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The nonlinear algorithm of adaptive identification
THE EOBLIl%AR ALGORITHM 07 ADAFTIViG IDEN'$fFICATION SH. LEWSHVILI , K. KRURDADZE Summary. In work is considered the generalized iterative algorithm of nonlinear adaptive model in distinction of the particular algorithm described in 141. The functioning of such class of algorithms, so as the class of linear al orithms [51, is determined with state of adaptive identification [f 1, functioning of statistic modelling system [3], the stochastic properties of identification of plants of metallurgic processes [Z]. This algorithm has self-adjusting properties and can successfully function during the identification of complex processing objects with time varying parameters..Some aspects of using such algorithms for identification problems in metallurgy are described. Keywords. Iterative algorithms; nonlinear structure; selfadjusting structure: adaptive Ldentification. The modern modellind theory and methodology [l] gives ua a chance to construct a model of a lot of practical modern plants. The most practical objects ofmodernity-separate states and its regions, subsy-stems. technologycal processes or enterprises, closed systems of ki*ing, construction of ~trophomorfic systems and others demands the construction of high effective identification systems. A lot of identification methods use measured input and output sequence of data without real time filtration. But, during the construction of identification systems of concrete plants, these methods cannot be used directhy for its nonstationarity of real systems. This nonstationarity is the result of stochastic varying in time ofdifferent parameters of complex objects. For identification such objects is required the extraction of information ( if there is a priori indefinability ) with the help of current observation analysis. In this case, the adaptive system with selfadjusting parameters is the suitable real object model. The considerable part of modern complex nonlinear object can be described as a streightened regression:
G =GHl,il,e
F:l(X!l,e)+
&2
.
H2,3.1,i2,e
F~l(X~l,e)*
(e=1,2,3,...),
(1)
where Fi,,Fi2,Fi3- known operators; c -model output; H, rH2 ,H3 - desired adaptive model parameters; X' - input values. If X1 is the past value of one of the same input, then ve have dynamic object with one input; if X1 indicates different inputs of real object, then there ia the static with many inputs; and, at la1) USSR-380094,Tbilisi,Saburtalo at., carp. 3, flat 51. 2) USSR-380059,Tbilisi,Digomi,corp. 12, flat 15. 340
at, if structure (1) haa as different inputs as their prehystory we shall have the dinamic model of plant with the lQt
of inputs,
The structure represented in such case (1) can be considered as linear structure to the desired parameters and if can be repressented as: HaFa (X11,
(2)
. where H - model parameters; Fa(X') - known operators; pa - total quantity of model parameters. If we indicate F,(X') by Xa, we shall have instead (2)
T
=& =
*axa
(3)
l
SO, the problem of nonlinear plant identification comes to the comperativelly easily solved problem-receiving the streightened regression (3). Thus, the problem is stated forobtaining the estimates K of the incomplete information due to the presence ofnoncontrolled input actions and measurement errors. Besides, it is assumed that the plants parameters vary in time. Suppose, that plant output is Ye= e +N,, where Ii, components are a time noncorrelated influences [4,5], statisfying the following conditions: ~~~~~/x~=~,~~~~~~~I,n ps ~~=(~~,~~*,.~‘+~~), K is simbol of expectation and I is unit matrix, Let we have at our disposal n rtumber of sample (Y,,X,), the matrix (XTX) has full rank equal to p, KT and KT+, are parameters estimation vectors on T and T+l iterative step and vector disadjustment P-Y-X$; we may construct the lagrange function FfK,L)=(KT+, -~~T~T~~~~-K)+L P, If our objective is to minimize F(K,L), for any iterative step in presence W matrix of weighted parameter (for simplicity we shall assume, that 1';is (PXP) diagonal matrix), then v/e shall have [3+6]: ii af,T+? =Kat tT +~Ya+'tPT+&
a2=t
X:2 Te,)-'Xaf T+l, , ’
(4)
Ca-l=1,2,...,p), (T=O,t,2,...), where Ka, T+,,K -current estimations of Ha; , al,T_ kT+,=YT+, a2=, ~~a2,~32,T+1 - di-kius~ent;
-25
%+1 -
current output of plant. For index of adequateness of model to real system we applied dispersion ratio k= nKTXTXx,((n-p)YTY)-'. (5) Though algorithm (4) obtained according to structure (1) are convenient, there are certain circumstanceswhich make us look for other structures of system models. The thing is that in the process of real complex systems identification (as nonlinearity degree increases) the model achives a steep rise in its dimensionality. If A designa-
341
les the nonlinearity degree and B the number of static (for example) system inputs (when the system is dynamic with a single input, B is the maximum number of lag variable of dynamic plant) the dimensionality of the model will make the following Ci+B=(A+B>J/AJBJ. It is evident that in case of real system identification the model dimensionality will turn out to be too large. Besides the dimensionality of the model is related to sample value n in the following way
n=
A
=(10+30)cA+B.
