User`s guide

Choice of LTI Model
10-13
Note also that the eigenvectors have changed.
[vc,dc] = eig(Ac)
vc =
–0.5017 0.2353 0.0510 0.0109
–0.8026 0.7531 0.4077 0.1741
–0.3211 0.6025 0.8154 0.6963
–0.0321 0.1205 0.4077 0.6963
dc =
10.0000 0 0 0
0 5.0000 0 0
0 0 2.0000 0
0 0 0 1.0000
The condition number of the new eigenvector matrix
cond(vc)
ans =
34.5825
is thirty time s larg er.
Thephenomenon illustrated aboveis not unusual. Matricesin companion form
or control labl e/obs erva ble canonical form (lik e
Ac) typicall y have
worse-conditioned eigensystems than matrices in general state-space form
(like
A). This me ans t hat their ei genv alues and eigenve ctor s are more se nsi tive
topert urbatio n.The problem genera llyg etsf arw or se forhigher-ord ers ystems .
Working with high-order transfer function mo dels and conv erting the m bac k
and forth to state space is numerically risky.
In summary, the main numerical problems t o be aware of in dealing with
transfer function m odels (and hence, calcula tions involving po lynomia ls) are:
• The potentially large range of numbers leads to ill -conditioned problems,
especial ly w hen s uch mod els are link ed t oge ther gi ving hig h-ord er
polynomials.