# The approximator of monotone functions

(57) Abstract:

The invention relates to automatic control and computer engineering and can be used in specialized computing and information-measuring systems. The aim of the invention is to improve the accuracy due to the implementation of piecewise quadratic dependence. The approximator comprises a generator of pulses of controlled frequency divider, pulse counter, the counter function, the voltage Converter in the time interval, key, code Converter in the frequency, connected in series meters length and number of the polling station and the controlled frequency divider is made in the form of series-connected code tires multiplexer and Converter code in frequency. 1 Il. The invention relates to automatic control and computer engineering and can be used in specialized computing and information-measuring systems, automatic control systems during playback and the calculation method of piecewise-quadratic approximation of monotone functions, the argument which presents an analog value.Known approximator of monotone functions, the argument which presents an analog vineuse device, the adder and the register [1].This approximator is also characteristic of the low fidelity of functional dependencies, due to the method of piecewise-linear approximation.The closest technical solution, selected as a prototype, is an approximator of monotone functions, the argument which presents an analog value containing the pulse generator is connected through serially connected controlled frequency divider, a balancing counter and digital to analog Converter to the first input of the comparison element, the second input is connected to the information input of the approximator, and the output of the comparison element connected to the control input record output register, setting the inputs of which are connected to the bit outputs of the counter function in parallel to the inputs of the code Converter comprising a decoder plot number, the block memory generator of high frequency pulses, the elements AND, OR, the pulse counter, the trigger register and shaper [2].One of the drawbacks of the prototype is limited scope due to play only a monotonically increasing functions. Drua, as in counterparts, using the method of piecewise-linear approximation.The aim of the invention is to improve the fidelity due to the implementation of piecewise quadratic approximation.The drawing shows a functional diagram of the approximator.The approximator of monotone functions contains counter 1 function counter 2 pulse and the transducer 3 code in the frequency of the controlled divider 4 frequency, made in the form of series-connected code tires multiplexer 5 and inverter 6 code in the frequency, connected in series counter 7 length of plot and counter 8 plot number, key 9, the pulse generator 10 and the inverter 11 voltage in the time interval, the input argument 12 approximator, entry 13 run, the first 14 and second 15 and third 16 installation inputs approximator.The playback function, the range of change of argument of which is divided into m equal length sections, each of which is approximated by a polynomial of second degree, for any i-th section of the approximation can be represented in the formY = a aU

_{x}-U aU

_{x}-U (1) and U

_{x}U 1im , where Y is play;

U

_{x}- the voltage of the argument;

U, U - neprarastumete the i-th site, moreover, the signs of polarity before coefficientsa and a can be different, but the same for all m plots combinations except those in which the range of variation of the voltage U

_{x}argument is the inverse polarity of the first or second derivatives of the function play (1), i.e., is considered only a playback function that is monotonic and has no points of inflection.It should be noted that the split parts of the range of variation of the voltage U

_{x}argument must be provided not only, as noted above, the equality of their length U

_{x}< / BR>

U

_{x}= U - U= const (2) but subject to the nodes of the approximation function play (1) did not undergo rupture, i.e., to conform to the equality

a= a+ Y

_{i-1}(3)

Y

_{i-1}= aU

_{x}a U

^{2}

_{x}(4)

Loop playback and compute monotone functions represented by the formula (1), begins with receipt on bus 13 run approximator pulse that triggers the control input transducer 11 voltage in the time interval (PNU) and producing the record code of ostavaca ia from the first 14 and second 15 installation inputs in counter 1 function and scetchy what you approximator account code ostavaca ia should be the cutting edge of trigger pulse, and start PNEW 11 - back or write these codes should be made in advance before the triggering pulse, the operation setting of counters 1 and 2 to its original state. At the output of PNEW 11 is formed of the voltage pulse duration T which is proportional to the voltage U

_{x}the argument at the input 12 of the approximator, coming to the information input PNEW 11

T = k U

_{x}, (5) where k is the conversion factor of PNEW 11.Given the continuity and linearity of the conversion of the voltage U

_{x}argument in the time interval can be considered as a fair expression

T

_{i}= kU

_{x}-U (6)

T = k U

_{x}, (7) where T

_{i}the time transformation, proportional to the difference between the current and initial values of the voltage U

_{x}at the i-th site;

T - time transformation, proportional to the length of a section U

_{x}.The pulse output of PNEW 11, when the control input key 9, opens the latter, resulting in pulses from generator 10 pulses coming in on the information input key 9, arrive at the frequency inputs of the converters 3 and 6 of the code in the frequency (PCC).Output f PCC 3 at the i-th sector is determined by the expression

f= N, (8) where f

_{o}- cassidey code counter 2 pulse on the i-th segment.In each moment of time t in the interval of duration T