In practice sample is limited which leads to model dimensionality restriction. Thus, for full identification of nonlinear system it is necessary to find a new model structure different from structure (1). Let us, model of nonlinear regression curves represent in the following common form F(H,X). Suppose, that structure of function F(H,X) is known [7,8] and differentiate function in given region. 'Then, for the model F(H,X) structure, we may construct (analogically (4) algorithm and methodology adaptive identification algorithms [3+G]) following adaptive algorithm:
KT+,=%+‘W)
Dk “($XT+,
)‘(D,
F(KTXT+,) )PT+, ,
(6)
where DkF(KTXl,+,) - vector of partial differential; 'T+l= 'T+l - F(I$XT+,) - parameter of disadjustment; 1 - simbol of transpoze. In accordance of methodology of works [3+6] we shall have analogically following expression W(T)=(~(Y~+,)-D~)(LI(P~+,))-' ,
(71
where Dg simbol of dispersion of distorbance noise TJ. The above algorithms (4) and (6) were tested on real systems (economics, medicine, cement industry, agriculture and tube-rolling production). Here are some results of experimental study. For index of adequateness of model to real system we applied dispersion ratio (5) when p/n was approximately equal or less then 0.1. Export data of a real economic dynamic (with one input) system were analysed by algorithm (6) in form of time series. The obtained dispersion ratio was equal to 0.92. That shows the model to be practically adequate to the real system. Dispersion ratio obtained by the algorithm (4) was smaller E=O.8. This shows that nonlinear algorithm (6) has an advantage over the linear algorithm (4). The (6) algorithm were used for cement ball mill. By us was constructed a model for cement quality forecast on 30 day before. For different types of cements we had different dispersion ratio. For streighened regression and (4) algorithms (static and incomplete cubical model with (maximum) 7 inputs) variation boundary E was 0.83 + 0.95. For nonlinear regression [8,9] and (6) algorithm correspondingly variation boundary was 0.87 + 0.97.
342
The algorithm (6) wae applied for prediction of daily productivity (yieldf tea-leaf on one year before for weat region of georgian republic. The best dispersion ratio wa8 0.9. For
system of tube assembly (static nonlinear model with six in-
puts and two outputs) was applied (6) algorithms; dispersion ratio was 0.95. Literature [t] Eykhoff, P.: System identification (Parameter and state estimation), Univ. of Technology Eindhoven, the Netherlands, John Wiley & Sons, 1974. [2! Hultgren, R,, R, Orr, P. Anderson, K, Kelley:Selected values of the~ody~~c properties of metals and alloys. Univ. of California, Berkeley, John Wiley & Sons, Inc, 1963. 37 Imedadee, V.V., Lelashvili, Sh,: Investigation some algorithms t of stock. approx. for identif. of multidim. plants, IFAC sgrmp. of identif., Praque, 1967. II41 p~(uT!3opc, 3.: ~p~6~0go~i~oob ogropmf9 0ffp3p3ao. ~k35mod.j~fi0 ~;dme 0aopbf2, 1973. 3; I cjbo ji53ap, [5! Lelashvili, Sh.: Adaptive identification algirithms for real-time computer control. The 2nd IFAC/IFIP symp. on software for computer control, Praque, 1979. [6f \03(-@3oyn, 8.: &@?J"i&@mn ~~~~~~~~~oo~ d~d~~~o ct~@%%%h& iX%tilf$~~o~6~~~~o ~~o~~~~~o~ d~~~~~~e~~~.~~~~a~, p f, fi?6r, [TJ Thorn, R.: Stabifite structuralle ei morphogenese. New York,1972, [a] 'Zese, W.: Atomic transition probabilities. I?ucl. inst. and methods, v. 90, 1970.
