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A Statistically Optimal Explicit B°-Identification Algorithm for Nonlinear Distributed-Parameter Systems
©
IF .\( ~lo( l l ; t "l i ( ( :I)llInd \" il llill .... l .i l llll, lll i,l[ l \'i R . L·S'i R . I~L"li
( :l l p \rig lll
A STATISTICALLY OPTIMAL EXPLICIT BOIDENTIFICATION ALGORITHM FOR NONLINEAR DISTRIBUTED-PARAMETER SYSTEMS E. Sakalauskas 1"1/1/11/" "I I'ilY'lwlllwl T alllllm l P ,.,,/J/I'III,I 11/ I'.' 11 I'I'gl'l 1(,1 , 1.1111111111111 // ,4((IIil'lllY 11/ .\ (11'1/((',1. 1\.11/11/(/,1.
{'S.\ N
Abs tra c t . A s t able numerical differentiation tecnique was used to build an iden t ificat i on algorithm f or dis tributed-parameter s ystems (DPS) , whic h belongs to the e xplicit class and is easily i mple mented on a computer. Pa ra meter e s timates are fo und on a pre-chosen s ub-optimal project i on space by pro j ecting in put resid l;als on zero- order B-splines spa ce. The r es '.; 1t s o! a numerica 1 sim ul a t i on are pre sen t ed. Ke ywords. I dentification, distrib uted para meter system, partial differen t lal eq uation, random field, projection s pace, spline . Note, that the same feat ures of parameter esti mates are reflected in the performance criterion to be used in the identificati on . As sugge s t e d by Raj bman , Bogda nov, Kneller ( 1982 ) the re a r e two widely used performance criterions being usually homogeno us conve x f uncti ons of the input residual (eq uation resid ual ) and the output resid ual can be described by
INTRODUCTION We consider a distrib uted-para me t e r system (DPS ) as an input - ou t pu t s yste m which is describe d by t he follo wing partial differential e quation ( PDE ) :
Lee) u(x,t)
=f(x,t),
(1 )
where xc A=[X.,X",] spatial variable,tEA:: (to,tAl - time vari able,lJ(~,t) - s ystem output f u nction,f(~,t) - general s yste m input function incl uding initial and bo undary conditions ( But kovskij, 1977 ) ,L(9) - operator of partial derivatives, 9 - vector (vector-f unction ) of internal para meters of t he sys t e m in the s yste m operator .
Ql(B) :: Qt Wx,t,9)J,
l (l,t ,B) =L(9)u(ll,t) - f(x,t), (2 )
Qe(9):: Qe[e(I,t,B)], e(Jt,t,9):::: U(X,t)- uO{)(,t,9) I (3)
where ((It,t,S) - input resid ual, e(x,t,9) -output residual, u(lt,t) - s ystem output observa tion,uO(x,t,8) - model equa tion solution fo r a given vector 9 . The two resid uals are evidently interrelated by
The proble m of i dent if i cation is for mu lated as f ollows: f or given operator L(B) struc t ure within the acc uracy of El find es timates ~ of unknown coeff icients 9 in a given input f unction f(lt,t) and from observations of the noise-corrupted outp ut variable u()(,t).
(4 ) On this approach, the available identification methods may be devided into implicit ones, which give sol utions
9=
We know se veral s ur veys of the pro blem (Goodson and Po li s , 1975 ; ~u b r us ly , 1977 ; Red dy and ::taja mi, 1977 ; Chavent , 1979 ; Rajbma n , Bogdanov , ;
min Qe [e(x,t,9)] ,
e
(5)
and e xplicit ones, whi ch give solutions
9 = 111 Ql [l(ll,t ,8)] .
