Orbit and trajectory correction 81
changes in any other pattern result in an orbit shift that is orthogonal to u i ).
Therefore, if we want to make an orbit shift ∆x, we can first calculate the
decomposition of ∆x over the BPM modes, and then use the projection on
each mode to calculate the required corrector changes. In fact, this is exactly
what Eq. (3.23) stands for, which can be re-written as
θ =
min(M,N )
i=1
1
s i
v i (u
T
i ∆x),
(3.28)
where (u
T
i ∆x) is the dot product between u i and ∆x that represents the
projection of the desired orbit shift over the i’th SV mode.
In the case of N > M , SVD of the orbit response matrix finds only M
basis vectors for the N -dimensional corrector space. There is an extra N − M
dimensional subspace of the corrector space that has no impact to the BPM
space. In other words, any corrector changes in this subspace do not result
in an orbit shift observed on the BPMs. The basis vectors of this subspace
are similar to the basis vectors that have zero singular values in terms of not
being able to make orbit changes observable on the BPMs. Because corrector
changes in the extra subspace do not change the orbit, there is no point of
making such corrector changes. This is why we patch zeros to the end in
Eq. (3.25). For the same reason, we do not allow corrector changes along the
v i vectors corresponding to zero singular values. Hence we replace
1
si with 0
when s i = 0 in Eq. (3.25) and Eq. (3.28).
While in both cases corrector changes cannot cause an orbit shift on the
BPMs, there is a subtle difference between the case with a zero singular value
and the case with corrector changes in the N − M dimensional subspace. In
the latter case, there is a redundancy in the correctors for the set of BPMs. In
other words, there are too many correctors and thus there is no unique solution
to the orbit correction least-square problem. However, in the former case, with
a zero singular value, there is a true deficiency in the orbit correction system
such that we lack the ability to make orbit shifts in a certain pattern (i.e., the
corresponding u i vector).
Similarly, when a singular value is very small, our ability to make orbit
shifts in the corresponding BPM pattern is limited. For a small orbit shift in
the pattern, large corrector changes are needed, as is evident in Eq. (3.28) by
the
1
si coefficient. Sometimes a singular value is so small, the required corrector
changes for an orbit correction could exhaust the strengths of some correctors
and cause the orbit correction system to fail. To prevent such a scenario, it is
necessary to limit the corrector changes in the SV modes with small singular
values. This could be done by choosing a threshold for the singular values and
setting
1
si to zero in Eq. (3.28) for all singular values s i below the threshold.
In doing this we give up the attempt to make orbit shifts in certain patterns
in exchange for the stability of the orbit correction system. It is not so much
a loss because our ability to make such orbit shifts is limited in the first place.
changes in any other pattern result in an orbit shift that is orthogonal to u i ).
Therefore, if we want to make an orbit shift ∆x, we can first calculate the
decomposition of ∆x over the BPM modes, and then use the projection on
each mode to calculate the required corrector changes. In fact, this is exactly
what Eq. (3.23) stands for, which can be re-written as
θ =
min(M,N )
i=1
1
s i
v i (u
T
i ∆x),
(3.28)
where (u
T
i ∆x) is the dot product between u i and ∆x that represents the
projection of the desired orbit shift over the i’th SV mode.
In the case of N > M , SVD of the orbit response matrix finds only M
basis vectors for the N -dimensional corrector space. There is an extra N − M
dimensional subspace of the corrector space that has no impact to the BPM
space. In other words, any corrector changes in this subspace do not result
in an orbit shift observed on the BPMs. The basis vectors of this subspace
are similar to the basis vectors that have zero singular values in terms of not
being able to make orbit changes observable on the BPMs. Because corrector
changes in the extra subspace do not change the orbit, there is no point of
making such corrector changes. This is why we patch zeros to the end in
Eq. (3.25). For the same reason, we do not allow corrector changes along the
v i vectors corresponding to zero singular values. Hence we replace
1
si with 0
when s i = 0 in Eq. (3.25) and Eq. (3.28).
While in both cases corrector changes cannot cause an orbit shift on the
BPMs, there is a subtle difference between the case with a zero singular value
and the case with corrector changes in the N − M dimensional subspace. In
the latter case, there is a redundancy in the correctors for the set of BPMs. In
other words, there are too many correctors and thus there is no unique solution
to the orbit correction least-square problem. However, in the former case, with
a zero singular value, there is a true deficiency in the orbit correction system
such that we lack the ability to make orbit shifts in a certain pattern (i.e., the
corresponding u i vector).
Similarly, when a singular value is very small, our ability to make orbit
shifts in the corresponding BPM pattern is limited. For a small orbit shift in
the pattern, large corrector changes are needed, as is evident in Eq. (3.28) by
the
1
si coefficient. Sometimes a singular value is so small, the required corrector
changes for an orbit correction could exhaust the strengths of some correctors
and cause the orbit correction system to fail. To prevent such a scenario, it is
necessary to limit the corrector changes in the SV modes with small singular
values. This could be done by choosing a threshold for the singular values and
setting
1
si to zero in Eq. (3.28) for all singular values s i below the threshold.
In doing this we give up the attempt to make orbit shifts in certain patterns
in exchange for the stability of the orbit correction system. It is not so much
a loss because our ability to make such orbit shifts is limited in the first place.