This paper is concerned with obtaining the optimum system in the Weiner sense for the multipole system. Earlier literature has shown how to obtain the mean-square value of the error when the multipole system transfer function has been specified, but thus far no published work has shown how to solve the synthesis problem, in general, for this case. The principal reason that this problem has appeared to be impossible of analytic solution thus far for cross correlation between the inputs is based on the fact that the usual variational approach results in a set of untractable simultaneous integral equations involving many complicated cross products of the desired weighting functions and the variational functions. The synthesis problem for the system is first solved for the case in which there is no correlation between the inputs to the various terminals. The result for the optimum weighting functions in this case is presented in equation (24), and the resultant mean-squared value of the error is shown in equation (25). Following this, the far more complicated case of the synthesis problem when the inputs to all the various terminals are correlated is considered. In this case, a rather unique technique is utilized to avoid the difficulties inherent in the use of the usual variational techniques. Through the technique utilized in this paper, the usual set of untractable simultaneous integral equations is completely avoided, and instead a set of ordinary algebraic equations results. The set of equations for this case is shown in equation (64), and in matrix form in equation (65). The resultant solution for the optimum physically realizable transfer functions is shown in equation (77). It is also shown, as a check, that the solution for the case of correlated inputs reduces to the solution obtained for the case of uncorrelated inputs. The paper then concludes with an illustrative example for the more complicated case of correlated inputs. The possibilities of applications of the results of this paper to such fields as the guidance and control of astronautical vehicles, military fire control systems, bombing navigation systems, process control systems, automatic milling machines, air traffic control, nuclear reactor control, etc., are fairly evident.

### Published in:

Automatic Control, IRE Transactions on  (Volume:4 ,  Issue: 2 )

Nov 1959