SU714406A1 - Arrangement for solving differential simultaneous equations - Google Patents

Description

(54) DEVICE FOR SOLVING DIFFERENTIAL EQUATION SYSTEMS

one

The invention relates to the field of computer technology and can be applied to solve with high accuracy systems of ordinary nonlinear differential equations, for example, in identification systems and control of dynamic objects. ,

Devices are known for solving systems of ordinary non-linear differential equations 1, containing a control unit, a memory unit, controllable key units, multiplication units, adders, integrators, functional converters units. The known devices make it possible to find a solution to systems of first-order ordinary differential equations as a result of the preliminary determination of the vector of the unknowns derived and its subsequent integration.

A device 2 is known for modeling differential equations systems comprising a control unit, analog-digital and digital-analog converters, multiplication units, blocks of analog adders, integrators, hybrid functional converters.

sa, address decoder, memory block, write trigger, controllable key blocks. The known device makes it possible to find a solution of systems of ordinary nonlinear differential equations of the first order, and the integration, multiplication, summation of fast changing variables is carried out in

0 analog part of the device.

The main disadvantage of this device is the relatively low accuracy of the calculations. This is due to the presence of significant methodological and instrumental errors. Thus, the need for time-slicing of all or part of the components of the vector of the right-hand sides in the case of analogue integration leads to significant methodological error. Reducing the latter by increasing the frequency of quantization, or using complex analogue extrapolators leads either to an increase in the required speed of the part of the device that determines the quantized values of the vector of the right Parts, or to an increase in the instrumental error.

0

The purpose of the customization tool is to increase the accuracy of the device to solve differential equations. This is achieved by the fact that the device for solving systems of differential equations, containing a control unit, a multiplication unit, a memory unit, a unit of adders and two groups of elements, additionally contains a recurring computation unit, a group of multipliers, a comparison unit, three accumulators adders and a shift block, with the outputs of the multipliers of the group connected to the inputs of the sugator block whose outputs are connected to the first group of inputs of the memory block and to the first group of inputs of the multiplication block, the second group of inputs of the multiplying block connected to the first group of blockout outputs and the group of inputs of the first block of accumulating adders, respectively; the inputs of the HI group multipliers are connected to the second group of outputs of the memory block; the outputs of the first block of accumulating adders are connected to the first input of the recurrent computation block; first, second, third, third and the fifth outputs of the memory block are connected respectively to the second, third, fourth, fifth and sixth inputs of the block

recurrent calculations, the first and second outputs of which are connected to the inputs of elements AND of the first group, the outputs of the ksoryths are connected to the first

§H6du TpeTbefo. Apple accumulating adders and to the seventh input of the recurrent computation block, the second output of which is connected to the second group of memory flea inputs and to the BOMS input of the second block of adders, the first coch output is connected to the first shift input of the shift, the second and third inputs of which are connected to the sixth the output of the memory block, the second output of the second block accumulates accumulators of the adders and the input of the shift block are connected to the inputs of the AND elements of the second group, the first outputs of which are connected to the first group of inputs of the third block accumulating adders, the output of which is connected to the first input of the comparison block, the output of which is connected to the input of the control unit, the outputs of the elements AND the second group are connected to the second group of inputs of the third, block accumulating totalizers and the eighth input of the recurrent computation unit respectively to the first input of the memory block, the input of the first block of accumulating adders, the second input of the second block of accumulating adders, the fourth input of the shift block, the inputs of the first and first elements and

the second group, the input of the third group of accumulating adders and to the second input of the comparison block, the recurrent computation block contains a group of elements OR, two groups of multipliers, a group of adders, elements NOT, outputs of elements OR of a group of connectors to the first inputs of multipliers of the first group, second inputs which are connected to the fifth input of the block of recurrent calculations, the outputs of the multipliers of the second group are connected to the first inputs of the adders of the group, the second and third inputs of which are connected respectively to the outputs of the multipliers of the first group pty and third5 in: one unit, outputs of multipliers of the third group are connected to the second output of the block directly and through the elements NOT to the first output of the block, the sixth, seventh and eighth inputs of which are connected to the first, second and third inputs of the elements of OR group, the first and second inputs -group multipliers are connected respectively to the outputs of the group adders and the second block Jxprdy, the first and fourth inputs of which are connected to the first and second inputs of the multipliers of the second group.

