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Left and right spectral factorization.
[am,bm,cm,dm] = sfl(a,b,c,d) [am,bm,cm,dm] = sfr(a,b,c,d) [ssm] = sfl(ss) [ssm] = sfr(ss)
Description
Given a stabilizable realization of a transfer function G(s) := (A, B, C, D) with ,
sfl
computes a left spectral factor M(s) such that
Sfr
computes a right spectral factor M(s) of G(s) such that
Algorithm
Given a transfer function G(s) := (A, B, C, D), the LQR optimal control
u = -Fx = -R-1(XB + N)Tx stabilizes the system and minimize the quadratic cost function
[F,X] = lqr(A,B,Q,R,N) = lqr(A,B,-C'*C,(I-D'*D),-C'*D).Finally, to get the stable spectral factor, we take M(s) to be the inverse of the outer factor of
iofr
is used to compute the outer factor.
Limitations
The spectral factorization algorithm employed in sfl
and sfr
requires the system G(s) to have and to have no
-axis poles. If the condition
fails to hold, the Riccati subroutine (
aresolv
) will normally produce the message
WARNING: THERE ARE jThis happens because the Hamiltonian matrix associated with the LQR optimal control problem has j-AXIS POLES... RESULTS MAY BE INCORRECT !!
sfl
or sfr
to check whether
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