System Dynamics Shahram Shadrokh Closing the Loop: Dynamics of Simple Structures.

System Dynamics

Shahram Shadrokh

Closing the Loop: Dynamics of Simple Structures

FIRST-ORDER SYSTEMS

The simplest system that can generate exponential growth, and goal-seeking behaviors is the first-order, linear feedback system.

The order of a dynamic system or loop is the number of state variables, or stocks, it contains.

A first-order system contains only one stock.

Linear systems are systems in which the rate equations are linear combinations of the state variables and any exogenous inputs.

The term "linear" has a precise meaning in dynamics: in a linear system the rate equations (the net inflows to the stocks) are always a weighted sum of the state variables (and any exogenous variables, denoted Uj):

dS/dt = Net Inflow = a1Sl + a2S2 + . . . + anSn + b1Ul + b2U2 + . . . + bmUm

POSITIVE FEEDBACK AND EXPONENTIAL GROWTH

In a first order system, there is only one state variable (stock), denoted here by S.

In general, the net in flow is a possibly nonlinear function of the state of the system:

S = INTEGRAL(Net Inflow, S(0))

Net Inflow = f(S)

If the system is linear, the net inflow must be directly proportional to the state of the system:

Net Inflow = g S

Here the constant g has units of (1/time) and represents the fractional growth rate of the stock.

A phase plot is a graph showing the net rate as a function of the state of the system.


.

Net

Infl

ow

Rat

e(u

nit

s/ti

me)

State of the System (units)

1

g

00

UnstableEquilibrium

dS/dt = Net Inflow Rate = gS

Zero is an equilibrium of the system:

no savings, no interest income; no people no births.

However, the equilibrium is unstable: add any quantity to the stock and there will now be a small, positive net inflow, increasing the state of the system a bit.

0

128

256

384

512

640

768

896

1024

0

1.28

2.56

3.84

5.12

6.4

7.68

8.96

10.24

0 200 400 600 800 1000

Behavior

Sta

te o

f th

e S

yste

m (

Un

its)

Net In

flow

(un

its/time)

State of the System(left scale)

Net Inflow(right scale)

Time

In the general case where the phase plot of a state variable can be nonlinear, the state of the system will grow whenever the net rate is an increasing function of the stock.

An equilibrium is unstable whenever the slope of the net rate at the equilibrium point is positive.


Exponential growth: structure (phase plot) and behavior (time plot)

The fractional growth rate g = 0.7%/time unit. Initial state of the system = 1 unit. Points on plot show every doubling (10 time periods).

0

2

4

6

8

10

0 128 256 384 512 640 768 896 1024

StructureN

et In

flo

w (

un

its/

tim

e)

State of System (units)

t = 1000

t = 900

t = 800

t = 700

Paper Folding

Take an ordinary sheet of paper. Fold it in half. Fold the sheet in half again. The paper is still less than half a millimeter thick. If you were to fold it 40 more times, how thick would the paper be? If you folded it a total of 100 times, how thick would it be? Give your intuitive estimate, without using a calculator. Give your 95% upper and lower confidence bounds for your estimates

In fact, after 42 folds the paper would be 440,000 kilometers thick-more than the distance from the earth to the moon!

And after 100 folds, the paper would be an incomprehensibly immense 850 trillion times the distance from the earth to the sun!


Exponential growth over different time horizons

The state of the system is given by the same growth rate of 0.7%/time period in all cases (doubling time = 100 time periods).

0

1000

0 200 400 600 800 1000

Time Horizon = 10td

Sta

te o

f th

e S

yste

m (

un

its)

0

1 1030

0 2000 4000 6000 8000 10000

Time Horizon = 100td

Sta

te o

f th

e S

yste

m (

un

its)

0

2

0 2 4 6 8 10

Time Horizon = 0.1td

Sta

te o

f th

e S

yste

m (

un

its)

0

2

0 20 40 60 80 100

Time Horizon = 1td

Sta

te o

f th

e S

yste

m (

un

its)

The Rule of 70

td = ln(2)/g => td =70/(100g)


When the system is linear, the behavior is pure exponential decay.