The drawing shows a block diagram of a device for solving a system of differential equations.

The device contains a group of AND multipliers G, a control block 2, a memory block 3, a block of 4 s / mmators, a multiplication block 5, a block b of accumulating adders, a block of 7 recurrent calculations, a first group of 8 elements I, a block of 9 accumulators, a block 10 shift, the second group of 11 elements And, block 12 accumulating adders, block 13 of comparison.

Each block of group P of block 1 of the multiplication consists of m multipliers by the dimension of the vector of processed quantities, that is, the group and blocks

A multiply multiplies the matrix of multipliers 1 - 1, 2 - 1, ... m-i, 1 - 2,

2-2, I1- 2, 1 - (P -1), 2 - (p - 1), ..., P1- (P - 1), 1 - i, 2 -P ..., w - And. Block 4 adders consists of totalizers 4 - 1, 4 - 2, ...

4- Hi. Block 5 multiplication consists of P1 multipliers 5-1.5 -2, ...,

5- hi. All multiplying blocks, blocks of adders, accumulating adders, elements OR, elements NOT included in the block of recurrent calculations, storing registers have the same structure.

The operation of the proposed device can be illustrated by the example of solving systems of differential equations of the form:,. ,, ,,,:.

,. (, х (- (1) where X (t) jx. (t) J is the vector of unknown 5 ,, ,, о, B boq-i matrices of constant coefficients, (i) fj, (tU-vector right parts (t) (t is a non-linear vector-function, t-time, x (o) is the value of the Spri i-and, where W allows decomposition in the Taylor series in a neighborhood of some point X, i.e.) -s (yg + 4 (Xg) xU) „r. (T) -x-. (P). Shad-Chr. V Vr ((. (2). Here bpfxC-t) is the residual term of the row. HLis / oL) ) L To solve with the help of the proposed device, the system of differential equations (1) should be presented on the basis of the rules and Taylor transformation formulas (T-transformations) defining the relations waiting for the original xshi it; T and 9th xeni X () in accordance with the ratios 4 L - W &) ".. where is the symbol of the transition from X () to X and, conversely, xSh, W is a constant coefficient having the dimension i, 1 is an argument that takes integer values O, 1 ..., oo, in the form of corresponding values of the system of differential equations of T-images. For the general case, the radius of convergence of the row (3) can be less than the interval O, T of integration system (1), the last before moving to its T-image of m, must be pre-submitted, by subdividing the T, T into N p There are no subintervals of H T / N, in which series (3) is known to converge, in the form of a system of local equations; + Ax (yy)) - (C), x. (Obx. (IH where g (-c) (1H + g), 4 (t;) -. (X.,: U,) x. (- D) -xJ Γ (x -, V .fciHl il .-) j- "h -... P r., OitiH, i O, A, .., N-1 System (4) is obtained from the original differential system equations (1) as a result of the successive transfer of the origin of coordinates from point O to point, with T having the local time of the time varying from to TbH, and the relationship between the 1 (1 + 1) th local systems of equations of, the continuity condition x (i) - (according to (3), the system of T-equations representing - without taking into account the remainder of the series (2), the system of local differential equations (4)) takes id; ,, ,, .., (5) X (VM.Ha, x .. (i;) .... (V) ... X. {o) - NCiH), cv 1,2 ,. .., m ,; where YV- (- 1 H of discrete 5-6 function, representing with |, component of vector -i x (r :), discrete of function, representing to component) of vector (e) -rof “JK the discrete function representing the q / th component (t) of the vector f (b is the number of the last of the considered members of the row (2), and;. U (.O x (i (.ai), V, H -: x, uvx . (4 | ;;:. E - v - girt4 - -V l - Thy image; degree of original x -q. (tri - xgf Z -q,. The proposed device works as follows. According to (5), for kg of the 1st subinterval, the device integration interval O, T calculates a set of discrete vectors - "(| based on: : S. (X) -x) (1c) -... -V m -V11-.m) (o) x (, 1, ..., m, / i.e., determines the twofold discretion l (o) , X (), ..., X - (5J, where sHM is the maximum number taken into account in the integration subinterval. The number of discrete values actually taken into account in the subinterval and the step size H of the integration are controlled in the Device GGSNREEEBROT SHen Urpredel KM The npeoegiaidia Hrilfi (3):. | Vl). (MX .. (o) E; where is the given value, characterizing the required accuracy of the calculations. If the number of matching digits (bits) in the right and left parts of the expression (8) is not enough, the calculations are repeated to determine a larger number of discrete X (1 (libre with a reduced step H). If condition (8) is satisfied with satisfactory accuracy, then the device calculates the vector of the first discrete (1 + 1) th subinterval integration in accordance with the expression-.