ESTIMATION OF SPRING STIFFNESS

UNDER CONDITIONS OF UNCERTAINTY.

INTERVAL APPROACH

S. I. Kumkov

Institute of Mathematics and Mechanics Ural BranchRussian Academy of Sciences,

Ural Federal University, Ekaterinburg, Russiakumkov@imm.uran.ru

The 1st International Workshopon Radio Electronics & Information Technologies (REIT’2017)

March 15, 2017, IRIT–RTF, Ural Federal UniversityEkaterinburg, Russia

The aim of this presentation is to demonstrate

application of Interval Analysis methods to engineering

practical problem of estimating spring stiffness parameters

under conditions of uncertainty when components of the

parameter vector can not be estimated with guarantee

by standard methods of mathematical statistics.

The work was supported by Act 211 Government of the Russian Federation,Contract no. 02.A03.21.0006.1-07909.

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Topics of presentation.

Experiment on investigation spring properties.

Interval approach and its peculiarities.

Problem formulation and the basic procedures.

Results of estimation. Model example.

Conclusions.

References.

3

Experimental process and its model

The “global” process model as a reliable describing function of the springcompression S (mm) on the whole interval of the argument F (loadingforce, Newton) is given

S(F, a, s0) = aF + s0, F > 0, a < 0, s0 > 0, (1)

where a is the spring stiffness, mm/N; s0 is the initial spring length, mm.

Results of the experiment are presented as a collection–sample (havinglength N) of the argument Fn and the compression sn measurements

{Fn, sn}, n = 1, N. (2)

4

Measurements corruption

In the experiment, each nth measured value of the loading

force Fn and the spring compression sn are corrupted as

follows:n = 1, N,Fn = F ∗

n + en, |en| ≤ emax,sn = s∗n + bn, |bn| ≤ bmax,

(3)

where F ∗n and s∗n are the unknown true values under measuring;

en and bn are measuring errors with unknown probabilisticproperties but bounded in modulus by the values emax andbmax, correspondingly.

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Conditions for estimation and possiblea priori information

No probabilistic information on errors is known and the sampleis dramatically short: N ≈ 5 ∼ 7 measurements only.

Parameters a and s0 have to be estimated.

From theoretical estimations and previous experience, the followingapproximate (rough) a priori constraints on possible valuesof the coefficients could can be given; for example:

aap = [aap, aap], s0ap = [sap0 , sap0 ]. (4)

Here and further in the text, we keep at the standard notationsaccepted for the interval variables [9].

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Interval approach. Its peculiarities

Ideas and methods of the Interval Analysis Theory and Applicationsarose from the fundamental, pioneer work by L.V. Kantorovich [1].Nowadays, very effective developments of the theory andcomputational methods were created by many researchers, e.g. [2–4]and in Russia [5–8].

Special interval algorithms have been elaborated for estimating parame-ters of experimental processes in high-temperature chemistry, organicchemistry [10-15], investigation of metals [16], air traffic control, etc.

Remind that essence of this branch of numerical methods theory andapplication consists in estimation (or identification) of parametersunder bounded errors (noises or perturbations) in the inputinformation to be processed, and under complete absence ofprobabilistic characteristics of errors.

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Interval approach. The main definitions

Uncertainty set of each measurement (USM).

We consider the two-dimensional case. The argument F

and the spring compression are scalars. But both the argument

Fn and process sn measurements are corrupted with the bounds

emax and bmax on their errors. It is possible to show the

rectangular uncertainty set Hn

n = 1, N,Hn = [F n,F n]× [sn, sn],F n = Fn − emax, F n = Fn + emax,sn = sn − bmax, sn = sn + bmax.