(6)
This division is possible, beca use residual e{lt.,t,El) and criterion Qe are implicitly related to parameters a , and residual t(.,t , 9) and cri terion Qt are explicitly relate d to 9. Besi de s , f r om Sakala us kas ( 1983 ) a nd other publications, the implicit identification methods are more noise-resistant and more implementable for relatively small numbers of meas ured points, but require large computation efforts for repeated sol utions of the model equation (2). The explicit methods are, on the contrary, noise irresistant and require relatively large numbers of meassured points, but are easily computer implemented. The explanat i on is, that computation of LtQlu(",t) -1 ' 1
50
E. Sakalauskas
for a noise-corrupted u(j"t) is an ill-posed problem, and can only be solved on a large number of measured points. But criterion qt is an explici t function of 9 , and does not require large computation effonts for its minimization. Among the explicit methods of identification we separate a group of projection techniques, as a most developed one, in which problem (6 ) is replaced by an equivalent set of algebraic eq uations generated by an orthogonal projection operator PM: \.~(6.l\) HI! where 1.2,(AIA) a Hilbert space of functions with 4.A domain, and l. a(6IA):JH .. - N-dimensional projection space. Then, because of f(JI,t,9) € l. a (A .. A) estimates 9 of parameters e are solutions of the set of equations, which can be wri tten as
P {{)(,t,e) ==
" also group
o.
This includes the so called modular functions or momen t te c hniques CFairman and Shen, but in prac tice cannot be applided for the identification of nonlinear DPS. In this case integration by parts is impossible, and witho ut it the error rate is very large, because of the large error in H",t,Q) and of the free choce of the projection space. We suggest an identification algorithm for DPS, based on a noise-resistant computation oP derivations of' meas ured functions (Sa kala uska"' . 1984 : and on the choise of s ub-optimal projec tion space as zero-order B-splines (B-splines ) . By s Ub-optimal we mean that the errorso P estimated parameters are nearly minimal.
Let us sa y zl1.,t)
( 8)
where O==l8,,1: - N-dimensional vector of constant system parameters, all of which or a part of which are to be estimated, I('n('] - given bijec ti ve non-linear f unctions, D~E {~Zht", 'll"/rax'3t J ra"j '71'1.", raj "Ox , I } - opera tors of partial derivatives with Iu(x,t) = U(lC,tl, ~ f,7,.
=
Assume the outp ut variable U(lI.,t) is discretely measured with an additive noise, which is simulated by an ergodic random field t(t,t) with a zero mean and an absolutely integrated covariance function, tha t is u*{)(,t> + (lC,tJ, E{()(,t)}=O, (9 )
where U(lC,t) interpolant of the corresponding discrete values of the function, U*(lC,tl - exact output function, E - mathematical expectation. First step of the algorithm. We compute funchons C("[D,U(ll,t) • Vie apply a method of stable numerical differentiation (Sakala uskas,19 84 ) and determine partial derivati-
,
(10)
(11 )
To determine z(t) we use the Oalerkin projection method, which is well known as mathematical physics eq uations solution method (Marchook a nd Agoshkov, 1951 ) . The sought f unction z(t) is series expanded over a set of basic functions (~ . (t)} which i define a finite projection space K
(12)
z(tJ '" E z. S.(!) J=f J
J
This is equivalent to orthogonal projection of func tion z(i)€ Lz(A} on to Z I( with the help of an or thoprojector (13 )
After a series of standard considerations we arrive at a set of linear algebraic eq uations with respect to the vector 7. of the coefficients t Zj 1~ of expansion (12) I<
E
A j=1
We consider a class of DPS described by (1) with non-linear operators
= D. u(t,t) == 'ClalA(",t)j~ta
must be determined for a f ixed xe A. For a sake of shorthess we describe u(",t) and %(X,i) as u(t) and z(t) respectively, and ass ume initial conditions
f
IDENTIFICATION ALGORITHM
U(ll,t)==
ves D.U(ll,t) • The method is based on the formation of a regularization operator aft e r Tikhonov and Arsenin(1 979) by projection regularization as presente d for general operator e quation s by Khiamarik (1 98~
zJ'
B.{t)·B. (thH '" J
l
SD1 u(t) · B.(t)·Jt, (l4)
A
l
where L=1,K . To avoide differentiation of the noise-corrupted func tion u( t) in the right-hand side of (14) we perform a integration by parts of the right-hand side of (14 ) with the bo undary condition (11 ) . Then we may write
. z · Is .et) B.(t) cH =
~
j:-i J,. J
L
$u(t)·DfB . (t)dt ... u(t)B.(tJI A L L
t A
to'
(15 )
wi th i. =1,K which corresponds to (16 )
where r -Gramm'S matrix, d - vector of the right-hand side of (15). -i
The acc uracy of vector z=r d and of the corresponding function z(t) depends on the K t y pe of basic functions tB.(tll and on the L 1 dimension K of the projec tion space Z • In a previo us publication (Sakala uskas, \984) it was shown , that there exists a sub-optimal projection space with a basis of Bsplines. To determine the ~ -th derivatio~ such a s ub-optimal functions {BLCt)}: are the B-splines of the ~ -th order. By suboptimal we mean solutions with a nearly minimal error. The same publication give a determination of the regularization para-
51
:\ HO· ide ntifi cati o n .-\Igorit hill
me ter re la ted to the order K of space Z I( and of the regularization operator R(K)= PI
]+1
... i
(17)
where C~ 0.1 -constant,J4e{l,2,3,4}- number of monotonous intervals of the function under differentia tion, M~ 20 - number of discrete points of measurement, relative standart deviation measurement error in per cent, [.] - interger function, ~e{ .. }2.}- order of derivative. It is assumed "hat the number of measured points in a monotono us interval is not less than then 10.
