### 3.7. A GENERALIZED APPROACH FOR MODELING—THE PRINCIPLES OF CONSERVATION AND ANALOGY

Because of the large variety of control-system components that occur in practice, a generalized approach is useful for obtaining their mathematical model. Therefore, rather than pursue the presentation of further specific control-system components, this section will provide a generalized approach to deriving mathematical models.

There are several general principles that can be useful in serving as guides. The most important are the principle of conservation and the concept of analogous circuits.

#### A. Principle of Conservation

The principle of conservation is a very important guideline for the derivation of a mathematical model. A statement of this concept is that

In terms of rates, the principle of conservation is stated as follows:

Exactly what is being conserved depends on the application. However, this principle is usually used to establish a balance or inventory of mass, energy, momentum, or charge.

The principle of conservation has been used several times throughout this chapter. For example, let us reconsider the hydraulic motor and pump power transmission system illustrated in Figure 3.21 and the electric hot-water heating system of Figure 3.27. In the case of the hydraulic motor and pump power ...