(5)

8

Interval approach. The main definitions

Admissible value of the parameter vector (a, s0) and

corresponding admissible curve S(Fn, a, s0)

(a, s0) : S(Fn, a, s0) ∈ Hn, for all n = 1, N. (6)

Information Set (InfSet) is a totality of admissible values of

the parameters vector (a, s0) satisfying the system of interval

inequalities

I(a, s0) ={(a, s0) : S(Fn, a, s0) ∈ Hn, for all n = 1, N

}. (7)

9

Results of measuring and admissible dependence

set of measurementUncertainty

Admissible dependenceof the linear type with ,a s0

1H

kH

NH

sk

sn

sN

, mms

F, N

Fk

F1 Fn

FN

. . . . . .. . .

s1

s0

Fn n= F emax

__ Fn n= +F emax

_

sn n= +s bmax

_

sn n= s bmax

__

nH

nH

10

Tube of admissible process dependencies

Tube of admissible dependencies Tb(F ) is a totality of all

admissible values of the process, or a totality of admissible

dependences describing the process. For the linear model

S(F, a, s0) = aF+s0 and the information set I(a, s0), the tube

lower Tb(F ) and upper Tb(F ) boundaries are calculated

F ∈ [F1, Fn] :Tb(F ) = min(a,s0)∈I(a,s0) S(F, a, s0),Tb(F ) = max(a,s0)∈I(a,s0) S(F, a, s0).

(8)

11

Image of Tube of admissible dependencies

1H

kH

nH

NH

Upper boundary of the tube

s1

sk

sn

sN

, mms

F, N

Fk

F1

Fn

FN

. . . . . .. . .

Lower boundary of the tube

Tube of admissible dependences

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Problem formulation

Since of very short length of the measurementssample, absence of probabilistic characteristicsof the errors, and measurements uncertainty, it isimpossible to use (with any good reasoning)the standard statistical methods [17–19]).

It is necessary:on the basis of the Interval Analysis methods to builtthe Information set I(a, b, c) of admissible values(or the Set-membership) of coefficients a and s0consistent with the described data.

Note that by computing the information set of parameters and constructingthe tube, we find the united set of solutions for the system of intervalinequalities.

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Constructing Partial Information Sets. Two-dimensionaluncertainty sets (rectangles) with non-overlappinguncertainty intervals in the argument F

1) !! New definition of an admissible dependence

S(F, a, s0) ∈ Hn for at least one F ∈ [F n, tn], n = 1, N.(9)

2) Check the main condition

[F k,F k] < [F k+1,F k+1], k = 1, N − 1. (10)

3) For each pair with non-overlapping uncertainty intervals in F , formationof pair of the diagonals Dk and Dn completely describing the bunch ofadmissible dependencies passing through the uncertainty sets Hk and Hn

{S(F, a, s0)} = {S(F, a, s0) ∈(Hk, t ∈ [tk, tk] and Hn, t ∈ [tn, tn]

),

n = 2, N, k = n− 1, N − 1.(11)

4) Computation of corresponding collection of the Partial InformationSets

{Gk,n}, n = 2, N, k = n− 1, N − 1. (12)

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Constructing the Partial Information Sets.

b)

III ~ LII

IV ~ LIV

III

a, mm/N

, mms0

Gk n,

( , )a s0

( , )a smin 0max

( , )a smax 0min

, mms

sk

nH

sn

F, N. . .

Fk

Fn

. . . . . .

1 2

4

3

5

6

7

lines with the extremal

lines with intermediate

LIV

а)

LI

values of parameters ,s a0

( , )a smax 0min

LII

LIII

8

( , )a smin 0max

( , )a sII 0,II

( , )a sIV 0,IV

values of parameters ,a0

s

kH

kD

nD

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Computation of the resultant Information Set

There are several approaches for solving system of the interval

inequalities (7)

– classic linear programming methods [1], and many others,

– parallelotopes Fiedler M., et al [2], Hansen [3], Jaulin, et al

[4], Shary, Sharaya [5–7],

– the “stripes” method by Zhilin [8].Here, we apply special DIRECT method (see, Kumkov and

with co-authors [10–16]) that gives exact description of the

Information set I(a, s0) in the plane a× s0.It is performed in contrast to outer approximation of information

sets by the parallelotope approaches, for example, the powerful

algorithms SIVIA [4].

