78 Beam-based Correction and Optimization for Accelerators
and hence the residual vector will become
r = x 0 + Rθ − ˆ
x = Rθ − ∆x,
(3.12)
where ∆x ≡ ˆ
x−x 0 is the difference between the target orbit and the measured
orbit. The objective of the least-square problem, χ
2 , is a function of θ,
f (θ) = χ
2 = (θ
T R
T − ∆x
T )(Rθ − ∆x).
(3.13)
The condition for f (θ) to be at a minimum is for its derivatives with all
variables θ j to be zero, i.e.,
∂f
∂θ j
= 2
N
k=1
(R
T R) jk θ k − 2
M
i=1
R ij ∆x i = 0,
(3.14)
for j = 1, 2, · · · , N . This can be written in the vector form
∂f
∂θ
= 2R
T Rθ − 2R
T ∆x = 0.
(3.15)
The solution for θ is thus
θ = (R
T R)
−1 R
T ∆x.
(3.16)
3.3.1 Orbit correction with SVD
Eq. (3.16) can be used to calculate the desired changes to the corrector magnets for orbit correction when the orbit errors and the orbit response matrix
are known. The matrix R
T R is N × N in dimension. Eq. (3.16) gives a unique
solution for θ if and only if the inverse matrix of R
T R exists. In reality, this
might not be the case. For example, when the number of correctors is larger
than the number of BPMs, matrix R
T R has a rank M < N and does not
have an inverse matrix. Even if mathematically the inverse matrix exists, the
solution by Eq. (3.16) might not always be appropriate for orbit correction
since matrix R
T R could be nearly degenerate, i.e., some of its eigenvalues are
very close to zero. In such a case, the solution would be very sensitive to noise
in the orbit measurement and can result in large corrector changes in response
to small orbit errors. Therefore, a method to solve the least-square problem
in Eq. (3.13) with considerations of the realistic conditions is needed.
Singular value decomposition (SVD) [42] of the orbit response matrix provides the answer to the above challenge [21, 23]. The method based on SVD
to solve the least-square problem is very powerful and is widely applied to
orbit corrections in accelerators. The SVD of the orbit response matrix is in
the form of
R = USV
T ,
(3.17)
and hence the residual vector will become
r = x 0 + Rθ − ˆ
x = Rθ − ∆x,
(3.12)
where ∆x ≡ ˆ
x−x 0 is the difference between the target orbit and the measured
orbit. The objective of the least-square problem, χ
2 , is a function of θ,
f (θ) = χ
2 = (θ
T R
T − ∆x
T )(Rθ − ∆x).
(3.13)
The condition for f (θ) to be at a minimum is for its derivatives with all
variables θ j to be zero, i.e.,
∂f
∂θ j
= 2
N
k=1
(R
T R) jk θ k − 2
M
i=1
R ij ∆x i = 0,
(3.14)
for j = 1, 2, · · · , N . This can be written in the vector form
∂f
∂θ
= 2R
T Rθ − 2R
T ∆x = 0.
(3.15)
The solution for θ is thus
θ = (R
T R)
−1 R
T ∆x.
(3.16)
3.3.1 Orbit correction with SVD
Eq. (3.16) can be used to calculate the desired changes to the corrector magnets for orbit correction when the orbit errors and the orbit response matrix
are known. The matrix R
T R is N × N in dimension. Eq. (3.16) gives a unique
solution for θ if and only if the inverse matrix of R
T R exists. In reality, this
might not be the case. For example, when the number of correctors is larger
than the number of BPMs, matrix R
T R has a rank M < N and does not
have an inverse matrix. Even if mathematically the inverse matrix exists, the
solution by Eq. (3.16) might not always be appropriate for orbit correction
since matrix R
T R could be nearly degenerate, i.e., some of its eigenvalues are
very close to zero. In such a case, the solution would be very sensitive to noise
in the orbit measurement and can result in large corrector changes in response
to small orbit errors. Therefore, a method to solve the least-square problem
in Eq. (3.13) with considerations of the realistic conditions is needed.
Singular value decomposition (SVD) [42] of the orbit response matrix provides the answer to the above challenge [21, 23]. The method based on SVD
to solve the least-square problem is very powerful and is widely applied to
orbit corrections in accelerators. The SVD of the orbit response matrix is in
the form of
R = USV
T ,
(3.17)