinking and Modeling for a complex world McGraw Hill 2000Thinking and Modeling for a complex world, McGraw Hill, 2000

Stable equilibrium


Net

inflow

rat

e (u

nits/

tim

e)

Net Inflow Rate = - Net Outflow Rate = -dS

0

1

-d


Negative Feedback with Explicit GoalsNegative Feedback with Explicit Goals• In general, negative loops have

non‐zero goalls

• Examples:

• The corrective action determining net flow to the state of the system is : Net Inflow = f (S, S*)

• Simpplest formulation is: Net Inflow = Discrepancy/adjustment time = (S*‐S)/AT

AT: adjustment time is also known as time constant for the loop


Net inflow rate

Discrepancy (S* - S)

B

S* desired stateof the system

AT adjustment time

-

-

+

+

General Structure

dS/dt

S state of thesystem


Phase Plot for Negative Feedback with Non‐Zero Goal with Non Zero Goal

• In the phase‐plot, the net rate of change is a straight line with slope ‐1/AT

• The behavior of the negative loop with an explicit goal is also exponential decay, in which the state reaches equilibrium when S=S*

• If the initial state is less than the desired state, the net inflow is positive and the state increases (at a diminishing rate) until S=S*. If the initial state is greater than S*, the net inflow is negative and the state falls until it reaches S* Ref: Figure 8-10, J. Sterman, Business Dynamics: Systems

Thinking and Modeling for a complex world McGraw Hill 2000Thinking and Modeling for a complex world, McGraw Hill, 2000

Net

inflow

rat

e (u

nits/

tim

e)

1-1/AT

S*

Stable equilibrium


Net Inflow Rate = -Net Outflow Rate = (S* - S)/AT

0


Half‐LivesHalf Lives• Exponential decay cuts quantity remaining in half in fixed period of time

• The ‘half‐life’ is calculated in similar way as doubling time. • The system state as a function of time is given by:

State of Initial Gapsystem

S(t) = SS* − (SS* − SS0 )e− t /τ S(t) ( )e

• Thhe exponenti lial term ddecays ffrom 1 to zero as t tends to infinity.

• Half life is given by value of time th :

Gap Remaining

Desired State

• ‐4τ e‐4 = 0.02 1‐e‐4 = 0.98

Time Constants and Settlingg Time

• For a first order, linear system with negative feedback the system negative feedback, the system reaches 63% of its steady‐state value in one time constant, and reaches 98% of its steady state value in 4 98% of its steady state value in 4 time constants.

• The steady‐state is not reached The steady state is not reached technically in finite time because the rate of adjustment keeps falling as the desired state is approached. the desired state is approached.

Time Fraction of Initial Gap Remaining

Fraction of Initial Gap Corrected

0 0 1 1 1 00 e ‐0 = 1 1‐1 = 0

τ e ‐1 = 0.37 1‐e ‐1 = 0.63

2τ e ‐2 = 0.14 1‐e ‐2 = 0.87

3τ e ‐3 = 0.05 1‐e ‐3 = 0.95

5τ e ‐5 = 0.007 1‐e ‐5 = 0.993


1.0

0.8

0.6

0.4

0.2

00 1AT 2AT 3AT

Frac

tion o

f in

itia

l gap

rem

ainin

g

Time (multiples of AT)

1 -

exp

(-3/A

T)

1 -

exp

(-2/A

T)

1 -

exp

(-1/A

T)

exp(-1/AT)

exp(-2/AT)exp(-3/AT)


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Documents

Introduction to Engineering Systems, ESD.00 System Dynamics · Modes of Behavior Ref: Figure 4-1, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world,