Asymptotic Method for Solution of Identification Problem of the Nonlinear Dynamic Systems

Transcription

1 Flomat 3:3 8, Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: Asymptotc Method for Soluton of Identfcaton Problem of the Nonlnear Dynamc Systems FA Alev a, NA Ismalov a, AA Namazov a, NA Safarova a, MF Rajabov b, PB Besebay c a Insttute of Appled Mathematcs, Baku State Unversty, Z Khallov str3, Baku, Azerbajan b Insttute of Control Systems of ANAS, Baku, Azerbajan c Department of Hgher Mathematcs, S Sefulln Kazakh Agrotechncal Unversty, Astana, Kazakhstan Abstract A dynamc system, when the moton of the object s descrbed by the system of nonlnear ordnary dfferental equatons, s consdered The rght part of the system nvolves the phase coordnates as a unknown constant vector-parameter and a small number The statstcal data are taken from practce: the ntal and fnal values of the object coordnates Usng the method of quaslnearzaton the gven equaton s reduced to the system of lnear dfferental equatons, where the coeffcents of the coordnate and unknown parameter, also of the perturbatons depend on a small parameter lnearly Then, by usng the least-squares method the unknown constant vector-parameter s searched n the form of power seres on a small parameter and for the coeffcents of zero and the frst orders the analytcal formulas are gven The fundamental matrces both n a zero and n the frst approach are constructed approxmately, by means of the ordnary Euler method On an example the determnaton of the coeffcent of hydraulc resstance CHR n the lft n the ol extracton by gas lft method s llustrated, as the obtaned results n the frst approachng concdes wth well-known results on order of Introducton The problem of dentfcaton [3, 9, 3] of the dynamc systems [7, 8,, 6] has a lot of practcal applcatons, one of them s the applyng n the ol felds, for example, n the ol extracton [8, 3, ] Really, n the serve of ol by means of ppelnes, n the extracton of ol by the gas lft method or by the rod-pumpng settng and other methods - the determnaton of CHR [3, 6, 9], n layers - the determnaton of parameters of formaton of gas-lqud mxture [] and other, requres development of numercal methods for the solvng of correspondng problems of dentfcaton [8] of the dynamc systems In [9] the gradent method on the bass of Gramm-Schmdt orthogonalzaton for determnaton of CHR s gven In [4, 9] the asymptotc method s gven for determnaton of CHR n the frst approachng relatvely to small parameter, where a small parameter s accepted by nverse to the value of depth of the well It s shown, that f the use of ordnary Gramm-Schmdt method requres enough large machne tme, then an asymptotc method allows to calculate the soluton of CHR n the frst approachng analytcally and the numercal results concde Mathematcs Subject Classfcaton Prmary 49J5; Secondary 49J35 Keywords Coeffcent of hydraulc resstance, Least-squares method, quaslnearzaton, dentfcaton of the dynamc systems, asymptotc method Receved: 6 December 6; Revsed: 4 Aprl 7; Accepted: Aprl 7 Communcated by Ljubša DR Kočnac Emal addresses: f_alev@yahoocom FA Alev, nao@ramblerru NA Ismalov, atfnamazov@gmalcom, AA Namazov, narchs3@yahoocom NA Safarova, beseba@malru PB Besebay

