A new algorithm of parameter identification of equivalent circuit for electrical charge replacement is suggested. The approach is based on the solution of integral equation of the I type with respect to the function of indicial admittance, by which then determination of replacement circuit parameters is carried out. Application of smoothing splines and original regulating algorithm including kernel setting error of integration equation permits to obtain a stable algorithm of parameter identification. The investigation of algorithm shows high calculating efficiency and sufficient accuracy of parameter identification

Isaev, Yusup Niyazbekovich

Kolchanova, Veronika Andreevna

Kuleshova, Elena Olegovna

Litasov, V. А.

Voskoboinikov, Yu. Е.

Electronic archive of Tomsk Polytechnic University

In the case of using central reflector (fig. 2) the value of this relation in onegroup approximation is equalto 0,82 according to the data [6]. This result is not unexpected one and can be explained, first of all, by thefact that onegroup model does not include the slowneutron storage in the reflector due to slowing down fastneutrons.Solutions in twogroup approximation (fig. 4) showvisually the role of central reflector. The efficiency ofneutron trap permits us to increase the relation of slowneutron average flux to its maximum value up to 0,966.It should be noted that at transition to the effectiveboundary the neutron field distribution near the «coreexternal reflector» boundary is deformed. A more detailed consideration requires increase in system order andsignificant complication of computations.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 174Fig. 4. Distribution of neutron flux00,20,40,60,811,21,41,61,820 5 10 15 20Ɏ,ɨɬɧ.ɟɞ. 6,94x, ɫɦ21,2811,37ɥɨɜɭɲɤɚ ɚɤɬɢɜɧɚɹɡɨɧɚɷɮɮɟɤɬɢɜɧɚɹɞɨɛɚɜɤɚɎ1rɎ1Ɏ2r Ɏ2REFERENCE1. Perspective nuclear fuel cycles and reactors of new generation /V.I. Boyko, D.К. Demyanyuk, F.P. Koshelev, V.N. Meshcheryakov,I.V. Shamanin, V.V. Shidlovskiy. – Tomsk: TPU press, 2005. – 490 p.2. Investigation nuclear reactors / G.А. Bat, А.S. Kochenov, L.P. Kabanov. – The 2nd edition. – Moscow: Energoatomizdat, 1985. –280 p., ill.3. Investigation nuclear reactors IRТТ: A manual in practical trainingand probation period / V.А. Varlachev, О.F. Gusarov et al. – Tomsk:TPU press, 2002. – 56 p.4. The bases of the theory anf calculation methods of nuclear reactors /G.G. Bartalomey, G.А. Bat, V.D. Baybakov, М.S. Alkhutov. – The2nd edition. – Moscow: Energoatomizdat, 1989. – 512 p., ill.5. Merrey R. Physics of nuclear reactors. – Moscow: Atomizdat, 1961.– 292 p.6. Tsyplonkov К.V., Kuzmin А.V. Influence of central reflector on distribution neutron flux in thermal reactor // Modern Techniques andTechnologies: Mat. of the XI Intern. Scientificpractical Conf. ofyoung scientists. – Tomsk: TPU press, 2005. – V. 2. – P. 484–486.Arrived on 13.06.20061. IntroductionOne of the most interesting from the physical point ofview and practically important trends in different fields ofengineering is barrier discharge. In particular, barrierdischarge is used in water purification, plasma technology, etching, etc. However, a strong spatial irregularity andshort durability of physical processes taking place in barrier discharge make it difficult to study this phenomenon.In characteristic description of electric discharge (barrier discharge, in particular) their description as objects ofelectric circuit is widely used [1]. The bases of such approaches are the replacement of electrophysical phenomenaby the phenomena taking place in electric circuit consisting of definite electric elements (resistance, capacity,and inductance). Such electric circuit will be called anequivalent circuit of electric discharge replacement.Investigating the discharge physics voltage U(t) andcurrent I(t) in circuit with discharge gap are available formeasuring. Therefore, there appears the problem of determination of electric discharge replacement circuitparameters with registered function values U(t), I(t). Infact, we have the problem of identification of replacement equivalent circuit.UDC 621.391REGULARISING ALGORITHM OF PARAMETER IDENTIFICATION OF ELECTRIC CHARGE EQUIVALENT CIRCUIT. P. I.Yu.Е. Voskoboinikov*, Yu.N. Isayev, V.А. Litasov*, V.