Practical applications of computer technologies are considered. Estimation of parameters of the aluminum corrosion process is investigated. In practice, estimation of parameters of experimental chemical processes is very actual and important for researches for necessary organization of corresponding technological processes by correct choice of the parameters. But usually, determination of the process parameters is hampered by their complicated internal structure and incompleteness of input information for processing. The paper considers application of the interval analysis procedures to estimation of parameters of an experimental chemical process under conditions of corruption, uncertainty of errors (corruption) probability characteristics, and short sample of measurements. For these reasons, the standard statistical approach can be applied only formally; especially, it becomes impossible to determine accurately the con dential intervals for parameters of the process. Under such conditions, namely methods of the interval analysis can work reliably and give exact set of the admissible values of the parameters to be estimated. In this paper, existing interval analysis procedures were advanced and arranged for a concrete experimental process and its data. As a comparison, approximate estimations of parameters have been calculated by the standard statistical approach. It is shown that the standard approach gives very rough and, often, practically senseless estimations of the process parameters.
of ordinary di erential equations of the rst approximation and analytical functions.
The vector of parameters, i.e., the activity coe cients, comprises: KNT;AL of transformation of nitrate in the direct reaction with aluminum, KNT;T of the own nitrate thermal decomposition, and KAL;Ox of aluminum transformation into the aluminum oxide by free oxygen. Moreover, aluminum oxide partially (with the coe cient KF ) precipitates as a lm on the aluminum surface but residuary part of oxide sediments in the molten environment and does not participate further in reactions. The coe cients of the reagents activity (of creation and transformations) and redistribution of the aluminum oxide between the oxide lm and the powder sediment have to be found and the latter coe cient KF is also of interest.
But estimation of these coe cients is strongly hampered by complexity of the process description, fatally short length of measurement samples, and complete absence of any probabilistic characteristics of errors in the measurements. Measuring is implemented only at the beginning t = 0 of the process (the initial concentration of nitrates and mass of metallic aluminum before the corrosion test) and at its termination t = tf (the nal concentration of nitrates and nal mass of the metallic specimen with the oxide sediment on it).
Estimation of the set of admissible values of the coe cients is performed in
the following way. On the basis of the measurements, the uncertainty intervals
of the direct or indirect measurements (both at the beginning and the end of
the process) of the components are built using the prescribed bounds on the
measuring errors. The sought-for set is a totality only of such parameters values,
for which the integral curves of the process components pass through the
mentioned intervals at the beginning and termination instants of the experiment.
This excludes application of the standard statistical methods [
The goal of this work is in further development of a new approach for processing the mentioned experimental data. The paper has the following structure. In Section 1, the process model is described and the estimation problem is formulated. Section 2 is devoted to the main Interval Analysis ideas applied. Section 3 shortly describes structure of the estimation algorithm. Section 4 presents results of numerical simulation of the estimation problem. 2
Model of the process and problem formulation
Earlier, the corrosion process was investigated in [
The original description of chemical reactions (of the rst approximation) is a) 2N aN 03 + 2Al N a2O + Al2O3 + 2N O; b) 2N aN 03 2N aN O2 + O2; c) 4Al + 3O2 2Al2O3:
On the basis of this description, the following mathematical model (the rst approximation) of the process was built:
t 2 [0; tf ]; _
N T = A_L =
KNT;ALN T AL
KNT;T N T; KNT;ALN T AL
O_x = KNT;T N T
OxAl(t) = AL(0) AL(t);
F lm(t) = KF OxAl(t); M (t) = AL(t) + F lm(t); (1) (2) (3) (4) (5) (6) (7) where, t is time, the independent argument with termination instant tf ; N T (t) is the nitrate current concentration, wt%; AL(t) is the current mass of the aluminum electrode, grams; OxAl(t) is the current total mass of the oxide, grams; Ox(t) is the current oxygen mass; F lm(t) is the current mass of the lm precipitated on the surface of the metal aluminum electrode, this value is a part of the whole oxide mass; M (t) is the auxiliary variable representing the current mass of metal AL and the oxide Al2O3 lm precipitated on the metal surface; KNT;AL, KNT;T , and KAL;Ox are the activity coe cients (of corresponding physical dimensions; here and later, their dimensions are omitted for simplicity of description); KF is a deal coe cient of the lm mass. The oxide part OxAl(t) F lm(t) goes out of the reaction as a sediment and does not participate in the further processes.
