Orbit and trajectory correction 85
BPM i. Additional weight factor could be applied to emphasize the importance
of the orbit at certain locations.
In terms of the residual vector, the objective function can now be written
f (θ) = r
T W
T Wr,
(3.30)
with the diagonal matrix
W = diag(w 1 , w 2 , · · · , w M ).
(3.31)
Following the same process leading up to Eq. (3.16), the desired corrector
change is found to be
θ = (R
T W
T WR)
−1 R
T W
T W∆x.
(3.32)
If we define the weighted orbit response matrix,
R w = WR,
(3.33)
the solution can be rewritten as
θ = (R
T
w R w )
−1 R
T
w W∆x.
(3.34)
Using R w and W∆x in place of R and ∆x, respectively, Eqs. (3.17)-(3.28)
can be used for the weighted BPM case.
In the case of N ≤ M where all singular values of R w are greater than zero
and are used in the calculation of the pseudo inverse matrix, the solution to
the correctors, θ, is not changed by the weighting factors. However, if N > M ,
or if not all singular values of R w are used, then the weighting factors can
change the solution to put more emphasis on the BPMs with higher weights.
One application of weighting BPMs in orbit correction is to make large
local bumps. To make an orbit bump on a single BPM, the target orbit,
∆x, is set to zero on all BPMs except the target BPM, which is set to the
desired value. Eq. (3.34) can then be used to calculate the corrector changes.
If the adjacent BPMs are very close to the target BPM, it could be difficult
to make an exact local bump. There are times when we only want to make
a large bump at the target, and are not very concerned of the small orbit
distortion on the nearby BPMs. In such cases, we can reduce the weights on
the adjacent BPMs. Figure 3.7 shows an example, in which we want to make
a 1 mm horizontal orbit bump on one of the BPMs in SPEAR3. If all BPMs
have the same weights, the solution found with 50 SVs requires kick angles
as large as 0.25 mrad. Because the maximum kick angle for one corrector is
1.5 mrad, the maximum bump would be 6 mm. However, if we set the weights
of the four nearby BPMs (two on each side) to 20% of the other BPMs, the
maximum required kick angle is only 0.125 mrad; the bump on the target
BPM can now reach 12 mm.
It is worth pointing out that in the above example, if all 56 SVs are used
in the calculation, there will be no difference between the solutions of the
no-weight and with-weight cases. It can make a difference if the weights of the
nearby BPMs are set to zero, though.
BPM i. Additional weight factor could be applied to emphasize the importance
of the orbit at certain locations.
In terms of the residual vector, the objective function can now be written
f (θ) = r
T W
T Wr,
(3.30)
with the diagonal matrix
W = diag(w 1 , w 2 , · · · , w M ).
(3.31)
Following the same process leading up to Eq. (3.16), the desired corrector
change is found to be
θ = (R
T W
T WR)
−1 R
T W
T W∆x.
(3.32)
If we define the weighted orbit response matrix,
R w = WR,
(3.33)
the solution can be rewritten as
θ = (R
T
w R w )
−1 R
T
w W∆x.
(3.34)
Using R w and W∆x in place of R and ∆x, respectively, Eqs. (3.17)-(3.28)
can be used for the weighted BPM case.
In the case of N ≤ M where all singular values of R w are greater than zero
and are used in the calculation of the pseudo inverse matrix, the solution to
the correctors, θ, is not changed by the weighting factors. However, if N > M ,
or if not all singular values of R w are used, then the weighting factors can
change the solution to put more emphasis on the BPMs with higher weights.
One application of weighting BPMs in orbit correction is to make large
local bumps. To make an orbit bump on a single BPM, the target orbit,
∆x, is set to zero on all BPMs except the target BPM, which is set to the
desired value. Eq. (3.34) can then be used to calculate the corrector changes.
If the adjacent BPMs are very close to the target BPM, it could be difficult
to make an exact local bump. There are times when we only want to make
a large bump at the target, and are not very concerned of the small orbit
distortion on the nearby BPMs. In such cases, we can reduce the weights on
the adjacent BPMs. Figure 3.7 shows an example, in which we want to make
a 1 mm horizontal orbit bump on one of the BPMs in SPEAR3. If all BPMs
have the same weights, the solution found with 50 SVs requires kick angles
as large as 0.25 mrad. Because the maximum kick angle for one corrector is
1.5 mrad, the maximum bump would be 6 mm. However, if we set the weights
of the four nearby BPMs (two on each side) to 20% of the other BPMs, the
maximum required kick angle is only 0.125 mrad; the bump on the target
BPM can now reach 12 mm.
It is worth pointing out that in the above example, if all 56 SVs are used
in the calculation, there will be no difference between the solutions of the
no-weight and with-weight cases. It can make a difference if the weights of the
nearby BPMs are set to zero, though.