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 2 I 0 2

I. Kempf ∗ , M. Abbott, L. Bobb, G. B. Christian, Diamond Light Source, Oxfordshire, UK
G. Rehm, Helmholtz-Zentrum für Materialien und Energie, Berlin, Germany
S. Duncan, University of Oxford, Oxford, UK
Abstract
The electron beam stability is critical for 4th generation
light sources. As opposed to 10 % of beam size up to 140 Hz
at Diamond, advances in detector speed and resolution at
Diamond-II increase the stability requirements to 3 % up to
1 kHz. This paper presents a novel control methodology forthe fast orbit feedback at Diamond-II, which will stabilise
the beam using two arrays of 252 slow and 144 fast correctors and 252 beam position monitors at 100 kHz. In contrast
to existing approaches that separate slow and fast feedback
loops, our approach is based on a two-matrix factorisation
called the generalised singular value decomposition (GSVD),
which decouples the system into 144 two-input modes controlled by slow and fast magnets and 108 modes controlled
by slow magnets only. The GSVD-based controller is implemented in the existing Diamond storage ring using acentralised communication architecture, such as planned
for Diamond-II. We present results from the Diamond storage ring and simulation, which confirm that the proposed
approach meets the target specification for Diamond-II.

INTRODUCTION
The fast orbit feedback (FOFB) at Diamond Light Source(Diamond) attenuates disturbances of the electron beam in
the storage ring. Disturbance sources include ground andgirder vibrations, power supply and RF noise, and transients
induced by insertion device (ID) gap changes. At Diamond,the FOFB uses 173 electron beam position monitors (BPMs)
measuring both horizontal and vertical positions, and 172
vertical and 172 horizontal correctors to perform global orbit
correction of the electron beam at 10 kHz. The FOFB reduces the root-mean square (RMS) deviation of the electron
beam to within 10 % of the beam size up to a closed-loop
bandwidth of 140 Hz, i.e. the frequency at which disturbances are attenuated by 3 dB. For Diamond-II, advances
in beamline technology require the closed-loop bandwidth
to be increased to 1 kHz, while the new multi-bend achromat lattice will reduce the beam size [1], resulting in theincreased beam stability requirements summarised in Table 1(top).
To meet these requirements, the FOFB hardware and software for Diamond-II have been redesigned, increasing thespecifications as shown in Table 1 (bottom). Firstly, thesampling frequency is increased to 100 kHz and the overalllatency reduced to 100 µs. Secondly, the number of BPMs
∗ Corresponding author: idris.kempf@eng.ox.ac.uk. Also at University of

Oxford. This work was supported in part by Diamond Light Source andin part by the Engineering and Physical Sciences Research Council (EPSRC) with a Collaborative Awards in Science and Engineering (CASE)
studentship.

Table 1: Top: Beam size, relative and absolute orbit stability
requirements at standard straight source points. Bottom:
FOFB specifications. Tables adapted from Ref. [1].
Parameter