А. Kolchanova, H.О. KuleshovaTomsk Polytechnic University*Novosibirsk State ArchitectureBuilding UniversityEmail: voscob@mail.ru, isaev_yusup@mail.ruA new algorithm of parameter identification of equivalent circuit for electrical charge replacement is suggested. The approach is basedon the solution of integral equation of the I type with respect to the function of indicial admittance, by which then determination of replacement circuit parameters is carried out. Application of smoothing splines and original regulating algorithm including kernel settingerror of integration equation permits to obtain a stable algorithm of parameter identification. The investigation of algorithm shows highcalculating efficiency and sufficient accuracy of parameter identification.In the works [2, 3] the parameter values are along indicial admittance g(t). In its turn, the function g(t) is defined as a solution of the Volterra integral equationconvolution of the I genus, which is an illdefined specifiedproblem [4, 5]. However, in these works a number of significant aspects are not taken into account, they are connected with the solution of this illdefined specified problem, which affected the accuracy of parameteridentification negatively. One can refer to such aspects: insufficient stability of the involved algorithm differentiatedby noise function U(t) when calculating integral equationkernel; neglecting random error of specified equation kernel both at the stage of regularising solution constructionand at choosing the regularization parameters.Therefore, in the present work a stable algorithm ofidentification of replacement circuit parameters, basedon regularizing algorithm of solving integral equationconvolution with inaccurately specified kernel [6] andinvolving the aspects mentioned above fully has beensuggested.Setting up the problemIf the voltage in the circuit is impulse, the transitionin discharge gap in terms of U(t) and I(t) is convenientto describe by means of the Duhamel integral [7]. If thevoltage U(t) is a finite function, i.e. out of the interval[0,TU] it is turned out to zero, then the following expression is true for the current in the circuit(1)where g(t) is the indicial admittance. As a rule, the valueU(0)=0 hence, we obtain the Volterra integral equationconvolution of the I genus:(2)The function g(t–τ) is called the integral equationkernel, I(t) is the right part of the equation.The integral equation (2) is necessary to be solvedrelative to the function g(t), which is an illdefined specified problem, and then by the function g(t) define theparameters of the replacement equivalent circuit ofelectric discharge.Thus, the problem of parameter determination ofthe replacement equivalent circuit includes the following stages:Stage 1. Calculation of derivative on thechanged (with errors) magnitudes of the function U(t).Stage 2. Solution of the integral equation (2) relative to the function g(t).Stage 3. Determination (probably by the form of thefunction g(t)) of the replacement equivalent circuitstructure and function parameterization g(t).Stage 4. Estimation of the function parameters g(t)and calculation of values of resistance, capacity and inductances forming the replacement equivalent circuit.The solution of the formulated determination isconsidered with the following assumptions:1. The function U(t) is different from zero in the interval(0,TU] (i.е. it is finite) and is measured in this interval at themoments tj=Δ.j, j=0,1,...,NU–1, where NU=ent[TU/Δ]+1, Δis the sampling increment, is the integer part of the realnumber z. The changed values U~j admit the statement(3)where ζj are random quantities with the expectationfunction M(ζj)=0, dispersion D(ζj)=δζ2 and showing thevoltage measurement errors.2. The function g(t) is different from zero in the interval [0,Tg].Supposing these, the function I(t) is finite with the interval [0,TI], of the determination interval, where TI=TU+Tg.3. The function I(t) is measured in the interval [0,TI] atthe moments tj=Δ.j, j=0,1,...,NI–1, where NI=ent[TI/Δ]+1.The measured magnitudes assume the statement(4)where ηj are random magnitudes with numerical characteristics M(ηj)=0, D(ηj)=δη2.In short, let us dwell on the solution algorithms ofeach stage of the involved problem of