Herr, Werner

Miles, J

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SL/Div Reps OCR Output5 August, 1994Geneva, SwitzerlandLondon, nited Kingdom, 27.6-1.7.94Presented at the Fourth Eurogean Particle Accelerator Conference (EPAC 94),programming.Chebyshev norm. In both cases the computation is carried out using the Simplex method of linearmeasure the distance from the goal. In one case the I,-norm is chosen, while in the other it is the loo orapproaches the goal as close as possible. Two algorithms are presented which differ only in how theyon the principle that even when the desired goal is unreachable, it should be useful to find a solution whichA new look at the problem is presented in which constraints appear quite naturally. The method is basedcertain constraints.spectrum. In both cases some difficulties can be expected if, in addition, the solution is required to satisfyminimise the orbit deviations in a meansquared sense or suppress unwanted frequencies in the orbitThe goal of most popular closed orbit correction a1goriLhms is to find a set of correctors which eitherAbstractW. Herr, ]. MilesGoal Programming Algorithms for Closed Orbit CorrectionCERN-SL-94-63IIIIIIIIIIIIIIIIIIUIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII Bw ” 3 *‘CERN LIBRAMES, GENEVA CERN SL/94-63 (AP)CERN ~ SL DIVISIONEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHi:l. OCR Outputcorrecting the closed orbit subject to (optional) constraints\|··|Ii = > jlnlFollowing the usual notations, the general problem ofthat the problem is formulated in a particular way. is defined as :ble to compute a unique solution at each iteration provided lation is more recent [11, 12, 13]. Recall that the I1-normdipoles (a weak condition in practice) it should be possi values is smaller with any other l,·norm. The LP formua machine with suitably distributed BPM’s and correction accuracies or outliers, for the effective weight given to suchessentially uncorrected machine. It will be shown that in important in the analysis of data which includes gross inis the need to reduce peak-to-peak orbit excursions in an appears to originate with Laplace and is considered to belikely to occur during machine startup or commissioning, The idea of setting approximation problems in this normproduce greater returns on the others. Another scenario,2.1 The I1 solutionrificing the requirements on any one constraint will tend toconiicting. Trade-offs must be made in the sense that sacfor the SPS extraction system.combined or compared. It is also clear that they could beoptimum septa and bumper currents and septa positionscommensurate, which means that they cannot be directlysolution. A later paper [10] describes the computation ofrest of the machine. These constraints are not necessarilya cyclotron and compares LP results with a least squaresware limits and the orbit deviation is over thecontrol. An early reference [9] deals with field trimming inthe correction dipole currents do not exceed their hardgorithm This approach is not unknown in acceleratorparticular section ofthe machine, while at the same timesolved either in primal or dual form using the Simplex alcould be that the orbit must follow a specific path over areformulated as extended linear programs (LP) which areoccasions when more may be required. For example, itmerical approximation literature [6, 7]. The problem isand a beam-dynamics point of view. However, there arewhich are well known in the management science and nucases this is certainly adequate from both an operationsorbit correction problem can be solved using algorithmsimise the rms orbit deviation at the BPM ’s ...". In mostWith an appropriate choice of norm, the constrained closedor goal which can be stated in simple terms as "... minmonly used correction strateges assume a single objective 2 TWO ALGORITHMSimplementation details [1, 2, 3]. In all of these the comthe subject of closed orbit correction and the associatedther.A large number of papers have already been written onis well documented [4, 5] and will not be considered furremainder of the paper. The case p = 2 i.e. least squares,1 STATEMENT OF THE PROBLEMsystem is overdetermined, and this will be