Specifications

11 Reliable Computations
11-10
very little. This is true ingeneral. Different roots have different sensitivities to
different perturbations. Computed roots may then be quite meaningless for a
polynomial, particularly high-order, with imprecisely known coefficients.
Finding all the roots of a polynomial (equivalently, the poles of a transfer
function or the eigenvalues of a matrix in controllable or observable canonical
form) is often an intrinsically sensitive problem. For a clear and detailed
treatment of the subject, including the tricky numerical problem of deflation,
consult [6].
It is therefore preferable to work with the factored form of polynomials when
available. To compute a state-space model of the transfer function
defined above, for example, you could expand the denominator of , convert
the transfer function model to state space, and extract the state-space data by
H1 = tf(1,poly(1:20))
H1ss = ss(H1)
[a1,b1,c1] = ssdata(H1)
However, you should rather keep the denominator in factored form and work
with the zero-pole-gain representation of .
H2 = zpk([],1:20,1)
H2ss = ss(H2)
[a2,b2,c2] = ssdata(H2)
Indeed, the resulting state matrix a2 is better conditioned.
[cond(a1) cond(a2)]
ans =
2.7681e+03 8.8753e+01
and the conversion from zero-pole-gain to state space incurs no loss of accuracy
in the poles.
format long e
[sort(eig(a1)) sort(eig(a2))]
ans =
9.999999999998792e-01 1.000000000000000e+00
2.000000000001984e+00 2.000000000000000e+00
3.000000000475623e+00 3.000000000000000e+00
3.999999981263996e+00 4.000000000000000e+00
Hs
()
H
Hs
()