s; -
be interpreted as the estimates of mathematical expectations of random fields f:(""t) with weight functions "'... (x,t). Consequen tly dispersions {bo",.,) depend on the type of weight functions 'I'... (",t). It has been observed (Vilenkin, 1979) that the sub-optimal weight function which give nearly minimal estimate dispersion of mathematical expectation is function't'("~t) == 1/30 A , where 6 ,j. - the lengths of in tervals A and A • We suggest weight functions h ... (",t)\~ to be chosen as piece-wise constant functions which will be sub-optimal. The basis of the piecewise constant functions will be foomed by a set of zero-order-B-splines (B -splaines) (Zavialov, Kvasov, Mirosnitchenko, 1980). Then both the solution error of (18) and the obtained parameter e~ timates errors are also nearly minimal.
The first algorithm step is thus performed and in this way D~ \A()t,t), 1::.f;6 , as well as C(n[D~u(""t)], n:1,N are determined. Algorithm step two. We solve (7). For the determination of {en\~we choose a set of M linearly independent functions {'I'", (", t)}~ in projection space H... Then on the condition of orthogonal projection '(",t,e) onto HII in space 1. 2(A-A) eq. (7) has a corresponding set of algebraicAequations linear with respect to vector 9
.. Sf 1: 9..
All ",,4
'l"[D~u(x,t)].
'f'... (X,t)dlCdt=
Sf f(X,t)· t ... (lC,t)
4/0
at dt
(18)
where L== 1,N • Then we describe the ma trix of the equations by A={a"m}' Solution accuracy of eq. ( 18) depends on the computation accuracy of function I(n[D~U(lC,t)] and on the type of weight functions {'I'", (x,t)}~ • In the suggested algorithm computation of partial derivatives their error is formed by two alternative errors: interpolation error(regular component) £1(lC,O and initial da ta noise trans~er error (random component) £.. (",0 , that ~
1S
=D~u(",t)-J)~l.t(~,t) == G1(X,O- f/~It), (19) the assumption of E\ t:~(x,t)} =0 • Error
£("',t)
with E. ()t,t) is then transformed by non-linear functions ~,,[.]. As the result, the elements of the set of e quations ( 16 ) matrix are disturbed, that is a =0- +oa ,Ga =ffIOCt.(l,t))t (x,tldlldt, (20) n",
a:", -
""'"'"
4 It. '"
NUMERICA L SIMULA. TION RESULTS To check the performance of the suggested algorithm the following DPS model was studied 8\ 'a'louV.,t)/1Jt1.
where ICE.4=[O,ll, teA=[O,ll, f(t.,t)=2tf 1 Cn(H")·si.nfft + ~<.si.nad)/(1+]()~ Exact values of parameters are 91•=i, 91*=2 . The corresponding exact solution is u·(,(,t)= 20 t + (n(i-tt)·!>in1ft. An output function U('(,t) was formed by piece-wise linear interpolation of its discretely measured values U(~L,t.i) wi th L:: i,M, ,j::: 1,N t • The number of measured points was M. =30, Ht =50 and U(lCut.j):: U*(lCi)tj) + ('j
(23)
where ~.. - random sequence with a zero .~ mean and a standard devia tion b~ . A set of piece-wise constant functions {'I'... (J(,t)l~ was compiled as a set of BO-splines {B~(](,t), S;(IC,t)} on a grid of nodes X n, :{X O,"ll A}' nt:{to,tllt,,} with supplements suppS:(JC,tl:: [lCO,X1]JCfto,t1] and supp B;(IC,t)=["nXA],,[t1~tAJ. Relative errors £9 of estimates of parameters dependent on standard devia tion where
e
f. - .... 1 e -e£{ ••••z}
e
0;,
13 - 9*' I a*1
• ·100;1"
,..0 _
~
Qt -ll.Il-.,t>D
nu(x,t)1 = rAA I I u"(JC,f) d" cH ]~/z
..