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Resultant Information Set; two-dimensionalparameter vector

The set is constructed by direct intersection of collection ofthe Partial Information Sets as follows:

U

I G=k n,( , , , ) ( , , , )a e b a e bmax max max max

n = N2,

_

k = n 1_

s0 s0 (13)

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Image of the resultant Information Set

. . .

. . .

a, mm/N

s0

_

s0_

a_

a priorirectangle

a_ acnt

s0,cnt

aapr s0,apr

s0

U

I G( , , , ) =a e bmax max k n,

n = N2,

_

k = n 1_

, mms0

Gk n, ( , )a s0

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Analysis of sample consistency of input sample

But intersection of all Partial Information Set can be empty.

This means that the actual corruption in measurements are

larger that the given values of the bound emax and bmax.

Simultaneously, it denotes on possible presence of outliers in

the measurements of the sample under procession.

Nowadays there are reliable technique for estimation of actuallevel of corruption in the sample by special functional [6,7].We apply elaborated direct method of variation of the bounds’emax and bmax values [10–16].

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Constructing the Tube of admissible dependencies

The tube of admissible dependencies Tb(t) is a totality of

all admissible values of the dependence on the process. For

linear model S(F, a, s0) of the process and the information set

I(a, s0), the tube boundaries are calculated as follows:

F ∈ [F1, FN ] :Tb(F ) = min(a,s0)∈I(a,s0) S(F, a, s0),Tb(F ) = max(a,s0)∈I(a,s0) S(F, a, s0).

(14)

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Results of estimation. Model example. Information Set

LSQM-point

0.11_

0.10_

0.09_

115

120

125

True point

mm/Na,

0.12_

, mms0

( , )a s0* *

( , )a s0~ ~

( , )a s0cnt cnt

Central point

Unconditional minimal outer

a_ a

_

s0

_

s0_

box-estimate [ , ] [ , ]a_ a

_

s0

_

s0_

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Results of estimation. Tube of admissible dependencies

F, N

392 588 784 980196

20

40

60

80

100 +3s

LSQM-line

Tube of admissible

3s_

Lower boundaryof tube

Upper boundaryof tube

dependencies

s, mm

True dependenceand true valuesTrue

dependenceMeasurements anduncertainty sets5 measurements

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Comparison with the standard statistical approach

The LSQM-curve and point-wise estimation of parameters

a, s0 and their practically meaningless “cloud-built” intervals

are available by only formal application of standard statistical

procedures [17–19].

Results of application of the LSQM-method are also shown

in the previous figure.

It is seen that the tube constructed by the described interval

method is essentially narrower than the rough “corridor” ±3σ.

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ConclusionsThe Interval Analysis methods was applied to estimation of springparameters in the compression process under conditions of absenceof probability data for the measuring errors.

Important case was investigated when errors are both in the loading forcemeasurements and in ones of the spring compression (length).

Investigations are fulfilled on the basis of the wide used Hooke’s law withthe linear dependence of spring compression vs the loading force.

It was shown that under mentioned conditions Interval Analysis approachgives guaranteed estimation of the process parameters and betterestimation of the tube of admissible dependencies.

Moreover, simulation results show that using simultaneously, the intervaland standard statistical approaches complement each other; and thisallows one to perform more detailed analysis and qualitative comparisonof the estimation results.