Diamond

Diamond-II

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

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

Closed-loop BW
Latency
BPMs
Correctors≥ 1 kHz≤ 100 µs
252
252 slow/144 fast

140 Hz700 µs173
172

H: horizontal, V: vertical, BW: bandwidth

is increased to 252. Finally, to achieve the 1 kHz closedloop bandwidth, the open-loop bandwidth, i.e. the overallcorrector bandwidth, is increased from 500 Hz to ≥ 5 kHz
by introducing two types of correctors: 252 slow correctors
producing a deflection of 1 mrad, and 144 fast correctors
producing a deflection of 20 µrad [1].
While using two types of correctors allows the closed-loop
bandwidth to be increased, it prohibits the existing FOFB
control algorithm to be reused, which is based on diagonalising the dynamics using the modal decomposition andcontrolling the single-input single-output (SISO) modal dynamics. Although extensions of modal decomposition havebeen proposed in [2, 3], they leave the decoupling processunspecified for systems with fewer fast than slow correctors.
For Diamond-II, we propose a joint design method based on
the generalised singular value decomposition (GSVD) [4]to decouple the system into sets of two-input single-output(TISO) and SISO systems. It can be shown that the generalised modal decomposition of a system with two corrector
types is closely related to the modal decomposition of a system with one corrector type, which allows techniques to be
carried over to the system with two corrector types, including regularisation gains that account for a large condition
number of the orbit response matrix (ORM).
Based on the assumption that the bandwidths of the slow
and fast correctors differ significantly, other approaches split
the control problem into two separate feedback loops: one
feedback loop for the slow correctors that may be operated
at a lower sampling/actuation frequency, and a separate feedback loop for the fast correctors. Such a separation is implemented in most synchrotrons that use separate sets ofslow and fast correctors [5–7], but interactions at intermediate frequencies can require the introduction of a frequency
deadband between the slow and fast loops [8]. Dependingon the disturbance spectrum, this approach can lead to significant performance degradation [6]. One common way

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to avoid introducing a frequency deadband is to subtract
the predicted effect of the slow correctors from the feedback signal for the fast correctors [5]. Another solution is toperiodically subtract the DC gain from each fast corrector(individually) and to import these values into the slow feedback loop, thereby shifting the low-frequency action fromthe fast correctors to the slow correctors [9]. However, thisapproach neglects the coupling between slow and fast correctors and relies on a SISO analysis of the combined slow andfast loops. As in the case of systems with one corrector type,large condition numbers of the ORM limit the closed-loop
bandwidth [10], and neglecting the coupling may require
further reduction of controller gains [11]. None of these approaches – the frequency deadband method [5], the periodic
DC method [9], or combinations of those [11, 12] – provide
a means of jointly investigating the stability, performanceand robustness properties of the combined feedback loops.
This paper presents the controller design for Diamond-II
using preliminary corrector models, and summarises the
GSVD-based controller [13]. To estimate the FOFB performance at Diamond-II, predictions of the Diamond-II disturbance [14] are used for obtaining worst-case estimatesof the beam displacement. To validate these theoretical expectations, the algorithm is implemented and tested on theexisting Diamond storage ring.

DIAMOND-II CONTROLLER DESIGN

𝑦(𝑠) = 𝑅s 𝑔s (𝑠)𝑢 s (𝑠) + 𝑅f 𝑔f (𝑠)𝑢 f (𝑠) + 𝑑 (𝑠),(2)where 𝑅s ∈ R 𝑛𝑦 ×𝑛s , 𝑅f ∈ R 𝑛𝑦 ×𝑛f , 𝑛 𝑦 = 𝑛s = 252 and𝑛f = 144. The preliminary corrector models 𝑔s (𝑠) and 𝑔f (𝑠)include the frequency responses of the DAC, ADC, signal
processing filters, power supplies, magnets, and copper vessel for slow correctors and stainless steel vessel for fast correctors, resulting in 2nd- and 3rd-order transfer functions.
Note that since 𝑛s = 𝑛 𝑦 , the FOFB can also be operatedusing slow correctors only, or fast correctors substituted forslow ones.
To reduce the latency in the FOFB loop, the FOFB communication network will be changed from a distributed topology
to a centralised topology [1], and the centralised controller isplanned to be implemented on the FPGA of a White Rabbit
switch v4 (WRSv4). The time delay of 𝑔 (·) (𝑠) is assumed tobe 100 µs, which is a worst-case upper bound of the latency
estimates for Diamond-II. The latency estimate includes
39.35 µs for signal processing (DAC, ADC, BPM), 8 µs forcommunication, 20 µs for FOFB computations, and 30 µs to
account for uncertainty in the fast corrector model.