-hh .- ,,,. : - ...... .., .... ... ..., .H ° (, derived from the properties of relations (3), (4), (5), after which the vectors are determined discrete (+ -1) i4i (2), ..., X (S) The calculations are repeated in the same way until the end of the system integration interval O, T is reached (1). The device starts with the input of control signals from the block; - the first input of block 3 of memory and storage unit of block 9. When these signals are received, 6 read from block 3 of memory and write 9 ycjiOBi ft. temporarily from block 3 of memory to sixth in: x: one of block 7 of recurrent computations of n HoSo vector is missing). 1. According to the signals from control unit 2, installation is made in blocks accumulating adders. The receipts at the first input of the .3 memory block correspond to the corresponding xyngsho of the block; the control reads from block 3 to the first inputs of the first block 1 multiplying (1-1, 2-1, ..., I1-1) of group VI of blocks 1 TyyoT "heryy - 1yoTo" and the coefficients i (Want), and on the second inputs of the block - read the unit code. When a block arrives from block 2, the unit code is read to the first input of the block 3 1ЖШ ЖШ п8ш1ёДйгд on the second entry 5 with a simple 11. ШШгаа5ШгШШ: ШвийШййГ1 of block 2 to the input of block 6 of guide adders, the zero code initially set in it is summed with the vector y: CXg-i) received from the outputs of multiplication block 5. 2. From the memory block 3, the .Haif first inputs of the first multiplication unit 1 (1-1, .2-1, ..., "-) value of the vector Z (1 {) for the current value of k are read. For the second inputs, the unit code . The value of the vector (Xg-) is read into the second inputs of the multiplication unit 5 from the memory unit 3. According to the control signals from block 2 in block b of accumulating adders, the value of the vector v (Xg.-) and the vector (xg -., -), Zi M coming from the outputs of block 5 multiplication is carried out, i.e. first two terms). 3. From the data block for the first inputs of the multiplication block group h, the values of the vectors are read (f.}, For the second inputs are the values of the vectors Z-CI -Of-k s p and for the first block O, for the second (.- 1 and t Thus, at the outputs of block 4 adders, the value of the vector Zj t is formed in accordance with the expressions (6). According to the control signals from block 2, the obtained current value (k) is written into block 3. On the second. ryovhd1Y block 5 multiply reads the ratio of the estimated value of vfCi HP and the resulting value of the next term (x.jZCK) / P -c output block and the 5 multiplication is fed to the inputs of the block b accumulating adders, where it is summed with the previous values (1). The described operations are repeated starting at Gt-3, m times, i.e., the current value of Y is calculated., - (( ) According to control signals from, 2 is an inverted vector. (O) discrete output from block 10 through block 11 elements And goes to a group of inputs of the third 5 block 12 accumulating adders, According to control signals from block. 2 at the input of block 12, the zero code and the value of the vector t 1 (o) discrete entered initially into it are summed up. From block 3 of memory, the second third, fourth and fifth inputs of block 7 of recurrent subtractions are considered, respectively, the values of H / (), F, -Ck), tB), CA), i.e. The current value of vector A is. (kH) discretes that are from the second output of block 7 recurrent; the calculations are fed to the second group of inputs of memory block 3, the first input of the second block of accumulating adders 9 and the group of inputs of the first block of 8 AND elements. The vector Xdt + l is written to the memory block 3 and summed up, in accordance with (9 ), with the sum of the preceding vectors accumulated in the block 9, the decree. At the same time, the vector X-Ck + l) through the first block of 8 elements And arrives at the first group of inputs of block 12, where the current value of the left side of condition (8) is calculated and condition (8) is