assumed in thein the case of the SPS (SppS) and LEP, m > n, i.e. thelinear programming. dipole limitations can be incorporated in (1.3). Note thatcomputation is carried out using the Simplex method of used to express orbit path requirements while correctionother it is the lm or Chebyshev norm. In both cases the 1,,-norms, p = 1,2 and oo. The constraints (1.2) can bethe goal. In one case the I1-norm is chosen, while in the quite arbitrary, though the discussion will be limited towhich differ only in how they measure the distance from unknown correction dipole strengths. The norm isthe goal as close as possible. Two algorithms are presented ings and the n~dimensional column vector Is contains theable, it should be useful to find a solution which approaches The m·dimensiona1 column vector y contains BPM readthe principle that even when the desired goal is unreachstraints appear quite naturally. The method is based onar, : cos(];1.; — pj] + 1rQ)A new look at the problem is presented in which contion is required to satisfy certain constraints.some d.iHicu1ties can be expected if, in addition, the solu where the m x n transfer matrix A has elements :unwanted frequencies in the orbit spectrum. In both cases(1-3)Ehthe orbit deviations in a mea.n—squared sense or suppress(1.2)subject to Chrithms is to find a. set of correctors which either(1-1)= ||U··Ak||Minimise ||1·||The goal of most popular closed orbit correction algocan be stated u :AbstractCH-1211 Geneva 23CERN SLW. Herr and J . MilesGoal programming algorithms for closed orbit correctiontion (2.5), the terms ?k serve merely to indicate priorities, ISR-MA/73-17, CERN, Geneva, (1973). OCR OutputMachines Using a. Limited Number of Magnets, CERNwhere I is an m x m identity matrix. In the objective func·[4] B. Autin and Y. Marti; Clored Orbit Correction of A.G.u’.v'Schemes for PEP, PEP-Note 318, PEP, (1979).u,v[3] A. Chao et al.; On·Line Orbit and Dispersion Correctionuuvr.Methods, A293 (1990) 305-307.and k',I•"for the CERN SPS and LEP, Nuclear Instruments andIz — u — v (2.9)[2] D. Brandt et al.; A N ew Closed- Orbit Correction ProcedureE(h’ - B") + u' — v, = (2.8)ence, Chicago, (1989), IEEE Conf. Proc. 3, (1989) 1681.C(h’— Is") + u, - u' = (2.7)[1] W. Herr et ol.; Proc. IEEE Particle Accelerator Confersubject to A(I•’ - k") + u - u = (2.6)Plz + ¢gU; + Vgvt (2.5) 4 REFERENCES(1.1) can be restated as the following ansatz :software package for PC’s [14].Using this definition the closed orbit correction problemoptimisation in the 1,,,,-norm are being conducted using allrllm = msrlrel ational. Small scale ofiiine experiments using prioritisedthey are not fully tested and remain only partially opernorm is defined as : bit correction package COCU [2] of the SPS and LEP butand they must be dealt with correctly. The Chebyshev routines are now incorporated into the standard closed orpeak-to-peak excursions are not considered to be outliers of very limited strength at high energy. The FORTRANdistribution) is best carried out in this norm, for the large bit in the SppS low—B insertions which contained correctorswith sharply defined extremes (as typihed by the uniform in the ll case. This was used successfully to correct the orThe analysis of data with errors drawn from distributions ment and included the possibility of handling constraintsTRAN versions [15, 16] running in the NODAL environ2.2 The Im solutionAlgol procedures [12]. These were later replaced by FORlider using codes which were NODAL [19] adaptations of(which is not strictly necessary).norms (unconstrained) were performed on the SppS colstraints. The remaining constraints ensure nonnegativityEarly closed orbit correction experiments in the ll and 1,,0as possible to satisfy as closely as possible the goal conthe objective function (2.1) which must be made as small 3 IMPLEMENTATION AND STATUSdimensional row vector c = [1, 1, ..., 1] is used to constructnonnegative n-dimensional column vectors k’,h”. The m·design of digital filters.tions. The variable k = h' — Is" is defined in terms of twohas been examined [17, 18] and has applications in theis used to transform the inequality constraints into equaimation subject to either general or restricted