## Numerical Spectral Estimation of Control Parameters for Electromechanical System

Fourier transformation (FT) is the mathematical basis for presentation ofa temporary or spatial signal (or some model of this signal) to its performance in frequency domain. The statistics methods play animportant role in the spectral analysisfor the signalshaving noise or casual character. If the basic statistical characteristics of a signal were known precisely or they could without a mistake be defined on a final interval of this signal, the spectral analysis would represent by itself branch of exact science. However, itis possible to receive some rating of its spectrum on a unique piece of a signal.

Now essential contribution to development of spectral analysis methods is rendered by means of effective algorithms intended for discrete FT, which are its numbers of versions.

Similarly, the occurrence of fast FT has sharply increased efficiency of classical spectral estimate methods, and development of fast computing algorithms (Cooley-Tukey, Vinogradov, with time or frequency decimation, gradient adaptive procedure of the quickest descent, recursive adaptive procedure of the least squares and others) create new spectral estimate methods. It promotes many spectral estimate methods in real time and appreciably is a research work which is consequence of fast algorithm realizations for adjustment of linear parametrical models on a method of the least squares to readout of the data.

The degree of resolution improvement and/or reliability increase for spectral estimates is defined by conformity to the chosen model of analyzed process and opportunity of measured data approximation or infinite auto correlation sequence on the basis of the number of model parameters.

The set of problems is connected with regression analysis for non-homogeneous and correlated observations. For diagnostics of similar situations the so-called “analysis of residuals” and for sampling action of acceptable estimates the "weighing" information operation is usually used. In the cases of weight heterogeneity, it is defined on the basis of dispersion analysis. However, in the case of correlated processes - by some time functions. However gathering of necessary information requires large volume of experiments and huge expenses for processing of results. In such situation, the analysis of processing data for various EMS of technological installations has shown that a successful and virtually realizable approach is analysis of residuals, in spite of it being not entirely formalized.

The spectral estimate of power density (SEPD) procedure as the most important characteristic for spectral estimate of EMS parameters can be carried out by three various equivalent ways on the base of : covariance functions, compact functions, or filtration - squaring and function averaging.

In the first case, it is necessary to take FT from previously calculated covariance function and to subtract it from the average value. Such (infinite) FT usually exists, even if (infinite) FT of initial stationary casual process does not exist. This approach gives bilateral SEPD, designated as S(f) and determined for function f from ]-∞;+∞[, i.e. Wiener-Khintchine task, including two sub-tasks - estimates for covariance function and spectral density through FT, is solved.

In the second case, an approach is based on finite FT realizations of initial process, i.e. consists in FT on a final interval by means of procedures of fast FT

_{} (1)

In the third case a calculation approach means application of analog devices (analog spectral analyzers).

Spectral density function in consideration of spectral estimate weight displacement for all three cases from action of white noise agrees is set by the formula

_{}, (2)

where G - expressed in terms of displacement of weight spectral density of indignation acting in system, f_{n} - natural frequency of undamped fluctuations, ζ- system attenuation factor.

The maximal value G_{yy}(f) is reached on resonant frequency _{}. At 2ζ«1 second derivative of the function G_{yy}(f) in a point f=f_{r} is

_{}, (3)

where B_{r} - passband on a half-level energy in the field of a spectral resonant maximum, i.e. _{}.

Then the normalized accuracy error looks as

_{}, (4)

where B_{e} - bandwidth.

The expression (4) is convenient for maximal displacement estimates of spectral density of processes on an EMS output, and as well as it is possible to conclude, that for receiving of small frequency resolution displacement B_{e} should be less passband B_{r}. At EMS designing it is recommended to accept _{} giving a small systematic mistake_{}.

To make spectral estimate at regulation and control of EMS parameters, it is necessary to choose such realization length, which would provide it in real time with the given degree of accuracy. The formulas connecting realization length to an estimate mistake of each parameter contain usually values, unknown before the experiment’s end, and their direct use as a basis for a choice of realization length is not always justified.

Nevertheless at estimate of EMS parameters it is possible to make assumptions which are acceptable from the model accuracy point of view and to receive practically acceptable formula connecting a realization length and an estimate mistake. From the formula for the normalized average square of a mistake

_{}. (5)

It is visible, that the casual part of the normalized root-mean-square mistake of spectral estimates (square root from value (5) gives the normalized root-mean-square mistake e) depends only on realization length T and from frequency resolution B_{e} (in the assumption, that spectral density varies a little within the limits of a frequency band width B_{e} and in the assumption that the process is Gaussian). It means that realization length required for reception of such spectral estimate with a given normalized casual mistake is defined as

_{}, (6)

where B_{e} - known parameter of a design method instead of a process parameter. It is necessary only to choose the frequency resolution B_{e}.

However, as researches which have been carried out on the basis of the applied program package MATLAB (Spectral Analysis using FFTs) have shown that the essential role in shift definition is value G_{x}"(f)determining spectrum "irregularity" within the band limits of B_{e}. Moreover, a systematic part of a spectral estimate mistake depends, first of all, on the frequency resolution B_{e}. This situation strongly has a place in multi-mass elastic EMS as such system has several physical resonant systems. If it is supposed that within the limits of band width B_{e} concerning each resonant natural frequency the spectrum of entrance process is constant, the spectrum of EMS reaction near to each resonant frequency will look like _{}, where c - a constant, and |H(f)|- amplitude-versus-frequency characteristic for n considered elastic masses.

