Specifications

Choice of LTI Model
11-9
A major difficulty is the extreme sensitivity of the roots of a polynomial to its
coefficients. This example is adapted from Wilkinson, [6] as an illustration.
Consider the transfer function
The matrix of the companion realization of is
Despite the benign looking poles of the system (at –1,–2,..., –20) you are faced
with a rather large range in the elements of , from 1 to . But
the difficulties don’t stop here. Suppose the coefficient of in the transfer
function (or ) is perturbed from 210 to ( ).
Then, computed on a VAX (IEEE arithmetic has enough mantissa for only
), the poles of the perturbed transfer function (equivalently, the
eigenvalues of ) are
eig(A)'
ans =
Columns 1 through 7
–19.9998 –19.0019 –17.9916 –17.0217 –15.9594 –15.0516 –13.9504
Columns 8 through 14
–13.0369 –11.9805 –11.0081 –9.9976 –9.0005 –7.9999 –7.0000
Columns 15 through 20
–6.0000 –5.0000 –4.0000 –3.0000 –2.0000 –1.0000
The problem here is not roundoff. Rather, high-order polynomials are simply
intrinsically very sensitive, even when the zeros are well separated. In this
case, a relative perturbation of the order of induced relative
perturbationsof the order of in some roots. But some of the roots changed
Hs
()
1
s 1+
()
s 2+
()
... s 20+
()
-------------------------------------------------------------
1
s
20
210s
19
... 20!+++
-----------------------------------------------------------
==
AHs
()
A
0 1 0 ... 0
0 0 1 ... 0
::..:
0 0 ... . 1
20!– . ... . 210–
=
A 20! 2.4 10
18
×≈
s
19
Ann
,()
210 2
23–
+ 2
23–
1.2 10
7–
×≈
n 17=
A
10
9–
10
2–