Decoupling the Diamond-II System

Background: The Existing System
The electron beam dynamics at Diamond are modelled
by a cross-directional system with one corrector type [10]:𝑦(𝑠) = 𝑅𝑔(𝑠)𝑢(𝑠) + 𝑑 (𝑠),tolerance and allowing source point control of bending magnet beamlines. The FOFB will use 𝑛s = 252 slow correctors
with a bandwidth of ∼ 200 Hz and 𝑛f = 144 fast correctors
with a bandwidth of ∼ 8 kHz. The electron beam dynamics
for the system with two corrector types are(1)

where 𝑅 ∈ R 𝑛𝑦 ×𝑛𝑢 is the ORM, 𝑠 ∈ C the Laplace variable,𝑢(𝑠) ∈ C 𝑛𝑢 are the corrector currents (inputs), 𝑦(𝑠) ∈ C 𝑛𝑦
the BPM signals (outputs) and 𝑑 (𝑠) ∈ C 𝑛𝑦 the disturbances.
The coupling between the planes is negligible so the horizontal and vertical orbit corrections are treated as separate control problems. The stable transfer function 𝑔(𝑠) ∈ C captures
the temporal dynamics of the correctors and is modelled as𝑔(𝑠) := 𝑎/(𝑠 + 𝑎)e− 𝜏𝑑 𝑠 , where 𝑎 := 2𝜋 × 700 rad s−1 is themagnet bandwidth and 𝜏𝑑 := 900 µs the time delay [10]. At
Diamond, the controller is designed in modal space, i.e. thethin singular value decomposition (SVD) 𝑅 = 𝑈Σ𝑉 T is substituted in Eq. (1) and decoupled as 𝑦ˆ (𝑠) = Σ𝑔(𝑠) 𝑢(𝑠)
ˆ + 𝑑ˆ(𝑠),T
Tˆ
:= 𝑉 𝑢(𝑠), 𝑑 (𝑠) := 𝑈 T 𝑑 (𝑠). In
where 𝑦ˆ (𝑠) := 𝑈 𝑦(𝑠), 𝑢(𝑠)ˆmodal space, the multi-input multi-output (MIMO) control
problem is simplified to 𝑛 𝑦 decoupled SISO problems.

The Diamond-II System
For Diamond-II, the beam will be monitored by 𝑛 𝑦 = 252
BPMs, with primary BPMs located upstream and downstream of each ID. The front-end X-ray BPMs are integrated into the 100 kHz data stream so that primary (electron) BPMs can be substituted for front-end X-ray BPMs
by providing the corresponding ORM, increasing the fault

Analogous to the SVD for systems with one corrector
type, the GSVD [4] factorises 𝑅s and 𝑅f as 
Σs 0 TΣ𝑅s = 𝑋𝑈 ,𝑅f = 𝑋 f 𝑈fT ,
(3)
0 𝐼 s
0
where 𝑋 = [𝑥 1 . . . 𝑥 𝑛𝑦 ] ∈ R 𝑛𝑦 ×𝑛𝑦 with det(𝑋) ≠ 0,is the shared matrix of generalised output modes, Σ (·) =diag(𝜎(·) ,1 , . . . , 𝜎(·) ,𝑛f ) ≻ 0 with (·) = {s, f} are the matri2 + 𝜎 2 = 1,
ces of generalised singular values that satisfy 𝜎s,𝑖f,𝑖and 𝑈 (·) ∈ R 𝑛 (·) ×𝑛 (·) with 𝑈 T(·) 𝑈 (·) = 𝐼 are the matrices ofgeneralised input modes. In contrast to the standard singularvalues that measure the gain in the direction of the singular
vectors, the generalised singular values measure the relative
gain of slow and fast correctors in the direction of 𝑥𝑖 . It canbe shown that the matrices 𝑋 and 𝑅 = [𝑅s 𝑅f ] share thesame standard singular values and left singular vectors [13],relating the modal decomposition to the generalised modal
decomposition. For Diamond-II, the condition numbers are
𝜅(𝑋) = 4404 for the vertical and 𝜅(𝑋) = 2371 for the horizontal planes, and the generalised singular values satisfy
1 < 𝜅(Σs ) < 2 and 1 < 𝜅(Σf ) < 2.
Substituting the GSVD in Eq. (2) and left-multiplying
with 𝑋 91 decouples the system with two corrector types as


