THE CHAPTER 6. SOME PROBLEMS of SECONDARY
PROCESSING
of a MEASURING INFORMATION
MIS, in essence, are intended for determination of final performances of object. Measured parameter in itself it is not enough. Therefore process of measurements consists of several
stages. Usually them subdivide into three stages:
- primary processing;
- secondary processing;
; - analysis of an obtained information.
The primary processing usually provides reductions of a measuring information on the given parameter to a physical kind. For want of it the methodical errors of a measurement, nonlinearity
of gauges and process of scaling are taken into account. For example, for want of determination of a velocity of a flight vehicle it is necessary to take into account errors introduced by
receiver of air pressure (RAP), nonlinearity of the gauge and to reduce it in a unit of measurements. Besides the kill of a parameter and deleting of anomalous measurements is made.
For the second time processing usually imply characterization on a totality of the measured parameters. For example, maximal velocity of rise is determined on a number of parameters, such as
flight vehicle height, velocity, transhipment, temperature, pressure and humidity of atmosphere etc.
The analysis of a totality of characteristics is made with the purpose of a decision making on object as a whole and in the present work will not be considered. Historically MIS
developed as follows:
- MIS on the basis of various, including optical automatic,
recorders. For want of it the information processing was made largely manually;
-
Emerging the registrars on a magnetic fillet and stationary COMPUTERS has allowed essentially to automate process, but all process of information processing was carried out already after
realization of measurements in stationary conditions.
- the emerging of microprocessor
engineering has predetermined the tendency on transposition first of all primary, and then and secondary processing on object and fulfilment of processing in real time of realization of
measurements. Besides it has allowed to be advanced and in the field of systems of automatic control.
However, accommodation of means of processing on object always collides with a problem of restrictions on weigh and dimensions, especially in the field of aircraft, space researches
and number other. Therefore it is important to organize process of information processing in such a manner that this processing would require a minimum of resources of microprocessor
computing sites, first of all a volumes of storage devices (RAM and ROM), and the accounts were executed so fast, that the process of information processing was executed in real time. This
largely is promoted by optimization of construction of architecture of a system, about what the talk will be in the appropriate chapter. Here will are considered some problems of
optimization of process of calculations and required volumes of memories.
Except for
usual logic operations and elementary mathematical transformations the process of processing of a measuring information requires use various special and transcendental functions. To special
functions concern of a different kind of function for graduating and testing of a primary measuring converter (PMC), and also some other of dependence. Let's consider some from
them:
1. The linearization is made in the event that the physical principle of a
measurement of the given parameter causes difference of static performance (CP) of a primary measuring converter from linear.
In this case operation of a linearization consists of summation to actual CP of a single-error correction Dy1,
where Dy1
there is a function from x (Fig. 38). Dy1 = f(x)
there is a function a constant for given PMC. Therefore for a linearization given PMC the reminder in memory of the computing device (ÂÓ) of this function is necessary.
|
Fig.38 Fig.39 |
The introduction graduating corrections is caused by that the design realization RMC causes emerging tool errors, which expreses as constant for given RMC and can be expressed as function Dy2=f(x) (Fig. 39). For want of it Dy1 and Dy2 can be incorporated by summation. Thus, for want of to primary processing the summation graduating corrections Dy representing function Dy = f(x) is made which owes is stored in memory MIS.
By multiplication of outcomes of measurements on factor of process of scaling result an outcome of a measurement in a physical kind, that is expressed in the stipulated units of
measurements. These operations together with kill, deleting of anomalous measurements, interpolation also make primary processing of a measuring information.
The secondary processing consists of fulfilment of mathematical transformations of a kind: X = f(p1 . . . pn; M1 . . . Mm),
where Õ - researched performance of object; ði - measured parameters; Ìm - factors.
The character of factors can be double - constants factors and factors being function one, or several parameters. For example, the physical constants enter into a solution as constants factors and functions are not. Other factors, for example parameters of standard atmosphere, error of a method and etc. express as function. For example, function Vâ = f(M), where Vâ - air velocity, M - number of the Mach (Fig. 40)
![]() |
Fig.40 |
The total number of a special functions (SF) for a solution by that, or other problem changes in different limits. However, it is roughly possible to accept, that their number is comparable to number of parameters and, as a rule, more. Point figure we, certainly give we can not, and in it there is no necessity, as and so it is clear, that the problem of a storage SF is enough important, that it here to consider.
2. The mathematical processing of outcomes of measurements concludes in itself a number of logic, arithmetical, transcendental operations and other transformations. For want of it most labour-consuming are the transcendental operations (sin, cos, tg, ctg, ln, lg, ex and other.). In calculus mathematics in application to the COMPUTER the submission of such functions as numbers, that is information to arithmetical operations is recommended. However it requires large machine time and hinders information processing in real time.