The equation for the net rate of change of the stock is

Net Inflow = -Net Outflow = -dSwhere d is the fractional decay rate (its units are 1/time).

S(t) = S(0) exp( -dt)

NEGATIVE FEEDBACK AND EXPONENTIAL DECAY

.

1

-d

State of the System (units)0

StableEquilibrium

Net Inflow Rate = - Net Outflow Rate = - dS

Net

Infl

ow

Rat

e(u

nit

s/ti

me)

Exponential decay: structure (phase plot) and behavior (time plot)

The fractional decay rate d = 5%/time unit. Initial state of the system = 100 units.

-5

0

0 20 40 60 80 100

Structure

Ne

t In

flo

w (

un

its/

tim

e)

State of System (units)

t = 0

t = 10

t = 20

t = 3 0

t = 40

0

50

100

0

5

10

0 20 40 60 80 100

Behavior

Net In

flow

(un

its/time)

State of the System(left scale)

Net Inflow(right scale)

Time

The exponential decay structure is a special case of the first-order linear negative feedback system.

As discussed in chapter 4, all negative feedback loops have goals. In the case of exponential decay, such as the death rate and depreciation examples, the goal is implicit and equal to zero. In general, however, the goals of negative loops are not zero and should be made explicit.


In the general case, the corrective action determining the net inflow to the state of the system is a possibly nonlinear function of the state of the system, S, and the desired state of the system, S*:

Net Inflow = f(S, S*)

The simplest formulation, however, is for the corrective action to be a constant fraction per time period of the discrepancy between the desired and actual state of the system:

Net Inflow = Discrepancy/AT = (S* - S)/AT


B

Net InflowRate

SState of

the System

+

S*Desired State of

the System

-

ATAdustment Time

-

dS/dt

Discrepancy(S* - S)

B

NetProduction

Rate

Inventory

+

+

DesiredInventory

-

InventoryShortfall

B

Net HiringRate

Labor

+

DesiredLabor Force

-

LaborShortfall

dS/dt = Net Inflow RatedS/dt = Discrepancy/ATdS/dt = (S* - S)/AT

Net Production Rate = Inventory Shortfall/AT = (Desired Inventory - Inventory)/AT

Net Hiring Rate = Labor Shortfall/AT = (Desired Labor - Labor)/AT

General Structure

Examples

+

+

ATAdustment Time

-

ATAdustment Time

-

0

100

200

0 20 40 60 80 100

Sta

te o

f th

e S

yste

m (

un

its)

The discrepancy can be negative, as when there is excess inventory; in this case the net inflow is negative and the state of the system falls.

The phase plot for the system shows that the net inflow rate to the state of the system is a straight line with slope ( -1/AT) and equals 0 when S = S*.


.

1

-1/AT

Net

Infl

ow

Rat

e(u

nit

s/ti

me)

0

S*State of the System

(units)

StableEquilibrium

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

The goal is 100 units. The upper curve begins with S(0) = 200; the lower curve begins with s(0) = 0. The adjustment time in both cases is 20 time units.

Time Constants and Half-Lives

Just as exponential growth doubles the state of the system in a fixed period of time, exponential decay cuts the quantity remaining by half in a fixed period of time.

S(t) = S* - (S* - S(0))exp( -t/AT)

The product (S* - S(0))exp( -t/AT) is therefore the current gap remaining between the desired and actual states.

The half-life is given by the value of time, th which satisfies

0.5 = exp(-th/AT) = exp(-dt)

th = AT ln(2) = In(2)/d = 0.70AT = 70/(l00d)


What is the behavior of a first-order system when the net rate of change is affected by both types of loop?

