Orbit and trajectory correction 77
In the earlier days of accelerators, the ability to control beam orbit was
limited for reasons such as the lack of orbit monitors and corrector magnets and the lack of effective orbit correction schemes. Global orbit correction
could only reduce the overall orbit distortion to a certain level. Local orbit
correction was often used to steer the beam at important locations such as the
interaction region for a collider, or the source points of light sources. Presently
accelerators are equipped with many high precision BPMs and fast and well
regulated corrector magnets. The calculation of corrector strengths from the
measured orbit errors in global orbit correction with the singular value decomposition (SVD) has been very successful. The beam orbit can often be
precisely controlled on target by a global orbit feedback system. Creating a
local orbit bump can be as easy as specifying the desired orbit and leaving the
rest to the orbit feedback. Local orbit bump can also be easily created using
the orbit response matrix with the SVD method. Therefore, we will be focused
on the SVD-based global orbit correction method in the following. Early day
global orbit correction methods and local orbit correction are discussed briefly
toward the end of the section.
There are typically multiple BPMs and multiple correctors, e.g., with M
BPMs and N correctors. All three situations, M > N , M = N , or M < N ,
are possible. When there are more BPMs than correctors (i.e., M > N ), the
system is over-constrained; the orbit target may not be met on all BPMs. As
will be discussed later, because of the potential degeneracy in the system, even
when M ≤ N , it is possible that the orbit errors will not be completely eliminated. Therefore, in general, the objective of orbit correction is to minimize
the difference between the measured beam orbit and the orbit target in the
least square sense, i.e., to minimize
χ
2 =
M
i=1
(x i − ˆ
x i )
2 ,
(3.8)
where x i and ˆ
x i are the measured orbit and the target orbit on BPM i, respectively. Defining the corresponding column vectors x and ˆ
x and the residual
vector,
r = x − ˆ
x,
(3.9)
the objective function can be expressed as
χ
2 = r
T r.
(3.10)
When the orbit is measured to be x 0 , we need to find the desired changes
to the corrector magnets, θ, to minimize the objective function, χ
2 . Here θ
is a column vector with N elements, whose j’th element, θ j , represents the
desired change of kick angle on corrector j. After the correction θ is applied
to the machine, the orbit will change. Using the orbit response matrix, the
new orbit is predicted to be
x = x 0 + Rθ,
(3.11)
In the earlier days of accelerators, the ability to control beam orbit was
limited for reasons such as the lack of orbit monitors and corrector magnets and the lack of effective orbit correction schemes. Global orbit correction
could only reduce the overall orbit distortion to a certain level. Local orbit
correction was often used to steer the beam at important locations such as the
interaction region for a collider, or the source points of light sources. Presently
accelerators are equipped with many high precision BPMs and fast and well
regulated corrector magnets. The calculation of corrector strengths from the
measured orbit errors in global orbit correction with the singular value decomposition (SVD) has been very successful. The beam orbit can often be
precisely controlled on target by a global orbit feedback system. Creating a
local orbit bump can be as easy as specifying the desired orbit and leaving the
rest to the orbit feedback. Local orbit bump can also be easily created using
the orbit response matrix with the SVD method. Therefore, we will be focused
on the SVD-based global orbit correction method in the following. Early day
global orbit correction methods and local orbit correction are discussed briefly
toward the end of the section.
There are typically multiple BPMs and multiple correctors, e.g., with M
BPMs and N correctors. All three situations, M > N , M = N , or M < N ,
are possible. When there are more BPMs than correctors (i.e., M > N ), the
system is over-constrained; the orbit target may not be met on all BPMs. As
will be discussed later, because of the potential degeneracy in the system, even
when M ≤ N , it is possible that the orbit errors will not be completely eliminated. Therefore, in general, the objective of orbit correction is to minimize
the difference between the measured beam orbit and the orbit target in the
least square sense, i.e., to minimize
χ
2 =
M
i=1
(x i − ˆ
x i )
2 ,
(3.8)
where x i and ˆ
x i are the measured orbit and the target orbit on BPM i, respectively. Defining the corresponding column vectors x and ˆ
x and the residual
vector,
r = x − ˆ
x,
(3.9)
the objective function can be expressed as
χ
2 = r
T r.
(3.10)
When the orbit is measured to be x 0 , we need to find the desired changes
to the corrector magnets, θ, to minimize the objective function, χ
2 . Here θ
is a column vector with N elements, whose j’th element, θ j , represents the
desired change of kick angle on corrector j. After the correction θ is applied
to the machine, the orbit will change. Using the orbit response matrix, the
new orbit is predicted to be
x = x 0 + Rθ,
(3.11)