Also the selection of realization length at spectral density estimate is important for two reasons: first, in many EMS the spectrum represents the most important parameter of stochastic process; secondly, in comparison with estimates of other parameters the spectral estimate determination with the given mistake value requires the most rigid requirements to realization length.

As the construction modern EMS is based on use of semi-conductor engineering and taking into account a work microprocessor speed and necessary determination accuracy of spectral estimates, it is necessary to establish a fact of impossibility of their determination in real process time. For overcoming this difficulty a number of spectral estimate methods were examined based on process pattern performances (frequently such methods are named parametrical) and adaptation of length of a researched area vector. As a result of this, sequential estimates are more preferred for EMS than unitized estimates because they allow similar adaptations by simpler means.

The analysis of existing algorithmic design shows that the block estimate methods are more expedient for carrying out of estimates when volume of the available data is strongly limited. It is possible to apply a lot of consecutive estimation methods at presence of longer recordings of the data for autoregressive parameter updating in process of each new readout acquisition from a system gathering data in real time. Then on the basis of these updated parameters it is possible to build the new diagram of a spectral estimate. Such methods are appropriate for tracking electromechanical signals with parameters which are slowly varied in time. The methods of consecutive estimation in time are sometimes named as adaptive algorithms because they are constantly adapted for EMS signal characteristics, even if they are changed.

As SEPD is formally defined by a number of values of auto correlation function, the SEPD estimate task on finite data aggregate belongs to a number of badly caused, or incorrect, tasks.

In classical spectrum analysis the most essential values of this function are estimated at correlation shifts varied from zero up to some maximal value of M. Attempts to reduce this task to one easier and practically to soluble task have resulted in use of various new models of SEPD.

Such class consists of the models of temporary series which give rational functions of SEPD (autoregressive, random signals, and autoregressive – sliding average models). These models are parameterized by some final and, fortunately, small set of coefficients. As a result, less displaced spectral estimates with higher resolution are obtained if, certainly, the used model describes the analyzed data. As a matter of fact, an approach consists in construction of EMS parameter model of its processes on the basis of the given experimental observation. In these models the analyzed process is considered as a process on an output of linear system with frequency characteristic H(f), which has the white noise on the input. Frequency characteristic, in this case, is set as

_{}, (7)

where z=ехр(-j2πfΔt) - z-transformation, and the model of process is described by the difference equation

_{}. (8)

If all coefficients b_{k} are equal to zero, the model is named as a sliding mean model, and the approach is named as spectral analysis by estimates of highest likelihood.

If all factors а_{1}, except for а_{0}, are equal to zero, model is named autoregressive, and the approach is named as spectral analysis on the basis of maximum entropy estimates).

If both а_{1}and b_{k} are distinct from zero, we have the autoregressive - sliding average model, and the approach has no special name. The terms of "sliding average" originally was described by Slutsky (he used the term "sliding summation") and "linear autoregressive" described by Yul for models of a temporary series and invented by the Swedish mathematician Wold. Moreover, it is necessary to note that the consecutive algorithms used for autoregressive estimation of parameters are divided into two categories: first, algorithms on a basis of gradient approximation such as the least-average-squares method, and, second, least-squares recursive algorithms providing higher characteristics in comparison with the former algorithm. However, such characteristics are achieved by means of additional computer power.

Also at creation of models it is necessary to investigate the question of choosing a filter, which, as a rule, is unknown, and in practice usually is tested some orders of model. Moreover, the criterion here is such - if the order of model is chosen too small, the strongly smoothed spectral estimates are received; if too large - the resolution is increased, but there are false spectral peaks in an estimate. Therefore, in other words, autoregressive spectral estimation of the model degree choice is equivalent to the compromise between the resolution and a variance (dispersion) size for classical methods of spectral estimates. Final model degree for data received from real unknown processes carries subjective character and there is no universal technique of a choice yet. Thus, criteria are expedient for using only for a choice of an initial model degree. They provide good results in case of artificial autoregressive signals synthesized on the basis of PC, but their results in case of the valid data depend on the precision of the simulation with the help of autoregressive processes.

On the basis of above mentioned approaches, the SEPD estimates of primary currents for unified block thyristor converters were carried out for the 6-pulses rectifier for tracking 5-meter dish EMS at an assumption of instantaneous current switching in converter phases. A block diagram recursive adaptive procedure of the least squares and software for realization of such estimates were developed on the basis of the FAST RLS package. At the same time, an algorithm represents the following order of actions: reception of new readout of the data x[n+1], transition from a vector Α(p,Ν)to Α(p,Ν+1)without solving the matrix normal equation directly (using only the valid scalar sizes describing a square of a mistake of a linear prediction forward and back), a gain factor with creation of complex parameter arrays for a linear prediction forward and back, and gain factors.

The results were received on the basis of given algorithm of digital spectral estimation for output current signal at thyristor control for various values of integer scalar appropriate to last (on time) data readout and an allowable level of white noise measured of the data. First of all, the analysis of the data has shown the lowest level of noise has appeared at value ξ=0.705at a variation of an attenuation constant (ζ)from 0.1 up to 0.9 and an allowable level of white noise (p). It testifies that the given value of attenuation constant is closest to known optimum at which the greatest speed of algorithm convergence and maximal spectral resolution is provided. Secondly, at constant values ζand p the acceptable error (less than 5% from signal amplitude) can be received at not less than eight values of vector data length.

V.Arkadyev, S.Arkadyev