0.2 Dynamics  (Page 3/3)

Classifications

There are a number of rather common classifications of systems that prove useful. The two mostimportant are given here in terms of an input-output description.

• A A system is called linear if, and only if, the following two conditions hold. In an input $x_1$ causes an output $y_1$ , and an input $x_2$ causes an output $y_2$ , then an input which is the sum of two inputs, $x_1+x_2$ , must cause an output $y_1+y_2$ . This is called superposition. If the input $x_1$ is scaled by an arbitrary value $a$ , then the resulting output must also be scaled by the same value $a$ .

• B A system is called time-invariant or stationary if, and only if, the following is true for arbitrary $t$ .

$\text{If}x\left(t\right)\to y\left(t\right)\text{then}x(t+T)\to y(t+T)$ .

Feedback

A particular structure [Luenberger 1979] whichis so important that it warrants special discussion has the feature that the output affects the input. This is illustrated by thefollowing figure.

Feedback is often part of naturally occurring systems and it also is often a part of constructed systems. Themost common feedback system is probably the thermostat that uses a measured temperature to feedback a controlling signal to a heaterin an oven or room heating system. The filling mechanism in the tank of a toilet uses a float to feedback a measure of the waterlevel to control the input valve. A person's blood sugar level is controlled by a complicated biological feedback system. The powersteering of a car, the auto-pilot of an airplane, and the control of a satellite rocket are all examples of feedback.

An interesting model using feedback can be used to describe a bank savings account. Here the output can be theamount of interest earned which is then fed back and added to increase the account. This feedback is called compounding, andresults in the rapid exponential growth of an account.

A similar model will be used to describe a population where the feedback signal is the number of people addedby births less the number of deaths. This forms the basis of the exponential predictions of population growth, and we will exploreit in detail later.

The basis of the free marketplace is based on feedback through price changes to cause the supply to follow thedemand.

While feedback is a useful concept, its effects become more difficult to predict as the systems become morecomplex. A simple example illustrates one problem. Consider a person adjusting the temperature of his shower by the hot and coldvalves. If there is a time delay introduced by a length of pipe between the valves and the shower head, the person will overcontrol. If the water is initially too hot, he will turn on the cold water, but because of the delay, no effect is immediately feltso more cold water is turned on. This continues until finally the now very cold water reaches the shower head, whereupon the personstarts the same procedure of increasing the hot water. This oscillation will continue until the person "gets smart" and allowsfor the delay. A similar problem can occur in college education because of the four-year delay between the choice of a major andthe graduation to a job.

A bit of reflection begins to show the complicated nature of a social system will involve multi-variablenonlinear systems with time delays and multiple feedback loops.