the parameteridentification of the replacement equivalent circuit.2. Calculation of derivative by the changes values of voltageThe kernel of the integral equation (2) is a derivative of voltage U(t). It is known that differentiation procedure is an illdefined specified problem (inparticular, minute errors can cause any great mistake inderivative).For stable differentiation of the function U(t), specified by the changed at the moments tj values U~(tj) as anapproximation for U(t) let us take smoothing cubic spline(SCS) Sλ(t). Recall that cubic polynomial is called smoothing cubic spline, meeting the conditions:1. In each interval [tj,tj+1] Sλ(t) has the following form(5)where tj<t<tj+1.2. The function Sλ(t) has the continuous second derivative on the whole segment [0,TU].Calculations of the coefficients aj, bj, cj, dj of SCS(which depend on the smoothing parameters λ) are described in the works in details [5, 8] and for their unique calculation let us adopt the edge conditions of the form:(6)These conditions correspond to typical form of the voltage impulse U(t) (see fig. 1). One can show that SCS withthe conditions (6) returns minimum to the functional(7)2 2010( ( )) ( ( ) )U uj j jT Njf t dt p f t U−=′′ + ⋅ −∑∫ λ(0) 0; ( ) 0.US S Tλ λ′ ′= =2 3( ) ( ) ( ) ( ) ,j j j j j j jS t a b t t c t t d t tλ = + − + − + −( )dU tdτ( ) , 0,1,..., 1,j j j II I t j Nη= + = −( ) , 0,1,..., 1,j j UU U j j Nζ= Δ + = −( )dUdττ0( )( ) ( ) .t dUI t g t ddττ ττ= −∫0( )( ) (0) ( ) ( ) ,t dUI t U g t g t ddτ τ ττ= + −∫Mathematics and mechanics. Physics75among all the functions f(t) with integrable by the squareof the second derivative and meeting the condition (6).After calculation of the SCS coefficients the first derivative S'λ(t) (which is the estimation for the derivative ) can be found by the formula(8)where tj<t<tj+1.The main problem in plotting the SCS is the choiceof the smoothing parameter λ, which can change in therange from 0 (smoothing spline becomes interpolated,crossing the values U~j, i. е. S0(tj)=U~j) to ∞ (SCS becomesa straight line). If λ turns out to be small, the highfrequency constituents are present in the spline, whichis explained by the errors ζj, particularly developing inthe spline derivatives in the form of highfrequency oscillations. If this parameter is too large, the plotted spline appears to be «oversmoothing» and leading and descending edges of the impulse U(t) are strongly smoothedin it, which affects the accuracy of the first derivativecalculation negatively.One can point out the two approaches to the choiceof parameter λ: estimation of λ from minimum condition by smoothing rootmeansquare error [5] and thechoice λ by the spline accuracy characteristics [9]. Letus dwell on the second approach, more appropriate forthe solved differentiation problem U(t).In this approach the smoothing spline is referred toas an output signal of some filter (spline filter), to the input of which the digital string consisting of changes values U~j of the function U(t) comes. In such interpretationthe smoothing properties of spline can be defined by itshardware function hλ(t), that characterises a hard error ofsmoothing and differentiating: the less the function«width» is, the less the hard error is. As a numerical characteristics of hardware function its width is taken Δh(λ):Physical interpretation of this characteristic is rathersimple for the differentiation problem: in smoothingspline and its derivative the function constituents U(t)and the derivative U'(t) are kept with small amplitude distortion, if their width is more than that of the hardwarefunction hλ(t). Specifying «limit» size Δlim of constituents,those are to be kept in spline, the value λ of can be determined from the solution of nonlinear equation:(9)The hardware function hλ(t) is calculated with theformulaFrequency characteristic of spline is defined by theexpression [9]whereare the weighting coefficients of the functional, Δ is thesampling increment.Let us demonstrate the presented approach to thechoice of smoothing parameter λ by the results of thefollowing calculation experiment. In figure, а, the graphof function is shown (curve 1) and its «exact» derivative (curve 2). The function values are multiplied by( )d U tdt0 1 22 2 22 6 4; ; ;3 6 jp p pq q q p pλ λ λΔ Δ= + = − = =Δ Δ Δ220 1 2( )sin( )2 1 cos( )2 ,2 cos( ) 2 cos(2 )2HTT q q qλ ωωωωω ω ω=⎛ ⎞⎜ ⎟ ⎡ ⎤− Δ= ⎜ ⎟ ⎢ ⎥Δ + Δ + Δ⎣ ⎦⎜ ⎟⎝ ⎠1( ) ( ) .2i th t H e dωλ λ ω ωπ∞−∞= ∫( ) .h limλΔ = Δ0( )( ) .