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References1. Kantorovich L.V. On new approaches to computational methods and processing theobservations // Siberian mathematical journal. 1962, III, no. 5, pp. 701–709.2. Fiedler M., Nedoma J., Ramik J., Rohn J., and K. Zimmermann. Linear optimizationproblems with inexact data. Springer-Verlag., London. 2006.3. Hansen E., G.W. Walster. Global Optimization using Interval Analysis. Marcel Dekker,Inc., New York. 2004.4. Jaulin L., Kieffer M., Didrit O., and E. Walter. Applied Interval Analysis. Springer-Verlag, London. 2001.5. Shary, S.P. Finite–Dimensional Interval Analysis. Electronic Book, 2014,http://www.nsc.ru/interval/Library/InteBooks

6. Shary S.P. and I.A. Sharaya. Raspoznzvaniye razreshimosti interval’nykh uravneniyi iego prilozheniya k analizu dannykh // Vichslitelnye tekhnologii. (2013), 8, no. 3, pp.80–109.7. Sharaya I.A. Dopuskovoye mnozhestvo resheniyi interval’nykh lineyinykh system uravneniyiso svyazannymi coeffitsientami // in Computational Mathematics, Proc. of XIV BaikalInternational Seminar-School “Methods of Optimization and Applications”. Irkutsk, Baikal,Russia, 2 – 8 July, 2008. Irkutsk, ISEM SO RAS. (2008), 3, pp.196–203.8. Zhilin S.I. Simple method for outlier detection in fitting experimental data underinterval error // Chemometrics and Intelligent Laboratory Systems. (2007), 88, pp.6–68.9. Kearfott R. B., Nakao R. B., Neumaier A., Rump S. M., Shary S. P., and vanHentenryck P.: Standardized Notation in Interval Analysis. Comput. Technologies, 15,no. 1. 7–13 (2010).10. Redkin A.A., Zaikov Yu.P., Korzun I.V., Reznitskikh O.G., Yaroslavtseva T.V., andS.I. Kumkov. Heat Capacity of Molten Halides // J. Phys. Chem. B, (2015), 119: 509–512.

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References11. Kumkov S.I. and Yu.V. Mikushina. Interval Approach to Identification of CatalyticProcess Parameters // Reliable Computing. (2013), 19: 197–214.12. Arkhipov P.A., Kumkov S.I., et.al. Estimation of plumbum Activity in Double systemsPb–Sb and Pb–Bi // Rasplavy. (2012), no. 5, pp.43–52.13. Kumkov S.I. and Yu.V. Mikushina. Interval Estimation of Activity Parameters ofNano-Sized Catalysts // Proceedings of the All-Russian Scientific-Applied Conference“Statistics, Simulation, and Optimization”. The Southern-Ural State University, Chelyabinsk,Russia, November 28–December 2. (2011), pp. 141–146.14. Kumkov S.I. Processing the experimental data on the ion conductivity of moltenelectrolyte by the interval analysis methods // Rasplavy. (2010), no. 3, pp.86–96.15. Potapov A.M., Kumkov S.I., and Y. Sato. Procession of Experimental Data onViscosity under One-Sided Character of Measuring Errors // Rasplavy (2010), no. 3,pp. 55–70.16. Gladkovsky S.V. and Kumkov S. I. Application of approximation methods to analysisof peculiarities of breaking-up and forecasting the break-resistibility of high-strength steel// Matematicheskoe modelirovanie sistem i protsessov // Sbornik nauchnykh trudov,Permskii Gos. Tekhnicheskii Universitet, perm, 1997, no. 5, pp. 26–34 (in Russian).17. GOST 8.207-76. The State System for Providing Uniqueness of Measuring. DirectMeasuring with Multiple Observation. Methods for Processing the Observation Results.–M.: Goststandart. Official Edition.18. MI 2083-93. Recommendations. The State System for Providing Uniqueness ofMeasuring. Indirect Measuring. Determination of the Measuring Results and Estimationof their Errors. –M.: Goststandart. Official Edition.19. R 40.2.028–2003. Recommendations. The State System for Providing Uniqueness ofMeasuring. Recommendations on Building the Calibration Characteristics. Estimation ofErrors (Uncertainties) of Linear Calibration Characteristics by Application of the LeastSquare Means Method. –M.: Goststandart. Official Edition.

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Thanks for attention

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