12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

MODIFIED FAST ORBIT FEEDBACK CONTROLLER FOR DISTURBANCEATTENUATION IN LONG STRAIGHTS FOR DIAMOND-II ∗S. Banerjee1,† , I. Kempf1,2 , M. Abbott1 , L. Bobb1 ,1 Diamond Light Source, Didcot, United Kingdom
2 University of Oxford, Oxford, United Kingdom

Abstract
At Diamond Light Source, the fast orbit feedback (FOFB)
uses one array of correctors and the controller is designed
using the internal model control (IMC) structure. The
Diamond-II upgrade will introduce an additional array offast correctors and a new controller that is designed using
the generalised modal decomposition, increasing the overall closed-loop bandwidth from 140 Hz to 1 kHz. Althoughsimulation results have shown that the resulting beam displacement is within specification in all straights, they havealso shown that the performance on long straights is limited,particularly in the vertical plane. In this paper, the controller
is tuned in order to increase the FOFB performance in longstraights by introducing a mode-by-mode regularisation parameter. The performance of the controller beyond 1 kHz is
assessed using new disturbance data and a new measurement
noise model, showing that the Diamond-II performance criteria are met, even in the presence of measurement noise.

INTRODUCTION
At Diamond Light Source (Diamond), the fast orbit feedback (FOFB) attenuates disturbances in the storage ring andreduces the root-mean square deviation of the electron beam
to ≤10 % of the beam size up to 140 Hz in both planes. The
FOFB uses 173 beam position monitors (BPMs) and 172
identical corrector magnets, and the control algorithm isbased on the internal model control (IMC) structure, whichis naturally amenable to systems with large time delays. The
controller is designed using the modal decomposition, which
decouples the multi-input multi-output (MIMO) systems into
sets of single-input single-output systems using the singularvalue decomposition (SVD) of the orbit response matrix(ORM) [1].
Due to advances in detector speed and resolution at
Diamond-II [2], the beam stability requirements are raised to
3 % of the beam size up to 1 kHz, such as shown in Table 1.
The increased closed-loop bandwidth in turn requires to introduce an additional corrector type. The FOFB at DiamondII will use 252 slow correctors with a bandwidth of ∼ 200 Hz
and 144 fast correctors with a bandwidth of ∼ 8 kHz. To
accommodate the fast correctors, the Diamond FOFB controller has been extended and adapted to the Diamond-II
configuration [3]. Analogous to Diamond, the MIMO system is decoupled into sets of two-input single-output (TISO)
and SISO systems using the generalised singular value decomposition (GSVD), which is a two-matrix factorisation∗ Work supported by Diamond Light Source† Corresponding author: shohan.banerjee@diamond.ac.uk.

Table 1: Beam Size, Relative and Absolute Orbit Stability
Requirements at Standard Straight Source Points [2].
Parameter

DiamondBeam size H/V
123 µm/3.5 µm
Rel. stability
10 % up to 100 Hz
Abs. stability H/V 12 µm/0.35 µm

Diamond-II
30 µm/4 µm
3 % up to 1 kHz
0.9 µm/0.12 µm

H: horizontal, V: vertical, BW: bandwidth

technique [4]. The decoupled systems are then embedded in
the IMC structure and the controllers designed using midranging control [5]. In addition, a tunable regularisation
matrix is included in the feedback loop to avoid large control
inputs and increase the robustness of the controller.
As part of the Diamond-II design, the performance ofthe new control algorithm has been assessed using estimates of the Diamond-II disturbance [6]. Although thishas shown that the Diamond-II controller meets the beamstability requirements, it has also shown that the performancefor primary BPMs on the long straights of the new multibend achromat (MBA) lattice is worse than on the standard
straights and mid straights. To reduce this performance difference, this paper extends the approach presented in [3] byintroducing mode-based weights for the regularisation matrix, increasing the control effort for primary BPMs on longstraights. The performance of the FOFB is then reassessed,
considering both disturbance and a new measurement noisemodel.

DISTURBANCEThe Diamond-II disturbance data presented in Ref. [6] hasbeen extended from 1 kHz to 2.44 kHz. The data includes
power spectral density (PSD) estimates of ground and girder
vibrations, RF and power supply noise, and is scaled using
the local beta function to obtain the disturbance profile ateach BPM, such as shown in Fig. 1 for the upstream primary
BPM of the first standard straight, which is comparable toprimary BPMs in other straight.
Note that the PSDs from Fig. 1 lack phase information,
which complicates the analysis of the MIMO system. Since
the ORMs are ill-conditioned, most of the measured disturbance is concentrated in the modes associated with large
singular values of the ORMs. However, this is not reflectedin the PSDs from Fig. 1, and therefore impacts the performance estimation of the MIMO system.

