80 Beam-based Correction and Optimization for Accelerators
and replacing s
−2
i
with zeros in the above for any s i = 0. Then Eqs. (3.23)(3.24) can be used to calculate the solution, θ, in the general case.
The solution Eq. (3.23) can be better understood if we examine the SVD
of matrix R closely. The SVD of R can be rewritten in the expanded form
R =
min(M,N )
i=1
s i u i v
T
i ,
(3.26)
where u i and v i are the i’th column in matrix U and V, respectively. Each
term in the summation corresponds to an SVD mode, which consists of the
singular value, s i , and the u i and v i vectors. The u i vector represents the pattern over all BPMs and the v i vector represents the pattern over all correctors.
The calculation of the predicted orbit shift by a given corrector variation can
be written as
∆x = Rθ =
min(M,N )
i=1
s i u i (v
T
i θ),
(3.27)
where (v
T
i θ) is the dot product of the two vectors v i and θ. This dot product
is a scalar value that represents the projection of the corrector changes to the
i’th SV mode. Eq. (3.27) indicates that the projection v
T
i θ, multiplied by the
vector s i u i , gives the orbit changes due to the i’th SV mode. Since all the u i
and v i vectors are normalized to ||u i || = ||v i || = 1, where || · || represents the
Euclid norm of a vector, the amount of orbit changes with a given projection
to the SV mode is determined by the singular value, s i . A large singular value
means the corrector pattern corresponding to the v i vector is very effective
in changing the orbit (yet only producing changes to BPMs by the pattern as
given by the corresponding u vector). A small singular value means the SV
mode is not effective in making orbit changes. In the extreme case, when the
singular value is zero, the projection of corrector changes in that mode does
not cause any orbit change on the BPMs at all. For a small or zero singular
value, the effects of corrector changes with the particular pattern tend to
cancel, resulting in small or no net orbit shifts on the BPMs.
The SVD of the orbit response matrix reveals that with a given set of orbit
correctors, it is easy to make orbit changes in some patterns, while it is difficult
or impossible to make orbit changes in some other patterns. The orbit response
matrix can be seen as a map from the N -dimensional corrector space to the
M -dimensional BPM space. A vector in the corrector space, θ, is mapped to a
vector in the BPM space, ∆x. The column vectors in the V matrix represent
an orthogonal basis of the corrector space, while the column vectors in the
U matrix represent an orthogonal basis of the BPM space. Corrector changes
along the basis vector v i result in an orbit shift only along the u i vector.
The corresponding singular value, s i , represents the effectiveness of making
orbit changes in this mode. Conversely, an orbit shift in the u i pattern can
only be made through corrector changes in the v i pattern (since corrector
and replacing s
−2
i
with zeros in the above for any s i = 0. Then Eqs. (3.23)(3.24) can be used to calculate the solution, θ, in the general case.
The solution Eq. (3.23) can be better understood if we examine the SVD
of matrix R closely. The SVD of R can be rewritten in the expanded form
R =
min(M,N )
i=1
s i u i v
T
i ,
(3.26)
where u i and v i are the i’th column in matrix U and V, respectively. Each
term in the summation corresponds to an SVD mode, which consists of the
singular value, s i , and the u i and v i vectors. The u i vector represents the pattern over all BPMs and the v i vector represents the pattern over all correctors.
The calculation of the predicted orbit shift by a given corrector variation can
be written as
∆x = Rθ =
min(M,N )
i=1
s i u i (v
T
i θ),
(3.27)
where (v
T
i θ) is the dot product of the two vectors v i and θ. This dot product
is a scalar value that represents the projection of the corrector changes to the
i’th SV mode. Eq. (3.27) indicates that the projection v
T
i θ, multiplied by the
vector s i u i , gives the orbit changes due to the i’th SV mode. Since all the u i
and v i vectors are normalized to ||u i || = ||v i || = 1, where || · || represents the
Euclid norm of a vector, the amount of orbit changes with a given projection
to the SV mode is determined by the singular value, s i . A large singular value
means the corrector pattern corresponding to the v i vector is very effective
in changing the orbit (yet only producing changes to BPMs by the pattern as
given by the corresponding u vector). A small singular value means the SV
mode is not effective in making orbit changes. In the extreme case, when the
singular value is zero, the projection of corrector changes in that mode does
not cause any orbit change on the BPMs at all. For a small or zero singular
value, the effects of corrector changes with the particular pattern tend to
cancel, resulting in small or no net orbit shifts on the BPMs.
The SVD of the orbit response matrix reveals that with a given set of orbit
correctors, it is easy to make orbit changes in some patterns, while it is difficult
or impossible to make orbit changes in some other patterns. The orbit response
matrix can be seen as a map from the N -dimensional corrector space to the
M -dimensional BPM space. A vector in the corrector space, θ, is mapped to a
vector in the BPM space, ∆x. The column vectors in the V matrix represent
an orthogonal basis of the corrector space, while the column vectors in the
U matrix represent an orthogonal basis of the BPM space. Corrector changes
along the basis vector v i result in an orbit shift only along the u i vector.
The corresponding singular value, s i , represents the effectiveness of making
orbit changes in this mode. Conversely, an orbit shift in the u i pattern can
only be made through corrector changes in the v i pattern (since corrector