Abstract An n -dimensional servo system is defined as a combination of n simple servo-mechanisms with coupled error-measuring devices. The system is to reproduce n inputs, and the controlling feedbacks consist of linear combinations of the differences between the n inputs and outputs. The analysis undertaken in this paper is to supply the designer of such systems with general performance criteria. They are essentially of three different types. First, error coefficient matrices are established which give a measure for the quality of reproduction of some typical inputs.In the second part the stability of n -dimensional servo systems is investigated with special emphasis on the possibilities of combining stable and unstable servos into stable systems by judicious coupling. There are several results to the effect that this is possible, and that the time delay of the response can be diminished in most cases where there is a sufficient number of design parameters (including the coupling coefficients) at the designer's disposal. In the third part, rms.-error and integrated-square error criteria are established for multidimensional servo systems. By developing expressions in which these errors appear formally identical with the corresponding errors for one-dimensional servos, it is shown how the design techniques developed for one-dimensional servos, such as attenuation-phase diagrams, can be applied to multidimensional systems whenever theoretical minimization of the errors is impossible or impractical.
M. Golomb, E. Usdin
Journal of The Franklin Institute-engineering and Applied Mathematics