where corresponding exact elements in ma triy. A. No,·! we refer to Kazakov (1969 ) and re place error '(',,[f(lt',t)] by statistically equivalent errors €~(",t) =jI()t,t)+ 7(lC,tJ , (21) where J1(",t)- regular component in the error, '1('" t) - cen tered random componen t. The disturbing elements {fa" ... } in ma trix A may
ez['Ou(x,t)hxi~ = Hx,t), (22)
+
Table 1. Relation of E. and O
b (%)
0
(24)
Si .
2
5
10
15
12.3
16.7
21.5
32.6
-" .'-
L Sakalauskas
CONCL:JSIONS The suggested stable m~thod of numerical differenti.a tion opens the way to rea] ize an explicit projection DPS identification algorihm. To improve noise resistance of the algorithm the projection space is chosen as a set of piece-wise constant functions being sub-optimal in the sense of nearily minimal errors of the obtained parameter estimates. The res ults of simulation on a typical eBmple suggest a good performance of the algorithm ~ith a JO ~u rela tive Ol.:tput error.
REF:::RENCRS Butkovskij,A.G. (977). General theory of structu:'a] schemes for distribu-ted parameter systems. In Str~lctural Theory of Distributed Svstems. Nauka, Moscow (in Russian). Chap. 2, pp. 9D B. Chavent,G. (1979). Identification of distrib uted parameters. In Preprints of the )-rd IFAC Symposium on Identification and System Parameter Estimation. Darmstadt, pp. 85-99. Fa irma n , P . \, ., and D.". ::;hen OnO). Parameter identification for a class of distrib u ted syste ms . Int. J. Control. Vo 1. 11, N 11, pp. 929-940. Gcodson,R.E., and M. P. Polis ( l nS) . I. survey of parameter identification in distrib uted systems. In Proc. IFAC 6-th Wo rld Congress. Boston, Cambrige, part lA, 1-12. Kazakov, I.E. (1969). Statistical methods in control s stems daSl n. Mashinostrcyenye, Moscow in Russian). Khiamarik,U.A. ( 1983). Projection methods for regularization of :inear ill-posed problems. In Preprints of Tartu University. Tartu university, Tart u (in RUSSian). Vol. 50, pp. 59 - 90 . Kubrusly,C.S. (1977). Distributed parameters system identification. A survey. Int. J. Control, Vol. 26 , N 4, pp.509535. Marchook ,G.I., and V.I.Agoshkov (1981 ) . Some projection method algorithms. In Introducti on to projection - grid methods. Nauka, Moscou (in Russian). Chap. I, pp. 2 1-9 6. Rajbman N.S., Bogdanov V.D., and Kneller D.B. (1982). Identification of distributed systems. Automatica and Telemekhanika (in RUSSian). N 6, pp. 5-)6. Reddy , ~ ., and V.Rajarni (1977 ; . A revie~ of the state and parameter estimation theory in distributed systems. J.Inst. Electr. and Telecom. Eng. Vol. 2), pp. 689-700. Sakalauskas,E.J. (1984). Galiorkin-Tikhonoy method in numerical differentiation problem. J. of Numerical Mathematics and Mathematic PhYSiCS (in Ru sian), N 11, pp.1742-l747. Tikhonov,A.N., and V.J.Arsenin (1979) Method of regularization of operator equations. In Ill-posed problems solution methods. Nauka, Moscow (in Russian). Chap. 2, pp. 5)-109. Vylenkin,S.I. (1979). Statistical processing of stochastic functions study data. Nauka, Moscow (in RUSSian).