(0)hh t dthλλλ∞Δ =∫2( ) 2 ( ) 3 ( ) ,j j j j jS t b c t t d t tλ′ = + − + −( )dU tdτBulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 176Figure. Calculation of the function U(t) derivatives510dU( t )U( t ),dt642240 0 02, 0 04, 0 06, t510dS ( t )dU( t ) ,dt dtO 64224t0 0 02, 0 04, 0 06,ɚ       ɛb100, so that the function would differ from zero in the figure scale. Curve 3 corresponds to derivative of integralspline S0(t), plotted by the changed (with error) valuesU~j, j=1,2,...,NU=240. Relative level of errors ζj is specified to be equal to 0,05. In figure, b, the graph of «exact» derivative and the derivative values calculated by interpolating cubic spline are shown. Onecan see significant oscillations of this derivative, characteristic for differentiating inexact specified function. In figure, b, the graph of derivative and the derivative values Sλ'(t), calculated by smoothing cubic splineare presented Sλ(t), see (8). A rather good (in comparison with the derivative of interpolating spline) agreement of these two derivatives is obvious.Smoothing parameter is chosen from the solution ofthe equation (9) at Δnp=5.10–3 s (sampling intervalΔnp=2,5.10–4 s). The magnitude is specified to be equal tothe half width of the leading edge of the voltage impulseU(t), that permits the derivative Sλ'(t) to keep «fine» detailsof derivative U'(t) (in particular in the range [0, 0.01]).Small oscillations Sλ'(t) of the derivative in the range [0.01,0.03] are at the second stage of interpretation as specifyingerrors of integral equation kernel and are taken into account in construction of regularising solution of this integral equation. For this purpose let us present the values ofspline Sλ'(t) derivative in the junction tj in the form:(10)The random magnitudes ξj reflect the errors in calculation of derivative on smoothing spline Sλ(t). If theerrors ζj of measurement U(tj) have the same dispersion,the random errors of differentiating have also dispersionD(ξj)=σξ2.( ) ( ) , 0,1,..., 1.j j Ujt tdS t U t j Ndtλξ=′ = + = −( )dU tdτ( )dU tdτMathematics and mechanics. Physics77REFERENCES1. Samoylovich V.I., Gibalov К.V., Kozlov V.К. Physical chemistry of barrier discharge. – Moskow: Moscow State University Press, 1989. – 360 p.2. Isayev Yu.N., Kolchanova V.А. Algorithm of parameter determination of electric replacement circuit of ozoniser under the influence ofpulse voltage // Bulletin of the Tomsk Polytechnic University. –2006. – V. 309. – № 1. – P. 59–65.3. Isayev Yu.N., Kolchanova V.А., Khohlova Т.Е. Determination of dipole parameters under the influence of pulse voltage // Electrichestvo. – 2003. – № 11. – P. 64–67.4. Tikhonov А.N., Arsenin V.Ya. Methods of solution of illdefinedproblems. – Moscow: Nauka, 1986. – 285 p.5. Voskoboynikov Yu.Е., Preobrzhenskiy N.G., Sedelnikov А.I.Mathematical treatment of experiment in molecular gas dynamics. –Novosibirsk: Nauka, 1984. – 238 p.6. Voskoboynikov Yu.Е., Litasov V.А. Regularizing algorithm of nonparametric identification at inaccurate initial data // NSTU Scientific Proceedings. – 2005. – № 2(20). – P. 33–45.7. Bessonov L.А. Theoretical bases of electrical engineering. – Moscow: Gardariki, 1999. – 638 p.8. Zavialov Yu.S., Kvasov B.I., Miroshnichenko V.L. The methods ofspline functions. – Moscow: Nauka, 1980. – 321 p.9. Voskoboynikov Yu.Е. Frequency approach to the estimation of accuracy of smoothing and differentiating experimental data on the basisof smoothing splines // Avtometriya. – 1986. – № 1. – P. 38–43.Arrived on 18.07.2006

Regularising algorithm of parameter identification of electric charge equivalent circuit

A new regularizing algorithm of function calculation of indicial admittance in discharge gap equivalent circuit using stable differentiation and integral equation algorithms which allow for errors of initial data is suggested. Application of the least perfect square method at error modelling for the function parameters of indicial admittance is an additional way of «smoothing» modeling error of regularizing solution

Voskoboynikov, Yu. Е.