 
Σs 0
Σ
𝑦˜ (𝑠) =𝑔 (𝑠) 𝑢˜ s (𝑠) + f 𝑔f (𝑠) 𝑢˜ f (𝑠) + 𝑑 (𝑠), (4)0 𝐼 s0where 𝑦˜ (𝑠) := 𝑋 91 𝑦(𝑠), 𝑢˜ (·) (𝑠) := 𝑈 T(·) 𝑢 (·) (𝑠) and 𝑑˜(𝑠) :=𝑋 91 𝑑 (𝑠). The decoupled systems shows that for 𝑛f < 𝑛s =MO2I02
2

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Υ

Q(s)

u(s)

P(s)

+

y(s)

+

Γ

Model

P̄(s)

+
−

Figure 1: IMC structure in original space with additional
compensators Υ and Γ.
𝑛y , only a subset of the disturbances is attenuated by slow and
fast correctors, i.e. those lying entirely in span 𝑥1 , . . . , 𝑥 𝑛f .

TISO and SISO Controllers
Equation (4) reads row-wise as 𝑦˜ 𝑖 (𝑠) = 𝑃˜𝑖 (𝑠) 𝑢˜ 𝑖 (𝑠) +˜𝑑𝑖 (𝑠), where for 𝑖 ∈ Itiso := {1, . . . , 𝑛f }, 𝑃˜𝑖 (𝑠) :=[𝜎s,𝑖 𝑔s (𝑠), 𝜎f,𝑖 𝑔f (𝑠)], 𝑄˜ 𝑖 (𝑠) := [𝑞 s,𝑖 (𝑠), 𝑞 f,𝑖 (𝑠)] T , and𝑢˜ 𝑖 (𝑠) := [𝑢 s,𝑖 (𝑠), 𝑢 f,𝑖 (𝑠)] T , and for 𝑖 ∈ Isiso :={𝑛f + 1, . . . , 𝑛s }, 𝑃˜𝑖 (𝑠) := 𝑔s (𝑠), 𝑄˜ 𝑖 (𝑠) := 𝑞 s,𝑖 (𝑠), and𝑢˜ 𝑖 (𝑠) := 𝑢˜ s,𝑖 (𝑠). Although any control approach including
PID control could be applied, the internal model control(IMC) structure is used [15], which is naturally amenable totime delays and also used at Diamond. The IMC structure isshown in is shown in Fig. 1 in original space, where Υ and Γ
are matrices that will be considered in the following section.
For the remainder of this paper, the plant model is assumed tobe accurate, i.e. 𝑃𝑖 (𝑠) = 𝑃¯𝑖 (𝑠) and 𝑃˜𝑖 (𝑠) = 𝑃˜¯𝑖 (𝑠), so that thecontrol inputs are 𝑢˜ 𝑖 (𝑠) = −𝑄˜ 𝑖 (𝑠) 𝑑˜𝑖 (𝑠). Substitutingin the
row-wise dynamics yields 𝑦˜ 𝑖 (𝑠) = 1 − 𝑃˜𝑖 (𝑠) 𝑄˜ 𝑖 (𝑠) 𝑑˜𝑖 (𝑠).
First, the desirable closed-loop dynamics are defined as𝑦˜ 𝑖 (𝑠) = 1 − 𝑇(·) (𝑠) 𝑑˜𝑖 (𝑠) for 𝑖 ∈ I(·) ,(5)for (·) = {tiso, siso} and where the complementary sensitivities must satisfy 𝑇(·) (0) = 1 for a zero steady-state
error. Secondly, the controller structure is simplified as𝑘˜ (·),𝑖 × 𝑓 (·) (𝑠) for (·) = {s, f}, so that equating Eq. (5) andthe row-wise dynamics yields
𝑞 (·) (𝑠) := 𝑘˜ (·),𝑖 𝑇(·) (𝑠)/𝑔 (·) (𝑠),(6)for (·) = {s, f} and where 𝑇s (𝑠) := 𝑇siso (𝑠) and 𝑇f (𝑠) :=𝑇tiso (𝑠) − 𝑇siso (𝑠), and 𝑘˜ s,𝑖 := 1/𝜎s,𝑖 and 𝑘˜ f,𝑖 := 1/𝜎f,𝑖 for
𝑖 ∈ Itiso , and 𝑘˜ s,𝑖 = 1 for 𝑖 ∈ Isiso . The controller in Eq. (6)splits the control effort between slow and fast correctors andis also referred to as mid-ranging controller [16].
For Diamond-II, the corrector models 𝑔s (𝑠) and 𝑔s (𝑠) are
2nd- and 3rd-order transfer functions with a time delay of𝜏𝑑 = 100 µs, so the complementary sensitivities are chosen as 𝑇(·) (𝑠) := 𝜔 (·) /(𝑠 + 𝜔 (·) ) × 𝐻2, (·) (𝑠)e−𝑠 𝜏𝑑 , where(·) = {tiso, siso} and 𝐻2, (·) (𝑠) is a second-order Butterworth filter [15] with a cutoff frequency of 500 Hz for SISO
and 10 kHz for TISO systems.

