Proceedings of IBIC2012, Tsukuba, Japan

TUPA34

INVERSE RESPONSE MATRIX COMPUTATION FOR THE STORAGE
RING SLOW ORBIT FEEDBACK CONTROL: SYNTHESIZEDTOPOLOGICAL INVERSE COMPUTATIONJ. M. Lee#, J. Y. Huang*, C. Kim, Accelerator Division, Pohang Accelerator Laboratory,POSTECH, 790-784, Korea
For the storage ring orbit feedback control, the inversed
SVD (Singular Value Decomposition) method is normallyworking under stabilized orbit situation, but less effectiveat relatively large orbit fluctuation. To overcome such
numerical drawback, we investigated the alternative
feedback control based on the solution tracking algorithm.
Using our novel STIC (Synthesized Topological InverseComputation), we simulated the formation of residual
tune orbit under the closed orbit dynamics response
matrix, measurable relationship between BPM-data
readout and corrector-MPS setting. By placing empiricalevidence of beam based alignment measurement, weachieved remarkable numerical fidelity on our STIC
feedback algorithm, especially ascribing the topologicalimportance of inverse response matrix computation. We
demonstrated the STIC-inherent triggering behaviour andadaptive pattern-notched feedback stability.

INTRODUCTION

Even though LOCO (linear optics closed orbit)
configurations are so complicated, entire physics of beam
dynamics feature can be extracted from the measured
response matrix. In this study, we introduced the PLS-II
response matrix, even not so much matured since
upgrading commission year 2012. [3] Proper matrix
refinement can be made both numerically and empirically,
according to our self-consistent manner of filtering out
the uncertainty of measurement errors escaping frombeam dynamics constraints. We are deeply investigating
our advanced feedback algorithm, Orbit-STIC, practicallyin collaboration with world-wide accelerator community.

NUMERICAL PROCEDURESAs depicted in Fig. 1, the closed orbit feedback
algorithm can be expressed with the following equations:
=∆∆

∆ =∆
Practically useful for SOFB (slow orbit feedback)with assistance of FOFB (fast orbit feedback), theinversed SVD manipulation [1] is not fully acceptablebecause a type of consecutive instability noise irreversibly
accumulates in the beam trajectory deviation. On theother hand, a novel numerical algorithm – emerging froma topological math approach – can lead to numerical selfconsistency and solution tracking, dramaticallysuppressing ill-posed instability problems associated withnumerical truncation residuals. This approach, known as a
singularity regularization method, makes it feasible tocompute a feature-invariant inverse matrix and systemmatched de-noising filter. For deep investigation of theclosed orbit feedback control, we applied our novel recipe
of inverse computation, namely STIC(SynthesizedTopological Inverse Computation) [2]. Fundamentally,
this Orbit-STIC feedback is similar to the numericalcomputation of the inner origin function ascribing a typeof complex systems. In our other study [2], we described
the singularity inhibitor algorithm with essential
topological nature engaged in the symplectictransformation (e.g. S-matrix, satisfying A3=SA1 with
S2=-I). Because this inverse math algorithm deals with thephase-component feature, commonly required is thecomplete phase-space relationship, e.g. the horizontal andvertical coupling term in the closed orbit response matrix.
___________________________________________# jaymin@postech.ac.kr* huang@postech.ac.kr∆ =−∆+ (∆∆==+( −∆

( )

∆

∆

=[( )

=−

=( )+]

(3))

( )

(1)
(2)

∆

∆

( ))

− ∆∆ )

–

∆ ==

(∆

∆

(4)

∆

(5)

−

(6)
(7)

∆

= ( + ) −+( )

∆∆

(8)

Eqs (1)-(4), include the basic recursive formula for thefeedback control. Eqs (5)-(7), are the modified formalism
based on Eq(4). This is efficient and convenient way ofstabilizing the iteration process. For the STIC feedback inFig. 2, as → , it is proper to get the stabilized BPMsolution (xo). In contrast, for the SVD method, as → ,the feedback speed is retarded and eventuallyextinguished. (Refer to Table 1.) Typically, when theparticular orbit solution is given as xp, the correctorMPS setting is simply computed according to Eq (8). Two
ISBN 978-3-95450-119-9

