Reflect Poles of Rational Matrices
reflect_poles.RdGiven a square, rational matrix \(x(z)\) and a set of poles of \(x(z)\),
the function reflect_poles constructs an allpass matrix \(U(z)\)
such that \(x(z)U(z)\) has poles which are the reciprocals of the
selected poles and the other poles are not changed.
Also the zeroes are not changed, i.e. \(x\) and \(xU\) have the same zeroes.
Arguments
- x
Square, rational matrix object (with real coefficients).
- poles
Complex or real vector of poles to be reflected. It is assumed that for complex conjugated pairs of poles, only one is contained in this vector.
- ...
Not used.
- tol
(double) Tolerance parameter used for the checks.
- check_poles
If
TRUEthen the procedure checks that the given poles are poles of the rational matrix x(z).
Value
Rational matrix object which represents the rational matrix
\(x(z)U(z)\) with the reflected poles. The object is of the
same class as the input object x.
Note
The procedures are only implemented for rational matrices with real coefficients.
Therefore complex poles occur in complex conjugated pairs and such pairs are
jointly reflected to ensure that the result again has real coefficients.
Note however, that the argument poles must only contain one element
of the pairs to be reflected.
In some degnerate cases the procedure(s) may run into numerical problems:
For matrices in rmfd form,
the procedure assumes that the matrix has no poles at \(z=0\).
For the matrices in statespace form, poles
on (or close to) the unit circle are not allowed. In addition multiple
poles may lead to unreliable results.
See also
The procedure uses the helper functions blaschke
and make_allpass. For reflecting zeroes,
see reflect_zeroes.
Examples
# ###################################################################
# rational matrix in statespace form ('stsp' object)
set.seed(12345)
# create random (2,2) rational matrix in state space form
# with state dimension s=5
( x = test_stsp(dim = c(2,2), s = 5) )
#> statespace realization [2,2] with s = 5 states
#> s[1] s[2] s[3] s[4] s[5] u[1]
#> s[1] 0.5855288 -0.2761841 -0.7505320 1.4557851 0.6121235 0.49118828
#> s[2] 0.7094660 -0.2841597 0.8168998 -0.6443284 -0.1623110 -0.32408658
#> s[3] -0.1093033 -0.9193220 -0.8863575 -1.5531374 0.8118732 -1.66205024
#> s[4] -0.4534972 -0.1162478 -0.3315776 -1.5977095 2.1968335 1.76773385
#> s[5] 0.6058875 1.8173120 1.1207127 1.8050975 2.0491903 0.02580105
#> x[1] -1.8179560 0.3706279 0.2987237 -0.4816474 1.6324456 1.00000000
#> x[2] 0.6300986 0.5202165 0.7796219 0.6203798 0.2542712 0.00000000
#> u[2]
#> s[1] 1.1285108
#> s[2] -2.3803581
#> s[3] -1.0602656
#> s[4] 0.9371405
#> s[5] 0.8544517
#> x[1] 0.0000000
#> x[2] 1.0000000
# poles of x(z)
( x_poles = poles(x) )
#> [1] 0.3228713+0.0000000i -0.3326091+0.0000000i 0.0091529-0.8604103i
#> [4] 0.0091529+0.8604103i -4.0175418+0.0000000i
# reflect all unstable poles (inside the unit circle) ###########
# note: for complex zeroes, select only one of the complex conjugated pair!
x1 = reflect_poles(x, poles = x_poles[(abs(x_poles) < 1) & (Im(x_poles) >= 0)])
r_poles = x_poles
r_poles[abs(r_poles) < 1] = 1 / r_poles[abs(r_poles) < 1]
(x1_poles = poles(x1))
#> [1] 0.012362-1.162105i 0.012362+1.162105i -3.006533+0.000000i
#> [4] 3.097209+0.000000i -4.017542+0.000000i
j = match_vectors(r_poles, x1_poles)
all.equal(r_poles, x1_poles[j])
#> [1] TRUE
# Check that the transformation matrix U (x1 = x %r% U) is all-pass
all.equal(zvalues(x) %r% Ht(zvalues(x)), zvalues(x1) %r% Ht(zvalues(x1)))
#> [1] TRUE
set.seed(NULL)
# ###################################################################
# rational matrix in RMFD form
set.seed(12345)
# create random (2 x 2) rational matrix in RMFD form with degrees (2,1)
( x = test_rmfd(dim = c(2,2), degree = c(2,1)) )
#> ( 2 x 2 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 2, deg(d(z)) = q = 1
#> left factor d(z):
#> z^0 [,1] [,2] z^1 [,1] [,2]
#> [1,] -0.2841597 -0.1162478 0.3706279 -0.7505320
#> [2,] -0.9193220 1.8173120 0.5202165 0.8168998
#> right factor c(z):
#> z^0 [,1] [,2] z^1 [,1] [,2] z^2 [,1] [,2]
#> [1,] 1 0 0.5855288 -0.1093033 0.6058875 0.6300986
#> [2,] 0 1 0.7094660 -0.4534972 -1.8179560 -0.2761841
# poles of x(z)
( x_poles = poles(x) )
#> [1] -0.492635-0.6478764i -0.492635+0.6478764i 1.045832-0.6704744i
#> [4] 1.045832+0.6704744i
# reflect all unstable poles (inside the unit circle) ###########
# note: for complex zeroes, select only one of the complex conjugated pair!
x1 = reflect_poles(x, poles = x_poles[(abs(x_poles) < 1) & (Im(x_poles) >= 0)])
r_poles = x_poles
r_poles[abs(r_poles) < 1] = 1 / r_poles[abs(r_poles) < 1]
(x1_poles = poles(x1))
#> [1] -0.7436752-0.9780254i -0.7436752+0.9780254i 1.0458317-0.6704744i
#> [4] 1.0458317+0.6704744i
j = match_vectors(r_poles, x1_poles)
all.equal(r_poles, x1_poles[j])
#> [1] TRUE
# Check that the transformation matrix U (x1 = x %r% U) is all-pass
all.equal(zvalues(x) %r% Ht(zvalues(x)), zvalues(x1) %r% Ht(zvalues(x1)))
#> [1] TRUE
set.seed(NULL)