It was revealed [
That is why for adequate description of the process in the mathematical model (1){(7), a special important variable M (t) is introduced that describes the total current mass of the aluminum electrode and lm on its surface. Value of this variable can be measured at the termination instant.
Remark 1: Since of description of the process in the di erential form (2){ (4), values of the components during the process and at the termination instant can be presented only by numerical integration for each value of the coe cients KNT;AL, KNT;T , and KAL;Ox.
Measured data. Measurements at the initial (t=0) instant: N Tmes;0 with bounded additive error emax;NT , ALmes;0 with bounded additive error emax;AL, Oxmes;0 = 0; OxAlmes;0 = 0; F lmmes;0 = 0; Mmes;0 = ALmes;0:
Measurements at the termination ( nal t = tf ) instant: N Tmes;f with bounded additive error emax;NT , Mmes;f with bounded additive error emax;AL.
Values of AL(tf ), Ox(tf ), OxAl(tf ), and F lm(tf ) are not measured. No probabilistic information on measuring errors is known. There are no a priori bounds on possible values of the coe cients.
The problem is formulated as follows: to built the set (i.e., the Information Set or Set-membership) of admissible values of the activity coe cients KNT;AL, KNT;T , KAL;Ox, and to estimate a collection of intervals for the coefcient KF consistent with the described data.
Note once more that because of incomplete observability of the process phase
coordinates, fatally short length of the measurements sample (only two
measurements), absence of probabilistic characteristics of errors, and measurements
uncertainty, it is impossible to use standard statistical methods (see, [
Interval approach to solving the problem Since probabilistic information on errors is unknown, only uncertainty intervals of measurements can be constructed:
H NT;0 = [N Tmes;0 emax;NT ; N Tmes;0 + emax;NT ]; H AL;0 = [ALmes;0 emax;AL; ALmes;0 + emax;AL]; H NT;f = [N Tmes;f emax;NT ; N Tmes;f + emax;NT ];
H M;f = [Mmes;f emax;AL; Mmes;f + emax;AL]:
Note: here and in the sequel, notations of the interval variables recommended
by the Standard [
After termination of the process, an interval of the coe cient KF is calculated as
KF = (H M;f
ALf )=OxALf ; (12) where ALf and OxALf are values of the variables at the terminal instant tf computed by integration of the describing di erential equations system (1)-(7).
Ideas and methods of the Interval Analysis Theory and Applications arose
from the fundamental, pioneering work by L.V. Kantorovich [
In application to the problem under consideration, essence of Interval Analysis Methods consists in estimation (or identi cation) of a process parameters under bounded measuring errors in the input information and under total absence of probabilistic characteristics of the errors.
The estimation results are represented in the form of so-called the
Information Set or Set-membership that approximate (see, for instance, [
In practice and, especially, in the case of small dimension of the parameters vector to be estimated, the inner grid approach is more preferable since its computational simplicity and smaller computations in comparison with both the outer peeling and inner{box approach.
In our problem, it is just the case: vector of the main parameters is only three-dimensional: KNT;AL, KNT;T , and KAL;Ox. So, the inner grid approach was used.
De nition. A point (KNT;AL, KNT;T , KAL;Ox) of the parameters space KNT;AL KNT;T KAL;Ox is admissible and an initial point (N T0; AL0) from the box HNT;0 HAL;0 is admissible if the corresponding computed point (N Tf ; ALf ) at the termination instant belongs to the box HNT;f HAL;f .