satisfied. If condition (8) is satisfied, The corresponding code coming from the outputs of block 13 to the control input block 2 causes the last control signal to be transmitted to the second control key block 10, with the result that the vector vector obtained in accordance with expression (9) comes from the second output block 9 accumulating adders to the eighth input block 7 p Therefore, in the next cycle of operation of the device, the vector X.,. (o) .C-Rl arrives at the eighth inputs of the recurrent calculation unit 7. Instead of the vector X (q7), the information is shifted to last (shift of the vector X., - (o)), and in block 10 the vector X- (o) is written. condition iM-T reaching the end of the interval, O, T is fulfilled, then the calculations are stopped, otherwise. the next {-i +1) th device operation cycle is performed - as described above, starting from step 1. If condition (8) is not fulfilled, then by the control signals from block 2 in the next cycle of the device operation at the inputs of block 12 from the first outputs of block 7 of recursive calculations, the inverted value of the vector X | (k discrete with the first block of 8 elements) And the sign of the sum of the k-s am-total in the sum -C-1) x (. rejection (8): control signals from block 2, the calculations are repeated as described above, beginning with point 2, but from the second of the outputs of block 7 of recurrent calculations through the first block of 8 elements. And the vector (k + l) instead of the vector (I) comes to its seventh input. I continue to work similarly: depending on the results of checking the conditions (8), the transition to the next the subinterval of integration, or the computation of the next vector in the current subinterval of the discrete unknowns vector. The use of new elements .-, recurrent computation block, group AND multiplication blocks, cumulative-shifting block, accumulating adders block and cumulative estimating translates unit and the availability of new binding elements allow zey.meysdu based jxecTKoro controlling - in accordance with the expression (8) - to improve the accuracy of the results in each computing cycle by increasing the number of discrete oppedel e yx or reducing integration step. Formula of the invention 1, A device for solving systems of differential equations, comprising: a control unit, a multiplication unit, a memory unit, a unit of adders and two groups of elements AND, that is, and with the aim of increasing accuracy, it additionally contains a block of recurrent calculations, a group and multipliers, a shift block, three blocks of accumulating adders and a comparison block, with the outputs I1 of the multipliers block of the group connected to the inputs of the block of adders, the outputs of which are connected to the first group of inputs of the memory block and to p The first group of inputs of the multiplier unit, the second group of inputs of the multiplier unit are connected to the first group of outputs of the memory unit and the group of inputs of the first block of accumulating adders, respectively, the outputs of the group multipliers are connected to the second group, the outputs of the memory unit, the outputs of the first block of accumulating sum. The mators are connected to the first input of the recurrent computation block; the first, second, third, fourth, and fifth outputs of the memory block are connected to the second one, respectively. The third, fourth, fifth, and sixth inputs of the recurrence block of the first calculations, the first and second inputs of which are connected to the inputs of the AND elements of the first group, the outputs of which are supported. The second input to the third input of the third block of accumulating adders and the seventh input of the recurrent calculation block which is connected to the second group of inputs of the memory block and to the first input of the second block of accumulating adders, the first output of which is connected to the first input of the shift block, the second and third inputs of which are connected to the sixth output the memory block, the second output of the second block accumulating adders and the output of the shift block are connected to the inputs of elements AND of the second group, the first outputs of which are connected to the first group of inputs of the third block of accumulating adders, the output of which is connected to the first input of the comparison block whose output is connected to the input

