Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach

ESTIMATION OF SPRING STIFFNESS
UNDER CONDITIONS OF UNCERTAINTY.
INTERVAL APPROACH
S. I. Kumkov
Institute of Mathematics and Mechanics Ural Branch
Russian Academy of Sciences,
Ural Federal University, Ekaterinburg, Russia
kumkov@imm.uran.ru
The 1st International Workshop
on Radio Electronics & Information Technologies (REIT’2017)
March 15, 2017, IRIT–RTF, Ural Federal University
Ekaterinburg, 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.
2

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 spring
compression S (mm) on the whole interval of the argument F (loading
force, 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 (having
length 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 probabilistic
properties but bounded in modulus by the values emax and
bmax, correspondingly.
5

Conditions for estimation and possible
a priori information
No probabilistic information on errors is known and the sample
is dramatically short: N ≈ 5 ∼ 7 measurements only.
Parameters a and s0 have to be estimated.
From theoretical estimations and previous experience, the following
approximate (rough) a priori constraints on possible values
of the coeﬃcients could can be given; for example:
aap = [aap, aap], s0
ap = [sap
0 , sap
0 ]. (4)
Here and further in the text, we keep at the standard notations
accepted for the interval variables [9].
6

Interval approach. Its peculiarities
Ideas and methods of the Interval Analysis Theory and Applications
arose from the fundamental, pioneer work by L.V. Kantorovich [1].
Nowadays, very effective developments of the theory and
computational 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, organic
chemistry [10-15], investigation of metals [16], air traffic control, etc.
Remind that essence of this branch of numerical methods theory and
application consists in estimation (or identification) of parameters
under bounded errors (noises or perturbations) in the input
information to be processed, and under complete absence of
probabilistic characteristics of errors.
7

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 measurement
Uncertainty
Admissible dependence
of the linear type with ,a s0
1H
kH
N
H
sk
sn
sN
, mms
F, N
FkF1 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 T b(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 T b(F) and upper T b(F) boundaries are calculated
F ∈ [F1, Fn] :
T b(F) = min(a,s0)∈I(a,s0) S(F, a, s0),
T b(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
FkF1 Fn FN
. . . . . .. . .
Lower boundary of the tube
Tube of admissible dependences
12

Problem formulation
Since of very short length of the measurements
sample, absence of probabilistic characteristics
of the errors, and measurements uncertainty, it is
impossible to use (with any good reasoning)
the standard statistical methods [17–19]).
It is necessary:
on the basis of the Interval Analysis methods to built
the Information set I(a, b, c) of admissible values
(or the Set-membership) of coeﬃcients a and s0
consistent with the described data.
Note that by computing the information set of parameters and constructing
the tube, we ﬁnd the united set of solutions for the system of interval
inequalities.
13

Constructing Partial Information Sets. Two-dimensional
uncertainty sets (rectangles) with non-overlapping
uncertainty 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, formation
of pair of the diagonals Dk and Dn completely describing the bunch of
admissible 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 Information
Sets
{Gk,n}, n = 2, N, k = n − 1, N − 1. (12)
14

Constructing the Partial Information Sets.
b)
I
II ~ 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 ,a 0s
kH
kD
nD
15

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].
16

Resultant Information Set; two-dimensional
parameter vector
The set is constructed by direct intersection of collection of
the 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)
17

Image of the resultant Information Set
. . .
. . .
a, mm/N
s0
_
s0_
a
_
a priori
rectangle
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
18

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 actual
level 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].
19

Constructing the Tube of admissible dependencies
The tube of admissible dependencies T b(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] :
T b(F) = min(a,s0)∈I(a,s0) S(F, a, s0),
T b(F) = max(a,s0)∈I(a,s0) S(F, a, s0).
(14)
20

Results of estimation. Model example. Information Set
LSQM-point
0.11_
0.10_
0.09_
115
120
125
True point
mm/N
a,
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_
21

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 boundary
of tube
Upper boundary
of tube
dependencies
s, mm
True dependence
and true valuesTrue
dependence
Measurements and
uncertainty sets
5 measurements
22

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.
23

Conclusions
The Interval Analysis methods was applied to estimation of spring
parameters in the compression process under conditions of absence
of probability data for the measuring errors.
Important case was investigated when errors are both in the loading force
measurements and in ones of the spring compression (length).
Investigations are fulﬁlled on the basis of the wide used Hooke’s law with
the linear dependence of spring compression vs the loading force.
It was shown that under mentioned conditions Interval Analysis approach
gives guaranteed estimation of the process parameters and better
estimation of the tube of admissible dependencies.
Moreover, simulation results show that using simultaneously, the interval
and standard statistical approaches complement each other; and this
allows one to perform more detailed analysis and qualitative comparison
of the estimation results.
24

References
1. Kantorovich L.V. On new approaches to computational methods and processing the
observations // Siberian mathematical journal. 1962, III, no. 5, pp. 701–709.
2. Fiedler M., Nedoma J., Ramik J., Rohn J., and K. Zimmermann. Linear optimization
problems 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 i
ego prilozheniya k analizu dannykh // Vichslitelnye tekhnologii. (2013), 8, no. 3, pp.80–
109.
7. Sharaya I.A. Dopuskovoye mnozhestvo resheniyi interval’nykh lineyinykh system uravneniyi
so svyazannymi coeffitsientami // in Computational Mathematics, Proc. of XIV Baikal
International 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 under
interval 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 van
Hentenryck 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., and
S.I. Kumkov. Heat Capacity of Molten Halides // J. Phys. Chem. B, (2015), 119: 509–
512.
25

References
11. Kumkov S.I. and Yu.V. Mikushina. Interval Approach to Identification of Catalytic
Process Parameters // Reliable Computing. (2013), 19: 197–214.
12. Arkhipov P.A., Kumkov S.I., et.al. Estimation of plumbum Activity in Double systems
Pb–Sb and Pb–Bi // Rasplavy. (2012), no. 5, pp.43–52.
13. Kumkov S.I. and Yu.V. Mikushina. Interval Estimation of Activity Parameters of
Nano-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 molten
electrolyte 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 on
Viscosity 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 analysis
of 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. Direct
Measuring with Multiple Observation. Methods for Processing the Observation Results.
–M.: Goststandart. Official Edition.
18. MI 2083-93. Recommendations. The State System for Providing Uniqueness of
Measuring. Indirect Measuring. Determination of the Measuring Results and Estimation
of their Errors. –M.: Goststandart. Official Edition.
19. R 40.2.028–2003. Recommendations. The State System for Providing Uniqueness of
Measuring. Recommendations on Building the Calibration Characteristics. Estimation of
Errors (Uncertainties) of Linear Calibration Characteristics by Application of the Least
Square Means Method. –M.: Goststandart. Official Edition.
26

Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach (20)

Recently uploaded (20)

Estimation of Spring Stiffness Under Conditions of Uncertainty. Interval Approach