Michael Golomb

A theory of multidimensional servo systems

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 designers 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.

Interpolation Operators as Optimal Recovery Schemes for Classes of Analytic Functions

Suppose that the only information we have about a function f is that it belongs to a certain class ℬ (usually a ball in a normed space) and that it takes on given values at some finitely many points x1,...,xn of its domain (or that some other finitely many linear functionals l1,..,ln have given values at f). Suppose we are to assign a value to f(x) where x does not belong to the set {x1,...,xn} (or to l(f) where l is not in the span of {l1,...,ln}). The value α* for f(x) is considered optimal if for any other assignment α there is some fα ∈ ℬ with fα(xi) = f(xi) (i = l,...,n) for which the error |α − fα(x)| is at least as large as |α* − f(x)|. If moreover a function s* ∈ ℬ can be found such that s* evaluated at x gives the optimal value α* and this is so for every x in the domain of the functions f then s* is considered an optimal interpolant (or extrapolant) for these functions.

Bounds for solutions of nonlinear differential systems

by m a n y authors ([3], [9], [10], [11], [12], [13].; for more complete bibliography see [4], [aa], [6]), usually based on the assumption that the nonlinear terms, represented by F(t, x (t)) in (t.1), decrease faster than the linear terms as [[ xH--~0 . and have the effect of small perturbations of the solution to the linear part. In this article bounds are established for solutions of systems (t . t) with various kinds of nonlinearities, some that increase more slowly than the linear terms when ]]x![-+oo and others that decrease faster than the linear terms when H xl]--~0. In each case the bounds are derived from bounds to the general solution of the linear part assumed as known. In the case of (relatively) slowly increasing nonlinear terms the bounds hold for solutions with large initial values, in the case of (relatively) rapidly decreasing nonlinear terms the bounds are valid for solutions with small initial values. The bounds reflect at least the growth of the various terms involved and are rather sharp for some equations. They are developed from the solution to a certain nonlinear integral inequality [equation (2.6)]. This solution [see (2.17) and (3.18)] is an extension of a result for the corresponding linear integral inequality, which has been used frequently for the purpose of estimating the growth of solutions of differential systems and was first formulated as an explicit lemma by R. BELLMAN ([lJ; see also [4], p. 35, [7 3 , [14]). 2. Large Initial Values The.sys tems considered are of the form

Variations on a Theorem by Archimedes

This equation was first formulated and proved by Archimedes (see [1]) by his method of exhaustion. In a recent note [2] it was shown that T 2) analogues of these results. This is not quite an elementary exercise since the quadratures involved cannot be carried out explicitly.