User`s guide

10 Reliable Computations
10-14
• 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
extraordinary arithmetic range difficulties of polynomial coefficients, as
illustrated in the “Transfer Function” section. The transformation from state
space to zero-pole-gain is stable, although the handling of infinite zeros can
sometimes be tricky, and repeated roots can cause problems.
If possible, avoid repeated switching between different model representations.
As discussed in the previous sections, when transformations between models
are not numerically stable, roundoff errors are amplified.