343
G =GHl,il,e
F:l(X!l,e)+
&2
.
H2,3.1,i2,e
F~l(X~l,e)*
(e=1,2,3,...),
(1)
where Fi,,Fi2,Fi3- known operators; c -model output; H, rH2 ,H3 - desired adaptive model parameters; X' - input values. If X1 is the past value of one of the same input, then ve have dynamic object with one input; if X1 indicates different inputs of real object, then there ia the static with many inputs; and, at la1) USSR-380094,Tbilisi,Saburtalo at., carp. 3, flat 51. 2) USSR-380059,Tbilisi,Digomi,corp. 12, flat 15. 340
at, if structure (1) haa as different inputs as their prehystory we shall have the dinamic model of plant with the lQt
of inputs,
The structure represented in such case (1) can be considered as linear structure to the desired parameters and if can be repressented as: HaFa (X11,
(2)
. where H - model parameters; Fa(X') - known operators; pa - total quantity of model parameters. If we indicate F,(X') by Xa, we shall have instead (2)
T
=& =
*axa
(3)
l
SO, the problem of nonlinear plant identification comes to the comperativelly easily solved problem-receiving the streightened regression (3). Thus, the problem is stated forobtaining the estimates K of the incomplete information due to the presence ofnoncontrolled input actions and measurement errors. Besides, it is assumed that the plants parameters vary in time. Suppose, that plant output is Ye= e +N,, where Ii, components are a time noncorrelated influences [4,5], statisfying the following conditions: ~~~~~/x~=~,~~~~~~~I,n ps ~~=(~~,~~*,.~‘+~~), K is simbol of expectation and I is unit matrix, Let we have at our disposal n rtumber of sample (Y,,X,), the matrix (XTX) has full rank equal to p, KT and KT+, are parameters estimation vectors on T and T+l iterative step and vector disadjustment P-Y-X$; we may construct the lagrange function FfK,L)=(KT+, -~~T~T~~~~-K)+L P, If our objective is to minimize F(K,L), for any iterative step in presence W matrix of weighted parameter (for simplicity we shall assume, that 1';is (PXP) diagonal matrix), then v/e shall have [3+6]: ii af,T+? =Kat tT +~Ya+'tPT+&
a2=t
X:2 Te,)-'Xaf T+l, , ’
(4)
Ca-l=1,2,...,p), (T=O,t,2,...), where Ka, T+,,K -current estimations of Ha; , al,T_ kT+,=YT+, a2=, ~~a2,~32,T+1 - di-kius~ent;
-25
%+1 -
current output of plant. For index of adequateness of model to real system we applied dispersion ratio k= nKTXTXx,((n-p)YTY)-'. (5) Though algorithm (4) obtained according to structure (1) are convenient, there are certain circumstanceswhich make us look for other structures of system models. The thing is that in the process of real complex systems identification (as nonlinearity degree increases) the model achives a steep rise in its dimensionality. If A designa-
341
les the nonlinearity degree and B the number of static (for example) system inputs (when the system is dynamic with a single input, B is the maximum number of lag variable of dynamic plant) the dimensionality of the model will make the following Ci+B=(A+B>J/AJBJ. It is evident that in case of real system identification the model dimensionality will turn out to be too large. Besides the dimensionality of the model is related to sample value n in the following way
n=
A
=(10+30)cA+B.