MOP04108 Feedback Systems and Beam Stability

119

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 1

PSD (µm2 /Hz)

CONTROLLER
The electron beam dynamics at Diamond-II are𝑦(𝑠) = 𝑅s 𝑔s (𝑠)𝑢 s (𝑠) + 𝑅f 𝑔f (𝑠)𝑢 f (𝑠) + 𝑑 (𝑠),

10−4
SS

10−7

LS

100

MS

101
102
Frequency (Hz)103

Vertical Direction

10−2
PSD (µm2 /Hz)

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 1

Horizontal Direction

10−1

10−5
SS

10−8

100

LS

MS

101
102
Frequency (Hz)103

Figure 1: Estimated power spectral densitiy (PSD) of theuncorrected horizontal and vertical disturbance for DiamondII at primary BPMs on long straights (LS), standard straights(SS), and mid straights (MS). The values are averaged over
upstream and downstream BPMs.

PSD (µm2 /Hz)

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

10−5

(1)

where the 𝑠 is the Laplace variable and the subscripts s andf refer to slow and fast. The vectors 𝑦(𝑠), 𝑢 (·) (𝑠) and 𝑑 (𝑠)represent the beam position, the corrector inputs, and the
disturbances, respectively, and the matrices 𝑅s ∈ R𝑛𝑦 ×𝑛s
and 𝑅f ∈ R𝑛𝑦 ×𝑛f with 𝑛 𝑦 = 𝑛s = 252 and 𝑛f = 144 theORMs for the slow and fast correctors, which are obtained
from selecting the corresponding columns of a larger ORM
𝑅 ∈ R𝑛𝑦 × (𝑛s +𝑛f ) . The scalar transfer functions 𝑔s (𝑠) and𝑔f (𝑠) model the temporal dynamics of the correctors, andinclude the frequency responses of the DAC, ADC, signal
processing filters, power supplies, magnets, and copper vessel for slow correctors and stainless steel and copper vessel
for fast correctors.
The IMC structure used for Diamond-II is shownin Fig. 3, where 𝑃(𝑠) := [𝑅s 𝑔s (𝑠) 𝑅f 𝑔f (𝑠)] is the¯:= [ 𝑅¯s 𝑔¯ s (𝑠) 𝑅¯f 𝑔¯ f (𝑠)] the plant model,
plant, 𝑃(𝑠)𝑄(𝑠) := diag(𝑄 s (𝑠), 𝑄 f (𝑠)) the IMC filter, 𝑢(𝑠) :=[𝑢 s (𝑠) T 𝑢 f (𝑠) T ] T , and 𝑛(𝑠) is the measurement noise. Notethat 𝑛(𝑠) affects 𝑦(𝑠) when the control loop is closed, i.e.
when 𝑢s(𝑠) and 𝑢f (𝑠) in Eq. (1) are replaced with a control
law that uses the measured position 𝑦(𝑠) + 𝑛(𝑠).
The IMC filters invert the plant and are defined as
𝑄 (·) (𝑠) := 𝑃†(·) (𝑠)𝑇(·) (𝑠), (·) = {s, f}, where it is assumed¯
that 𝑃(𝑠)= 𝑃(𝑠) and 𝑇(·) (𝑠) are tunable transfer functions designed in the companion paper [8]. The matrixΥ ∈ R (𝑛s +𝑛f ) ×𝑛𝑦 is used to accommodate systems with fewer
slow than fast correctors, and the matrix Γ ∈ R𝑛𝑦 ×𝑛𝑦 is theregularisation matrix that is defined asΓ := (𝑅𝑅 T + 𝜇𝐼) 91 𝑅𝑅 T
= 𝑈 diag

10−6

100

101
102
Frequency (Hz)103

Figure 2: Power spectral densitiy (PSD) obtained from the
Diamond-II measurement noise model.