Thus, SF include graduating performance PMC, factors and single-error corrections functionally dependent from parameters and transcendental functions. They can be realized or is table, that requires large volumes of memory, but are fast calculated, or as factors of ascending power serieses, that requires much less volumes of memory, but requires large time for calculations. Besides the expansion in numbers of stochastic functions is extremely inconvenient. The problem is, what to find a method of realization SF, which would required a minimum volumes of memory and a time for calculations.
If to consider SF as certain stochastic functions, it is possible to raise the question about their digitization, or quantization on argument. For want of it, applying a piecewise linear interpolation, it is possible to use a mathematical means considered above, namely formula:
____ Dx = 3Örmin D and Dx = 50tgamax /2D. |
Here range of function y is accepted for unit, D in % from this range; rmin - minimum radius of a curvature; amax - maximum angle of declination of function.
The choice of the formulas is made on the following criterion: if rmin³1 that is selected the first formula, and if rmin<1 - that second. With the help of of these formulas the interval Dx is determined. Having divided a range of modifications (õ) on this interval we shall receive number of intervals and, therefore, and number of significances (ài) SF were in memory. The determination of current significance of function is made under the formula: y = ai + bix, where ai - nearest significance of tabulared significances of function to significance, current it, from an origin of coordinates, bi = (ài+1 - ai)/Dx, where Dx a pitch of a partition of argument of function.
All this correctly in the event that the interval Dx is uniform. As a rule rmin on different intervals variously, therefore such submission of function is redundant. The adaptive approach, that is modification of intervals Dx depending on a curvature allows to reduce number of intervals, but for want of it the algorithm of calculation becomes complicated and required memory size is increased, as it is necessary to enter into the table not only significances of function on samples, but also significance of intervals between them, and, in the end, any prize we do not receive, therefore it is possible to recognize application of the adaptive approaches in this problem hardly expedient.
As is known from a trigonometry and algebras all transcendental functions can be reduced in conjugate transformations: cos x - arccos x and ex - ln x. Here it is possible to accept two solutions: or to tabulate all four, or only two initial and to use these tables and for reconversions by a method based on recurrent iteration.
|
Fig.41 |
Let's assume, that f (x) represents some curve (à) (Fig. 41). For want of it to us is known y(x), it is necessary to define x. Let's give y0 on an input x, we shall receive y1. Magnitude y0-y1=D1y. Let's consider, that D1y linearly is connected with D1x, then we shall define y2=y1(x1+D1y). Further we shall define D2y=y0-y2 and we shall define y3(x)=f(y2+Dy2) and so on. The common member of a recurrent number: ffn(xn)=f[xn-1+Dfn-2(xn-1)]. The recurrent row for want of n®¥ converges. For want of Ïðè Dfn(xn) smaller, than specific allowable error the process of search ceases and (õ) it is considered found. The practical approbation of this method shows, that there are enough of several pitches to find a unknown quantity significance of function.
It is possible search to organize on other. Namely: a search area to divide in halves and to set anyone õ. If the area on the first sign coincides specific function, in a unknown quantity number is put in the first category 1. If is not present, zero. Half is further divided, in which there is a function still on two parts and the operation is repeated. The process proceeds before filling all code of address. It is possible to name it diatomical as process. Advantage of a method in more precise determinancy in expended time. But the previous method is on the average more economical in 2-3 times.
It is necessary to mark, that use of the tables of direct functions for deriving return probably only in the event that the direct function is unequivocally determined, that is to a given data y there corresponds one significance x. Of a number of functions be characteristic can that the position on an axes x of their maximum curvature does not coincide a position of an extremum. In this case account of number of intervals can be made under the formula:
_____ Dx = Örmin(x)D ´ cos(arctg |x’max| M |
Where Ì - factor which is taking into account a parity of scales on x and y. The accounts conducted because of of an actual material, show, that the number of intervals for creation of the tables, is estimated in tens, sometimes in hundreds but, anyway, volume the informations have less, than for want of drawing up of the tables of functions without application of a linear interpolation.
THE CHAPTER 7. PROBLEMS of the THEORY
of FEEDBACK In APPLICATION To MIS
The measurements are made, it is natural, that on their basis to accept any solutions. If these solutions are made by the person and influences as an operator object, we have a system with manual management. If the solutions are accepted by what that technical system - we have an automatic control system. Both in that and in the other case, if the parameters of that object are measured, which because of of these measurements is controled - we have a system with feedback.
|
Fig.42 |
The management efficiency depends on all elements of a system, including measuring systems. But special significance in this case there is a feedback. Generally in appendix to complicated objects the structure of feedback (ÎÑ) looks like shown on a Fig. 42. On object renders influence a population of some external effects F. The information about outcomes of this effect through feedback (ÈÈÑ) arrives on a resolver (S), where develops with continued effect, is treated and again acts on an input of controlled object. Resolver can be both person (operator), and some automatic system, and sometimes and jointly. The amount of parameters inspected with the help of ÈÈÑ and amount of effects affixed to object is not determined. The complexity of such structure is those, that the solution of a problem of optimization of a control system represents exclusive complexity. There is a mass of various theoretical operating time and practical approaches. Nevertheless, the search of the new theoretical concepts is still farly from being completed and if we shall try to offer still what that of idea, it will not be worse from it.