REFERENCES1. Witt W., Loffler M. The electromagnetic GunCCloser to WeaponSystem Status // Military Technology. – 1998. – № 5. – P. 80–86.2. Nosov G.V. Generation of powerful current pulses by electric sources with alternative inductance // Bulletin of Tomsk PolytechnicUniversity. – 2005. – V. 308. – № 7. – P. 68–70.3. Tatur Т.А. Bases of electromagnetic field theory. – Moscow: Vysshaya Shkola, 1989. – 271 p.4. The theory of electric devices / Edited by Prof. G.N. Alexandrov. –Moscow: Vysshaya Shkola, 1985. – 312 p.5. Kalantarov P.L., Tseytlin L.А. Inductance calculation: Referencebook. – Leningrad: Energoatomizdat, 1986. – 488 p.6. Electro engineering manual: in 3 vol. V. 1. General questions. Electro engineering materials / Edited by Prof. of MEI V.G. Gerasimovet al. – Moscow: Energoatomizdat, 1985. – 488 p.7. Zhelezny V.B., Lebedev А.D., Plekhanov А.V. Influence of railgunarmature on acceleration dynamics // II Seminar on Dynamics ofHighCurrent Arc Discharge in Magnetic Field: Materials. – Novosibirsk, 4–6 December, 1991. – Novosibirsk: Thermal Physics Institute Press, 1992. – P. 16–32.8. Galanin М.P., Lebedev А.D., Lototskiy А.P., Milyaev К.К. Heatand electromagnetic processes at the contacts of electrodynamic accelerator // Preprint of M.V. Keldysh Applied Mathematics Institute of RAS. – 2000. – № 42. – 32 p.Received on 04.12.2006Mathematics and mechanics. Physics691. Regularising algorithm for calculation of indicial admittance function In integral equation (1) of the first part of the given paper [1] the intergrand is substituted by its estimation Sλ'(t) of smoothing cubic spline derivative. It is necessary to find the solution of this equation, i. e. indicialadmittance g(t). The solution of such equation is knownto be illposed problem and to calculate stable solutionone must use special regularisation methods [2, 3].In the work [4] regularising algorithm for impulsefunction identification in stationary dynamic system(kernel of integral equation) in which input and outputsignals of identified system are known with random error has been suggested. Use of the discrete Fourier transformation (DFT) and algorithm of the fast Fouriertransform (FFT) stipulates high calculation efficiencyof regulating algorithm. Without repeating the construction of this algorithm, we are giving the basic calculation relations, adapting them for the problem of function recovery g(t) and for designations used in this paper.The calculation algorithm g(t) can be presented bythe following steps [4]:Step 1. Forming periodic (with N period) sequences:Step 2. Calculation of sequence elements(1)where i=√–––1.Step 3. Calculation of DFT sequence coefficients{ip(j)} (direct DFT):(2)Step 4. Determination of DFT coefficients (denotedas {Gpα(l)}) of regularizing solution (the calculated relations are presented below).Step 5. Calculation of periodic solution (inverseDFT from the sequence{Gpα(l)}):(3)Step 6. Formation of Ngdimensional vector gα according to the rule:where Ng=NI–NU+1. If the following condition is metthen projection gα j of vector gα are taken as values of solution regulation gα(t) in the nodes tj=j.Δ, j=0,1,...,Ng–1.Note that in calculations (1–3) the FFT algorithm isused which reduces the number of operations by the or1,U IN N N≥ + −( 1), 1,..., ,j p gg g j j N= − =α α102( ) ( ) exp( ), 0,..., 1.Np plig j G l l j j NNα απ−== = −∑101 2( ) ( ) exp , 0,..., 1.Np pjiI l i j lj l NN Nπ−=⎛ ⎞= − = −⎜ ⎟⎝ ⎠∑102( ) ( ) exp , 0,..., 1,Np pjiD l d j lj l NNπ−=⎛ ⎞= = −⎜ ⎟⎝ ⎠∑( ), 0,..., 1;( )  0, , 1,..., 1,( ) , 0,..., 1;( )0, , 1,..., 1.IpI IUpU UI j j Ni jj N N NS j j Nd jj N N Nλ⎧ ⋅Δ = −⎪= ⎨ = + −⎪⎩⎧ ′ ⋅Δ ⋅Δ = −⎪= ⎨ = + −⎪⎩( )dUdττUDC 621.391REGULARISING ALGORITHM OF PARAMETER IDENTIFICATION OF ELECTRIC CHARGE EQUIVALENT CIRCUIT. PART II.Yu.Е. Voskoboynikov*, Yu.N. Isayev, V.А. Litasov*, V.А. Kolchanova, Е.