MEASUREMENT NOISEThe FOFB at Diamond-II will also use new BPMs [7],and this reassessment includes a new measurement noisemodel. The PSD of the measurement noise is assumed
to√2 /𝜔2 + 𝛽2 , where 𝛼 := 0.5 nm/ Hz and
be 𝑁 (𝜔) :=𝛼√
𝛽 := 2 nm/ Hz are chosen based on the BPM specifications
for Diamond-II. Figure 2 shows the resulting PSD, which isused for both horizontal and vertical directions. Comparing
with Fig. 1, the PSD of the measurement noise is several
orders of magnitude lower than the PSD of the disturbancefor frequencies below 100 Hz, of similar magnitude between
100 Hz and 1 kHz, and represents a significant contribution
from 1 kHz on.

𝜎12
𝜎12 + 𝜇

𝜎𝑛2𝑦
,...,

𝜎𝑛2𝑦 + 𝜇

!

𝑈T

(2)

where 𝜇 ≥ 0.1 the regularisation parameter and the SVD
𝑅 = 𝑈Σ𝑉 T was substituted for 𝑅. Given 𝑇(·) (𝑠), the choiceof 𝜇 influences both closed-loop bandwidth and corrector
demand. A large value of 𝜇 (𝜇 ≫ 𝜎𝑛𝑦 ) reduces both closedloop bandwidth and corrector demand, but a small valueof 𝜇 can yield large corrector inputs and even an unstableclosed-loop [9]. The choice 𝜇 = 0.1 results in a maximum
fast corrector demand of 276 mA, which is below the limitof 920 mA [2].

MODE-BY-MODE REGULARISATIONAs for Diamond, the Diamond-II ORMs are illconditioned, i.e. the ratio of largest to smallest singularvalue is large. Because the IMC filters invert the plant, thecorrector inputs are proportional to 𝑅 †(·) , producing large
corrector inputs for disturbances in the direction of standard
singular vectors associated with small singular values of𝑅 †(·) . While the regularisation matrix Γ is used to reduce the

MOP041120

08 Feedback Systems and Beam Stability12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

Table 2: Targets and expected IBMs up to 1 kHz for Diamond-II at primary BPMs on long straights (LS), mid straights(MS), and standard straights (SS). The values are averaged over upstream and downstream BPMs.
Noise

Horizontal IBM (µm)

𝝁𝒊TargetVertical IBM (µm)

LS

MS

SS

LS

MS

SS

1.20

0.90

0.97

0.23

0.14

0.18

FOFB OFF
Upper bound from Eq. (6)Simulation
Upper bound from Eq. (6)Simulation

No
No
No
No
No

0.1
0.1
𝑓 (𝜃 𝑖 )𝑓 (𝜃 𝑖 )

0.68
0.48
0.17
0.48
0.17

0.46
0.23
0.06
0.23
0.06

0.61
0.35
0.08
0.35
0.08

0.25
0.25
0.17
0.26
0.17

0.20
0.09
0.05
0.09
0.05

0.21
0.12
0.07
0.12
0.07

FOFB OFF
Upper bound from Eq. (6)Simulation
Upper bound from Eq. (6)Simulation

Yes
Yes
Yes
Yes
Yes

0.1
0.1
𝑓 (𝜃 𝑖 )𝑓 (𝜃 𝑖 )

0.69
0.50
0.18
0.50
0.18

0.47
0.26
0.08
0.26
0.08

0.61
0.38
0.09
0.38
0.09

0.26
0.26
0.18
0.26
0.18

0.21
0.12
0.07
0.12
0.07

0.22
0.14
0.09
0.15
0.09

d(s)

Controller
−

Υ

Q(s)

u(s)

Plant

P(s)

+
+

+

n(s)

Γ

Model

P̄(s)

y(s)

+

+
−

Figure 3: IMC structure with corrector setpoints 𝑢(𝑠), disturbances 𝑑 (𝑠), beam positions 𝑦(𝑠), and measurement noise𝑛(𝑠).
corrector inputs, it also reduces the overall bandwidth andtherefore the performance at primary BPMs. To improve theperformance at primary BPMs, Γ from Eq. (2) is redefined
as
!
𝜎𝑛2𝑦
𝜎12
Γ := 𝑈 diag 2,..., 2
𝑈T,
(3)
𝜎1 + 𝜇1
𝜎𝑛𝑦 + 𝜇 𝑛𝑦
where 𝜇1 , . . . , 𝜇 𝑛𝑦 ≥ 0 are mode-based regularisation parameters. The mode-based regularisation parameters aredefined as 𝜇𝑖 = 𝑓 (𝜃 𝑖 ) with
𝜇𝑖 :=𝜇max − 𝜇min
(𝜃 𝑖 − 𝜃 min ) + 𝜇min ,𝜃 max − 𝜃 minfor modes that form a small angle with 𝑤 and hence increasing the controller gain for these modes. For the following
analysis, all primary BPMs are prioritised using Eq. (4), andthese results are compared against a regularisation matrix
with 𝜇𝑖 = 0.1 for all modes.
In general, large controller gains will result in a highclosed-loop bandwidth, whereas small controller gains will
result in a low closed-loop bandwidth. Although the regularisation matrix limits the corrector setpoints, it also reduces
the bandwidth for modes associated with small singular values, and therefore the overall controller performance. However, if most of the disturbance is concentrated in modes
associated with large singular values, the regularisation matrix does not significantly impact the controller performance,such as it is the case for Diamond. Note that this is not thecase for the disturbance data from Fig. 1, which lacks phase
information.

