First, a sampled-data system which can be described by a linear time-variant vector differential equation X(t)=A(t)X(t) in each step interval T₀ is considered. The step interval T₀ is defined as T₀=T/m, where T is the sampling period and m is a positive integer. Generally, this equation cannot be solved analytically. So, regarding A(t) as fixed in each step interval T₀ and introducing the time shift operator δ(·), we can solve the equation numerically. That is, X(t)=A_{n, i}X(t), X(nT+iT₀⁺)=B_{n, i}(T₀)X(nT+iT₀^-)+C_{n, i}(T₀)X(nT^-), for nT+iT₀<t≤nT+(i+1)T₀, n=0, 1, 2, …, i=0, 1, 2, …, m-1, where B_{n, i}(T₀) and C_{n, i}(T₀) involve δ(·).
A sampled-data system with dead time can be analyzed similarly as a sampled-data system without dead time by introducing the time shift operator.
Next, a sampled-data system which can be described by a nonlinear time-variant vector differential equation in each step interval T₀ is considered.
In this case, by introducing the variable equivalent gain l_{n, i} of a nonlinear element, it can be solved similarly as in the linear case.
The error of the approximate solution is considered in chapter 4. It is proved that the error can be made arbitrarily small by choosing sufficiently small T₀.

View full abstract

A new method of non-contact speed measurement is described. The basic principle of the method is to observe the natural irregularity of a moving object through a set of parallel slits, which forms a kind of spatial filter in that it picks up particular spatial-frequency components from the irregularity. Then, the object motion generates a narrow-band random signal whose center frequency is proportional to the object speed.
In this paper, the spectral density function is evaluated based on the assumption that the irregularity is two-dimensional random process, and the dependence of filtering characteristics on parameters such as width, length, interval and the number of slits is analyzed; the results imply that the set of parallel slits acts as a narrow band spatial filter.
For a quantitative evaluation of characteristics, the concept of half-value-width and selectivity is introduced, and the change of characteristics is discussed when there exists a transverse component of object velocity.
A proto-type speed measuring system based on this principle has been developed. It consists of a spatial filter and a center-frequency estimator which is based on the principle of first-zero searching of auto-correlation function. Static and dynamic errors have been analyzed by means of equivalent transfer functioh representation including inherent noise related to the band-width of the signal.
This system was applied to the speed measurement of some models similar to actual materials. Experimental results show that the analysis is reasonable and that this method is easily applicable to the speed measurement of various materials.

View full abstract