Linear optics measurement and correction - I 117
It is interesting to note that when the weight factors are the same for
all SV modes, i.e., λ i =
√
λ for i = 1 through P , Eq. (4.48) is equivalent
to Eq. (4.41) (because I = V
T V). In other words, when the weights are
identical across all terms, adding constraints through individual parameters
and through SV modes produces the same results.
4.2.8 Application of constrained fitting for optics correction
As an illustration of the constrained fitting method and also as a demonstration of optics correction with orbit response matrix fitting, in the following
we show an example of applying the method to the SPEAR3 storage ring
experimentally.
In the test example, the setpoints of a number of QF and QD quadrupole
magnets were intentionally changed to introduce optics errors to the machine.
Orbit response matrix and dispersion data were taken and fitted with the
constrained fitting method. The off-diagonal elements of the orbit response
matrix are included in the fitting. In addition to the BPM and corrector
gains and 78 quadrupole parameters, BPM coupling coefficients, corrector
rolls, and 13 skew quadrupoles are also fitted in order to account for the
cross-coupling between the two transverse planes and the vertical dispersion.
There are a total of 547 fitting parameters. The constraints are applied to
the individual quadrupole and skew quadrupole parameters using the scaled
Levenberg-Marquardt scheme.
Figure 4.6 (a) shows the SV spectrum with the weighting factor set to
three levels, λ = 0 (no constraints), 0.001, and 0.1. With the constraints given
at a level of λ = 0.001, the SVs at the low end (e.g., the last 10 SVs) are
substantially increased, yet without significant relative changes to the other
SV modes. When the weights are increased to λ = 0.1, however, most of the
SVs are significantly increased, leaving only about 15 relatively unchanged.
While the modifications made with λ = 0.001 are essential to prevent excursions to the under-constrained (low-SV) modes, the changes to the SV modes
by λ = 0.1 may have been too large for the solution to quickly converge.
Figure 4.6 (b) shows the convergence history of the normalized χ
2 vs.
the Euclid norm of the
∆K
K vector for several λ levels (with the additional
case of λ = 1 × 10
−5 ). The λ value for each case is fixed in all iterations.
The Euclid norm of
∆K
K serves as an indication of the stray into the underconstrained directions. Without constraints (λ = 0), the algorithm converges
in four iterations, reaching χ
2 = 2382 with ||
∆K
K || = 0.20. Its first iteration
goes beyond the final solution in terms of the ∆K excursion. The case with
λ = 1 × 10
−5 similarly takes a large first stride in ∆K before returning to a
lower level at the second iteration. After it has reached χ
2 = 2382 at the third
iteration, it continues on with two more iterations, each resulting in a tiny
reduction of χ
2 with a sizable step in ∆K. ||
∆K
K || for the case with λ = 0.001
does not have an initial overshoot. After χ
2 comes down to 2388 on the third
iteration, the fitting algorithm continues to converge with small steps, reaching
χ
2 = 2383 on the eighth iteration with ||
∆K
K || = 0.066. When the weight is
It is interesting to note that when the weight factors are the same for
all SV modes, i.e., λ i =
√
λ for i = 1 through P , Eq. (4.48) is equivalent
to Eq. (4.41) (because I = V
T V). In other words, when the weights are
identical across all terms, adding constraints through individual parameters
and through SV modes produces the same results.
4.2.8 Application of constrained fitting for optics correction
As an illustration of the constrained fitting method and also as a demonstration of optics correction with orbit response matrix fitting, in the following
we show an example of applying the method to the SPEAR3 storage ring
experimentally.
In the test example, the setpoints of a number of QF and QD quadrupole
magnets were intentionally changed to introduce optics errors to the machine.
Orbit response matrix and dispersion data were taken and fitted with the
constrained fitting method. The off-diagonal elements of the orbit response
matrix are included in the fitting. In addition to the BPM and corrector
gains and 78 quadrupole parameters, BPM coupling coefficients, corrector
rolls, and 13 skew quadrupoles are also fitted in order to account for the
cross-coupling between the two transverse planes and the vertical dispersion.
There are a total of 547 fitting parameters. The constraints are applied to
the individual quadrupole and skew quadrupole parameters using the scaled
Levenberg-Marquardt scheme.
Figure 4.6 (a) shows the SV spectrum with the weighting factor set to
three levels, λ = 0 (no constraints), 0.001, and 0.1. With the constraints given
at a level of λ = 0.001, the SVs at the low end (e.g., the last 10 SVs) are
substantially increased, yet without significant relative changes to the other
SV modes. When the weights are increased to λ = 0.1, however, most of the
SVs are significantly increased, leaving only about 15 relatively unchanged.
While the modifications made with λ = 0.001 are essential to prevent excursions to the under-constrained (low-SV) modes, the changes to the SV modes
by λ = 0.1 may have been too large for the solution to quickly converge.
Figure 4.6 (b) shows the convergence history of the normalized χ
2 vs.
the Euclid norm of the
∆K
K vector for several λ levels (with the additional
case of λ = 1 × 10
−5 ). The λ value for each case is fixed in all iterations.
The Euclid norm of
∆K
K serves as an indication of the stray into the underconstrained directions. Without constraints (λ = 0), the algorithm converges
in four iterations, reaching χ
2 = 2382 with ||
∆K
K || = 0.20. Its first iteration
goes beyond the final solution in terms of the ∆K excursion. The case with
λ = 1 × 10
−5 similarly takes a large first stride in ∆K before returning to a
lower level at the second iteration. After it has reached χ
2 = 2382 at the third
iteration, it continues on with two more iterations, each resulting in a tiny
reduction of χ
2 with a sizable step in ∆K. ||
∆K
K || for the case with λ = 0.001
does not have an initial overshoot. After χ
2 comes down to 2388 on the third
iteration, the fitting algorithm continues to converge with small steps, reaching
χ
2 = 2383 on the eighth iteration with ||
∆K
K || = 0.066. When the weight is