RESULTS AND DISCUSSION
Because the PSDs from Fig. 1 lack phase information, theperformance of the controller is estimated using two different
methods. First, assuming that the measurement noise PSD
𝑁𝑖 (𝜔) and the disturbance PSD 𝐷 𝑖 (𝜔) are uncorrelated, thePSD 𝑌𝑖 (𝜔) of BPM 𝑖 can be upper-bounded bywhere 𝜇min := 0.01 ≤ 𝜇𝑖 ≤ 𝜇max := 0.1, 𝜃 min := min𝑖 𝜃 𝑖 ,𝜃 max := max𝑖 𝜃 𝑖 , and𝜃 𝑖 := acos |𝑤 T𝑈𝑖 |/∥𝑤∥ 2 .
(5)
The variable 𝜃 𝑖 describes the acute angle between 𝑈𝑖 , the𝑖th left singular vector of 𝑅, and a vector 𝑤 ∈ R𝑛𝑦 with ones
at indices of interest and zeroes otherwise. For example, toprioritise primary BPMs, 𝑤 can be assigned ones at indicescorresponding to primary BPMs, lowering the value of 𝜇𝑖(6)

𝑌𝑖 (𝜔) ≤ 𝑌𝑖 (𝜔)| 𝐷 + 𝑌𝑖 (𝜔)| 𝑁 ,

(4)
where

𝑌𝑖 (𝜔)| 𝐷 :=

𝑛𝑦
∑︁

!2
√︁

|𝑆𝑖𝑘 (j𝜔)| 𝐷 𝑘 (𝜔)𝑘=1

𝑌𝑖 (𝜔)| 𝑁 :=

𝑛𝑦
∑︁

!2
√︁

|𝐼 − 𝑆𝑖𝑘 (j𝜔)| 𝑁 𝑘 (𝜔)𝑘=1

and 𝑆𝑖𝑘 (j𝜔) refers to element on row 𝑖 and column 𝑘 of thesensitivity, i.e. the transfer functions from disturbance and

MOP04108 Feedback Systems and Beam Stability

121

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 1

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 1

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

noise to the output, which is defined as
𝑆(𝑠) := 𝐼 − 𝑃(𝑠)𝑄(𝑠)Υ(𝐼 + (Γ − 𝐼)𝑃(𝑠)𝑄(𝑠)Υ) 91 Γ. (7)Second, the PSDs from Fig. 1 can be used to sample atime series for each BPM by adding a random phase, and thetime series can be used in simulations. Although the resulting performance does not represent a strict upper bound, theresults can be interpreted as a worst-case scenario. Because
the Diamond-II disturbance does not reflect the characteristic distribution in modal space, the regularisation matrix
significantly reduces the overall performance.
The integrated beam motions (IBMs) up to 1 kHz obtained
from the upper bound using Eq. (6) and the simulations for
𝜇𝑖 = 𝑓 (𝜃 𝑖 ) and 𝜇𝑖 = 0.1 are summarised in Table 2, which
also shows the corresponding beam stability target. The firstpart of the table shows the performance without noise, sothat FOFB OFF corresponds to the IBM of the disturbance.
For both choices of 𝜇𝑖 , the simulation yields significantly
better results than the upper bound from Eq. (6). Except for
the long straights in the vertical direction, the upper bound is
below target, suggesting that the Diamond-II stability specifications are met. Although the primary BPMs are prioritised,
the choice 𝜇𝑖 = 𝑓 (𝜃 𝑖 ) does not improve the performance,
but increases the upper bound from Eq. (6), and leaves thesimulation results unchanged. On one hand, this is related to
the lack of phase information for the disturbance data from
Fig. 1. On the other, these results suggest that decreasing 𝜇𝑖
below 𝜇max does not significantly affect the regularisation
gains from Eq. (3) and the closed-loop bandwidth, i.e. both𝜇max and 𝜇min in Eq. (4) would need to be lowered further toyield a performance increase on primary BPMs. However,this would also increase the corrector demand above the
Diamond-II limit of 920 mA.
The second half of Table 2 shows the performance in thepresence of measurement noise, so that FOFB OFF corresponds to the IBM obtained after adding 𝐷 (𝜔) and 𝑁 (𝜔).
Compared to the case without noise, the IBM for FOFB
OFF is ∼ 0.01 µm larger. As for the case without noise, thesimulations yield significantly better results than the upperbound from Eq. (6), and the different choices of 𝜇𝑖 do not
impact the performance. Except for the long straights thatare 0.03 µm above target in the vertical direction, the upperbounds from Eq. (6) is within target.