In the present work it is offered to update some problems of feedback. The researches are carried out on the simplified model of the one-channel amplifier with ÎÑ, which outcomes we shall try to distribute and to more general cases.
1. Concept "the feedback" with reference to electrical and electronic circuits, apparently, should not call vaguenesses, after large period of time after appearance of the monography G.Bode both plenty of the scientific and educational literature.
However controversy round problems, connected to feedback, proceeds. For example, "... Pertinently to underline subjectivity of a term feedback" [14]. There is no unity in definition of concept ÎÑ and its essence as a whole, availability and kind ÎÑ (positive or negative, on a current or on voltage parallel or sequential) [15]. "The errors in the description of properties of intensifying circuits with feedback, in main, are caused by formal use of a mathematical means and they have received so broad distribution, that there is no necessity to refer to the concrete references. These errors could remain unnoticed only because until recently radio engineering systemswere fulfilled, as a rule, experimentally, and their mathematical account was executed approximately and played an auxiliary role. However introduction of microelectronics and, as a corollary, the development of methods of mathematical modelling and computer designs of radio engineering circuits, so requires clarification of many existing submissions about amplifiers with feedback and purification in technical and, in particular, to educational literature"[16].
That fact especially rushes in eyes, that the approach to the analysis of systems of a various type (linear, nonlinear, impulse, digital etc.) is rather various. The methods of application of classical methods of the analysis differ and depending on that, the radio engineering system or system of automatic control is considered.
In the present work the attempt of development of such method of the analysis of systems with ÎÑ is done which would not have restrictions, are superimposed on a system with ÎÑ in the monography Baude, that is method not using of the differential equations. By such method it would be possible to analyze more broad class of systems, it would have the greater universality. Really, the linear differential equation can describe only linear system, which is idealization of the actual system. Therefore application of a method of the analysis based on the linear differential equations, to actual systems can reduce in errors, and for much nonlinear systems with ÎÑ the application it generally is impossible [17].
2. Among concepts of the theory of feedback most of all divergence in works of the various authors is connected to concept "a sign of feedback". This concept is determined as follows more often: "The feedback can be positive or negative depending on that, the target variable with its source magnitude is summarized or is deducted it." [17]. The similar definition is resulted and in many other works. There is also following definition): "... The amplifiers with feedback have positive feedback, when the external amplification will increase for want of simultaneous increase of influence of parameters of a lamp, or has negative feedback, when the amplification drops for want of simultaneous respective diminution of influence of modifications in a lamp." [18]. There are also other definitions, and sometimes definitions of a sign of feedback is not resulted at all.
The application first from the above-stated definitions is inconvenient, as in practice of development of an equipment usually it is meant, that the sign of feedback is a structural
indication of the concrete device and does not depend on a kind and performances of a source signal. Same it is meant and in the second definition. According to the first definition, the
sign ÎÑ depends on a kind and performances of a signal. Is valid if to give on an input of the system with ÎÑ a signal as sinusoidal voltage, at the influence of a reactivity of a
circuit ÎÑ the target signal will have a phase shift concerning source. This phase shift for want of modification of frequency of a source signal varies and for want of closure of an
outline ÎÑ the target variable can for want of one significances of frequency develop with source, for want of other - is deducted. As on an input always there is a determined or casual
signal (strictly speaking, with an infinite bandwidth) and the phase-frequnsy characteristics of the device with ÎÑ is always nonlinear, any actual system with ÎÑ, according to the first
definition, is simultaneously the system as with positive, and negative connection.
Such
definition of a sign ÎÑ would be desirable to have which would reflect a structure of the system and did not depend on a kind and performances of a source variable. the Second
definition, though and is not precise enough, links a sign ÎÑ with a structure of the amplifier with ÎÑ, however it the defect is, that the definition represents a corollary, influence
of a sign ÎÑ, instead of it an essence.
As a rule, in appendices of concept ÎÑ it is meant,
that the sign ÎÑ is an indication of a structure of the system and depends, for example, on number of active elements and polarity of connection of transformers. As the concept of a sign
ÎÑ is one from main in theory feedback, before to proceed to an exposition of a main material it is necessary to stay on definition of this concept. With the purpose of installation of
unequivocal connection of a sign ÎÑ with a structure of the device the concept "inversion" is used.
