SU129032A1 - The method of reproducing the functions of one or two variables and the device for implementing the method - Google Patents

Description

In the known analog computing devices, to reproduce the functions of one or two variables specified in the form of charts or tables, diode functional converters are used, built on the method of summing up broken lines, or using the function representation of two variable functions of one variable. However, the existing methods of constructing functional converters do not allow directly and independently adjusting and modifying the functions of the function, which limits the use of these methods.

The proposed method of reproducing functions of one or two variables, based on piecewise approximation, allows building functional converters for a wide class of functions and increasing their reproduction accuracy, for which the function is formed from a number of elementary functions (lines, surfaces) and the maximum or minimum of them are using logic circuits "OR and" I. Voltages that vary continuously are applied to the inputs of these circuits, and voltages that are equal to the largest or smallest of the input voltages are applied to the output voltage.

A device for carrying out the proposed method differs from the known ones in order to reproduce a triangular function from each of the two input variables it contains functional transducers connected to the multiplicator units in a matrix pattern, and the outputs of the latter are connected to an adder.

Let us consider the possibility of representing a given nonlinear function of one continuous variable (u) by its logical construction from a series of elementary functions x, (i), y (i). . . z (u) (in what follows, for simplicity, we will consider them linear). Curves that satisfy the condition F (and) О or the condition F (and) O can be approximated by the tangent lines, respectively, as follows: (),; with fCwj O, precede / (“) - min Ai (wX JCJ (M :). l:„ (m). Ri® | m next general meaning of these functions:.,; -. “ ) r-o («); / (V.) An example of constructing a function of the form T {u) is shown in FIG. 1. Curves with several extremes can be synthesized using the above functions and an auxiliary curve A (i), and its arrival through the points F (u) 0. For example, if a curve has three maxima and two minima, then it can be logically expressed as follows: (M) - min {max FI (II}, F (u), P, (h, A (tt}, D (and ), F. (ii)} max {min I /, r "), LD"), A (and, A ("). F, (ii F, (u). In the same way, you can logically form a function with any the number of extrema. Below we consider the possibility of representing a given function of two variables (x, y) by its logical construction from a series of elementary functions of two variables f (x, y); f2 (x, y); fz (x, y) In the simplest case we assume that these elementary functions are linear, then e Each of f (x, y) 2 (x, y) a x + b2y + c; / c (xl ;, g /) aa + cc + c is geometrically represented as planes. By performing logical operations between such elementary by planes, a piecewise-plane approximation of a given surface can be made.As an example, Fig. 2 shows a surface in the form of a pyramid, standing on the x, y plane. f2 (x, y); fz (x, y), then the entire surface as a whole is expressed in a logical form: P (x, y) max {tp1 fiX, y),, y), f. (x, y 0. In the general case, when the function is two the variables are given as a number of points in the space x, y, z, it is necessary to construct a piecewise-plane approximation of a given surface, this can be done by connecting these points. As a result, triangles will be formed (Fig. 3). If now between the formed planes we perform the logical operations "max and" min, then we get a faceted surface approximating a given function g the above variables reveal the advantage of the proposed method over the known summation method for the case of non-monotonic functions. A large part of such curves in the summation method is formed as a small difference of large magnitudes, which can lead to unstable operation of the circuit. since each point of the resulting curve is created independently (formed only by one of the elementary functions). The following describes a variant of constructing a device that implements the proposed method of reproducing functions of two variables, in which the applications Zi, T., Z, are specified directly. . . . At the same time, the intervals x and & y between the given abscissas and ordinates are constant.

Interpolation is carried out as follows. The current value of Z is inside the quad with aplicates Zi, j; Zi. Z, + i, y + i is defined as the sum of the specified items. At the same time, the weight is proportional to the product of the current x and y coordinates of the given aplicate diametrically with respect to the height.

In this way

,,) (/-3,.)-f 2 /, y +, (/ - “J Ru (+ yu“ / (1-8y) + Z, + b, 4, a, (1)

Such interpolation leads to the necessity of the formation of the functions a, p, 1-a, 1-p and the sum of the products of these functions.

Consideration of the transition from one interval to another shows that these functions must be "triangular, that is, linearly increasing in the previous interval, linearly decreasing in the subsequent interval and equal to zero in all other intervals.

