The strategy for Automated Beam Steering and Shaping (ABS) in the PS complex is to use theoretical response matrices calculated from an optics database. The main reason for this is that it enforces a certain understanding of the machine optics. A drawback is that the validation of such a matrix can be a lengthy process. However, every time a correction is made using an ABS program, a partial measurement of the response matrix is effectively performed. Since the ABS programs are very frequently used, the full matrices could thus be measured on an almost daily basis, provided this information is retained. The information can be used in two ways. Either the program passively logs the data to be analysed off­line, or the information is directly fed back to the matrix, which makes the program 'learn' as it executes. The data logging provides a powerful machine debugging tool, since deviations between the measured and theoretical matrices can be traced back to incorrect optical parameters. The 'learning' mode ensures that the correction will always converge. Implementation details and simulation results are discussed

Jansson, A

Lindroos, M

Lindroos, Μ

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"Intelligent" Automatic Beam Steering and Shaping

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN { PS DIVISIONCERN/PS 2000-019 (OP)’INTELLIGENT’ AUTOMATIC BEAM STEERING AND SHAPINGA. Jansson1) and M. LindroosAbstractThe strategy for Automated Beam Steering and Shaping (ABS) in the PS complexis to use theoretical response matrices calculated from an optics database. The mainreason for this is that it enforces a certain understanding of the machine optics. Adrawback is that the validation of such a matrix can be a lengthy process. However,every time a correction is made using an ABS program, a partial measurementof the response matrix is effectively performed. Since the ABS programs are veryfrequently used, the full matrices could thus be measured on an almost daily basis,provided this information is retained. The information can be used in two ways.Either the program passively logs the data to be analysed off-line, or the informationis directly fed back to the matrix, which makes the program ’learn’ as it executes.The data logging provides a powerful machine debugging tool, since deviationsbetween the measured and theoretical matrices can be traced back to incorrectoptical parameters. The ’learning’ mode ensures that the correction will alwaysconverge. Implementation details and simulation results are discussed.7th European Particle Accelerator Conference, 26-30 June 2000, Vienna, AustriaGeneva, Switzerland28 July 20001) CERN and Manne Siegbahn Laboratory, Stockholm, Sweden1 INTRODUCTIONDue to the complexity of large accelerators, computer support is needed for ecientoperation. In the PS complex, software is developed for this purpose within the contextof Automated Beam Steering and Shaping[1] (ABS).The approach taken is that every relation between machine parameters and mea-sured quantities can, if suciently small corrections are considered, be treated within theframework of linear perturbation theory. The rst order relation between changes in theparameters ci and changes in the measured quantities mi are given by the response matrixA (also called the correction matrix), dened asA  c = m. (1)This response matrix is in general a function of the parameters. If this matrix is known,optimal corrections can be calculated from the errors in the measured quantities. Ratherthan inverting the matrix, which can be numerically unstable, a special correction algo-rithm called Micado[2] is used to calculate c from the measured errors m. This algorithmsolves the matrix equation approximately, using only a few of the most eective correctorsavailable.The matrix can be obtained in two ways: either by direct measurement, or as alinearization of a theoretical machine model. In the PS complex, the second method hasbeen chosen, since the creation and maintenance of an accurate theoretical model enforcesa thorough understanding of the machines. However, the way to verify that the theoreticalmodel correctly describes the machine is usually to measure the response matrix andcompare it to the theoretical one. This can be a long and cumbersome process.2 LEARNING CORRECTION SYSTEMS2.1 ’Passive learning’ - Data CollectionWhen the ABS programs are used, measurements and subsequent corrections areiterated until the optimum is found. In the process, a lot of measurement data passesthrough the program. Presently, this data is lost once it has been used to calculate acorrection. This is a waste of valuable information. If the information was retained, enoughmeasured data could be collected in a typical ABS program session to reconstruct theentire response matrix. This can be implemented by a simple log le. Since the programcollects information about the machine, but does not use it, this could be called a ’passivelearning’ mode.2.2 ’Active learning’ - Direct FeedbackIn the current implementation, the ABS programs are static. Given a certain error,they will always propose the same correction. This is not a problem as long the iterativecorrection converge, but this may not always be the case, for example in closed orbitcorrections if the tune of the machine is dierent from the model tune.The obvious next step from the passive data collection is to enable the program toanalyse the data online and modify the response matrix accordingly, if it should be foundto be incorrect.A program in ’active learning’ mode should identify and retain useful informationand use it to reconstruct as much as possible of the response matrix. The reconstructedparts should then be substituted into the current response matrix. A calculated matrixcould be used in the beginning of an ABS program session, but it would be gradually1Databasecorrection Oraclemonitorscorrectorsmatrixsettingsmeasurementsmeasurementssettingsnew matrixmatrixoptics datacalibrationsMeasurement application ABS generic user interfaceABS correction packageData analysis packageMatrix calculation packagemeasurementsMathematicaControl System (Unix)Figure 1: Schematic structure of the ABS application programs. The dashed line showwhere modications are needed. Since the modications are almost fully localised to theMathematica part, they could be implemented easily.transformed into an up-to-date measured matrix. At the end of the session, the two ma-trices could be compared in order to identify dierences between the model and the realmachine. Thus, the learning approach is not contrary to the underlying idea of using theo-retical matrices, but rather provides a simple and powerful tool to verify their