Claims (2)

Claim 1. A device for solving systems of differential equations, comprising: a control unit, a multiplication unit, a memory unit, an adder unit, and two groups of AND elements, characterized in that, in order to improve accuracy, it further comprises a recursive calculation unit, a group and units of multipliers , shear block, three accumulate blocks -. adders and a comparison unit, the outputs of the m blocks of the group multipliers connected to the inputs of the adder block, the outputs of which are connected to the first group of inputs of the memory block and to the first group of inputs of the block, multiplication, the second group of inputs of the multiplication block is connected to the first group of outputs of the memory block and group of inputs of the first block of accumulating adders, respectively, the outputs of m multipliers of the group are connected to the second group. of the outputs of the memory block, the outputs of the first block of accumulating adders are connected to the first input of the block of the river rent computing, the first, second, third, fourth and fifth outputs of the memory block are connected respectively to the second, third, fourth, fifth and sixth inputs of the recursive computing block, the first and second inputs of which are connected to the inputs of the elements of the first group, the outputs of which are connected to the first input of the third block of accumulative adders and to the seventh input of the block of recurrent calculations, the second output of which is connected to the second group of inputs of the memory block and the first input of the second block of accumulative adders, the first the output of which is connected to the first input of the shift block, the second and third inputs of which are connected to the sixth output of the memory block, the second output of the second block of accumulating adders and the output of the shift block are connected to the 'inputs of the elements AND of the second group, the first outputs of which are connected to the first group of inputs of the third block accumulating adders, the output of which is connected to the first input of the comparison unit, the output of which is connected to the input of the control unit, the output of elements of the second group are “connected” to the second group of inputs of the third block flowing adders and to the eighth input of the recursive block of calculations. The outputs of the control unit are connected respectively to the first input of the memory block, the input of the first block of accumulating adders, the second input of the second block of accumulating adders, the fourth input of the shift unit, the inputs of the elements And the first and second groups, the input of the third groups accumulating summatorb and to the second input of the comparison unit. ^L 2. The device p.p. .1, t. This is because the 'block'; of recurrence calculations contains ^{: a} group of OR elements, two groups of multipliers, a group of adders, NOT elements, and the outputs of the elements OR groups are connected to the first inputs of the multipliers of the first group, the second input 1 l of which connected to the fifth input. block recurrent. calculations, the outputs of the multipliers of the second group are connected to the first inputs of the adders of the group / second. and the third inputs of which are connected respectively to the outputs of the multipliers of 5 lei of the first group and the third input of the block, the outputs of the multipliers of the third group are connected directly to the second output of the block and through the elements NOT to the first output of the block , the sixth, seventh and eighth inputs of which are connected to the first, second, third inputs of the elements of the OR group, the first and second inputs of the multipliers of the third group are connected respectively to the outputs of the adder Group 15 and the second input unit, first and fourth inputs of which are connected to first and second inputs of the multipliers of the second group. -Sources of information, 20 taken into account during the examination 1. USSR copyright certificate No. 383085, cl. G 06 G 7/34, 1973. 2. Copyright certificate of the USSR. "481041, cl. G 06 G 7/34, 1975 25 (prototype). TSNIIIPI Order 9290/47 Circulation 751 Subscribed —------------ _________ _____, ..... · Branch of IPY '’Patent', Uzhhorod, st. Project, 4