De nition allows one to sift out directly such points (KNT;AL, KNT;T , KAL;Ox) and (N T0, AL0) that are not consistent with the given experimental data. 4
Structure of the main algorithms and steps of its performing are illustrated in Fig. 1. The algorithms are implemented in the following steps:
Kj Sthteepi1n:vfeirnsdeinpgrotbhleeminiotifaltrgaunessfesrby
(NTmes,0, ALmes,0) (NTmes,f, ALmes,f)
true point initial guess
Kj .S.ts.eipo.n3.a:l.ing.trri.odd.“uw.cii.nthg.oa.ut.ther.rere.e-s.dei.mrv.een”.-. ..................... ..................... ........... ......... ..................... ..................... ..................... ..................... ..................... ..................... Ki
Kj
Kj
Step 2: building the outer box around the initial guess (Kj,...,Ki)ing true point initial guess
Step 4: building the inner grid approximati.on o.f I(K) .......
..........
...... ......
................. ....................................... ............ .
........
Ki
King,i Ki King,i Ki Fig. 1. Structure of the main algorithms and steps of its performing; values of the initial guess coe cients are marked by the ing{index; parameters shown on the axes are illustrative Step 1: nding the initial guess (KNT;AL; KNT;T ; KAL;Ox)ing, i.e., the initial admissible point of coe cients inside the information set; Step 2: constructing the auxiliary outer guaranteeing box of coe cients; Step 3: introducing the three{dimensional grid on the auxiliary box of coe cients; Step 4: building the inner grid approximation of the information set I(KNT;AL; KNT;T ; KAL;Ox) of coe cients. 5
Numerical results are presented in Figs. 2 and 3. Computation was performed with the model prescribed true coe cients of values KNT;AL = 0:001, KNT;T = 0:001, KAL;Ox = 0:001, and KF = 0:1 and with the initial values N Tmes;0; ALmes;0. The true model values N T0 = 10 and AL0 = 0:3 are pointed with arrows.
NT(t)
Model true coefficients: KN*T,AL = 0.001 K*NT,T = 0.001
KA*L,Ox = 0.001 K*F = 0.1
The general picture (Fig. 2) shows decreasing the nitrate concentration N T (t) (dash-two-points curve). Also, the initial N Tmes;0; ALmes;0 corrupted measurements and ones N Tmes;f , Mmes;f at the termination instant are marked by crosses.
Processes on the components AL(t); M (t); ALOx(t); Ox(t), and F lm(t) are given in the lower part in Fig. 3 and in zoomed images in Fig. 5. It is seen both active decreasing the aluminum mass (since corrosion) and increasing the oxide and lm masses.
ALmes,0 = Mmes,0 {AL*0}
1 ALmes,0 = Mmes,0 NTmes,f Ox(t) t, hour AL(t) Flm(t) 4
ALOx(t) M(t)
Mmes,f t, hour 5 1 2
3
Picture of the measured data is given in Fig. 4 and 5. Here, the true model values are marked, the measurements and calculated values N Tmes;f ; Mmes;f (at the termination instant) are pointed by arrows; uncertainty boxes (sets) H NT;0 H AL;0 and H NT;f H M;f of measurements are drawn by rectangles. Integration of system (1)-(7) with true coe cients gives the values N Tf ; Mf .
Note that this point lies inside the uncertainty box H NT;f H M;f that proves admissibility of taken values of the coe cients and the initial position N Tmes;0; ALmes;0. Consider steps of the suggested algorithm.
AL and M
true point NTf , Mf t = tf HNT,f
HM,f true point measurement NTmes,f , Mmes,f and its uncertainty box
9 10 Fig. 4. Picture of the measured data 11
NT, wt% Step 1. The initial guess of admissible values of coe cients (KNT;AL = 0:001046; KNT;T = 0:001163; KAL;Ox = 0:001)ing (13) are found as a solution of an usual inverse problem of dynamics with the initial (left) value of the process N Tmes;0, Mmes;0 and the termination (right) value N Tmes;f ; Mmes;f .
