MATLAB CONTROL SYSTEM TOOLBOX 9 Especificaciones descargar pdf (Pagina 259)

100

101

Choice of LTI Model

11-13

The condition number of the new eigenvector matrix

cond(vc)

ans =

34.5825

is thirty times larger.

The phenomenon illustrated aboveisnotunusual.Matrices in companionform

or controllable/observable canonical form (like

Ac) typically have

worse-conditioned eigensystems than matrices in general state-space form

(like

A). This means that their eigenvalues and eigenvectors are more sensitive

toperturbation.Theproblemgenerally getsfarworseforhigher-order systems.

Working with high-order transfer function models and converting them back

and forth to state space is numerically risky.

In summary, the main numerical problems to be aware of in dealing with

transfer function models (and hence, calculations involving polynomials) are:

•The potentially large range of numbers leads to ill-conditioned problems,

especially when such models are linked together giving high-order

polynomials.

•The pole locations are very sensitive to the coefficients of the denominator

polynomial.

•The balanced companion form produced by

ss, while better than the

standard companion form, often results in ill-conditioned eigenproblems,

especially with higher-order systems.

The above statements hold even for systems with distinct poles, but are

particularly relevant when poles are multiple.

Zero-Pole-Gain Models

The third major representation used for LTI models in MATLAB is the

factored, or zero-pole-gain (ZPK) representation. It is sometimes very

convenient to describe a model in this way although most major design

methodologies tend to be oriented towards either transfer functions or

state-space.

In contrast to polynomials, the ZPK representation of systems can be more

reliable. At the very least, the ZPK representation tends to avoid the

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