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3. Modeling


The design of a control system typically requires a delicate interplay between fundamental constraints and trade-offs. 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


  • 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
Input-output 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.