Zavjalov,U.S., B.I.~vasov, and V.L.~yro shnychenko ( 1900) . f,Jetho ds of sp11ne-f'unctions. Nauka, Moscow (in Russian) •
IF .\( ~lo( l l ; t "l i ( ( :I)llInd \" il llill .... l .i l llll, lll i,l[ l \'i R . L·S'i R . I~L"li
( :l l p \rig lll
A STATISTICALLY OPTIMAL EXPLICIT BOIDENTIFICATION ALGORITHM FOR NONLINEAR DISTRIBUTED-PARAMETER SYSTEMS E. Sakalauskas 1"1/1/11/" "I I'ilY'lwlllwl T alllllm l P ,.,,/J/I'III,I 11/ I'.' 11 I'I'gl'l 1(,1 , 1.1111111111111 // ,4((IIil'lllY 11/ .\ (11'1/((',1. 1\.11/11/(/,1.
{'S.\ N
Abs tra c t . A s t able numerical differentiation tecnique was used to build an iden t ificat i on algorithm f or dis tributed-parameter s ystems (DPS) , whic h belongs to the e xplicit class and is easily i mple mented on a computer. Pa ra meter e s timates are fo und on a pre-chosen s ub-optimal project i on space by pro j ecting in put resid l;als on zero- order B-splines spa ce. The r es '.; 1t s o! a numerica 1 sim ul a t i on are pre sen t ed. Ke ywords. I dentification, distrib uted para meter system, partial differen t lal eq uation, random field, projection s pace, spline . Note, that the same feat ures of parameter esti mates are reflected in the performance criterion to be used in the identificati on . As sugge s t e d by Raj bman , Bogda nov, Kneller ( 1982 ) the re a r e two widely used performance criterions being usually homogeno us conve x f uncti ons of the input residual (eq uation resid ual ) and the output resid ual can be described by
INTRODUCTION We consider a distrib uted-para me t e r system (DPS ) as an input - ou t pu t s yste m which is describe d by t he follo wing partial differential e quation ( PDE ) :
Lee) u(x,t)
=f(x,t),
(1 )
where xc A=[X.,X",] spatial variable,tEA:: (to,tAl - time vari able,lJ(~,t) - s ystem output f u nction,f(~,t) - general s yste m input function incl uding initial and bo undary conditions ( But kovskij, 1977 ) ,L(9) - operator of partial derivatives, 9 - vector (vector-f unction ) of internal para meters of t he sys t e m in the s yste m operator .
Ql(B) :: Qt Wx,t,9)J,
l (l,t ,B) =L(9)u(ll,t) - f(x,t), (2 )
Qe(9):: Qe[e(I,t,B)], e(Jt,t,9):::: U(X,t)- uO{)(,t,9) I (3)
where ((It,t,S) - input resid ual, e(x,t,9) -output residual, u(lt,t) - s ystem output observa tion,uO(x,t,8) - model equa tion solution fo r a given vector 9 . The two resid uals are evidently interrelated by
The proble m of i dent if i cation is for mu lated as f ollows: f or given operator L(B) struc t ure within the acc uracy of El find es timates ~ of unknown coeff icients 9 in a given input f unction f(lt,t) and from observations of the noise-corrupted outp ut variable u()(,t).
(4 ) On this approach, the available identification methods may be devided into implicit ones, which give sol utions
9=
We know se veral s ur veys of the pro blem (Goodson and Po li s , 1975 ; ~u b r us ly , 1977 ; Red dy and ::taja mi, 1977 ; Chavent , 1979 ; Rajbma n , Bogdanov , ;
min Qe [e(x,t,9)] ,
e
(5)
and e xplicit ones, whi ch give solutions
9 = 111 Ql [l(ll,t ,8)] .
