User`s guide

10 Reliable Computations
10-8
Choice of LTI Model
Now turn to the implications of the results in the last se ction on the linear
modelingtechniques usedfor control engineering.The ControlSystem Toolbox
includes the following types of LTI models that are applicable to discussions of
computational reliability:
•Statespace
• Transfer function, polynomial form
• Transfer function, factored zero-pole-gain form
The following subsections show that state space is most preferable for
numerical computations.
State Space
The state-space representation is the most reliable LTI model to use for
computer analysis. This is one of the reasons for the popularity of “modern”
state-space control theory. Stable computer algorithms for eigenvalues,
frequency response, time response, and other properties of the
quadruple are known [5] and implemented in this toolbox. The state-space
model is also the most natural model in MATLAB's matrix environment.
Even with state-space models, however, accurate results are not guaranteed,
becauseof theproblemsof finite-word-lengthcomputer arithmeticdiscussedin
the last section. A well-conditioned problem is usually a prerequisite for
obtaining accurate results and makes it important to have reasonable scaling
of the data. Scaling is discussed further in the “Scaling” section later in this
chapter.
Transfer Function
Transfer function models, when expressed in terms of expanded polynomials,
tend to be inherently ill-conditioned representations of LTI systems. For
systems of ord er grea ter than 10, or with very large/small polynomial
coefficients, difficulties can be encountered with functions like
roots, conv,
bode, step, or c onve rsio n functions like ss or zpk.
ABCD
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