2 FA Alev et al / Flomat 3:3 8, wth the sought soluton to the order Comng from these results, the authors of [3, 5] generalzed the results obtaned n [4] for multdmensonal case Further, usng the methods of quazlnearzaton [4] and least-squares [7] the computatonal algorthm s suggested for solvng of general dentfcaton problem of the dynamc systems for determnaton of the constant vector-parameter, whch can be used for fndng the CHR n the lft and the parameters for formaton of gas-lqud mxture n the layers of the wells n the ol producton The results of calculatons show that n smplest case, when the sought unknown constant vectorparameter - the subject to determnaton s scalar, then the method of Gramm-Schmdt requres hours n the smple example wth exactness 8 order Therefore, makes the sense to develop an asymptotc method for solvng of dentfcaton problem of the dynamc systems, when the small parameter s ncluded to the rght part of correspondng dfferental equatons, where on the example of the ol felds t s the nverse value of depth of the wells In ths work t s assumed, that some seres of ntal and fnal values statstcal data of phase coordnate of nonlnear ordnary dfferental equaton to the rght part are ncluded the small parameter and unknown constant vector are gven It s requred to fnd the small paremeter and unknown constant vector n such way that ther solutons n the end pont concded wth certan exactness wth statstcal data Usng the methods of least-squares and quazlnearzaton s gven an teratve scheme for the constructon of asymptotc solutons n the frst approach relatvely to the small parameter The results are llustrated on the example of the ol extracton by gaslft method for determnaton of CHR, where the small parameter ε s accepted nverse to the value of depth of the well Also for the coeffcents of asymptotc expressons on ε the analytcal formulas are gven, where the lnear dfferental equaton s changed by the dscrete approxmaton usng ordnary Euler method The gven numercal results for the values of CHR, whch are dffers on order of from the [, 3, 9] can be used as a good ntal approach for teratve schemes [3] Problem Statement Let a nonlnear ordnary dfferental equaton ẏ x = f y x, α, ε and some sets of ntal and fnal values of the n-dmensonal phase vector yx are gven y k = y k, y k l = y lk, k =,, N Here α s a constant unknown vector, ε s a small parameter, f s the n-dmensonal functon s dfferentable on y, α, ε It s requred to fnd such vector-parameter α = α, that the end value y l, α, εthe soluton of equaton wth an ntal value y k exactly enough concded wth y lk k =,,, N Such problem further we wll call the dentfcaton of the dynamc systems Let the ntal approach y x, α be gven Usng the method of quazlnearzaton [3, 7] we present the equaton n the lnear form relatvely y x, α and ε, n the followng form ẏ x = A y x, α εa y x, α y x B y x, α εb y x, α α C y x, α εc y x, α, 3 where A y x, α, B y x, α, C y x, α, are the results of decomposton of Taylor n the In the spatal case [] the moton of the equaton descrbng by the hyperbolc equatons the problem s reduced the soluton of the correspondng problem, where the dscrete model s consdered

3 frst approach and determned n the followng form FA Alev et al / Flomat 3:3 8, A y x, α = f y y, α,, A y x, α = f yε y, α,, B y x, α = f α y, α,, B y x, α = f αε y, α,, C y x, α = f y, α, f y y, α, y f α y, α, α, C y x, α = f ε y, α, f yε y, α, y f αε y, α, α 4 Now we present the soluton of lnear equaton [9] n the followng form y t =, t ε, y, t ε, α, t ε,, 5 where, t,,, t, are determned from the next lnear dfferental equatons, t, = A y x, α, t,,,, = E,, t, = A y x, α, t, A y x, α, t,,,, = 6 and t, =, t, ε, t, s the fundamental matrx of the homogeneous equaton 5 n the frst approach relatvely small parameter ε, and n, t,, n, t,, n =, from 5 are determned n the followng form [5], t t, =, t, =, t, =, t, = t t t, τ, B y, α dτ,, τ, B y, α, τ, B y, α dτ,, τ, C y, α dτ,, τ, C y, α, τ, C y, α dτ 7 Further for the solutons 5 n the pont x = l and for the fnal values from we construct the functonal N I = y k l y lk A y k l lk y, 8 k= where A s the well-known weght matrx, y k l s the soluton of the equaton 5 at statstcal data y k = y k from Thus, f we can choose such α, that the functonal 8 get the mnmum value, then we n fact provde the closeness of the soluton yx, α, ε n the pont x = l wth y l k = y from lk