О. KuleshovaTomsk Polytechnic University* Novosibirsk State ArchitecturalBuilding UniversityEmail: voscob@mail.ru, isaev_yusup@mail.ruA new regularizing algorithm of function calculation of indicial admittance in discharge gap equivalent circuit using stable differentiationand integral equation algorithms which allow for errors of initial data is suggested. Application of the least perfect square method at errormodelling for the function parameters of indicial admittance is an additional way of «smoothing» modeling error of regularizing solution.der of 2–3 in comparison with «direct» calculation ofcorresponding sums. It permits the construction of regularizing solution of some hundreds or even thousandspoint «length».Let us consider the calculation of DFT coefficients{Gpα(l)} at step 4. Suppose that ηj current registration errors has zero average and variance ση2, but errors ξi ofderivative calculation has also zero average and varianceσξ2. Then the coefficients Gpα(l) are found from the nonlinear equation(4)where α is the regularization parameter, θ=σξ2/ση2 is thevariance relation, DpC(l) is the magnitude, complexconjugate to Dp(l). The elements of Qp(l) sequence are formed according to the rule:where Δω=2π/(NΔ) is the discretization step in frequency domain. The function Q(ω) can be interpreted asfrequency characteristic of stabilizing functional [2, 3]:it has to be a monotone increasing function andQ(ω)→∞ at ω→∞. If the order of regularization is r,then at sufficiently high frequencies ω the asymptoticform is true Q(ω)≈ω2r.To calculate the solution G *pα(l) of nonlinear equation at fixed parameter let us turn to the scheme of simple iteration(5)«Start point» G (0)pα(l) is given as G (0)pα(l)=Ip(l),l=0,1,...N–1. The condition of iteration cease has the view(6)Calculation experiment has shown that not more than5–8 iterations are necessary to meet the condition (6).Choice of regularization parameter included in (4),(5) is the main problem of regularization algorithm construction in practice. At aconservative values α in solution gα the noise terms conditioned by the ηj, ξj errors arepresent. At α pessimistic values the g(t) informative constituents are removed from the gα solution (solution is«oversmoothed»). Therefore, as α regularizing parameter it is desirable to takevalue αopt, giving the minimum ofrootmeansquare error determined by the functional:where g–j+ is the vector projection g–+, being the normalpseudosolution at precise initial data of u(t), i(t), M[•]is the expectation operator in terms of gα random vectorsensemble gα haracteristics of g(t) function, which areunknown in practice. Therefore, we restrict our consideration to estimation calculation αW for αopt.Introduce the functions [4]:(7)(8)where γ=1/α. As the value αW is accepted 1/γW, where γW=γ (n+1), but the value γ (n+1) meetsthe condition:(9)where ϑm,β/2, ϑm,1–β/2 are the quantiles of χ2distributionwith m=NI–1 freedom degrees of significance level β/2and 1–β/2 correspondingly (as a rule, β=0,05). Thesequence {γ n>0} is produced by iteration procedure:One can show that if the norm square ||I~||2 of I~vector meets the condition and γ (0)<<1   (usuallyγ≈10–10), then value γ (n+1) meeting (9) is found. Calculationof such value does not require more than 4–5 iterations.The investigations made [4] have shown high efficiency of estimation αW. Estimation of regularising solution constructed at α=αW does not exceed that of optimalsolution constructed at α=αW by more than 15...20 %.Note that function RW(γ), R'W(γ), includes, variancesσξ2, ση2. Usually in practice the variance values are unknown (it is particularly the case with variance σξ2 of error in voltage derivative calculation in terms of smoothing cubic spline).To overcome this difficulty the following estimationsof variance are suggested:• ση2: (10)• σξ2: (11)l2 21/ ( / 2 ) .2Lpl LUD D N lN Lξξξξ =−Δ= +∑l 2 2( / 2 ) ;2Lpl LIND I N lN Lηηηη =−= +∑22 , / 2mI >ηβϑσ( )( 1) ( ) (0)' ( )( ) , 1,2,...; 1.( )nn n WnWR m nRγγ γ γγ+ −= − = ( 1), / 2 ,1 / 2( ) ,nWm mRβ βϑ γ ϑ+ −≤ ≤2 2 ,UINC NNσ= ⋅ ⋅Δ22 *2222 *122 * 20( ) ( )( ) (1 ( ) ) ( )( )( ) (1 ( ) ) ( ),( ( ) )W Wp p ppN p p pl pdR R NdD l G l Q lI lD l G l Q lC G lαασ ξ α ηγ γγθγ θσ σ−=′ = = − ×⋅ + ⋅ ⋅⎡ ⎤+ + ⋅⎢ ⎥⎣ ⎦×+∑2*222 *122 * 20( )(1 ( ) ) ( )( )( ) (1 ( ) ) ( );( ( ) )Wp ppNp p pl pRG l Q lI lD l G l Q lNC G lαασ ξ α ηγθγ θσ σ−==+ ⋅ ⋅+ + ⋅= ⋅+∑21CKO( ) [ ( ) ],gjNjjM g gαα +== −∑11 22( 1) ( )01 2( )0( ) ( )0.01.