Tuning for Expected Disturbances
The controller is tuned based on the estimated power spectral density (PSD) of the disturbance for Diamond-II, which

100
10−4
10−8

Horizontal

100

Vertical

101102Frequency (Hz)103

Figure 2: Estimated power spectral density (PSD) of theuncorrected vertical disturbance for Diamond-II at the upstream primary BPM on the first standard straight [14].
is shown in Fig. 2 for the upstream primary BPM on the firststandard straight [14]. The PSD includes contributions fromground vibrations, girder motion, power supply ripple andRF noise.
The closed-loop bandwidth of the slow correctors, 𝜔siso ,is limited by rate constraints of the power supplies, but must
be chosen high enough to deliver low-frequency disturbanceattenuation across all 𝑛 𝑦 directions. According to Fig. 2, thePSD drops by several orders of magnitude above 100 Hz,so 𝜔siso is chosen as 𝜔siso = 2𝜋 × 100 rad s−1 . Note that if
𝜔siso is chosen larger than the open-loop bandwidth of 𝑔s (𝑠),the controller will result in large corrector gains above theopen-loop bandwidth.
Although a large 𝜔tiso yields a large overall closed-loop
bandwidth, in the presence of large time delays it can cause
disturbances to be amplified by a large sensitivity peak, i.e.
large values of 𝑆max := max 𝜔 |𝑆tiso (j𝜔)|. Moreover, large
𝜔tiso may cause disturbances at high frequencies (i.e. above
the closed-loop bandwidth) to be amplified, and make thecontroller prone to instabilities [15]. A bandwidth of 1 kHz
for 𝑆tiso (𝑠) results in 𝑆max = 6 dB at 5 kHz, so the bandwidthis reduced to 600 Hz (𝜔tiso = 2𝜋 ×2 krad s−1 ), giving 𝑆max =4.12 dB at 2.2 kHz. Note that with a time delay of 70 µs
instead of 100 µs, the peak could be reduced to 2.4 dB, or
the closed-loop TISO bandwidth increased to 1 kHz with
𝑆max = 4.12 dB.

Structure in Original Space
After designing the TISO and SISO controllers for thedecoupled systems, the modal controllers in Eq. (6) areconcatenated and mapped back to original space to obtain the control inputs as 𝑢 (·) (𝑠) = −𝑄 (·) (𝑠)𝑑 (𝑠) with
𝑄 (·) (𝑠) := 𝐾 (·) 𝑞 (·) (𝑠) and 𝐾 (·) := 𝑈 (·) 𝐾˜ (·) 𝑋 91 . Substituting the control inputs in Eq. (2) yields the closed-loop
dynamics in original space:𝑦(𝑠) = 𝐼 − 𝐼𝑇siso (𝑠) − 𝑋tiso 𝑋 91𝑇f (𝑠) 𝑑 (𝑠),

(7)



where 𝑋tiso := 𝑥1 , . . . , 𝑥 𝑛f , 0, . . . , 0 . Since 𝜅(𝑋) =𝜅(𝑅) ≫ 1, the inverse 𝑋 91 in 𝐾 (·) produces large controller
gains in directions associated with the small singular valuesof 𝑅. Moreover, the term associated with the 𝑛f < 𝑛s fast
correctors in Eq. (7) amplifies certain disturbance directions
when 𝑋 is not orthogonal.