4 3 Calculaton of the Gradent of the Functonal 8 FA Alev et al / Flomat 3:3 8, To obtan the formula for the gradent from the functonal 8 frst we puty l from 5 nto 8: k = N k= y {[ y, I = N ε, α, ε,, ε ], y A lk k= [, ε, y, ε, α, ε, y lk]} = {[ y, A, y y, A, α y, A,, A,, A,, A,, Ay lk α, A, y α α α α, Ay lk, A, y, A, α, Ay lk y lk A, y y A, y lk y A, y lk Ay lk lk] ε [ y, A, y y, A, α y, A,, A,, A,, A, y α α, A,, A,, A,, A, y y y α α α, A,, A,, A,, Ay lk y lk α y y, Ay lk α, A, α, A, y α, Ay lk, A, y, A, y, A, α A, y ] y A, α y A, lk lk = = N { y, A, y y, A, α y, A, k= y, A,, Ay lk α, A, α α, Ay lk, Ay lk y Ay lk lk ε [ y, A, y y, A, α y y, A, α y, A, y, Ay lk α, A, α, Ay lk, A,, A, α, Ay lk]} = {, A, y y, Ay lk y, A, y = N k=, A,, Ay lk y Ay lk lk, A,, A, α, Ay y lk 9 ε y, A, y, A, y y y, Ay lk, A,, Ay lk, A, y, Ay lk, A, y, A, y, Ay lk, A, α, A, ε, A }, α, A,, A,, A, Now we take the dervate on an unknown constant vector α from 9 where I α s determned n the followng form: I α = N ε, A, y, Ay lk, A, k=, A,, A, y, A,, Ay lk, A, ε y, A,, A, α

5 FA Alev et al / Flomat 3:3 8, We equate to zero the frst dervate Iα and search α n the form y t = For determnaton α and α we have the next algebrac equatons: N, A, y, Ay lk, A,, A, α =, k= N {, A, y, A, y, A, k=, A,, A, α, A, α After solvng the equatons relatvely to and α, correspondngly, we have : α = N { k=, A,, A,, Ay lk } = y, Ay lk, A, }, 3 α = N k= {, A,, Ay lk, A,, A, y, Ay lk, A,, A,, A,, A, y y N, A, 4 k= }, A, Thus calculatng α n the frst approach on ξ we obtan: α α εα Summarzng the above t s possble to present the computatonal algorthm [] for solvng of the problem of dentfcaton,, 8 Algorthm The nonlnear functon δ from, the ntal y k and fnal y k l, the gven weght matrces A, the ntal approaches y x, α and small number α are formng f y y, α,, f α y, α,, f y y, α,, f α y, α,, f yz y, α,, f αz y, α, are calculatng 3 A y x, α, A y x, α, B y x, α, B y x, α, C y x, α, α are formng from 3 4 The fundamental matrces, t,, y l are calculatng [9] at the ntal,, = E, α from 6 5 By usng of the fundamental matrces, α α εα, y l,, t, are calculatng the ntegrals 6 α and α are calculatng from 7, t, we accept y x as the ntal teraton If t s satsfed, we go to the step Otherwse the calculaton process s stopped Here t s assumed that, A, and, A, are exst If these condtons are not exst, we can use the regularzaton method [5]

6 FA Alev et al / Flomat 3:3 8, Ordnary Algorthm of Euler for Solvng and the Approxmate Formulas for, t,,, t,,, t,,, t,, t,,, t,, We note that at calculatng α over the algorthm one of the dffcultes s the procedure of fndng of, t,,, t, and, t,,, t,,, t,,, t, However by help of approxmate methods t s possble to restore them Now dscretzng on the step the equaton 3 on the frst Euler method [8, ], we obtan: y = E A y, α εa y, α y k B y, α εb y, α α k C y, α εc y, α 5 It s now possble to express y l through an ntal condton y Frst we show ths expressons for y, y y = E A Aε y B εb α C εc, y = E A ε E A A y [E A B B ε E A B B A B ] α [E A ] C C ε E A ε E A C C A C ] Let for y l are true the relatons y l = E A N N E A N ε y [ E A N B ε E A N B N E A N A B ] α ε E A N C ε E A N C N E A N A C By mathematcal nducton we wll prove easly, that y l = E A N N E A N ε y [ E A N B ε E A N B ] N E A N A B α ε E A N C ε E A N C N E A N A C 6 Now from 6 we defne the approxmate formulas for j, l Note that the fndng of α, α from 4 makes dffculty from calculatons of fundamental matrces n t,, n t,, n t, n =, from the system of lnear dfferental equatons 5 Therefore n the next pont by means of ordnary method of Euler we dscretze the equaton 5 and restore approxmately the fundamental matrces n t,, n =, =,, n the frst approach As s shown from 3-4 for renewal of α, α t s necessary to take nto account the hgher j, l The relaton 6 allows to fnd them Therefore comparng 6 wth 5 we have:, = E A N,,,,,, = N E A N, = E A N B, = E A N B N E A N A B, = E A N C, = E A N C N E A N A C 7 Thus t s smpler to calculate α, α from 3, 4 by means of approxmate formulas

