LOCO with Constraints and Improved Fitting Technique

LOCO has been a powerful beam-based diagnostics and optics control method for storage rings and synchrotrons worldwide ever since it was established at NSLS by J. Safranek. This method measures the orbit response matrix and optionally the dispersion function of the machine. The data are then fitted to a lattice model by adjusting parameters such as quadrupole and skew quadrupole strengths in the model, BPM gains and rolls, corrector gains and rolls of the measurement system. Any abnormality of the machine that affects the machine optics can then be identified. The resulting lattice model is equivalent to the real machine lattice as seen by the BPMs. Since there are usually two or more BPMs per betatron period in modern circular accelerators, the model is often a very accurate representation of the real machine. According to the fitting result, one can correct the machine lattice to the design lattice by changing the quadrupole and skew quadrupole strengths. LOCO is so important that it is routinely performed at many electron storage rings to guarantee machine performance, especially after the Matlab-based LOCO code became available. However, for some machines, LOCO is not easy to carry out. In some cases, LOCO fitting converges to an unrealistic solution with large changes to the quadrupole strengths {Delta}K. The quadrupole gradient changes can be so large that the resulting lattice model fails to find a closed orbit and subsequent iterations become impossible. In cases when LOCO converges, the solution can have {Delta}K that is larger than realistic and often along with a spurious zigzag pattern between adjacent quadrupoles. This degeneracy behavior of LOCO is due to the correlation between the fitting parameters - usually between neighboring quadrupoles. The fitting scheme is therefore less restrictive over certain patterns of changes to these quadrupoles with which the correlated quadrupoles fight each other and the net effect is very inefficient {chi}{sup 2} reduction, i.e., small {chi}{sup 2} reduction with large changes of {Delta}K. Under effects of random noise, the fitting solution tends to crawl toward these patterns and ends up with unrealistically large {Delta}K. Such a solution is not very useful in optics correction because after the solution is dialed in, the quadrupoles will not respond as predicted by the lattice model due to magnet hysteresis. We will show that adding constraints to the fitting parameters is an effective way to combat this problem of LOCO. In fact, it improves optics calibration precision even for machines that don't show severe degeneracy behavior. LOCO fitting is essentially to solve a nonlinear least square problem with an iterative approach. The linear least square technique is applied in each iteration to move the solution toward the minimum. This approach is commonly referred to as the Gauss-Newton method. By using singular value decomposition (SVD) to invert the Jacobian matrix, this method has generally been very successful for LOCO. However, this method is based on a linear expansion of the residual vector over the fitting parameters which is valid only when the starting solution is sufficiently close to the real minimum. The fitting algorithm can have difficulties to converge when the initial guess is too far off. For example, it's possible for the {chi}{sup 2} merit function to increase after an iteration instead of decrease. This situation can be improved by using more robust nonlinear least square fitting algorithms, such as the Levenberg-Marquardt method. We will discuss the degeneracy problem in section 2 and then show how the constrained fitting can help in section 3. The application of Levenberg-Marquadt method to LOCO is shown in section 4. A summary is given in section 5.