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System Design  Index  Book Contents 
Chapter 3
3. Modeling
Preview
The design of a control system typically requires a delicate
interplay between fundamental constraints and tradeoffs. To accomplish
this, a designer must have a comprehensive understanding of how the
process operates. This understanding is usually captured in the form of
a mathematical model. With the model in hand, the designer can proceed
to use the model to predict the impact of various design choices. The
aim of this chapter is to give a brief introduction to modeling.
Specific topics to be covered include
 how to select the appropriate model complexity
 how to build models for a given plant
 how to describe model errors.
 how to linearize nonlinear models
It also provides a brief introduction to certain commonly used
models, including
 state space models
 high order differential and high order difference equation models
Summary
 In order to systematically design a controller for a particular
system, one needs a formal  though possibly simple  description of
the system. Such a description is called a model.
 A model is a set of mathematical equations that are intended
to capture the effect of certain system variables
on certain other system variables.
 The italized expressions above should be understood as follows:
 certain system variables: It is usually neither
possible nor necessary to model the effect of every variable on
every other variable; one therefore limits oneself to certain
subsets. Typical examples include the effect of input on output,
the effect of disturbances on output, the effect of a reference
signal change on the control signal, or the effect of various
unmeasured internal system variables on each other.
 capture: A model is never perfect and it is therefore
always associated with a modeling error. The word capture
highlights the existence of errors, but does not yet concern
itself with the precise definition of their type and effect.
 intended: This word is a reminder that one does not
always succeed in finding a model with the desired accuracy and
hence some iterative refinement may be needed.
 set of mathematical equations: There are numerous
ways of describing the system behavior, such as linear or
nonlinear differential or difference equations.
 Models are classified according to properties of the equation they
are based on. Examples of classification include:
Model attribute

Contrasting attribute

Asserts whether or not ...

Single input single
output 
Multiple input multiple
output 
...the model equations
have one input and one output only 
Linear 
Nonlinear 
... the model equations
are linear in the system variables 
Time varying 
Time invariant 
...the model parameters
are constant 
Continuous 
Sampled 
...model equations
describe the behavior at every instant of time, or only in
discrete samples of time 
Inputoutput 
State space 
...the models equations
rely on functions of input and output variables only, or also
include the so called state variables 
Lumped parameter 
Distributed parameter 
...the model equations
are ordinary or partial differential equations 
 In many situations nonlinear models can be linearized around a
user defined operating point.
