Ideas for Fast Accelerator Model Calibration*

With the advent of a simple matrix inversion technique, measurement-based storage ring modeling has made rapid progress in recent years. Using fast computers with large memory, the matrix inversion procedure typically adjusts up to 103 model variables to fit the order of 105 measurements. The results have been surprisingly accurate. Physics aside, one of the next frontiers is to simplify the process and to reduce computation time. In this paper, we discuss two approaches to speed up the model calibration process: recursive least-squares fitting and a piecewise fitting approach. 1 CONVENTIONAL APPROACH Some of the first accelerator model fitting routines were based on numerical adjustment of model parameters to make simulated orbit perturbation data match the measured data [1]. Originally known as the 'GOLD' method [2], model calibration concentrated on one section of the lattice at a time. With the GOLD method, the operator first identified parts of the lattice where the model agreed with the measured data. The next step was to vary model parameters near any discontinuities until the model-simulated trajectories matched the measurements. The power of the GOLD method is twofold: (1) hardware errors are quickly identified, and (2) once the model fields are found optical parameters can be predicted throughout. The same technique can applied to both perturbation orbit and absolute orbit measurements. 'Multi-track fitting' increases the accuracy of the result. With multi-track fitting, a set of orbit perturbations made by different dipole kicks provides many self-consistent constraints for the analysis. In effect, with each new trajectory the particle beam makes an independent probe of the spatial field structure of the lattice. Multi-track fitting was initially carried out with RESOLVE [3], a graphical interface program where the operator can interactively adjust parameters to find model and/or hardware errors. In the limit, multi-track fitting can include the complete corrector-to-bpm response matrix with many parameters in the model declared as variables. ______________________________ *work supported in part by Department of Energy Contract DE-AC03-76SF00515 and Office of Basic Energy Sciences, Division of Chemical Sciences. Unfortunately, the number of operations required to setup the calculation, the computation time, and interpretation of the results become unwieldy in a graphical interface environment. To speed up the process, the problem was linearized and statistically correlated solutions were found by matrix inversion in a dedicated code [4]. As with many system identification problems, the non-linear aspect was accommodated by iterative re-expansion around the operating point. The matrix inversion procedure evolved into a powerful tool that can accurately predict quadrupole strengths, BPM gains, corrector gains, and a variety of other model parameters [5-8]. Although the linearization technique is robust and accurate, the turn-around time required to make the measurements, calculate hundreds of closed orbits, and then invert the matrix can be many hours. In the following sections, we describe ideas for reducing the model calibration turn-around time. The goal is to predict the model soon after the measurements become available. 2 RECURSIVE LEAST-SQUARES In order of increasing complication, some of the more useful accelerator model calibration procedures are: I. Fit single orbits to verify corrector and bpm operation. II. Fit single quadrupoles or families in-plane. III. Fit quadrupoles, correctors, bpms in-plane. IV. Fit quadrupoles, correctors, bpms, coupled-plane. Depending on the objectives of the analysis, one or more of these procedures might be adequate. For large calculations, it is worthwhile to minimize the number of correctors and bpms needed to realize an accurate fit. Eliminating redundant measurements reduces the time to calculate the closed orbits and to invert the sensitivity matrix. One way to speed up the process would be to employ a recursive least-squares algorithm. With recursive least-squares, a 'batch' of data is initially processed to arrive at preliminary least-squares answer [9]. Subsequent data is used to update the previous model estimate iteratively. For storage ring model calibration, this translates into choosing a subset of orbit Presented at the 1997 Particle Accelerator Conference, Vancouver, British Columbia, Canada, May 12-16, 1997. perturbations from the response matrix (with appropriate corrector phases) to yield an initial estimate for the model. Each new set of orbit data is then used to update the previous solution on an iterative basis. In a fast on-line system, the model calculations of the perturbed orbits could be carried out in parallel with the initial measurements. Once a sufficient block of data is acquired, the model is fit with a set of variables chosen by the operator. The new model is used to calculate the next set of closed orbits while more measurements made. Depending on the goal of the analysis, subsequent iterations could include more subtle model variables.