Application of beam-based optimization 215
The SPEAR3 storage ring has 72 sextupole magnets, each of them has
a set of skew quadrupole coils. Only 15 of these skew quadrupoles are powered. Not using the two skew quadrupoles within the injection kicker bump
(see Section 8.2), there are 13 skew quadrupoles available for vertical emittance minimization. In simulation, skew quadrupole errors are added to 29
sextupoles, not including the 13 knobs. The coupling ratio (i.e., r = y // x ) is
0.88% initially. Assuming that for a 500 mA total beam current, the gas scattering lifetime is 40 hrs and the Touschek lifetime is 10 hrs for the coupling
ratio of 0.2%, the total lifetime can be calculated with
1
τ
[hr] =
1
40
+
I
10I 0
0.002
r
.
(8.4)
The current loss over ∆t is calculated with ∆I = −I
∆t
τ +
√
2σ I ξ, where σ I is
the error sigma of beam current measurement, ξ is a random number drawn
from the Gaussian distribution, N (0, 1). The random term is added to simulate
the measurement error. The objective function is calculated with Eq. (8.3),
using I 0 = 500 mA and scaled to the beam loss over one minute. The loss
rate for the initial lattice is −0.61 mA/min. Assuming σ I = 0.002 µA, and for
the beam current change measured over a period of ∆t = 6 seconds, the noise
sigma for the objective function is 0.03 mA/min.
For the application of the RCDS method, it is preferable to provide a conjugate direction set as the initial directions. Although the method has the
ability to build up a conjugate direction set from the convergence history, it
takes many iterations to replace all the directions. In online applications, there
is typically not enough time to wait for the conjugate direction set to emerge.
The initial conjugate direction set may be calculated with a model. The final
direction sets of past RCDS runs may also be used. For the vertical emittance
minimization problem, we use the Jacobian matrix of the orbit response matrix fitting with respect to the skew quadrupoles to calculate the conjugate
directions. Each column of the Jacobian matrix consists of the derivatives of all
orbit response matrix elements (including dispersion functions in both planes)
with respect to one skew quadrupole. The off-diagonal matrix elements and
the vertical dispersion dominate the response of skew quadrupoles. Because
the Jacobian matrix contains sufficient information of the dependence of the
distributions of the linear coupling and the vertical dispersion on the skew
quadrupoles, it can represent the functional dependence of the vertical emittance on the skew quadrupoles. By the use of singular value decomposition
(SVD) of the Jacobian matrix, J = USV
T , the conjugate direction set, given
by the column vectors in matrix V, can be found. The combined knobs represented by the directions in V are ordered by the sensitivity in changing the
linear coupling and the vertical dispersion, and in turn the vertical emittance.
The left plot in Figure 8.6 shows the singular values of the Jacobian matrix
and the square root of the coupling ratio caused by a fixed step in each knob.
The right plot shows the first two conjugate directions.
The SPEAR3 storage ring has 72 sextupole magnets, each of them has
a set of skew quadrupole coils. Only 15 of these skew quadrupoles are powered. Not using the two skew quadrupoles within the injection kicker bump
(see Section 8.2), there are 13 skew quadrupoles available for vertical emittance minimization. In simulation, skew quadrupole errors are added to 29
sextupoles, not including the 13 knobs. The coupling ratio (i.e., r = y // x ) is
0.88% initially. Assuming that for a 500 mA total beam current, the gas scattering lifetime is 40 hrs and the Touschek lifetime is 10 hrs for the coupling
ratio of 0.2%, the total lifetime can be calculated with
1
τ
[hr] =
1
40
+
I
10I 0
0.002
r
.
(8.4)
The current loss over ∆t is calculated with ∆I = −I
∆t
τ +
√
2σ I ξ, where σ I is
the error sigma of beam current measurement, ξ is a random number drawn
from the Gaussian distribution, N (0, 1). The random term is added to simulate
the measurement error. The objective function is calculated with Eq. (8.3),
using I 0 = 500 mA and scaled to the beam loss over one minute. The loss
rate for the initial lattice is −0.61 mA/min. Assuming σ I = 0.002 µA, and for
the beam current change measured over a period of ∆t = 6 seconds, the noise
sigma for the objective function is 0.03 mA/min.
For the application of the RCDS method, it is preferable to provide a conjugate direction set as the initial directions. Although the method has the
ability to build up a conjugate direction set from the convergence history, it
takes many iterations to replace all the directions. In online applications, there
is typically not enough time to wait for the conjugate direction set to emerge.
The initial conjugate direction set may be calculated with a model. The final
direction sets of past RCDS runs may also be used. For the vertical emittance
minimization problem, we use the Jacobian matrix of the orbit response matrix fitting with respect to the skew quadrupoles to calculate the conjugate
directions. Each column of the Jacobian matrix consists of the derivatives of all
orbit response matrix elements (including dispersion functions in both planes)
with respect to one skew quadrupole. The off-diagonal matrix elements and
the vertical dispersion dominate the response of skew quadrupoles. Because
the Jacobian matrix contains sufficient information of the dependence of the
distributions of the linear coupling and the vertical dispersion on the skew
quadrupoles, it can represent the functional dependence of the vertical emittance on the skew quadrupoles. By the use of singular value decomposition
(SVD) of the Jacobian matrix, J = USV
T , the conjugate direction set, given
by the column vectors in matrix V, can be found. The combined knobs represented by the directions in V are ordered by the sensitivity in changing the
linear coupling and the vertical dispersion, and in turn the vertical emittance.
The left plot in Figure 8.6 shows the singular values of the Jacobian matrix
and the square root of the coupling ratio caused by a fixed step in each knob.
The right plot shows the first two conjugate directions.