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Simple System Behavior Summary

 

 

by Nick Louca, September, 2009

(c) 2009 by Ed Seykota

 

 

In this paper I present a summary of the behaviors different simple first order (single Level) systems exhibit.  The intention is to enable the reader to develop an intuitive feel for system structure by observing system behavior.

 

I review the behaviors of a simple accumulation and drain system (one Level and one Rate with no feedback).  I also summarize the behaviors of First Order Negative Feedback and Positive Feedback systems and their behaviors.

 

1.     Simple Accumulation System

 

A Simple Accumulation and Drain System is a system with one Level and one Rate with no feedback.

 

 

Simple Accumulation and Drain System

One Level and one Rate with no feedback

 

 

Equations

.

Initial Value of Level (units) = a number (units)

Level (units)

= Level (units) + Flow Rate (units/time) * dt (time)

Flow Rate (units/time)

= a number (units/time)

Solution Interval, dt (time) = a number (time)

 

 

 

Simple Accumulation and Drain System

Different Behaviors of the Simple Accumulation and Drain System

 

See Excel file.

 

2.     First Order Negative Feedback

 

A First Order Negative Feedback System is a system with a single Level.  The First Order Negative Feedback System may have a one or two rates. 

 

A First Order Negative Feedback system may have a Flow Rate that flows into the Level only or it may have a Flow Rate that flows out of the Level only.  A First Order Negative Feedback system  may also have two flow rates, one that flows into the Level and one that flows out of the Level.

 

Notice that the system structure of a First Order Negative Feedback system includes:

  1. a Target or Goal for the Level;

  2. the Delta or Gap: the difference between the Target and the Level; and

  3. a Time Constant - the time it takes the Rate to move the Level to the Target.

Negative Feedback systems act to decrease the Delta (Gap).

 

 

 

First Order Negative Feedback System

Flow Rate into the Level

 

Equations

.

Initial Value of Level (units) = a number (units)

Level (units)

= Level (units) + Inflow Rate (units/time) * dt (time)

Inflow Rate (units/time)

= Gap (units) / Time Constant (time)

Gap (units) = Target (units) - Level (units)
Target (units) = a number (units)
Time Constant (time) = a number (time)
Solution Interval, dt (time) = a number (time)

 

 

 

First Order Negative Feedback System

Flow Rate out of the Level

 

 

Equations

.

Initial Value of Level (units) = a number (units)

Level (units)

= Level (units) - Outflow Rate (units/time) * dt (time)

Outflow Rate (units/time)

= Gap (units) / Time Constant (time)

Gap (units) = Level (units) - Target (units)
Target (units) = a number (units)
Time Constant (time) = a number (time)
Solution Interval, dt (time) = a number (time)

 

 

 

First Order Negative Feedback System From Different Initial Conditions

 

Note: The above chart illustrates the different types

of behaviors we might expect from a First Order

Negative Feedback system.

 

The Level tracks the Target = 50 units.

 

See Excel file.

 

3.     First Order Positive Feedback

 

Positive Feedback is a mode of behavior in which system elements promote each others' growth.  First Order Positive Feedback Linear Systems exhibit exponential growth.

 

The Rate flows into the Level and increases the Level.  The Rate also increases as the Level increases.  The  Positive Feedback loop is a self-reinforcing loop.  It acts to increase the Delta (Gap).

 

 

 

First Order Positive Feedback System

Flow Rate into the Level

 

 

Equations

.

Initial Value of Level (units) = a number (units)

Level (units)

= Level (units) + Inflow Rate (units/time) * dt (time)

Inflow Rate (units/time)

= Level (units) * Gain (units/units/time) or

= Level (units) / Time Constant (time)

Gain (units/units/time)

= a number (units/units/time)

= 1 / Time Constant (1 / time)

Time Constant (time)

= a number (time)

= 1 / Gain (time)

Solution Interval, dt (time) = a number (time)

 

 

3.1    Behavior from Different Initial Conditions

 

 

 

Positive Feedback

Behavior From Different Initial Conditions

 

see Excel file.

 

 

3.2    Behavior for Different Gains at a Positive Initial Value

 

 

 

Positive Feedback

Behavior for Different Gains at a Positive Initial Value

Note: when the Gain is negative the behavior is asymptotic decay.

 

See Excel file.

 

 

3.3    Behavior for Different Gains at Negative Initial Value

 

 

 

Positive Feedback

Behavior for Different Gains at a Negative Initial Value.

Note: Gain is negative the behavior is asymptotic growth.

 

See Excel file.

 

4.      System Structure Recognition

 

You might attempt to look over the below behaviors and think about the system structures that exhibit these types of behavior.

You might notice that the systems that exhibit these types of behaviors are not simple first order systems but more complex systems.  We examine some of these systems later on in this series.

Note: The illustration of these charts is for educational purposes  only.  The objective is to elicit thoughts about the rates, levels and the system structures that surround us in our daily lives.

 

Source: http://research.stlouisfed.org/

 

Source: http://research.stlouisfed.org/

 

Source: http://research.stlouisfed.org/

 

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