DEFINITION: inversion - such operation above function f (t), for want of which all component it of a spectrum change a phase on p.
The inversion happens in such systems, as, for example, transistor cascade with the common emitter, transformer for want of determineted connection of windings etc. However in the pure
state to receive inversion it is impossible, as it is always accompanied by distortion of a spectrum of a signal because the actual elements of devices always have jet component of the
performances as multiterminal networks.
|
Fig.43 |
Therefore, for example, the transistor intensifying cascade with the common emitter should be is represented as the block diagram (Fig. 43), where is the sign - (-) the inverter is designated, and [K (jw)] designates ability of the cascade to strengthen a signal and to change it a frequent spectrum. The availability of inversion characterizes a structure of the system and does not depend on performances of a source signal. The system, in which is made inverting, is usually named as the inverter. If the signal sequentially passes odd number of inverters, on an output it is received inverted, if even - not inverted.
DEFINITION: the feedback is positive, if the signal, go out on a closed loop of feedback, is not inverted and is negative, if the signal is inverted.
In a transmission factor of a circuit the availability of inversion is designated by the minus.
The given definition allows more strictly to justify existing separation of all systems(devices) with feedback on two classes - device with positive ÎÑ, auto generation, used for a realization, in qualities of regenerative amplifiers correcting circuits etc., and device with negative ÎÑ, widely used as in a radio engineering for stabilization of parameters of the circuits and deriving of a broad passband, and in system of automatic control and management.
3. As object of the analysis the electronic amplifier is taken, in which there is a passive part, which determines frequent and temporary performances of the amplifier, and the active elements, which, after the account the reactivities in a passive part, can be considered as without inertial, inverting, with an amplification factor A. Concerning nonlinearity and delay the clauses will be done if necessary. The circuits of the amplifier limited by a non minimal phase, ladder type.
As source effect the voltage if necessary expressed in the complex form] is accepted as function of time [u (t)] or
frequency [U (w)] [19]. In a Fig. 44 the block diagram of a direct circuit is represented. The passive part it is characterized by transfer function
:
|
Fig.44 |
For a passive circuit | |
1. The transfer function of a circuit, represented in a
Fig. 44 is equal
for want of inversion
and for want of it availability. The considered circuit is not mutual, as it includes an
active element. The signal not is transmitted is accepted, that from an output to an input. The circumscribed circuit is enveloped by a circuit of feedback with transfer function,
which is accepted not mutual [20], transmitting a signal only in one direction from an output on an input of a direct circuit (Fig. 45).
The
circumscribed circuit is enveloped by a circuit of feedback with transfer function, which is accepted not mutual [20], transmitting a signal only in one
direction - from an output on an input of a direct circuit (Fig. 45). It is known [19,11,21], that the transfer function of the amplifier with ÎÑ is removed from transfer functions direct
and return circuits by a solution of a set of equations and is determined by expression:
|
Ðèñ.45 |
=
(7.1)
The defect of such method of a conclusion of expression of transfer function of the amplifier with ÎÑ consists that it does not reflect an essence of process happening in the amplifier, that, in turn, hinders the analysis of obtained expression, including definition of stability of the amplifier with ÎÑ. The problem is put to develop such method of a conclusion of expression of transfer function of the amplifier with feedback, which would be deprived of these defects, and to justify it application for main kinds of systems with ÎÑ. In the mathematical plan the problem is put to develop such method of the analysis of systems with ÎÑ, which would not require application of the theory of the linear differential equations, as it has that defect, that, strictly speaking, can not be applied for the analysis of nonlinear systems, to which any actual system with ÎÑ concerns, though on occassion it the application and is justified.
4. Considering the block diagram (Fig. 45) of the amplifier with ÎÑ it is possible to make a conclusion, that the signal sent on an input of the circuit, bypasses a closed loop and
again moves on an input, whence for the second time bypasses a closed loop and again moves on an input, and so an infinite number of times.
On an input of the amplifier with ÎÑ addition of signals sent on an input as voltage, to the addition of signals sent on an input( as
voltage, to addition of signals sent on input as voltage, to addition of signals sent on an input as voltage, to is equal (Fig. 45):
.
The transfer function composed from a direct circuit and a circuit ÎÑ, is equal:
.
If the signal bypasses an outline twice, it equivalently to passing of a signal through quadripole with transfer function -
, if n of
time, with transfer function -
. Thus, the voltage on an output of the amplifier with ÎÑ is equal:
. (7.2)
Bat '
for want of
.
Or: .
Therefore: .
Or: .
(7.3)
It is known [22], that a limit of a row , if
< 1, equally
.
From here, if <1,
, or [19,11,23,24]:
.
(7.4)
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