(6)
This division is possible, beca use residual e{lt.,t,El) and criterion Qe are implicitly related to parameters a , and residual t(.,t , 9) and cri terion Qt are explicitly relate d to 9. Besi de s , f r om Sakala us kas ( 1983 ) a nd other publications, the implicit identification methods are more noise-resistant and more implementable for relatively small numbers of meas ured points, but require large computation efforts for repeated sol utions of the model equation (2). The explicit methods are, on the contrary, noise irresistant and require relatively large numbers of meassured points, but are easily computer implemented. The explanat i on is, that computation of LtQlu(",t) -1 ' 1
50
E. Sakalauskas
for a noise-corrupted u(j"t) is an ill-posed problem, and can only be solved on a large number of measured points. But criterion qt is an explici t function of 9 , and does not require large computation effonts for its minimization. Among the explicit methods of identification we separate a group of projection techniques, as a most developed one, in which problem (6 ) is replaced by an equivalent set of algebraic eq uations generated by an orthogonal projection operator PM: \.~(6.l\) HI! where 1.2,(AIA) a Hilbert space of functions with 4.A domain, and l. a(6IA):JH .. - N-dimensional projection space. Then, because of f(JI,t,9) € l. a (A .. A) estimates 9 of parameters e are solutions of the set of equations, which can be wri tten as
P {{)(,t,e) ==
" also group
o.
This includes the so called modular functions or momen t te c hniques CFairman and Shen, but in prac tice cannot be applided for the identification of nonlinear DPS. In this case integration by parts is impossible, and witho ut it the error rate is very large, because of the large error in H",t,Q) and of the free choce of the projection space. We suggest an identification algorithm for DPS, based on a noise-resistant computation oP derivations of' meas ured functions (Sa kala uska"' . 1984 : and on the choise of s ub-optimal projec tion space as zero-order B-splines (B-splines ) . By s Ub-optimal we mean that the errorso P estimated parameters are nearly minimal.
Let us sa y zl1.,t)
( 8)
where O==l8,,1: - N-dimensional vector of constant system parameters, all of which or a part of which are to be estimated, I('n('] - given bijec ti ve non-linear f unctions, D~E {~Zht", 'll"/rax'3t J ra"j '71'1.", raj "Ox , I } - opera tors of partial derivatives with Iu(x,t) = U(lC,tl, ~ f,7,.
=
Assume the outp ut variable U(lI.,t) is discretely measured with an additive noise, which is simulated by an ergodic random field t(t,t) with a zero mean and an absolutely integrated covariance function, tha t is u*{)(,t> + (lC,tJ, E{()(,t)}=O, (9 )
where U(lC,t) interpolant of the corresponding discrete values of the function, U*(lC,tl - exact output function, E - mathematical expectation. First step of the algorithm. We compute funchons C("[D,U(ll,t) • Vie apply a method of stable numerical differentiation (Sakala uskas,19 84 ) and determine partial derivati-
,
(10)
(11 )
To determine z(t) we use the Oalerkin projection method, which is well known as mathematical physics eq uations solution method (Marchook a nd Agoshkov, 1951 ) . The sought f unction z(t) is series expanded over a set of basic functions (~ . (t)} which i define a finite projection space K
(12)
z(tJ '" E z. S.(!) J=f J
J
This is equivalent to orthogonal projection of func tion z(i)€ Lz(A} on to Z I( with the help of an or thoprojector (13 )
After a series of standard considerations we arrive at a set of linear algebraic eq uations with respect to the vector 7. of the coefficients t Zj 1~ of expansion (12) I<
E
A j=1
We consider a class of DPS described by (1) with non-linear operators
= D. u(t,t) == 'ClalA(",t)j~ta
must be determined for a f ixed xe A. For a sake of shorthess we describe u(",t) and %(X,i) as u(t) and z(t) respectively, and ass ume initial conditions
f
IDENTIFICATION ALGORITHM
U(ll,t)==
ves D.U(ll,t) • The method is based on the formation of a regularization operator aft e r Tikhonov and Arsenin(1 979) by projection regularization as presente d for general operator e quation s by Khiamarik (1 98~
zJ'
B.{t)·B. (thH '" J
l
SD1 u(t) · B.(t)·Jt, (l4)
A
l
where L=1,K . To avoide differentiation of the noise-corrupted func tion u( t) in the right-hand side of (14) we perform a integration by parts of the right-hand side of (14 ) with the bo undary condition (11 ) . Then we may write
. z · Is .et) B.(t) cH =
~
j:-i J,. J
L
$u(t)·DfB . (t)dt ... u(t)B.(tJI A L L
t A
to'
(15 )
wi th i. =1,K which corresponds to (16 )
where r -Gramm'S matrix, d - vector of the right-hand side of (15). -i
The acc uracy of vector z=r d and of the corresponding function z(t) depends on the K t y pe of basic functions tB.(tll and on the L 1 dimension K of the projec tion space Z • In a previo us publication (Sakala uskas, \984) it was shown , that there exists a sub-optimal projection space with a basis of Bsplines. To determine the ~ -th derivatio~ such a s ub-optimal functions {BLCt)}: are the B-splines of the ~ -th order. By suboptimal we mean solutions with a nearly minimal error. The same publication give a determination of the regularization para-
51
:\ HO· ide ntifi cati o n .-\Igorit hill
me ter re la ted to the order K of space Z I( and of the regularization operator R(K)= PI
]+1
... i
(17)
where C~ 0.1 -constant,J4e{l,2,3,4}- number of monotonous intervals of the function under differentia tion, M~ 20 - number of discrete points of measurement, relative standart deviation measurement error in per cent, [.] - interger function, ~e{ .. }2.}- order of derivative. It is assumed "hat the number of measured points in a monotono us interval is not less than then 10.
