Goal Seeking Behavior
Chapter 3 discussed the two types of feedback loops that exist in the world: positive loops and negative loops. Positive loops generate exponential growth (or rapid decline) and negative loops produce goalseeking behavior. This section introduces and analyzes a first order negative feedback loop system.
Figure 1: Generic First Order Negative Feedback Loop System
Figure 1 presents a generic, first order, linear, negative feedback loop system. The model is initialized in equilibrium by setting the initial value of the stock equal to its goal, and then "shocked" out of equilibrium (at time=1) by a step function that changes the system's goal from ten to fifteen. This procedure  ie., starting a system dynamics model in equilibrium and then shocking it out  is widely used in system dynamics because it allows the modeler to see the "pure" behavior of the system in response to the shock. Moreover, it illustrates another important idea in system dynamics modeling: A system containing negative feedback loops will be in equilibrium only when all of its stocks are equal to all of its goals simultaneously.
The implication of this idea is very important. In the real world it is rare, if not impossible, to find actual systems (particularly actual social systems) that exist in states of equilibrium because actual systems contain many negative feedback loops that do not reach their goals simultaneously. Thus, system dynamicists often start their models in states of equilibrium, not because they feel that this reflects reality, but rather because it is a useful reference point for analyzing the "pure" behaviors of their models.
Figure 2: Simulation of the Generic First Order Negative
Feedback Loop System in response to a step increase at time=1
The behavior exhibited by the negative feedback loop system as shown in Figure 2 is goal seeking. When the goal is increased from ten to fifteen, a discrepancy between the stock and its goal is created. In response to this discrepancy, the system initiates corrective action by increasing its flow and smoothly raising its stock to a value of fifteen. The time it takes the system to reach its new goal is determined by the size of TimeCoef. The bigger TimeCoef is, the longer it takes for the system to adjust to its new goal. This is illustrated in Figure 3, where the negative feedback loop system of Figure 1 is simulated with three different values of TimeCoef (1, 2, and 3) in response to a reduction in the goal (opposite the case shown in Figures 1 and 2) from ten down to five with a step function at time=1.
(Click on image to run simulation)
Figure 3: Three Simulations of the Generic First Order Negative
Feedback Loop System give a step reduction at time=1
Exact Analytical Solution
To more precisely determine how long it takes a linear first order negative feedback loop system to reach its goal, the exact analytical solution to the model presented in Figure 1 must be studied. This solution is:
Stock_{t} = Goal + (Stock_{0}  Goal) * e^{(1 / TimeCoef)*t}

To study Equation 1, it is helpful to look at the special case when the system is seeking a goal of zero. Equation 2 presents this special case. It found by simply substituting goal values of zero into Equation 1.
Stock_{t} = Stock_{0} * e^{(1 / TimeCoef)*t}

One very important thing to note about Equation 2 is that it is very similar to Equation 1 of Exponential Growth, the exact analytical solution to a linear positive feedback loop system. Indeed, Equations 1 of Exponential Growth and Equation 2 differ only by the signs of their exponents. Equations 1 of Exponential Growth has an exponent of Coef * t and Equation 2 has an exponent of (1/TimeCoef) * t. Since Coef and 1/TimeCoef are both numbers, the exponents, and hence the equations, differ only by sign.
The reason that Equation 2 has a minus sign in front of its exponent is that it represents a system seeking a goal of zero. Thus, under the assumption that the system's stock contains positive values, the system will steadily lose units via its outflow until there is nothing left. This process is known as exponential decay.
For the more mathematically inclined reader, the derivation of Equation 1 is presented in the shaded box below.