( )Nn np plNnplG l G lG lα αα− +=−=⎡ ⎤−⎢ ⎥⎢ ⎥ ≤⎢ ⎥⎢ ⎥⎣ ⎦∑∑( )122 ( )( )( ) ( ),( ) (1 ( ) ) ( )0,1,...Cn pp pnp p pD lG l I lD l G l Q lnααα θ+ = ⋅+ + ⋅=( ), 0,...., / 2;( )(( ) ), / 2 1,... 1,pQ l l NQ lQ N l l N N⋅ Δ =⎧= ⎨ − ⋅ Δ = + −⎩ωω2 2( )( ) ( ),( ) ( ( ) 1) ( )0,1,..., 1,Cpp pp p pD lG l I lD l G l Q ll Nααα θ= ⋅+ + ⋅= −Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 270These estimations are based on the assumption that inthe neighbourhood of l=N/2 point (the point, relative towhich DFT coefficient modules is symmetric) DFT coefficients are conditioned only by specification errors ofinitial realizations. Amount of sampling, by which pointvariance estimates are calculated, equals to 2Lη+1, 2Lξ+1respectively. To specify Lη, Lξ one can suggest the relation: Lη=Lξ=(0,075...0,1)N. Thus, having acceptedLη=Lξ=0,1N, we obtain for N=256 the amount of sampling 51 (that permits for reliable estimation of variance).2. Parameterisation of conductivity function and parameter estimationAccording to the view of gα(t) regularising solutionconstructed in the previous paragraph one decides onthe view of parametric dependence g(t,P) for gα(t) approximation, it uniquely defines the structure of dischargeequivalent circuit. The argument P means the vectorconsisting of parameters of parametric dependence.Sufficiently universal approximation of indicial admittance is the function of the view(12)where P vector includes four parameters:Among them are the roots μ1, μ2 of circuit characteristic equation. The admittance function (12) corresponds to electric circuit of the second order, the circuitof which is presented in [5].To estimate the parameters pq, q=1,2,...,Q of g(t,P)function let use the leastsquare method of, i. е. estimations of plq are from the condition of functional minimum:and are the solution of system consisting of Q (in general case nonlinear) equations:(13)Parameters pq, q=1,2,...,Q are uniquely connectedwith (i. е. determined by) the parameters θ1,...,θS ofelectrical equivalent circuit (resistance, capacity, inductance magnitudes) and this connection is expressed inalgebraic relations of the following view:(14)Construction of those algebraic relations is made bythe analysis method of transient processes in the DCcircuits studied in details in the course of theoretical bases of electrical technology [6] and therefore these questions are not considered here.Having calculated the estimations plq, q=1,2,...,Q wesolve the nonlinear set of equations (14) (probably byusing the method of least squares) and find the estimation θls, s=1,2,...,S. At this point parameter identification of gas discharge equivalent circuit is completed.3. Result of calculation experimentTo support the efficiency of the identification algorithm suggested and to determine its accuracy capability a large calculation experiment has been carried out, itconsisting of the following.As a «true» function of indicial admittance g(t) thefunction (12) was taken with complex conjugate roots ofcharacteristic level: μ1=μ2*=–416,666–i.571,304. Thenthe admittance function g(t) was preset by the expression:or in general form by the formulaFor such admittance function the identification problem consists in estimation of parameters A, μ, ω, ϕ.Interval of g(t) function assignment was equal to[0, 0,015 sec]. The form of specified voltage is presentedin Figure а [1] and [0, 0,055 sec] is assignment interval.The current values I(t) were defined by the interval (1)[1] and Interval of I(t) function assignment was equal to[0, 0,070 sec]. The discretization step Δ was specifiedΔ=2,5.10–4 sec, it determined the following lengths ofdiscrete sequences: U=220, Ng=60, NI=280.The values U(tj) were distorted by normally distributed random numbers ζ j with zero average and varianceσζ2 defined by relative noise level δU:Similarly, the values I(tj) were distorted by normallydistributed random numbers ηj zero average and variance ση2, defined by relative noise level δI.Then, in terms of noised, in this