The design of nonlinear filters and control systems. Part 1

Abstract

The objectives of this paper are to provide a systematic analytical approach to the synthesis of continuous nonlinear filters and to apply the results to a variety of problems, e.g., filtering of signals from noise, characterization of nonlinear systems, and the design of compensation networks for control systems. The problems are illustrated in Fig. 1. The work starts from the concept of a functional as the mathematical representative of a system. The optimum possible functional for any particular problem is approximated by a finite number of Volterra kernels. A typical question which the paper attempts to answer is: “Given an input (signal plus noise) which is a sample function from a stationary, ergodic, random process and using the mean-square error criterion, what is the optimum filter consisting of a finite number of Volterra kernels to filter the signal from the noise?” 1 1 A filter which consists of h 0 , h 1 , h 2 , h 3 , ···, h n is called a filter of degree n and sometimes will be denoted as < h 0 , h 1 , h 2 , ···, h n >. Every kernel h j is called a kernel of order j . Its corresponding filter is a filter of order j . This type of question leads to a set of simultaneous integral equations for which an iterative method of solution is provided. Examples and applications to filtering and control are discussed. Part II of the paper will describe an experimental application of the theory to the characterization of the servomechanism associated with the pupil of the human eye. A measure of the complexity of the experiment will be developed and the applicability of the method to real problems will be discussed from this point of view.