Mathematical Modeling of Systems
Concept
Given a physical system, finding a model (mathematical representation) that accurately predicts the behavior (the output for a given input) of the system.
Key Points
To analyze and design control systems, we need quantitative mathematical models of systems. The dynamic behavior of systems can be described using differential equations.
If the equations can be linearized, then the Laplace transform can be used to simplify the solution.
The input/output relationship for linear components and sub-systems can be described in the form of transfer functions.
System Representation and Modeling
In order to analyze (and subsequently control) most dynamic systems, it is essential to attain a reasonable understanding of how a system functions. To achieve this objective, we formulate mathematical models that help us describe the behavior of systems.
Mathematical models generally serve two purposes:
a. They are used in conjunction with analytical techniques to develop control schemes for the systems that they represent.
b. They are used as a design tool in computer simulation studies. In this context, the model is used as the control object to test and on which to evaluate possible control schemes. This procedure is generally more efficient, cheaper and less time consuming than to test the control schemes on the actual system.
Here we will lay the groundwork for the formulation of mathematical models for energetic (dynamic) systems.
Modeling of dynamic systems: General approach
A dynamic (or energetic) system is a collection of energy storage elements, power dissipative elements, power sources, transformers and transducers. For successful modeling of energetic systems, it is important to know the characteristics of each of these elements.
The structure of dynamic systems
The dynamics of many physical systems results from the transfer, loss, and storage of mass or energy. Thus, one way of developing a model of such a system is to identify the flow paths and storage components of mass or energy and to describe quantitatively how these paths and components are connected. On the other hand, it is sometimes more appropriate to use another physical law, such as Newton’s laws of motion. A basic law that is used to model electrical systems is conservation of charge.
For a given system, we must decide what physical laws are appropriate to use and then develop the model using those laws. There are many other physical laws, but the systems we will treat can be described by the following laws:
1. Conservation of mass
2. Conservation of energy
3. Conservation of charge
4. Newton’s laws of motion
The physical laws alone do not provide enough information to write the equations that describe the system. Three more types of information must be provided; the four requirements are:
1. Appropriate basic physical laws.
2. Specific arrangement or way the system elements are interconnected.
3. Empirically based descriptions for some or all of the system elements.
4. Any relationships due to integral causality.