Denote these triangular functions - (x) and X (y). The general formula for approximating two variables with given applicants can be written in the following form:

/ j P

Z-. SZ, /., -, () X, .-, (y) - Z ,,, .-., X, .-, (A-) X;) - Z, + ,, /, (x) A, .., () - f (2) -} - Zi +, j +, x) lj (y).

At any value of henna, all the members of this sum, with the exception of four, for which 0, turn to zero. The remaining four members form the interpolation formula (1).

FIG. Figure 4 shows a structural scheme for modeling an interpolation formula.

In the diagram: / - functional transducers for reproducing triangular functions; 2 multiplication blocks; 3-elements for specifying the aplicat and 4-adder. The outputs of the functional converters are connected in a matrix scheme to the multiplication units, and the outputs of the latter are connected to an adder.

If appliqué functions are specified for abscissas and ordinates, then the entire device for approximating the function of two variables will require: n + m triangular functions, mn multiplicative blocks of two variables, the same number of elements to specify an item and one adder.

Circuit diagrams for obtaining triangular functions can be constructed on a logical basis, as shown above.

In order to reduce the number of multiplication schemes, a piecewise-plane approximation of the function is carried out by applying the logical operations of choosing the maximum or minimum of two quantities.

We set ourselves the task of implementing a piecewise-plane approximation of functions of two variables.

Therefore, the interpolation between the four aplicates

Zij, - Zi + i, j; Zi, j + and Z, i-i ,, - + i must also be piecewise-plane. Let, as in the case of the interpolation formula (1), the application for

the center of the quadrilateral Xcp is -i Xi.); (yi-yj + i) is

Medium-arithmetic application of its verbs, i.e.

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where For -,

№ 129032 - 4 1U + 1 (-M; + l

Piecewise plane interpolation is obtained by connecting straight lines of point 2, with points Z, at Z ;, y + / ,. 4 + lu and, +, In general, the piecewise-plane approximation of the function of two variables in this case is formed as a faceted surface, each face being a flat triangle, two of whose vertices are determined by adjacent given coordinates, and a third is the middle coordinate .

Piecewise planar approximation of a given function can be obtained by replacing the functions a, (1-а) р., (1-р) аи (1-а) (1-j3) with piecewise-plane approximations in the interpolation formula (1) these functions. When multiplying two variables, the function is represented by a 2 nd order surface. Consider the approximation of such a function by a faceted surface within the variation of the variables O and 1. O: p 1 and provided that at the points, (

. R -- ; approximation surface matches

with the values of the aPH function 7 of FIG. Figure 5 shows the coordinates of the coinciding points, the equations of the faces of the approximating surface and its plan view. On the edges of a faceted surface, it is shown which logical operations (max or min) should be performed between two adjacent faces.

It can be shown that the equation of the approximating surface can be expressed in the following form

. 5 J I mip (a.) + Max a- | -3- /), 0 I (-3)

Denoting the approximate multiplication by formula (3) by I (a, p) and substituting it into interpolation formula (1), we obtain the following expression for piecewise-plane interpolation.

Z Z ,, / 7tX, i (;) ,,. I (y) -fZi ,,., - i (-c), XXy) -rZ; + „,.).,., (Y) f

r2, nb; +, I), ChlH v (y) j

This expression is depicted as the surface shown in FIG. 6

In the piecewise-plane approximation of the function of two variables, the structural scheme shown in FIG. 4 remains the same; according to the multiplication device, they acquire other content.

An electrical multiplication circuit according to the formula (3) is shown in FIG. 7. Using diodes 5 to 6, the operation is performed to select the minimum of the two voltages of air. Resistances 7, 8 and 9 serve to form the sum of a + p-1. A diode W is used to compare this sum with zero. Resistance 11 is used to sum the results of two logical operations.

Independent variables are introduced into the device in the form of direct current voltages. The depicted function is obtained at the output of the device, also in the form of direct current voltage.

Subject invention

1. A method of reproducing functions of one or two variables, based on piecewise approximation, characterized in that, in order to extend the class of reproducible functions and increase the accuracy of their reproduction, the function is formed from a number of elementary

functions (lines, surfaces) and highlight the maximum or minimum of them using logical circuits "IDI and" I.

2. An apparatus for carrying out the method according to rG1, characterized in that, in order to reproduce the functions of the variables, it contains functional converters for reproducing a triangular function from each of the two input variables, multiplication blocks and an adder, and the outputs of the functional converters are connected in matrix form to the blocks multiply, the outputs of which are connected to the adder.

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