MO2I0208 Feedback Systems and Beam Stability

3

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PSD (µm2 /Hz)

d(s)
Plant

Mag. (dB)

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obtained from

0
−10
−20
100

µ=0

µ = 0.1

µ=1

10
0
−10
−20
100

|𝑌𝑖 (𝜔 𝑘 )| ≤𝑛𝑦∑︁|𝑆𝑖 𝑗 (𝜔 𝑘 )||𝐷 𝑗 (𝜔 𝑘 )|,

(9)

𝑗=1

101
102
Frequency (Hz)103(a) MIMO sensitivity.

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101
102
Frequency (Hz)

µ=1

103

(b) MIMO controller gains.

Figure 3: Minimum (dashed) and maximum singular valuesof the sensitivity 𝑆(j𝜔) for different values of 𝜇 (a), andcontroller gains for slow and fast (dashed) correctors (b).
In original space, two compensators, Υ ∈ R 2𝑛𝑦 ×𝑛𝑦 andΓ ∈ R 𝑛𝑦 ×𝑛𝑦 , are added to the IMC structure in Fig. 1, where𝑃(𝑠) := [𝑅s 𝑔s (𝑠) 𝑅f 𝑔f (𝑠)], 𝑄(𝑠) := diag(𝑄 s (𝑠), 𝑄 f (𝑠)),and 𝑢(𝑠) := [𝑢 s (𝑠) T 𝑢 f (𝑠) T ] T . The compensator Υ :=†[𝐼 ΥfT ] T with 𝑋 𝑋tiso
only applies to the case 𝑛f < 𝑛s andprojects 𝑑 (𝑠) onto 𝑥1 , . . . , 𝑥 𝑛f for the TISO systems [13].
The compensator Γ reduces the controller gain and isdefined as Γ := (𝑋 𝑋 T + 𝜇𝐼) 91 𝑋 𝑋 T , where 𝜇 ∈ R ≥0 is aregularisation parameter [13]. With Υ and Γ, the closedloop dynamics are 𝑦(𝑠) = 𝑆(𝑠)𝑑 (𝑠), where the sensitivity
𝑆(𝑠) is𝑆(𝑠) := 𝐼 − 𝑃(𝑠)𝑄(𝑠)Υ(𝐼 + (Γ−𝐼)𝑃(𝑠)𝑄(𝑠)Υ) 91 Γ. (8)The effect of Γ on the minimum and maximum singular values of 𝑆(j𝜔) is shown in Fig. 3a for 𝜇 = {0, 0.1, 1}.
The maximum singular values of the transfer functions from𝑑 (𝑠) to 𝑢 s (𝑠) and 𝑢 f (𝑠) are shown in Fig. 3b. As 𝜇 increases,
the overall bandwidth decreases, but the minimum singularvalues (dashed) remain unchanged. It can be shown that the
bandwidth is reduced for higher modes of 𝑅 = [𝑅s 𝑅f ] thatsatisfy 𝜎𝑖2 ≪ 𝜇. As most of the disturbances are input disturbances, most of the disturbances occur within the lower ordermodes, which justifies the use of the compensator Γ [13].
For Diamond-II, 𝜇 is chosen as 𝜇 = 0.1, yielding a fast corrector demand of 276 mA and 92 mA in the horizontal andvertical direction up to 1 kHz, which is below the maximum
demand of 920 mA.