way, values U~(tj),I~(tj) the regularizing solution gα(tj) was constructed according to algorithms in 3, 4 points of the given paper.Smoothing parameter λ was determined from the solution of nonlinear equation (4) [1] at Δnp=5.10–3 sec. Regularizing parameter was calculated from the condition(9), in this case it was suggested that variances σξ2, ση2were unknown, and instead of σξ2, ση2 (7), (8) their estimations were used in (10), (11).According to the constructed solution gα(tj) from theset of equations (13) the estimations were calculated Al,μl , ωl , ϕl . Accuracy of these estimations was determinedby the vector of relative errors:22max ( ).2U jU tζδσ ⎛ ⎞⋅⎜ ⎟= ⎜ ⎟⎝ ⎠( ) cos( ).tg t A e tμ ω ϕ⋅= ⋅ ⋅ ⋅ +416,666( ) 0,71 cos(571,304 1,059)tg t e t− ⋅= ⋅ ⋅ ⋅ +1 1 1 22 2 1 21 2( , ,..., );( , ,..., );( , ,..., ).SSQ Q Spppϕ θ θ θϕ θ θ θϕ θ θ θ=⎧⎪ =⎪⎨⎪⎪ =⎩# 1( ) 0;( ) 0.QJ PpJ Pp⎧∂ =⎪ ∂⎪⎪⎨⎪∂⎪ =∂⎪⎩#21( ) ( ( ) ( , ))gNj jjJ P g t g t P== −∑ α1 12 23 24 2,p ApPp Apμμ= =1 21 2( , ) ,t tg t P A e A eμ μ= +Mathematics and mechanics. Physics71Since vector projections δp are random, then bysampling of random vectors by volume the vector of average δp(n) relative estimation errors δp– has been calculated, the projections of which are equal to average valuesof corresponding vector projections δp(n). Vector δp(n)contains relative estimation errors  Al(n), μl (n), ωl (n), ϕl (n),that are constructed in terms of initial data U~ (n)(tj),I~(n)(tj), obtained by distortions of «accurate» values U(tj),I(tj) by nrealisations of errors in voltage j(n) and currentspecification ηj(n).In the table the values of vector δp– projections arepresented for different relative levels of δI=0,01; 0,10;δU=0,01; 0,10.It is seen that relative errors of identification dependweakly on error level of initial data. It can be explainedby the two reasons:• Application of stable algorithms for differentiatingand solving integral equations, including errors ofinitial data of the solved problem efficiently.• Application of least perfect square method in construction of estimations for indicial admittance function parameters, which is additional way of «smoothing» errorin construction of regularising solution gα(tj).Table. Relative errors of parameter identification of indicialadmittance functionComparing vector separate projections δp–, one cannote that A,μ parameters are estimated with a little more errors in comparison with ω, ϕ. It can be explainedby the error of regularising solution g(tj) in amplitude.Analysis of the table and results of other calculationexperiments allow us to conclude: the suggested identification algorithm identifies the parameters of indicialadmittance with acceptable accuracy for construction ofelectric discharge equivalent circuit.δU0,01 0,10δI0,010,0350,0310,0050,0070,1190,0960,0190,0120,100,1070,0880,0230,0110,1360,1120,0450,023lll//.//pA A Aμ μ μδω ω ωϕ ϕ ϕ−−=−−Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 272REFERENCES1. Voskoboynikov Yu.Е., Isayev Yu.N., Litasov V.А., Kolchanova V.А.,Kuleshova Е.О. Regularasing algorithm of parameter identificationof electric charge equivalent circuit. Part I. // Bulletin of the TomskPolytechnic University. – 2007. – V. 310. – № 1. – P. 79–82.2. Tikhonov А.N., Arsenin V.Ya. Methods of illposed problem solution. – Moscow: Nauka, 1986. – 285 p.3. Voskoboynikov Yu.Е., Preobrazhenskiy N.G., Sedelnikov А.I.Mathematical treatment of experiment in molecular flowfluid dynamics. – Novosibirsk: Nauka, 1984. – 238 p.4. Voskoboynikov Yu.Е., Litasov V.А., Regularising algorithm of nonparametric identification at inaccurate initial data // Scientific Bulletin of NSTU. – 2005. – № 2(20). – P. 33–45.5. Isayev Yu.N., Kolchanova V.А., Khohlova Т.Е. Determination oftwoterminal circuit parameters under the influence of impulse voltage // Electrichestvo. – 2003. – № 11. – P. 64–67.6. Bessonov L.А. Theoretical bases of electro mechanics. – Moscow:Gardariki, 1999. – 638 p.Received on 18.07.2006

Regularising algorithm of parameter identification of electric charge equivalent circuit

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