112 Beam-based Correction and Optimization for Accelerators
threshold in the calculation of the pseudo-inverse matrix. This approach may
work if there are only a few prominent near-degeneracy SV modes, whose
SVs are substantially lower than the other SV modes. However, it is common
that a large stretch of SV modes all have small SVs. These SV modes can
cause large deviations to the fitting parameters. Typically such a solution will
have large, unrealistic errors in the quadrupole gradient (e.g., with
∆K
K at or
above a few percent). Because of the large false gradient errors in the solution,
when it is applied for optics correction, it could actually increase the optics
errors in the machine. On the other hand, removing all the small SVs prevents
finding the quadrupole errors in the subspace spanned by the corresponding
V -vectors, which in turn will limit the precision of optics correction. Optimizing the cut-off threshold can help balancing the above two detrimental effects;
but it does not cure them since either keeping or dropping a small SV can
cause problems.
The initial optics correction for the SOLEIL storage ring is a classic example that demonstrates the dilemma of fitting orbit response matrix data
with the original LOCO algorithm [88]. There are 56 correctors in each of
the two transverse planes and a total of 120 BPMs in the SOLEIL ring. The
fitting lattice parameters include 160 quadrupole gradients. When fitting the
uncoupled orbit response matrix, there are 512 fitting parameters total. Since
there is no clear-cut step on the SV spectrum of the Jacobian matrix, the
cut-off threshold had to be found by painstakingly trying out many options.
The best option, with 410 SVs kept, predicted
∆K
K up to 6%, far exceeding
the expected gradient errors according to magnet calibration measurements.
The beta beating in the machine could not be brought below 5%.
Another approach to combat the near-degeneracy difficulty in orbit response matrix fitting is to reduce the number of fitting quadrupole parameters. By removing some quadrupole parameters that are correlated with other
quadrupoles, a set of quadrupole parameters with small correlations between
each other can be selected. This approach would work in some cases, but
could not address the problem in general. In essence, the approach of removing quadrupole parameters is the same as the approach of removing SV
modes: both apply a hard cut to reduce the dimension of the parameter space;
the only difference is how to select the dimensions to remove. It could be argued that the approach of removing SV modes is superior because the SV
modes are orthogonal and can remove the degeneracy with the least number
of dimensions.
Constraints over individual fitting parameters:
In order to use the orbit response matrix data to correct linear optics, a
new method is needed to derive a set of reasonable quadrupole errors that
can represent the optics errors in the orbit response matrix and are applicable for optics correction. The applicability requires the quadrupole errors
to contain only small strays to the SV modes with small singular values.
This can be achieved by modifying the Gauss-Newton method with additional constraints to the fitting parameters to prevent large excursions in the
threshold in the calculation of the pseudo-inverse matrix. This approach may
work if there are only a few prominent near-degeneracy SV modes, whose
SVs are substantially lower than the other SV modes. However, it is common
that a large stretch of SV modes all have small SVs. These SV modes can
cause large deviations to the fitting parameters. Typically such a solution will
have large, unrealistic errors in the quadrupole gradient (e.g., with
∆K
K at or
above a few percent). Because of the large false gradient errors in the solution,
when it is applied for optics correction, it could actually increase the optics
errors in the machine. On the other hand, removing all the small SVs prevents
finding the quadrupole errors in the subspace spanned by the corresponding
V -vectors, which in turn will limit the precision of optics correction. Optimizing the cut-off threshold can help balancing the above two detrimental effects;
but it does not cure them since either keeping or dropping a small SV can
cause problems.
The initial optics correction for the SOLEIL storage ring is a classic example that demonstrates the dilemma of fitting orbit response matrix data
with the original LOCO algorithm [88]. There are 56 correctors in each of
the two transverse planes and a total of 120 BPMs in the SOLEIL ring. The
fitting lattice parameters include 160 quadrupole gradients. When fitting the
uncoupled orbit response matrix, there are 512 fitting parameters total. Since
there is no clear-cut step on the SV spectrum of the Jacobian matrix, the
cut-off threshold had to be found by painstakingly trying out many options.
The best option, with 410 SVs kept, predicted
∆K
K up to 6%, far exceeding
the expected gradient errors according to magnet calibration measurements.
The beta beating in the machine could not be brought below 5%.
Another approach to combat the near-degeneracy difficulty in orbit response matrix fitting is to reduce the number of fitting quadrupole parameters. By removing some quadrupole parameters that are correlated with other
quadrupoles, a set of quadrupole parameters with small correlations between
each other can be selected. This approach would work in some cases, but
could not address the problem in general. In essence, the approach of removing quadrupole parameters is the same as the approach of removing SV
modes: both apply a hard cut to reduce the dimension of the parameter space;
the only difference is how to select the dimensions to remove. It could be argued that the approach of removing SV modes is superior because the SV
modes are orthogonal and can remove the degeneracy with the least number
of dimensions.
Constraints over individual fitting parameters:
In order to use the orbit response matrix data to correct linear optics, a
new method is needed to derive a set of reasonable quadrupole errors that
can represent the optics errors in the orbit response matrix and are applicable for optics correction. The applicability requires the quadrupole errors
to contain only small strays to the SV modes with small singular values.
This can be achieved by modifying the Gauss-Newton method with additional constraints to the fitting parameters to prevent large excursions in the