AL and M t = 0 HNT,0
HAL,0 measurement NTmes,0, ALmes,0
and its uncertainty box example of integration result with true values of coefficients true point t = tf
t = 0 measurement NTmes,0,ALmes,0
and its uncertainty box inverse problem gives the initial guess coefficients (“ing”) KNT,AL = 0.001046, KNT,T = 0.001163,
KAL,Ox = 0.001, and [KFlm] = [0, 0.174] 0.3 0.20 0.1
Step 2. Constructing an auxiliary outer box around the initial guess (KNT;AL; KNT;T , KAL;Ox)ing. The procedure consists of: { coordinate-wise variation of each coe cient under xed values of two other ones with veri cation of admissibility each changed value of the coe cient under variation; { variations are implemented for two directions to nd the minimal and maximal values on each coe cient, i.e., to nd approximate intervals of each coe cient.
Results of this procedure are shown in Fig. 6. The maximal upper points are marked by white squares; the minimal ones are represented by black squares. The following marginal boundary values were found:
KNT;AL = [0:000896; 0:00125]; KNT;T = [0:000618; 0:001738]; (14) KAL;Ox = [0; 0:004225]:
Remark 2: Because of independent coordinate-wise variations, the box found can have inadmissible points, for example, the box apices or, even, parts of the edges.
0.004225
KAL,Ox 0.0010
Step 3. Having boundaries (14), introduce grid in each of these intervals with some reasonable values of the grid step on each parameter. In model computations, the number of nodes in each grid was su ciently given as 51.
Step 4. The concluding operation is in simple direct veri cation of each node of the introduced three-dimensional grid. Only admissible nodes are included further into the inner part of the information set.
Results of constructing a cross-section of the inner approximation of the information set I(KNT;AL, KNT;T , KAL;Ox) for the xed value of the icoe cient KNT;T = 0:001163 is shown in Fig. 7.
The similar pictures are obtained for presentations of the initial grid and nal inner approximation of the cross-sections of the information set I(KNT;AL, KNT;T , KAL;Ox) for xed values of the coe cient KAL;Ox, and the initial grid and nal inner approximation of the cross-section of the information set I(KNT;AL, KNT;T , KAL;Ox) for xed valuesof the coe cient KNT;AL.
By the similar procedures, the whole informational set I(KNT;AL, KNT;T , KAL;Ox) for all points from the initial uncertainty box H NT;0 H AL;0 is built 0
0.000896 marginal pointwise cross-section
for KNT,AL= 0.00133 true point is inside 5 KAL,Ox cross-section for fixed value KNT,T = 0.001163 2 2 4 0 0 . 0 as a totality (see ideology and techniques of such representation in [9{11]) of all admissible cross-sections; its spatial image is shown in Fig. 8.
Underline that the results represented in Figs. 7, 8 have the guaranteed character since the true values of the coe cients KNT;AL = 0:001, KNT;T = 0:001, and KAL;Ox = 0:001 obligatory (by construction) lie inside the nal inner approximation of the information set I(KNT;AL, KNT;T , KAL;Ox) for all initial values N T (t = 0), AL(t = 0) from the initial uncertainty box H NT;0 H AL;0.
Analysis of experimental data has shown that for practical applications, the mentioned point-wise initial guess of type (13) with approximate estimating intervals of type (14) are su cient and acceptable. In contrast to now existing approaches to estimation of experimental process parameters under mentioned conditions of uncertainty, the interval approach allows one: { to built veri ed inner approximation of the information set for the activity coe cients; { to nd veri ed interval estimations of the important coe cient of the oxide lm precipitated on the aluminum surface for various values of admissible coefcients from the information set; { if necessary, to enhance the box (set) of the initial (or termination) measurements for some xed prescribed values of coe cients.