Page 310 - Mechanical Engineers' Handbook (Volume 2)

1 Rationale 301

The type of model used by the control engineer depends upon the nature of the system
the model represents, the objectives of the engineer in developing the model, and the tools
which the engineer has at his or her disposal for developing and analyzing the model. A
mathematical model is a description of a system in terms of equations. Because the physical
systems of primary interest to the control engineer are dynamic in nature, the mathematical
models used to represent these systems most often incorporate difference or differential
equations. Such equations, based on physical laws and observations, are statements of the
fundamental relationships among the important variables that describe the system. Difference
and differential equation models are expressions of the way in which the current values
assumed by the variables combine to determine the future values of these variables.
Mathematical models are particularly useful because of the large body of mathematical
and computational theory that exists for the study and solution of equations. Based on this
theory, a wide range of techniques has been developed specifically for the study of control
systems. In recent years, computer programs have been written that implement virtually all
of these techniques. Computer software packages are now widely available for both simu-
lation and computational assistance in the analysis and design of control systems.
It is important to understand that a variety of models can be realized for any given
physical system. The choice of a particular model always represents a trade-off between the
fidelity of the model and the effort required in model formulation and analysis. This trade-
off is reflected in the nature and extent of simplifying assumptions used to derive the model.
In general, the more faithful the model is as a description of the physical system modeled,
the more difficult it is to obtain general solutions. In the final analysis, the best engineering
model is not necessarily the most accurate or precise. It is, instead, the simplest model that
yields the information needed to support a decision. A classification of various types of
models commonly encountered by control engineers is given in Section 8.
A large and complicated model is justified if the underlying physical system is itself
complex, if the individual relationships among the system variables are well understood, if
it is important to understand the system with a great deal of accuracy and precision, and if
time and budget exist to support an extensive study. In this case, the assumptions necessary
to formulate the model can be minimized. Such complex models cannot be solved analyti-
cally, however. The model itself must be studied experimentally, using the techniques of
computer simulation. This approach to model analysis is treated in Section 7.
Simpler models frequently can be justified, particularly during the initial stages of a
control system study. In particular, systems that can be described by linear difference or
differential equations permit the use of powerful analysis and design techniques. These in-
clude the transform methods of classical control theory and the state-variable methods of
modern control theory. Descriptions of these standard forms for linear systems analysis are
presented in Sections 4, 5, and 6.
During the past several decades, a unified approach for developing lumped-parameter
models of physical systems has emerged. This approach is based on the idea of idealized
system elements, which store, dissipate, or transform energy. Ideal elements apply equally
well to the many kinds of physical systems encountered by control engineers. Indeed, be-
cause control engineers most frequently deal with systems that are part mechanical, part
electrical, part fluid, and/or part thermal, a unified approach to these various physical systems
is especially useful and economic. The modeling of physical systems using ideal elements
is discussed further in Sections 2, 3, and 4.
Frequently, more than one model is used in the course of a control system study. Simple
models that can be solved analytically are used to gain insight into the behavior of the
system and to suggest candidate designs for controllers. These designs are then verified and
refined in more complex models, using computer simulation. If physical components are
developed during the course of a study, it is often practical to incorporate these components