Theoretical Concepts

The correlation between correctors and monitors for the linear optics is established by superimposing the monitor reading pattern for every single corrector. The horizontal and vertical plane are treated independently assuming a small betatron coupling. The coefficients of the two resulting correlation matrices also called response matrices can be derived analytically from the machine model or from orbit measurements in the real machine. In the SLS storage ring these matrices have a dimension of 72x72 corresponding to the 72 correctors per plane built into sextupoles and 72 monitors adjacent to the corresponding sextupoles.

  

Figure 1: Response matrix for the vertical plane.

To turn this into a correction algorithm it is necessary to invert the matrices in order to get the corrector pattern as a function of a given monitor pattern. If the correlation matrix is a quadratic nxn matrix and has n independent eigenvectors this is easy to accomplish and one gets a unique solution for the problem. In reality the number of correctors and monitors can be reduced due to monitor failures and magnet saturations in such a way that the matrix is no longer quadratic and the solution is no longer unique.

  

Figure 2: Inverse of the response matrix shown in Figure 1 for the vertical plane.

A very flexible way to handle these scenarios offers the SVD algorithm [5]. This numerically very robust method minimizes the rms orbit and the rms orbit kick at the same time if the number of correctors is larger than the number of monitors while the rms orbit is minimized in the reverse case. Figure 1 depicts the 72x72 correlation matrix for the vertical plane as derived from the machine model. It should be noted that the coefficients of the matrix are highly correlated. As a result the inverse matrix in Figure 2 has the property that only the diagonal and their adjacent coefficients have significant values. In spite of the fact that the matrix contains the global correlation information of all correctors and monitors the nonzero coefficients gather around the diagonal of the matrix. Thus given a corrector only adjacent monitors determine its value.

  

Figure 3: First row of the ``inverse'' of the response matrix for 72 monitors and 72, 48 and 36 vertical correctors (``vcm'''s).

This is also true under the condition that the number of correctors used for the orbit correction is reduced. Figure 3 illustrates this for the first row of the nonquadratic ``inverse'' matrix which corresponds to the first vertical corrector. Successively the total number of vertical correctors (``vcm'''s) is reduced from 72 to 48 and 36 (every third/second corrector is disabled). For the latter case 12 adjacent coefficients are necessary to determine the value of the corrector instead of 3 in the 72 ``vcm'' case. The same statement holds for the removal of certain ineffective corrector combinations by zeroing the corresponding small weighting factors calculated by SVD. This has a direct influence on the implementation of the feedback.