s; -
be interpreted as the estimates of mathematical expectations of random fields f:(""t) with weight functions "'... (x,t). Consequen tly dispersions {bo",.,) depend on the type of weight functions 'I'... (",t). It has been observed (Vilenkin, 1979) that the sub-optimal weight function which give nearly minimal estimate dispersion of mathematical expectation is function't'("~t) == 1/30 A , where 6 ,j. - the lengths of in tervals A and A • We suggest weight functions h ... (",t)\~ to be chosen as piece-wise constant functions which will be sub-optimal. The basis of the piecewise constant functions will be foomed by a set of zero-order-B-splines (B -splaines) (Zavialov, Kvasov, Mirosnitchenko, 1980). Then both the solution error of (18) and the obtained parameter e~ timates errors are also nearly minimal.
The first algorithm step is thus performed and in this way D~ \A()t,t), 1::.f;6 , as well as C(n[D~u(""t)], n:1,N are determined. Algorithm step two. We solve (7). For the determination of {en\~we choose a set of M linearly independent functions {'I'", (", t)}~ in projection space H... Then on the condition of orthogonal projection '(",t,e) onto HII in space 1. 2(A-A) eq. (7) has a corresponding set of algebraicAequations linear with respect to vector 9
.. Sf 1: 9..
All ",,4
'l"[D~u(x,t)].
'f'... (X,t)dlCdt=
Sf f(X,t)· t ... (lC,t)
4/0
at dt
(18)
where L== 1,N • Then we describe the ma trix of the equations by A={a"m}' Solution accuracy of eq. ( 18) depends on the computation accuracy of function I(n[D~U(lC,t)] and on the type of weight functions {'I'", (x,t)}~ • In the suggested algorithm computation of partial derivatives their error is formed by two alternative errors: interpolation error(regular component) £1(lC,O and initial da ta noise trans~er error (random component) £.. (",0 , that ~
1S
=D~u(",t)-J)~l.t(~,t) == G1(X,O- f/~It), (19) the assumption of E\ t:~(x,t)} =0 • Error
£("',t)
with E. ()t,t) is then transformed by non-linear functions ~,,[.]. As the result, the elements of the set of e quations ( 16 ) matrix are disturbed, that is a =0- +oa ,Ga =ffIOCt.(l,t))t (x,tldlldt, (20) n",
a:", -
""'"'"
4 It. '"
NUMERICA L SIMULA. TION RESULTS To check the performance of the suggested algorithm the following DPS model was studied 8\ 'a'louV.,t)/1Jt1.
where ICE.4=[O,ll, teA=[O,ll, f(t.,t)=2tf 1 Cn(H")·si.nfft + ~<.si.nad)/(1+]()~ Exact values of parameters are 91•=i, 91*=2 . The corresponding exact solution is u·(,(,t)= 20 t + (n(i-tt)·!>in1ft. An output function U('(,t) was formed by piece-wise linear interpolation of its discretely measured values U(~L,t.i) wi th L:: i,M, ,j::: 1,N t • The number of measured points was M. =30, Ht =50 and U(lCut.j):: U*(lCi)tj) + ('j
(23)
where ~.. - random sequence with a zero .~ mean and a standard devia tion b~ . A set of piece-wise constant functions {'I'... (J(,t)l~ was compiled as a set of BO-splines {B~(](,t), S;(IC,t)} on a grid of nodes X n, :{X O,"ll A}' nt:{to,tllt,,} with supplements suppS:(JC,tl:: [lCO,X1]JCfto,t1] and supp B;(IC,t)=["nXA],,[t1~tAJ. Relative errors £9 of estimates of parameters dependent on standard devia tion where
e
f. - .... 1 e -e£{ ••••z}
e
0;,
13 - 9*' I a*1
• ·100;1"
,..0 _
~
Qt -ll.Il-.,t>D
nu(x,t)1 = rAA I I u"(JC,f) d" cH ]~/z
..