_{i}for any i-th plot output N counter 2 pulses on the code bus PCC 3, will be equal to

N= A f t (9) and

A= A (f T (10)

f= A, (11) where A,A - output codes of the counter 2 pulse corresponding to the initial moments of the approximation on i-1-th and i-th areas;

f, f is the output frequency of the controlled divider 4 frequency (UDC) i-1-th and i-th regions, respectively, equal to the output frequency PCC 6;

P is the number of bits PCC 6 equal to the number of bits of the multiplexer 5;

A - the output code of the multiplexer 5 to the i-th site. Signs polarity "+" or "-" in the expressions (9) and (10) are used depending on the mode in which to run counter 2 pulse - summing or subtractive accordingly, when the output code of the counter 2 pulse in the initial approximation on the first segment corresponds recorded it earlier initial setpoint A. Substituting the expression (9) in (8) and using the formula (11), we get

f t (12)

With the frequency of the output PCC 3 for the conversion time T

_{i}to the input of counter 1 function receives a number of pulses N

_{i}equal

N

_{i}= f dt (13) Substituting equation (12) in (13) and performing the integration with IP corresponding end of the play cycle and compute monotone functions, output code N

_{Y}counter 1 function will correspond to the number of pulses equal to

N

_{Y}= AN

_{i}(15) moreover, due to the smoothness of playback, characterized by the absence of a record of any numbers in this counter in the moments of passing nodes of approximation that would lead to an abrupt change (discontinuity) output code will be fair with regard to formula (7) equality, corresponding to the condition of continuity of playback functions (1) according to expressions (3) and (4)

A A N

_{i-1}(16)

N

_{i-1}= U

_{x}U

^{2}

_{x}, (17) where A,A - output codes of the counter 1 function corresponding to the initial moments of the approximation on i-1-th and i-th areas. Signs polarity "+" or "-" in the expressions (15) and (16) are used depending on the mode in which to run counter 1 function in summarizing or subtractive accordingly, when the output code of the counter 1 function at the initial approximation on the first segment corresponds recorded it earlier initial setpoint A.Given the expression (14), formula (15) we write in the form

N

_{Y}= A U

_{x}-U U

_{x}-U (18)

Comparing formulas (1) and (18), it can be noted that in respect of equalities

a= a (19)

a= a (20)

/SUB> counter 1 function from values of the function play (1), the argument of which is represented by a voltage

N

_{Y}= Y (22) where is a scaling factor.For convenience of reference of the calculation counter 1 function can be performed BCD, and the coefficient selected in multiples of ten.The equality (19) for a given is provided by selecting the value of the initial setting of A corresponding to coefficienta the first section of the approximation and implementation of the equalities (20) and (21) for each of the m areas, the need for which arises from the formulas(3), (4), (16) and (17).In turn, the equality (20) is given , k, f

_{o}n is ensured by choosing the value of the initial setting of A corresponding coefficient of afirst phase approximation, and execution for subsequent plots a specific ratio between the value of coefficienta the i-th segment and the values of the coefficients a & i-1-th site, which boils down to the following.Taking into account the equality (7), (11) and (20), the formula (10) we write in the form

a= a U

_{x}(23)

Using the equality (21), we obtain the relation

a= a 2 a

_{2i-1}U

_{x}(24) i.e., when divided into the sections of the band changed is antov ia a preceding i-1-th site. While it is easy to verify that compliance with the relation (24) is equivalent to requiring compliance with the nodes approximation of the continuity of the first derivative of the function play (1) by breaking it down into parts, i.e., the required equality

= (25)

Thus the division into parts of the range of variation of the voltage U

_{x}the argument should be made in the absence of the nodes of the approximation gap not only playback functions (1) and its first derivative, i.e., the above splitting should be done by spline approximation by polynomials of the second degree.With regard to compliance with the equality (21) is given , k, f

_{o}, n, p, it is ensured by the establishment of cisla input PKC 6 in the initial approximation of the i-th site, which is achieved as follows. In the beginning of the formation interval of time T pulses with a frequency f

_{o}begin to flow from the output of the key 9 to the input of the pre-reset (the reset circuit not shown) of the counter 7, the length of the period T

