The approximate calculation of a class of automatic systems with forced parameter optimization

Abstract

The paper considers: (1) two linear filters ϕ1 and ϕ2 connected in parallel, with an input signal Θ (t), and (2) closed-loop systems for adjusting the parameters Xi of filter ϕ2, which employ a search modulation Δxi (t) of these parameters. The self-adjustment network includes a detector (linear or square-law) for the output quantity of the filter ϕ2 – ϕ1, phase discriminators and averaging filters Wϕ. On the assumption that the amplitudes of the signals Δxi (t) are small, approximate (and, within the limits of the small-parameter method, entirely correct) expressions are found for the control actions in the self-adjustment networks in terms of the frequency characteristics of the filters ϕ1 and ϕ2 and the present spectra of the signals Δxi (t) and Θ (t). The differential equations derived for the selfadjustment processes take account of the limited memory and passband of the filters ϕl5 ϕ2 and Wϕ, and cover the case where the frequency characteristic of the filter is a functional of the signals Δxi (t). These equations lead readily to a number of necessary conditions for the stability of the self-adjustment process, and also to the desirability, with a high-frequency sinusoidal search signal Δxi (t), of using two phase discriminators with reference voltages in quadrature, so as to make use also of the phase modulation on the carrier signal. As a simplified mathematical abstraction, detailed consideration is given to the case of an almost periodic signal Θ (t) and a test signal similar in nature to white noise. Some attention is devoted also to quasistationary self-adjustment modes of operation. Illustrative examples are given of the calculation of actual systems with several adjustable parameters.