where corresponding exact elements in ma triy. A. No,·! we refer to Kazakov (1969 ) and re place error '(',,[f(lt',t)] by statistically equivalent errors €~(",t) =jI()t,t)+ 7(lC,tJ , (21) where J1(",t)- regular component in the error, '1('" t) - cen tered random componen t. The disturbing elements {fa" ... } in ma trix A may
ez['Ou(x,t)hxi~ = Hx,t), (22)
+
Table 1. Relation of E. and O
b (%)
0
(24)
Si .
2
5
10
15
12.3
16.7
21.5
32.6
-" .'-
L Sakalauskas
CONCL:JSIONS The suggested stable m~thod of numerical differenti.a tion opens the way to rea] ize an explicit projection DPS identification algorihm. To improve noise resistance of the algorithm the projection space is chosen as a set of piece-wise constant functions being sub-optimal in the sense of nearily minimal errors of the obtained parameter estimates. The res ults of simulation on a typical eBmple suggest a good performance of the algorithm ~ith a JO ~u rela tive Ol.:tput error.
REF:::RENCRS Butkovskij,A.G. (977). General theory of structu:'a] schemes for distribu-ted parameter systems. In Str~lctural Theory of Distributed Svstems. Nauka, Moscow (in Russian). Chap. 2, pp. 9D B. Chavent,G. (1979). Identification of distrib uted parameters. In Preprints of the )-rd IFAC Symposium on Identification and System Parameter Estimation. Darmstadt, pp. 85-99. Fa irma n , P . \, ., and D.". ::;hen OnO). Parameter identification for a class of distrib u ted syste ms . Int. J. Control. Vo 1. 11, N 11, pp. 929-940. Gcodson,R.E., and M. P. Polis ( l nS) . I. survey of parameter identification in distrib uted systems. In Proc. IFAC 6-th Wo rld Congress. Boston, Cambrige, part lA, 1-12. Kazakov, I.E. (1969). Statistical methods in control s stems daSl n. Mashinostrcyenye, Moscow in Russian). Khiamarik,U.A. ( 1983). Projection methods for regularization of :inear ill-posed problems. In Preprints of Tartu University. Tartu university, Tart u (in RUSSian). Vol. 50, pp. 59 - 90 . Kubrusly,C.S. (1977). Distributed parameters system identification. A survey. Int. J. Control, Vol. 26 , N 4, pp.509535. Marchook ,G.I., and V.I.Agoshkov (1981 ) . Some projection method algorithms. In Introducti on to projection - grid methods. Nauka, Moscou (in Russian). Chap. I, pp. 2 1-9 6. Rajbman N.S., Bogdanov V.D., and Kneller D.B. (1982). Identification of distributed systems. Automatica and Telemekhanika (in RUSSian). N 6, pp. 5-)6. Reddy , ~ ., and V.Rajarni (1977 ; . A revie~ of the state and parameter estimation theory in distributed systems. J.Inst. Electr. and Telecom. Eng. Vol. 2), pp. 689-700. Sakalauskas,E.J. (1984). Galiorkin-Tikhonoy method in numerical differentiation problem. J. of Numerical Mathematics and Mathematic PhYSiCS (in Ru sian), N 11, pp.1742-l747. Tikhonov,A.N., and V.J.Arsenin (1979) Method of regularization of operator equations. In Ill-posed problems solution methods. Nauka, Moscow (in Russian). Chap. 2, pp. 5)-109. Vylenkin,S.I. (1979). Statistical processing of stochastic functions study data. Nauka, Moscow (in RUSSian).
Zavjalov,U.S., B.I.~vasov, and V.L.~yro shnychenko ( 1900) . f,Jetho ds of sp11ne-f'unctions. Nauka, Moscow (in Russian) •