Population = INTEGRAL(Net Birth Rate, Population(0)) Net Birth Rate = BR – DR

In the linear case only three behaviors are possible.

MULTIPLE-LOOP SYSTEMS

.

0

Population

Bir

th a

nd

Dea

th R

ates

0

Death Rate

Net Birth Rate

Birth Rate

1

1b

1 b-d

0Time0

Po

pu

lati

on

Structure (phase plot) Behavior (time domain)

0

Population

Bir

th a

nd

Dea

th R

ates

0

Death RateNet Birth Rate

Birth Rate

0Time0

Po

pu

lati

on

0

Population

Bir

th a

nd

Dea

th R

ates

0

Death Rate

Net Birth Rate

Birth Rate

0Time0

Po

pu

lati

on

b < d

b > d

b = d

Exponential Decay

Exponential Growth

Equilibrium

-d

The behavior of the system is the sum, or superposition, of the behaviors generated by the individual loops.

Because the system is linear (b and d are constants), the dominance of the two loops can never change.

Superposition means linear systems can be analyzed by reduction to their components.

However, realistic systems are far from linear. The behavior of the linear population growth system shows why.

Because the dominance of the feedback loops can never change, the population can only grow forever, remain constant, or go extinct.

In real systems, there must be shifts in feedback loop dominance, and therefore there must be important nonlinearities in all real systems.

MULTIPLE-LOOP SYSTEMS

No real quantity can grow forever.

Every system initially exhibiting exponential growth will eventually approach the carrying capacity of its environment, whether that is the food supply for a population of moose, the number of people susceptible to infection by a virus, or the potential market for a new product.

As the system approaches its limits to growth, it goes through a nonlinear transition from a regime where positive feedback dominates to a regime where negative feedback dominates.

The result is often a smooth transition from exponential growth to equilibrium, that is, S-shaped growth.

In real systems, the fractional birth and death rates cannot be constant mustchange as the population approaches its carrying capacity.

Net Birth Rate = BR - DR = b(P/C)P - d(P/C)P

where the fractional birth and death rates band d are now functions of the ratio of the population P to the carrying capacity C.

NON LINEAR FIRST-ORDER SYSTEMS: S-SHAPED GROWTH

0

Fra

cti

on

al

Bir

th a

nd

De

ath

Ra

tes

(1/t

ime

)

Population/Carrying Capacity(dimensionless)

Fractional Birth Rate Fractional

Death Rate

Fractional Net Birth Rate

0 1

Next construct the phase plot for the system using these nonlinear fertility and life expectancy relationships. The birth and death rates are now curves given by the product of the population and fractional birth and death rates.


0

Bir

th a

nd

Dea

th R

ates

(in

div

idu

als/

tim

e)

Population/Carrying Capacity(dimensionless)

Birth Rate

Death Rate

Net Birth Rate

0 Stable EquilibriumUnstable

Equilibrium

Positive Feedback Dominant

•• (P/C)inf 1

Negative FeedbackDominant

First, note that the point P = 0 is an equilibrium, as in the linear system.

Since the fractional birth rate remains nearly constant when population is small relative to the carrying capacity, the birth rate (in individuals/time period) is nearly linear for P<<C.


0

1

0

1

Population

Net Birth Rate

(P/C)inf

tinf0 Time

Po

pu

lati

on

/Car

ryin

g C

apac

ity

Net B

irth R

ate

0

1

2

0

Population

Net Birth Rate

0 Time

Po

pu

lati

on

/Car

ryin

g C

apac

ity

Net B

irth R

ate

Positive feedback dominates whenever the rate of change of the state variable is increasing in the state variable, that is, as long as the slope of the net rate of change as a function of the state variable is positive.

Negative feedback dominates whenever the net rate of change is decreasing in the state variable, that is, as long as the slope of the net rate is negative.

Formal Definition of Loop Dominance

System Dynamics Shahram Shadrokh Closing the Loop: Dynamics of Simple Structures.

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