constraintsviolated. The column vector u' is a slack variable whicherations could be worthwhile. The question of 1,,,, approxwhich are also known as ”hard constraints” and cannot bethe Haar condition is not satisfied then these additional itEquations (2.3) and (2.4) define system constraints,priority iterations will yield anything useful. However, ifunderachievement while v represents overachievement.a unique solution [7] and it is unclear whether the loweru{,u; will always be zero The variable u representsfied. This means that the 'P; iteration of the LP will haveviation variables. At optimality at least one of the pairA will have full rank and the Hear condition will be satisv are m-dimensional column vectors which are called deNormally each n >< n submatrix of the transfer matrixconstraints” and may be violated if necessary. Here u andtively.fines goal constraints, which are also known as ”softu' and v' are merely slack and surplus variables, respecthat two kinds of constraints occur. Equation (2.2) degoals for obvious practical reasons. The column vectorseral this problem will not have a unique solution NoteBoth of these quantities are minimised as lower priorityhence it it entirely equivalent to minimising In genachievement while exceeding correction dipole limitations.u4+v¢ = Iue-val = lnl ing to meet orbit path requirements and v; denotes overThe column vector ul denotes underachievement in tryThe formulation of the problem is such that form.ficiency will result from solving the problem in its dualu, v LP is approximately proportional to m°n so increased efing Note that the computational difficulty of anand k', k"so that minimising z is completely equivalent toE(k’— k")+u’ = f (2.4)Ck/ _ kl!() d (2.3) Z Z ue + vasubject to A(k’— k") + u - v : y (2.2)Minimise e(u + v) (2.1) first priority. The formulation of the problem is such thatfollows that the scalar quantity z must be minimised withstated as :with “P; denoting highest priority, and so on [6, 14]. ItThen the closed orbit correction problem (1.1) can be reSystem for the SPS, CERN 78-07, CERN, Geneva, (1978). OCR OutputM. C. Crowley-Milling and G. C. Shering; The NODAL[I glshev Approximation, BIT, 17, (1977) 249-261.N. N. Abdelmalek; The Discrete Linear Restricted Chebyica, Winnipeg, (1975) 53-107.eds. B. L. Hartnell and H. C. Williams, Utilas MathematFourth Manitoba Conference on Numerical Mathematics,tions Subject to Bounds on the Variables, Proc. of theM. J. D. Powell; The Minimax Solution to Linear Equa[1-,]270.shev Norm, A.C.M. Trans. Math. Software, 1, (1975) 264Overdetermined System of Linear Equations in the ChebyI. Barrodale and C. Phillips; Algorithm {95: Solution of an[16]I,-norm, Comm. ACM, 17, (1974) 319-320.lution of an Overdetermined System of Equations in theI. Barrodale and F. D. K. Roberts; Algorithm 478: So[15]Jersey, (1991).System: for Business Plus", version 2, Prentice-Hall, NewY-L Chang and R. S. Sullivan; "QSOB+, Quantitative[14]Num. Anal., 10, (1973) 839-848.gorithm for Discrete l; Linear Approximation, SIAM J.I. Barrodale and F. D. K. Roberts; An Improved Al[I3]8, (1966) 295-306.L., Linear Approximation on a Discrete Set, Num. Math.,I. Barrodale and A. Young; Algorithms for Best L; and[I2]212.gression Analysis, J. Amer. Stat. Assoc., 54, (1959) 206H. M. Wagner; Linear Programming Techniques for Re[11]CERN SPS/AOP/78-20, CERN, Geneva, (1978).rents and Septa Positions for the SPS Extraction System,M. Gyr; Computation of Optimum Septa and Bumper Cur[10](1976) 103-110.Currents for the NRL Cyclotron, Particle Accelerators, 7,Least Squares Method in Determining the Best Trim CoilC. M. Davisson and R. B. Theus; Linear Programmimg vs[9]4°7 edition, Scientific Press, San Francisco, (1991).L. Schrsge; "LINDO, An Optimisation Modeling System",[8]Methods ”, Wiley, New York, (1980).G. A. Watson; ”Approximation Theory and Numerical[1]ment", 3'° edition, Prentice-Hall, New Jersey, (1989).G. D. Eppen et al.; ”Quantitative Concepts for Manage[6](1993).cur, Proc. Particle Accelerator Conference, Washington,V. Ziemann; Corrector Ironing in the SLC Final FoIs]

Goal programming algorithms for closed orbit correction

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