Expected Performance
Although the PSDs from Fig. 2 can be used to design the
TISO and SISO controllers, they lack the phase informationthat is required to describe the spatial distribution acrossthe storage ring needed for the MIMO analysis. However,an upper bound on the amplitude spectral density (ASD)|𝑌𝑖 (𝜔 𝑘 )| – the square root of the PSD – at BPM 𝑖 can bewhere 𝑆𝑖 𝑗 (𝜔 𝑘 ) refers to row 𝑖 and column 𝑗 of 𝑆(·) in Eq. (8)and 𝐷 𝑗 (𝜔 𝑘 ) to the disturbance at BPM 𝑗. Another upperbound is |𝑌𝑖 (𝜔 𝑘 )| ≤ ∥𝑌 (𝜔 𝑘 ) ∥ 2 ≤ ∥𝑆(𝜔 𝑘 ) ∥ 2 ∥𝐷 (𝜔 𝑘 ) ∥ 2 ,though this is always equally or more conservative thanthe bound in Eq. (9).
In addition, the performance can be estimated with simulations that use disturbances obtained from sampling theASDs in Fig. 2 with a random phase. Although these simulations do not yield strict upper bounds, they also lack thephase information, so that simulations can be interpreted as
an additional measure of worst-case performance.
The integrated beam motion (IBM) up to 1 kHz, i.e. thesquare root of the integral of the PSD, is summarised for all
primary BPMs in Table 2. The results show that the upperbound from Eq. (9) yields a conservative estimate that ismore than two times larger than the attenuation expectedfrom simulations. However, except for the primary BPMs of
the long straight in the vertical direction, the upper bound is
below the target deviation, indicating that the Diamond-II
specifications are met. For the long straights, the IBM is up
to 0.04 µm above target for the upper bound, but below target
in simulations. Since the PSD lacks phase information, theDiamond-II disturbance does not reflect the characteristic
modal distribution caused by input disturbances and the regularisation significantly impacts the controller performance.
However, given that most of the Diamond-II disturbances
are input disturbances, it can be expected that the targets will
also be met on the long straights.

TESTS ON THE EXISTING DIAMOND
STORAGE RING
As part of the FOFB design for Diamond-II, the GSVDbased controller has been implemented and tested on theexisting Diamond storage ring. The aim of these experiments is to evaluate the controller in practice, i.e. investigate
the FOFB with 𝑛f < 𝑛s and model uncertainty, and to test
new FOFB configurations, such as the centralised topology
used at Diamond-II. The control system is implemented ona VadaTech AMC540 – a board combining a Xilinx Virtex-7
FPGA with two Texas Instruments (TI) DSPs [17] – andinterfaced with the FOFB communication network. The
existing 24 computing nodes from all the cells in the storage
ring are bypassed, and the communication network reconfigured to form the centralised topology used at Diamond-II.
The FPGA on the AMC540 is used for signal routing, andthe DSPs are used to produce the corrector inputs for thehorizontal and vertical directions. During tests with the
AMC540, the time delay is increased from 700 µs to 900 µs
and the communication network additionally broadcasts setpoints for the correctors.
As the FOFB at Diamond uses only one corrector type, asubset of 𝑛𝑢 = 160 correctors and 𝑛 𝑦 = 96 BPMs is selected

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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 upstream (US) and downstream (DS) primary BPMs onlong straights (LS), mid straights (MS), and standard straights (SS). All values are in µm. The disturbance corresponds to
FOFB disabled, and the bound is obtained from Eq. (9).