Feedbacks and Beam Stability

1c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Abstract

TUPA34Proceedings of IBIC2012, Tsukuba, Japanc 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○numerical approaches, STIC and SVD, are significantly
different: (i) STIC algorithm works at the solution
tracking mode with valid xo-seed value in Fig. 1. If theinitial xo-setting is largely deviated from the real orbit
situation, prompt feedback is jeopardized. (ii) SVD
algorithm computes go directly from BPM-readout(current orbit). Even uncertainty of xp is not totally
rejected under -parameter adjustable feedback speed.
However, under the SVD feedback control, the orbit will
be excursive without smart feedback such as STIC
solution tracking. More specification will be describedlater with our numerical simulation results in Fig. 3,relevant to quantitative basis in Fig. 4 and Table 1.

Figure 1: Algorithm diagram of STIC feedback process.
) is represented as the equivalent
The inverse process (inverse algebra using two conjugate operators ( ( ) , ( ) )in Eqs (5)-(6). The solution tracking operator ( ( ) ) keeps
the solution orbit stabilized, while the filtering operator( ( ) ) plays a nullifying role against the amplification ofnoise components along repeated feedback loop.

Our numerical simulation results, illustrated in Fig. 3,
are deserved to develop the hybrid algorithm connecting
STIC and SVD. Within numerical simulation, it ispossible to trigger the effect of STIC solution feedback.
When triggered, this solution is surely valid. The resultant
STIC xo-solution is stabilized better compared to theSVD-case. This is a case of triggering at the extremely
stabilized orbit – statistically reachable instantaneously.

Figure 2: (a) Simulated residual tune orbit, statistical
average of random-number generated orbit fluctuation.
Horizontal/vertical tune orbit indicates the operationalstatus (x=15.28, y=9.18) of PLS-II. (b) Empiricalevidence is observed from typical beam based offset
measurement, the BPM arrangement data presumed as)( ))∆ (∆=( −.

RESULT AND DISCUSSIONAs shown in Fig. 2, numerical simulation results
indicate that the residual tune orbit - corresponding to xoin Fig 1 - exists in real beam orbit, as well as numerical
final solution converged via numerical iteration. This
residual tune orbit can be taken as an average of statisticalfluctuation as numerically simulated. Its amplitude wascompared to the so-called reference orbit, empiricallymeasured through BBA(Beam Based Alignment) method[4], as plotted in Fig. 2 (b). Numerical counterpart isapparently coincident to the empirical pattern, in case thatthe numerical inversion (R-1) is reasonably close to ourSTIC result. At least, it is a good indicator to recognize
the plausible betatron oscillation - the lattice tune
arrangement coupled with quadrupole magnet to BPM
position involving beam based offset errors. Normally,
the betatron oscillation is damping slowly after every topup injections, and is fluctuating within the boundedstability in beam dynamic aperture.

Figure 3: Comparison between STIC and SVD feedback
simulation. In the case (a2), the STIC solution tracking is
performed when the seed setting (x0 in Fig. 1) is takenfrom other feedback orbits, STIC-(a1) or SVD-(b1).
Appropriate parameter can control the triggering
threshold, associated with statistically distributed orbit
stability (blue color). Up to complete orbit tracking
situation, the extremely stabilized orbit can be established
via such an adaptive triggering operation.

