Modeling
In order to deal in a systematic and efficient way with problems involving time-dependent behavior, we must have a description of the objects or processes involved. We call such a description a model. Models are used to describe how the system works.
A model for enhancing our understanding of the problem can take several forms. A physical model, like a scale model, helps us to visualize how the components of the design fit together and can provide insight not obtainable from a blueprint (which is another model form). Graphs or plots are still another type of model. They can often present time-dependent behavior in a concise way. The model type we will use most frequently is the mathematical model, which is a description in terms of mathematical relations. These relations will consist of differential or difference equations if the model is to describe a dynamic system. Mathematical models generally require a “theory” or equations that describe the system’s boundaries and behavior through its input parameters. In short, if no theory exists, the model is not viable and the system is not controllable.
One of our aims here is to introduce a framework that allows the development of mathematical models for describing the time-dependent behavior of many types of phenomena: fluid flow, thermal processes, mechanical elements, and electrical systems, as well as some nonphysical applications. In this regard, it is important to remember that the precise nature of a mathematical model depends on its purpose. For example, an electrical resistor can be subjected to mechanical deformations if its mounting board is subjected to vibration. In this case, the force-deflection spring model could be used to describe the resistor’s mechanical behavior.

Block diagrams are often used to display the mathematical model in a form that allows us to understand the interactions occurring between the system’s elements

Analysis
A mathematical model represents a concise statement of our hypotheses concerning the behavior of the system under study. We can deal with the verification of the model in two ways. Verification by experiment or testing is ultimately required of all serious design projects. This is not always done at the outset of a study, however, especially if we are dealing with component types whose behavior is known to be described by a specific model on the basis of past experience.
Once we are satisfied with the validity of our chosen component models, they can be used to predict the performance of the system in question. Predicting the performance from a model is called analysis. For example, the current produced in a resistor by an applied voltage v can be predicted to be by solving the resistor model for the unknown variable i in terms of the given quantities v and R. Most of our mathematical models will describe dynamic behavior and will thus consist of differential equations.

Control
The successful operation of a system under changing conditions often requires a control system.
The term control refers to the process of deliberately influencing the behavior of an object in order to produce some desired result. The physical device inserted for this purpose is the controller or control system.

In the terminology of the control engineer, we say that the controller must respond satisfactory to changes in commands and maintain system performance in the presence of disturbances.
Modeling, analysis, and control of dynamic systems constitute a unified area of study.