correctness.3 PRACTICAL IMPLEMENTATIONThe ABS programs have several components[3] (see Fig. 1). The application pro-gram is congured from a database[4], that also contains the theoretical model of themachine. The actual computation of corrections, based on the measured data and thedatabase model, is made in a Mathematica kernel running as a sub-process to the ap-plication program. This modular structure makes it very easy to implement the ideaspresented above, since in principle only the Mathematica code needs to be modied. Datalogging can be enabled simply by dumping all measured data to a flat le for furtheranalysis.The code for the analysis of the logged data have been written in Mathematica andcould therefore be incorporated easily in the correction module in order to make it analysethe data on-line. The data from several corrections can be summarised asA  C = M (2)where each column of the matrices C and M corresponds to one applied correction. Thecrucial dierence from (1) is that now the matrix A is the unknown. While the equation isoften under-determined because certain correctors have not been changed, there is oftena sub-system which is solvable.The analysis module performs the analysis of this data in several steps. It rstsuppresses data that are too noisy. That is, corrections where the changes made are eitherzero or too small to give an eect above the noise level. This is important for the stabilityof the solution.In the second stage it, analyses the remaining data to see which parts of the responsematrix that can be reconstructed. This is done by studying the properties of the matrix2C. First, the matrix is reduced by removing linearly dependent columns. Then, by per-mutating rows and columns, the program identies the largest possible square sub-matrixC1, such that~C =C1 C20 C3. (3)The corresponding permutated matrix equation(A1 A2   C1 C20 C3=(M1 M2(4)can then be solved forA1  C1 = M1 (5)since the columns of the reduced matrix are linearly independent and thus the inverseexists.Having identied the solvable correctors, the program goes back to the full data set,and selects all measurements where only these correctors have been used. The resultingmatrix equation is similar to (5), but in general over-determined, and can be solved usinga least squares t.Finally, the calculated columns are substituted into the previous response matrix.4 TESTSThe performance of the learning algorithm, as compared to the standard ABS algo-rithm based on a static matrix, has been evaluated using simulations. In each simulation,initial errors in both the matrix and the corrector settings were randomly generated, andthe performance of the two algorithms studied as a function of the number of iterative cor-rections made. Measurement noise was also included. Two examples of such simulationsare shown in Figs. 2 and 3. It was found that the ’learning’ algorithm always convergedeventually, whereas with a static matrix it often diverged.It was noted, however, that if the initial matrix error is very large, the residualerror can initially grow signicantly before the learning algorithm starts to converge. Ina real machine this could lead to beam loss, and is thus not acceptable. This problem,appears mainly when corrections are made using many correctors. In such a situation, theoperator would cancel the correction and make a new attempt with fewer correctors.Another experience from the simulations is that the average number of iterativecorrections needed to reconstruct the full matrix is not much greater than the theoreticalminimum (which is given by the number of columns in the matrix). Since the correctionsconverge very rapidly once the correct matrix is obtained, this gives an estimate of themaximum number of corrections needed for convergence.5 EXTRACTING THE OPTICS INFORMATIONJust as the response matrix can be calculated from optical parameters, the opticalparameters can under certain circumstances be deduced from the matrix. For example,the matrix elements for beam steering in a transfer line are given byxi =(pβiβj sin(µi − µj) if µi > µj,0 if µi < µj.(6)30 2 4 6 8 10 12 14 16 18 20Iteration50100150200PercentFigure 2: Simulation results for a system of 10 correctors and 10 monitors. The solid lineshows the residual error in percent as a function of number of corrections performed,when using the ’learning’ algorithm. The dashed line shows the same curve using a staticmatrix. The bar chart indicates how many percent of the matrix entries that have beenupdated. In this case the initial matrix had large errors. After an initial excursion, thelearning algorithm converges, whereas the static matrix version does not.For such a response matrix of n correctors and m monitors, the matrix has n m entries,where roughly half are zero in case monitors and correctors are evenly distributed alongthe line. The matrix can be parametrised in terms of the value of the beta function atn + m locations, as well as n + m − 1 positions in betatron phase (the phase of the rstelement can be arbitrarily chosen).Since the two numbers scale very dierently, when the matrices are suciently large,the resulting system of equations is over-determined and thus the optical parameters canbe determined.Therefore, if the measured matrices resulting from each ABS session are logged,eectively a history of the machine optics is obtained. This could be useful for exampleto study the long term stability of the machine.6 CONCLUSIONSBy making ABS programs learn from previous corrections, a faster and more secureconvergence can be obtained.An implementation of such a learning system have been made in Mathematica, asan extension to the current ABS Mathematica package. and tested by simulations withvery good results.By logging the information collected by the program, statistics on the machineoptics, and its stability in time, is automatically obtained.40 1 2 3 4 5 6 7 8 9 10Iteration20406080100PercentFigure 3: Simulation performed and presented as in Fig. 2. In this case, the initial matrixerrors were small. Both algorithms converges, but in the process the learning algorithmmeasures the matrix, which can then be compared to the calculated one.References[1] B. Autin et al, \Emittance Preservation in the PS Complex", PAC’97, Vancouver,1997.[2] B. Autin and Y. Marti, \Closed Orbit Correction Using a Small Number of Magnets",CERN internal note, ISR-MA/73-17, 1973.[3] M. Arruat et al, \Generic Automated Beam Steering and Shaping Programs with anObject Oriented Approach", ICALEPCS’97, Beijing, 1997.[4] B. Autin et al, \Database for Accelerator Optics", ICALEPCS’97, Beijing, 1997.5

"Intelligent" Automatic Beam Steering and Shaping

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