CONCLUSION
With the aim of improving the FOFB performance at primary BPMs, a new regularisation matrix was defined using
mode-by-mode regularisation parameters. The regularisation parameters were reduced for those modes that impact
the primary BPMs most. The performance of the resulting
controller was evaluated using a conservative upper boundand in simulations, and compared against a controller thatuses a constant regularisation parameter. Two scenarios wereconsidered, one with measurement noise and disturbance,and one with disturbance only. The measurement noise wasmodelled based on the specifications for the new Diamond-II
electron BPMs.
The simulation results have shown that the Diamond-II
specifications are met on all straights for both scenarios,
whereas with the upper bound, the IBM on long straights
is 0.03 µm above target. For both simulations and upperbounds, the new mode-by-mode regularisation matrix has
not led to any performance improvements. It remains unclear
why, but it is suspected that the lack of phase information for
the Diamond-II disturbance data impacts the regularisationand the resulting performance.
On-going work at Diamond is reassessing the slow andfast corrector demand using the new disturbance data andthe new measurement model. In addition, the fast corrector
models will be updated once they are prototyped.
Future research could analyse the impact of the regularisation matrix in more detail, e.g. by considering a robust
performance criteria in modal space and or by modifying thecontroller dynamics in generalised modal space. In addition,
the results could be reassessed using disturbance data thatreflects the characteristic modal distribution.

REFERENCES[1] S. Gayadeen and S. R. Duncan, “Design of an electron beamstabilisation controller for a synchrotron,” Contr. Eng. Pract.,
vol. 26, pp. 201–210, May 2014.
doi:10.1016/j.conengprac.2014.01.012[2] M. G. Abbott et al., “Diamond-II technical design report,”
Diamond Light Source, Tech. Rep., Aug. 2022. https://www.diamond.ac.uk/Home/News/LatestNews/2022/
14-10-22.html[3] I. Kempf, P. J. Goulart, and S. Duncan, “Control of Two-Array
Cross-Directional Systems using the Generalised Singular
Value Decomposition,” arXiv,
doi:10.48550/arXiv.2308.08631[4] G. H. Golub and C. F. Van Loan, Matrix Computations, 4thed. The Johns Hopkins Univ. Press, 2013.
[5] S. Gayadeen and W. Heath, “An internal model control approach to mid-ranging control,” in Proc. IFAC Symp. Adv.
Contr. Chem. Process, Jul. 2009, pp. 542–547.
[6] I. P. S. Martin et al., “Orbit Stability Studies for the DiamondII Storage Ring,” in Proc. Int. Part. Accel. Conf. (IPAC’22),
Bangkok, Thailand, Jul. 2022, pp. 2602–2605.
doi:10.18429/JACoW-IPAC2022-THPOPT017[7] L. T. Stant et al., “Diamond-II Electron Beam Position Monitor
Development,” in Proc. IBIC’22, Kraków, Poland, Dec. 2022,
pp. 168–172. doi:10.18429/JACoW-IBIC2022-MO3C2[8] I. Kempf, M. Abbott, L. Bobb, G. B. Christian, G. Rehm, and
S. Duncan, “Fast Orbit Feedback for Diamond-II,” presented
at IBIC’22, Saskatoon, Canada, this conference.
[9] S. Gayadeen, S. R. Duncan, and G. Rehm, “Optimal controlof perturbed static systems for synchrotron electron beamstabilisation,” IFAC PapersOnLine, vol. 50, no. 1, pp. 9967–
9972, 2017, 20th IFAC World Congress.

MOP041122

08 Feedback Systems and Beam Stability