_{p}repetition of output pulses is equal to

T

_{p}= , (26) where k

_{1}- the conversion factor of the counter 7 section length. Period T

_{p}R>

k

_{1}= k f

_{o}U

_{x}(27) the Output pulses of the counter 7 section length with a period of T

_{p}fed to the input of counter 8 plot number, resulting in the latter in turn is mounted in one of m States, starting with the second. The installation of the counter 8 plot number in a first state corresponding to the first area, and, thus, further establishing the correspondence between the numbers of sites and States of the counter 8 plot number, is produced by the pulse reset (the reset circuit not shown) at the same time that counter 7 length of the site, and, that, before starting PNEW 11, for example, at the time of writing the initial ostavaca and A . Sequentially changed as you move from area to area code in combination with the output buses of the counter 8 plot number is supplied to control inputs of the multiplexer 5, resulting in the release of the latest in moments of transition from i-1 to i-th plot to set the required i-th section of cheslea from a number multiplexing with 16 third set input numbers A...A bus which is connected to information inputs of the multiplexer 5. Establishing the necessary chisel input PKC 6 may be performed not only by using multiplexor, in which in advance "stitched" the above numbers. For each of the substituting elements of the input code (ROM address), as well as to the multiplexer 5 is code output buses counter 8 plot number, and the output code code numbers A , obtained in the result of the operation of decoding, conversion or sampling.Thus the implementation of spline approximation function plays (1) leads to the establishment of the counters 1 and 2 primary ostavaca and A corresponding first section of the approximation, and establishing with the help of a multiplexer 5, coded entrance PCC 6 A corresponding to the i-th site, starting with the first. There is no need at the beginning of each of the following sections to write to the counters 1 and 2 chisel and A , as these numbers to code the outputs of these counters are set to automatically effect a continuous (no jumps) or pulse count corresponding to the continuity of playback functions (1) and its first derivative, which, as noted above, is typical of the spline approximation.The earlier mentioned modes of operation of the counters 1 and 2 on addition or subtraction are set in advance on the basis of the signs of polarity of the derivative fu is a plot of positive i.e., if the playback function (1) monotonically increasing, the counter 1 function must be installed in the addition mode, and if negative, the subtraction mode;

if the polarity of the works marks the polarity of the first and second derivatives at any without exception, the i-th section is positive, the counter 2 pulse must be installed in the addition mode, and if negative, the subtraction mode.Given that the approximator is considered playback function (1), in which the polarity of the first and second derivatives are constant throughout the range of variation of the voltage U

_{x}argument, including when U

_{x}=U, signs polarity of the above derivatives selecting operation modes of the counters 1 and 2 can be replaced by a more simple signs polarity before the coefficients a and funktsii play (1), each of which, as mentioned above, the same for any of the m areas, especially:

if coefficientsa the " + "sign, the counter 1 function must be installed in the addition mode, and if the sign" - " in the subtraction mode;

if the polarity of the characters works polarity before coefficientsa and pologitelno, the counter 2 pulse must ustanovlju of the sameness of character polarity before the same coefficients playback functions (1) on each of the m areas, the operation modes of the counters 1 and 2 on addition or subtraction, set in advance according to the above characters, is also the same in each of these areas.It should be noted that if the playback function (1) is presented in a more concise form

Y = b b U

_{x}b U

^{2}

_{x}(28) where b,b,b - constant coefficients, the relationship between the coefficients and the coefficients a,a, amono to Express ratios

b= a a U aU (29)

b= a-2 a U (30)

b= a (31) Given a directly proportional dependence of the duration T of the time interval of the voltage U

_{x}argument, it can be noted that on the first segment at T = 0 the voltage of the argument U=0. When this expression (29)... (31) for the first leg will take the form

b= a (32) b= a (33) b= a (34) i.e., for the first plot is the identity coefficient, according to which, as mentioned above, set the initial setpoint A, AI A number .Thus, due to new relations, the elements and the possibility of installing different combinations of operation modes of the counters 1 and 2, is provided with regard to scaling hardware implementation of play and compute monotone functions, such as increasing and decreasing, mypolaropposite, and the argument which may be represented by an analog quantity, such as voltage. In addition improves the fidelity function due to the implementation of piecewise-quadratic approximation, as more accurate than the piecewise linear used in the prototype. The APPROXIMATOR of MONOTONE FUNCTIONS, containing the pulse generator, controlled frequency divider, pulse counter and the counter function, characterized in that, to improve the accuracy by implementing a piecewise quadratic approximation, it contains the key code Converter in the frequency counter plot number, the counter section length and the voltage Converter in the time interval, and the output of the pulse generator is connected with the information input key, a control input connected to the inverter output voltage in the time interval, control and information inputs which are connected respectively with the input of the start and the input argument approximator, the output and the first installation, the input of which is connected respectively with the output information and the input of the counter function, a counting input connected to the output of the code Converter in the frequency, frequency input connected to vyhorela connected to the counting input of the counter plot number, the output of which is connected with the control input of the controlled frequency divider, the output of which is connected to the counting input of the counter pulses, the output information and the input of which is connected respectively with the information input of the code Converter in the frequency and the second installation entrance approximator, the third installation input connected to the information input of the controlled frequency divider.

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