ISBN 978-3-95450-119-9
2Feedbacks and Beam Stability

Proceedings of IBIC2012, Tsukuba, JapanTUPA34

In Fig. 4, showing the feature of inverse matrices, it isclearly manifested how STIC feedback works differently
from SVD feedback process. In case of SVD, ( ) isonly effective, but ( ) actually vanishes as indicated, i.e.
‖ ‖ = in Table 1. This belongs to a single pattern styleof feedback kernel. In contrast, STIC is a homology case combing doublet patterns ( ( ) and ( ) ) into one matrix.
With such distinctiveness, this STIC process is morepotential than SVD for theoretical computation andmodelling. As we computed the residual tune orbit in Fig.
2, we can achieve a great numerical confidence regardlessof noise fluctuation. Indeed, the entire details of closed
beam orbit can be evaluated from the measured response
matrix, e.g. a theme of accelerator simulation works. Ofcourse, as we promised, our STIC process will beprominent for a practical implementation of on-tracking
solution, especially fast feedback process competing
FOFB system. If so, our feedback system may have agreat benefit to establish top-up operation mode safely
and quickly – optimizing a fast betatron oscillation
damping. Also, we believe, this smart and effective
algorithm can be interfaced with RF tune frequency
adjustment. Subsequent feedback under noise statistics
can be traced with sensitively evaluating the disparity
covariance (maximum entropy) across orbital instability
situation, as pointed out in other study [2].

−Our simulated numerical distinctiveness is so
remarkable and thinkable for practical utility of feedback
system with modern computers of massive parallelarchitecture. Our STIC result – phase retrieval solution is basically dependent on math approach, which is being
revealed as a category of Bayesian statistics andmaximum entropy theory. Like a hologram, STIC matrix
itself possesses the phase-retrieval feature and its tracking(lock-in) stability. Inversed SVD matrix is improper
physically, losing such crucial topology-invariant features.
A new math approach is emerging nowadays to think
about a new instrumentation and challenging.

23.27

ACKNOWLEDGMENT
We would express thanks to Grant Bunker [2], encouraging
our Orbit-STIC studies through his experience on EXAFS-STICstudy. This work was supported by the Ministry of Education,
Science and Technology of Korea.

Table 1: Characteristics of Inverse Matrix
InverseOperator

Norm of operator algebra:
‖ ‖−

1.0

0.23

‖

− ‖
0.051

1.0

8e-7

0.004

4e-7

( )

1.23

0.30

0.002

0.25

( )

0.54

0.271.0

0.35

1.0

2e-14

0.018

2e-14

1.0

2e-140.018

2e-14

( . )

0.5

0.25

0.5

0.25

( )

0 ( meaningless, featureless)( )pinv(R)@matlab

1.0

7e-13

2e-14

Figure 4: Color-map representation for the storage ringorbit response matrix, [RHH,RHV; RVH,RVV]. For∆ = ∆ system, [Q,R; R-1,P] consists of complete
Hilbert four-space set with R (forward), R-1 (inverse),
Q=RR-1 (x-space projection) and P=R-1R (g-spaceprojection). With |intensity|0.65 adjusted image, red/bluecolor indicates positively/negatively valued. Refer tonumerical feature characteristics in Table 1.

1e-12

REFERENCES[1]K.H Kim et al, Proceedings of EPAC 2000 (Vienna,Austria) 1906-1908.
[2] J.M.Lee, D-S.Yang and G.B.Bunker, “EXAFS Phase
Retrieval Solution Tracking for Complex Multi-component
System”, presented at international XAFS-15, Beijing, 2228, July, 2012.
[3] C.Kim et al, J. Korean Physical Society, 59 (2011) 34-38[4] L.Liu et al, Proceedings of PAC09 (Vancouver, Canada),3717-3719.

ISBN 978-3-95450-119-9
Feedbacks and Beam Stability

3c 2012 by the respective authors — cc Creative Commons Attribution 3.0 (CC BY 3.0)
Copyright ○Whenever STIC process confirms the stabilized BPMreadout orbit at good confidence, we can put appropriate
setting values for corrector magnets with sufficient safety.
In Fig. 3 (b), eventually xo-solution can be obtained at
the minimal deviation - indicating submicron stability
desirable. In this way, it is possible to reach the maximum
stability with safety and quickness, and meanwhile orbital
drift problem will be removed.