In practice sample is limited which leads to model dimensionality restriction. Thus, for full identification of nonlinear system it is necessary to find a new model structure different from structure (1). Let us, model of nonlinear regression curves represent in the following common form F(H,X). Suppose, that structure of function F(H,X) is known [7,8] and differentiate function in given region. 'Then, for the model F(H,X) structure, we may construct (analogically (4) algorithm and methodology adaptive identification algorithms [3+G]) following adaptive algorithm:
KT+,=%+‘W)
Dk “($XT+,
)‘(D,
F(KTXT+,) )PT+, ,
(6)
where DkF(KTXl,+,) - vector of partial differential; 'T+l= 'T+l - F(I$XT+,) - parameter of disadjustment; 1 - simbol of transpoze. In accordance of methodology of works [3+6] we shall have analogically following expression W(T)=(~(Y~+,)-D~)(LI(P~+,))-' ,
(71
where Dg simbol of dispersion of distorbance noise TJ. The above algorithms (4) and (6) were tested on real systems (economics, medicine, cement industry, agriculture and tube-rolling production). Here are some results of experimental study. For index of adequateness of model to real system we applied dispersion ratio (5) when p/n was approximately equal or less then 0.1. Export data of a real economic dynamic (with one input) system were analysed by algorithm (6) in form of time series. The obtained dispersion ratio was equal to 0.92. That shows the model to be practically adequate to the real system. Dispersion ratio obtained by the algorithm (4) was smaller E=O.8. This shows that nonlinear algorithm (6) has an advantage over the linear algorithm (4). The (6) algorithm were used for cement ball mill. By us was constructed a model for cement quality forecast on 30 day before. For different types of cements we had different dispersion ratio. For streighened regression and (4) algorithms (static and incomplete cubical model with (maximum) 7 inputs) variation boundary E was 0.83 + 0.95. For nonlinear regression [8,9] and (6) algorithm correspondingly variation boundary was 0.87 + 0.97.
342
The algorithm (6) wae applied for prediction of daily productivity (yieldf tea-leaf on one year before for weat region of georgian republic. The best dispersion ratio wa8 0.9. For
system of tube assembly (static nonlinear model with six in-
puts and two outputs) was applied (6) algorithms; dispersion ratio was 0.95. Literature [t] Eykhoff, P.: System identification (Parameter and state estimation), Univ. of Technology Eindhoven, the Netherlands, John Wiley & Sons, 1974. [2! Hultgren, R,, R, Orr, P. Anderson, K, Kelley:Selected values of the~ody~~c properties of metals and alloys. Univ. of California, Berkeley, John Wiley & Sons, Inc, 1963. 37 Imedadee, V.V., Lelashvili, Sh,: Investigation some algorithms t of stock. approx. for identif. of multidim. plants, IFAC sgrmp. of identif., Praque, 1967. II41 p~(uT!3opc, 3.: ~p~6~0go~i~oob ogropmf9 0ffp3p3ao. ~k35mod.j~fi0 ~;dme 0aopbf2, 1973. 3; I cjbo ji53ap, [5! Lelashvili, Sh.: Adaptive identification algirithms for real-time computer control. The 2nd IFAC/IFIP symp. on software for computer control, Praque, 1979. [6f \03(-@3oyn, 8.: &@?J"i&@mn ~~~~~~~~~oo~ d~d~~~o ct~@%%%h& iX%tilf$~~o~6~~~~o ~~o~~~~~o~ d~~~~~~e~~~.~~~~a~, p f, fi?6r, [TJ Thorn, R.: Stabifite structuralle ei morphogenese. New York,1972, [a] 'Zese, W.: Atomic transition probabilities. I?ucl. inst. and methods, v. 90, 1970.
343