7 FA Alev et al / Flomat 3:3 8, We note that wth the help of 7 we can easly fnd for α, α from 3 and 4 the next expressons n an obvous form α = N k= { B N E A A E A N B N N B B E A Aylk N A E A N B N N } E A A B, y N E A A E A y E A 8 α = N { N N B E A A E A B k= N N N y B E A A E A B B E A B A N N N N E A A E A C B E A B A N N N N E A Aylk B E A A E A C N N E A A C N E A B N N E A A B A N E A B N N B E A A E A B N N N y E A A E A B B E A Aylk y N E A A E A N B N N } E A A B 9 We note that for the obtanng of more exact values t s necessary to make the dscretzaton on the method of Runge-Kutta and other Thus, summarzng the above results we can offer the next computatonal algorthm for fndng of concdence α α εα Algorthm The functon f y, x, α and the statstcal data y, y from and 3 =,,, N are gven l The dervatves f y, f α are calculatng 3 A, A, B, B, C, C are calculatng from 4 4,,,,,,,,,,, are calculatng from 7 or 7 5 α and α are calculatng by the formulas 3 and 4, correspondngly or 8, 9 6 α α εα are calculatng 7 The condton I > I s checked up If the condton s not satsfed, we go to the step Otherwse the calculaton process s stopped We note that formulas 8 and 9 allow approxmately to fnd α We consder the followng example 5 Example Let us consder the gas lft process for the ol producton where the moton equaton s descrbed by the followng nonlnear ordnary dfferental equaton [3, 6, 8, ] Q = aλ cρfq, Q = u, ε c ρ F Q where c >> ω c, except Q = ρω c F all values are constant, F s the cross-sectonal area of pump-compressor ppes, that s constant relatvely to axes

8 FA Alev et al / Flomat 3:3 8, Here t s assumed that the transton from the end of rng ppe through the layer to begnnng of the lft x = l s executed on the followng dfference equaton: Ql = γql γ Ql Q, γ Ql = δ 3 Ql δ δ, where γ, δ, δ, δ 3 are constant real numbers, whch are subject to determnaton For smplcty we suppose that parameters γ, δ, δ, δ 3 are known and t s requred to fnd the CHR λ c, ncluded to through αλ c Further some nomnal trajectory Q x and parameter α are searchng supposng that k th teraton s already done Lnearzng the equaton near these data we have where Q k x = A Q k, α k Q k x B Q k, α k α k C Q k, α k, A y x, α =, A y x, α = 4a c ρ 3 F 3 Q, B y x, α = ρf, B y x, α = ρ 3 F 3 c C y x, α = ρfa, C y x, α = a c ρ 3 F 3 Q Note that by help of relatons 7, 8 the matrces k, k x,, k x, are calculatng n the followng form k N j x, = E A Q k x, α k h B Q k x j, α k h j=n =N B Q k x N, α k h, k x, = N j=n j =N C Q k x N, α k h, E A Q k x, α k h C Q k x j, α k h where h s an enough small number, whch s the step of ntegraton Let the statstcal data, that are the results of measurng of debt Q on leavng wth the gven ntal n volume of gas are gven, e Q, Q, =, 5 are known n Table y l y l Then the functonal from 8 has the followng form: I = 5 Q Q l, = where Q l s the soluton of the equaton 3 for the ntal condtons Q Let the parameters of the equaton look lke as: at x < l : l = 485 m, s = 33m/s, ρ =,77k m 3, d = m, λ=, at l < x l : s = 85m/s, ρ = 7k m 3, d = 73 m, λ = 3 3

