Higher order feedback loops

One stock – first order systemMore stocks – higher orderBehavior

Second order negative feedback – oscillating systemSecond order positive feedback – exponential growth and oscillations (unstable)

Higher order feedback loops

Second order negative feedbackpendulum

Higher order feedback loops

Higher order feedback loops

Generalization

Higher order feedback loops

Higher order feedback loops

Second order positive feedback loop

Higher order feedback loops

Delays

Delays occur frequently in many systems.

When a pollutant is dumped into a river, it takes time to dissipate.

Delays are divided into two types:(a) Material delays - resulting from the time involved in processing materials.(b) Information delays - resulting from time in perceiving and acting upon information.

Material delay example

Construction of apartments in a large city.Builders construct apartments in response to the gap between the total number of apartments desired and available.

Start with 10,000 apartments. Increase to 15,000.Delay - completion time 2 years.

Material delay example

Apartments in Construction Total Number of Apartments

Initiation Rate Completion Rate

Completion Time

Desired Apartments

DifferenceInitiation Time

Graph 1 Table 1

Material delay example

Initiation time = 1Apartments in construction = 0Completion rate=Apartments in construction/Completion timeTotal number of apartments =10,000Desired apartments = 10,000 + STEP(5000,1)Difference= Desired apartments – Total number of apartmentsCompletion time = 2

Material delay example

Information delay example

Joe’s Cement Factory makes and sells cement blocks Monday through Friday. Each morning Joe has to decide how many blocks to make, and he relies on the average sales over the past five days in making his decision.

Information delay example

Cement blocks to make

Baking rate

Sales

Graph 1

Information delay example

Cement blocks to make = 100 + STEP (20,10)Sales = 100 + STEP (20,5)Making rate = SMTH (Sales, 5) – Cement blocks to make

Information delay example

Solution interval DT

System Principle 11Solution interval is in all level equations and no others

Solution interval DT

DTSolution intervalPeriod of measurementDelta timeTime step

DT is the time period in which the level is changed by the rate

DT in Vensim

The best size of TIME STEP is determined by the following considerations:

TIME STEP should allow test inputs to be accessed regularly.TIME STEP should allow data to be accessed with appropriate regularity.TIME STEP should be smaller than 1/3 of the shortest time constant in the model (not applicable with automatic step size adjustment in Runge-Kutta integration).TIME STEP should be smaller than the shortest period for which a significant change in model behavior is at all likely.

DT in Vensim

An inappropriately long TIME STEP leads to incorrect behavior. In general if you see oscillation with a frequency that is near to twice TIME STEP you should test TIME STEP to see if it has an appropriate value.If you are using Runge-Kutta integration with automatic step size adjustment, the third and forth considerations do not apply. Vensim will automatically determine how small it needs to make TIME STEP in order to achieve the desired accuracy or issue an error if it is unable to do so.