LS DS

MS US

MS DS

SS US

SS DS

LS US

LS DS

MS US

MS DS

SS US

SS DS

0.64
1.20
0.53
0.16

0.72
1.20
0.41
0.18

0.45
0.90
0.21
0.06

0.48
0.90
0.25
0.07

0.6
0.97
0.2
0.04

0.61
0.97
0.48
0.11

0.25
0.23
0.24
0.15

0.25
0.23
0.27
0.20

0.21
0.14
0.11
0.07

0.2
0.14
0.07
0.03

0.22
0.18
0.13
0.07

0.21
0.18
0.11
0.06

and represented as in Eq. (2) with 𝑔s (𝑠) = 𝑔f (𝑠) = 𝑔(𝑠)and 𝜏𝑑 = 900 µs. The correctors are divided into 𝑛s = 96
“slow” and 𝑛f = 64 “fast” correctors, and the controller
is designed using the GSVD-based approach with 𝜔tiso =2𝜋 × 176 rad s−1 and 𝜔siso = 2𝜋 × 50 rad s−1 . With thischoice of parameters, the sensitivity overshoot is 3.4 dB at
300 Hz.
The GSVD-based controller is compared with a “standard”
IMC controller that controls the same 𝑛 𝑦 = 96 BPMs and𝑛s = 96 correctors, but with all 𝑛𝑢 correctors tuned to aclosed-loop bandwidth of 2𝜋 × 176 rad s−1 . All algorithms
are evaluated on the storage ring at 300 mA beam current
and with wigglers disabled.
The horizontal IBMs measured at BPM 1 are compared inFig. 4, which also shows the disturbance recorded before theexperiments (with FOFB disabled) and the corresponding
simulated attenuation. As expected from the lower SISO
bandwidths, the GSVD-based controller performs worse
than the standard controller, but the results show that bothcontrollers yield similar closed-loop bandwidths. The performance differences between standard and GSVD-based
controllers are within 0.01 µm for frequencies below 10 Hz,
and decrease for higher frequencies, which is in line withtheoretical expectations. Comparing the simulation with thereal-world results, the simulations perform slightly worsein the horizontal plane, but better in the vertical plane. The
differences could be caused by various factors, including
model uncertainty or time varying disturbance spectra.

CONCLUSION
In this paper we have proposed a joint control method for
the FOFB at Diamond-II that will use slow and fast correctors. The performance of our method was evaluated using
PSD estimates of the disturbance at Diamond-II. We calculated a worst-case upper bound that meets the performance
criteria except for the primary BPMs in the long straightthat exceed the target by 0.002 µm (0.9 %) in the verticalplane. However, simulations using sampled disturbanceshow that this upper bound may be conservative. Once thefast correctors are prototyped, the actuator dynamics andthe latency will be updated, which is likely to result in better
performance estimates.

OFF
ON, GSVD-based

IBM (µm)

DisturbanceTargetBound
Simulation

VerticalLS US

Horizontal

ON, standard
SIM, GSVD-based

SIM, standard

100
10−1
10−2
100

101
Frequency (Hz)102

Figure 4: Measured (OFF, ON) and simulated (SIM) horizontal IBM for BPM 1 of the Diamond storage ring withFOFB disabled, standard and GSVD-based IMC.
To demonstrate our algorithm in practice, we integrated a
new computing node in the existing storage ring and implemented a GSVD-based controller, splitting the control effort
onto designated slow and fast correctors. The results fromtests on the storage ring showed that although only a subset ofthe correctors covered the full bandwidth, the performanceof the controller with two corrector types was nearly as goodas the one with one corrector type, and that the simulations
accurately reflect experimental measurements.
Even though the simulation and real-world results proved
the feasibility for Diamond-II, several research questions
remain. The mid-ranging approach requires inverting 𝑔s (𝑠)and 𝑔f (𝑠), but for Diamond-II the fast correctors may besuch that 𝑔f (0) = 0. The mid-ranging approach may therefore result in undesirable integrating behaviour for the fast
correctors. To avoid this problem, one solution would be
to invert only parts of 𝑔f (𝑠) and quantify the resulting performance loss. Alternatively, one could combine the generalised modal decomposition with a H2 or H∞ controller
design [15], which may also reduce the sensitivity peaks.
While upper bounds showed that most performance requirements for Diamond-II are met, simulations using sampled Diamond-II data and tests on the existing facility
showed that these upper bounds are conservative. To obtainless conservative bounds, future research could extend the
disturbance data using phase information, e.g. by aggregating the different disturbance sources and modelling their
spatial impact onto the storage ring.

MO2I0208 Feedback Systems and Beam Stability

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

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ISSN: 2 6 7 3 - 5 3 5 0
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08 Feedback Systems and Beam Stability