9 FA Alev et al / Flomat 3:3 8, Now we pass to mplementaton of the above algorthm The ntal value of CHR λ c we accept equal to Acceptng y t = and repeatng the procedure -5 from the algorthm we determne the values of λ c, y t After 44 teratons the followng result was obtaned: λ c 9834, that concdes wth λ from 3 wthn Note that such approach can be satsfactory to fndng of ntal teratons of ordnary gradent method, for fndng of CHR [3], lnearzng [] and other References [] FA Alev, The Methods of Solvng of Appled Problems of Optmzaton of Dynamc Systems, Baku, Elm, 989, 37 p [] FA Alev, NA Alev, NA Safarova, et al Sweep method for solvng the Roesser type equaton descrbng the moton n the ppelne, Appl Math Comput [3] FA Alev, NA Ismalov, Inverse problem to determne the hydraulc resstance coeffcent n the gas lft process, Appl Comput Math [4] FA Alev, NA Ismalov, On a problem of dentfcaton n lnear statonary case, Reports of NAS of Azerbajan [5] FA Alev, NA Ismalov, Optmzaton problems wth a perodc boundary-value condton and boundary control for gas-lftng wells, Nelnn Kolyvannya Nonlnear Oscllatons Translated n: J Math Sc [6] FA Alev, NA Ismalov, H Hacyev, MF Gulev, A method of determne the coeffcent of hydraulc resstance n dfferent areas of pump-compressor ppes, TWMS J Pure Appl Math [7] FA Alev, NA Ismalov, EV Mamedova, NS Mukhtarova, Computatonal algorthm for solvng problem of optmal boundarycontrol wth nonseparated boundary condtons, J Comput Syst Sc Int [8] FA Alev, NA Ismalov, NS Mukhtarova, Algorthm to determne the optmal soluton of a boundary control problem, Automat Remote Control [9] FA Alev, NA Ismalov, AA Namazov, Asymptotc method for fndng the coeffcent of hydraulc resstance n lftng of flud on tubng, J Inverse Ill-Posed Probl [] FA Alev, NA Ismalov, AA Namazov, MF Rajabov, Algorthm for calculatng the parameters of formaton of gas-lqud mxture n the shoe of gas-lft well, Appl Comput Math [] FA Alev, MM Mutallmov, NA Ismalov, MF Radzhabov, Algorthms for constructng optmal controllers for gaslft operaton, Automat Remote Control [] TA Anake, SA Bshop, OO Agboola, On a hybrd numercal algorthm for the solutons of hgher order ordnary dfferental equatons, TWMS J Pure Appl Math [3] N Bakhtzn, R Latypov Estmaton of order of lnear objects on expermental nformaton, Automat Remote Control [4] PE Bellman, PE Kalaba, Quazlnearzaton and Nonlnear Boundary Problems, Moscow, Mr, 968 [5] Y Huang, ZH Lu, R Wang, Quaslnearzaton for hgher order mpulsve fractonal dfferental equatons, Appl Comput Math [6] K Jblou, A survey of Krylov-based methods for model reducton n large-scale MIMO dynamcal systems, Appl Comput Math [7] LU Jan, The Least Square Method and Its Applcaton, Scence and Technology of West Chna, 7 [8] AM Mabrok, AM Haggag, IR Petersen, System dentfcaton algorthm for negatve magnary systems, Appl Comput Math [9] NI Mahmudov, MA McKbben, On approxmately controllable systems survey, Appl Comput Math [] AKh Mrzadjanzadeh, IM Akhmedov, AM Khasaev, VI Gusev, Technology and Technque of Ol Producton, M, Nedra, 986 n Russan