104 Beam-based Correction and Optimization for Accelerators
The initial BPM and corrector gains are all set to unity. The initial BPM and
corrector coupling coefficients are set to zero.
Suppose the residual vector corresponding to p 0 is r 0 , the residual vector
after the fitting parameters are changed to p = p 0 + ∆p can be linearly
expanded to give
r = r 0 + J∆p, with J ij =
∂r i
∂p j
,
(4.28)
where J is the Jacobian matrix of the residual vector with respect to the
parameter vector p, whose matrix elements are as defined in the above. Each
column of J represents the differential impact of the corresponding fitting
parameter to the residual vector. The objective function can also be expanded
around p 0 , which is approximately given by
f (p) ≈ f (p 0 ) + 2r
T
0 J∆p + ∆p
T J
T J∆p,
(4.29)
where we neglected the quadratic and higher order dependence of r on the
fitting parameters. At the present solution, p 0 , the gradient is 2J
T r 0 and the
approximate Hessian matrix is given by 2J
T J. Because the gradient of the
objective function, ∇ p f (p) = 2J
T r 0 + 2J
T J∆p, is zero at a minimum of
f (p), the solution for the step change toward the minimum from the present
solution is found to be
∆p = −(J
T J)
−1 J
T r 0 .
(4.30)
This solution is the same as Eq. (3.16), here with the Jacobian matrix in
place of the orbit response matrix. In fact, the orbit response matrix is the
Jacobian matrix of the beam orbit with respect to the corrector strengths.
Solving least-square problems by iteratively applying Eq. (4.30) is called the
Gauss-Newton method.
Similar to the orbit correction case, Eq. (4.30) is applicable only if the
square matrix J
T J is invertible. This condition is equivalent to require that
all singular values of J be nonzero, with the SVD of the Jacobian matrix
given in the form, J = USV
T . However, in general, this condition is not met.
Consider a simple case without roll and crunch errors to BPMs or correctors.
If the residual vector contains only the orbit response matrix terms (i.e., not
including the dispersion terms), the Jacobian matrix is degenerate with two
zero singular values, one for each transverse plane. The degeneracy comes
from the fact that if all corrector gains and all BPM gains of the same plane
are raised by a common factor, the measured orbit response matrix does not
change (since in R ij = ∆x i /∆θ j both the nominator and denominator change
by the same ratio). Including the dispersion terms in the residual vector alleviates the degeneracy because these terms are not dependent on the corrector
gains and RF frequency measurement is very accurate. It is more helpful in
the horizontal plane as the orbit shift due to horizontal dispersion is much
larger than the spurious vertical dispersion.
The initial BPM and corrector gains are all set to unity. The initial BPM and
corrector coupling coefficients are set to zero.
Suppose the residual vector corresponding to p 0 is r 0 , the residual vector
after the fitting parameters are changed to p = p 0 + ∆p can be linearly
expanded to give
r = r 0 + J∆p, with J ij =
∂r i
∂p j
,
(4.28)
where J is the Jacobian matrix of the residual vector with respect to the
parameter vector p, whose matrix elements are as defined in the above. Each
column of J represents the differential impact of the corresponding fitting
parameter to the residual vector. The objective function can also be expanded
around p 0 , which is approximately given by
f (p) ≈ f (p 0 ) + 2r
T
0 J∆p + ∆p
T J
T J∆p,
(4.29)
where we neglected the quadratic and higher order dependence of r on the
fitting parameters. At the present solution, p 0 , the gradient is 2J
T r 0 and the
approximate Hessian matrix is given by 2J
T J. Because the gradient of the
objective function, ∇ p f (p) = 2J
T r 0 + 2J
T J∆p, is zero at a minimum of
f (p), the solution for the step change toward the minimum from the present
solution is found to be
∆p = −(J
T J)
−1 J
T r 0 .
(4.30)
This solution is the same as Eq. (3.16), here with the Jacobian matrix in
place of the orbit response matrix. In fact, the orbit response matrix is the
Jacobian matrix of the beam orbit with respect to the corrector strengths.
Solving least-square problems by iteratively applying Eq. (4.30) is called the
Gauss-Newton method.
Similar to the orbit correction case, Eq. (4.30) is applicable only if the
square matrix J
T J is invertible. This condition is equivalent to require that
all singular values of J be nonzero, with the SVD of the Jacobian matrix
given in the form, J = USV
T . However, in general, this condition is not met.
Consider a simple case without roll and crunch errors to BPMs or correctors.
If the residual vector contains only the orbit response matrix terms (i.e., not
including the dispersion terms), the Jacobian matrix is degenerate with two
zero singular values, one for each transverse plane. The degeneracy comes
from the fact that if all corrector gains and all BPM gains of the same plane
are raised by a common factor, the measured orbit response matrix does not
change (since in R ij = ∆x i /∆θ j both the nominator and denominator change
by the same ratio). Including the dispersion terms in the residual vector alleviates the degeneracy because these terms are not dependent on the corrector
gains and RF frequency measurement is very accurate. It is more helpful in
the horizontal plane as the orbit shift due to horizontal dispersion is much
larger than the spurious vertical dispersion.