IDENTIFICATION OF MODELS OF EXTERNAL LOADS

Yuri Menshikov

Dnepropetrovsk University, Nauchnaja st.13, 49050 Dnepropetrovsk, Ukraine

Keywords: External load, models, identification.

Abstract: In the given work the problem of construction (synthesis) of mathematical model of unknown or little-

known external load (EL) on open dynamic system is considered. Such synthesis is carried out by special

processing of the experimentally measured response of dynamic system (method of identification). This

problem is considered in two statements: the synthesis of EL for single model and the synthesis of EL for

models class for the purposes of mathematical modelling. These problems are ill-posed by their nature and

so the method of Tikhonov's regularization is used for its solution. For increase of exactness of problem

solution of synthesis for models class the method of choice of special mathematical models (MM) is used.

The calculation of model of external load for rolling mills is executed.

1 INTRODUCTION

At mathematical modeling of real motion of open

dynamic systems is important the correct choice of

mathematical model of external load on system. The

most accessible information about EL is contained in

reactions of object on these loads which can be

measured experimentally simply enough from the

technical point of view. The determination problem

of size and character of change of EL based on

results of experimental measuring of object

responses has been called the problem of

identification of EL (Gelfandbein and

Кolosov,1972), (Ikeda, Migamoto and Sawaragi,

1976), (Menshikov, 1983). Such approach has some

advantages: the construction of model of EL is being

carried out on basis of objective information

(experimental measuring); the results of

mathematical modeling later on are good even in

case of the great inaccuracy of MM.

At construction of mathematical model of

concrete dynamic system the different authors use

the various simplifying assumptions. The whole set

of possible equivalent mathematical models of real

object (dynamic system) are being obtain in result

(Menshikov, 1985). So it will be useful to build the

common model of EL which is the best in some

sense for class of possible mathematical models of

real object (Menshikov, 1985). The statement of

such problem can have application in mathematical

modeling, systems control, in detection of faults and

so on.

2 STATEMENT OF PROBLEM

We shall suppose for simplicity that with the aid of

known internal interactions (for example, measured

experimentally) some subsystem of initial dynamic

system can be received at which is known one

variable status and all external loads except the

external load which is being investigated. If at a

subsystem two variable statuses are known, then this

subsystem is being replaced with more simple

subsystem, at which one known variable status

executes a role of known external load (Menshikov,

1983,1985,1994).

Let us suppose that the motion of the received

open dynamic subsystem is being described by

system of the linear ordinary differential equations

with constant coefficients

ZCXBX +=

&

,

ZFXDY +=

,

where

*

21

),..,.,(

n

xxxX = ,

*

21

),..,.,(

1

n

zzzZ = ,

*

21

),..,.,(

2

n

yyyY = ((.)* is the mark of

transposition ); z

1

= z is the researched external

load; B, C, D, F are matrixes with constant

coefficients appropriated dimensions, moreover D is

376

Menshikov Y. (2007).

IDENTIFICATION OF MODELS OF EXTERNAL LOADS.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 376-379

DOI: 10.5220/0001649303760379

Copyright

c

SciTePress

diagonal matrix containing only one not zero

element, F is diagonal matrix containing only first

zero element,

X

is the vector - function of status

variables,

Y

is the vector - function of observed

variables.

A problem of determination of scalar model z(t)

of EL in many cases can be reduced to the solution

of the linear integral equation Volltera of the first

kind (

Menshikov, 1983,1985,1994)

∫

δ

=τττ−

t

tudztK

0

)()()( ,

or

UuZzuzA

p

∈

∈

=

δδ

,, ; (1)

where Z,U are B- functional spaces,

p

A

: Z → U.

The function u

δ

is obtained from experiment with

a known error

δ

:

δ≤−

δ

U

T

uu ,

where u

T

is an exact response of object on real

external load.

We denote by

p

Q

,δ

the set of functions which

satisfy the equation (1) with the exactness of

experimental measurements with a fixed operator

p

A :

},:{

,

δ≤−∈=

δδ

U

pp

uzAZzzQ .

The set of

p

Q

,δ

is unbounded set in norm of space U

as

p

A is a compact operator (Тikhonov, Аrsenin ,

1979). Any function from

p

Q

,δ

is the good

mathematical model of external load. However not

all of them are convenient for further use in

mathematical modeling. Let the value some

continuous non-negative functional Ω[z], defined on

Z

1

(Z

1

is everywhere dense set in Z) characterizes a

degree of use convenience of functions from the set

p

Q

,δ

.

Let the function

p

z

∈

p

Q

,δ

satisfies the

condition:

][inf][

1,

zz

ZQz

p

p

Ω=Ω

∩∈

δ

. (2)

Function

p

z we shall name as the solution of

synthesis problem of EL model.

Furthermore there are no reasons to believe that

the function

p

z

will be close to real external load. It

is only convenient model of external load to use for

mathematical modeling later.

Let the operator

p

A depends on vector-

parameters of mathematical model

m

m

Rppppp ∈= ,),...,,(

*

21

. It is supposed that the

parameters of mathematical model are determined

inexact with some error

mippp

iii

,...,3,2,1,

ˆ

0

=≤≤ . Therefore, the vector-

parameters p can accept values in some closed

domain

m

RDp ⊂∈ . The operator

p

A in (1) will

correspond to everyone of vector-parameter

Dp

∈

and they form some class of operators

}{

pA

AK = .

Let's designate through h size of the maximal

deviation of the operators

p

A from

A

K .

Let's consider the extreme problem (2) of model

synthesis of EL

h

Qz

,

~

δ

∈

for class of models К

A

[2,3,4]. The set of the possible solutions for all

p

A

has the following form in this case:

},,:{

,

Z

U

pAph

zuzAKAZzzQ +δ≤−∈∈=

δδ

.(3)

Any function from

h

Q

,δ

brings about the

response of mathematical model, which coincides

with the response of real object with an error, which

takes into account an error of experimental

measurements and error of a possible deviation of

parameters of a vector

Dp ∈ . A problem of

finding of mathematical model

h

Qz

,

~

δ

∈

of external

load with is convenient for use later was called by

analogy to the previous problem by

a problem of

models synthesis for a class of models

(Menshikov,

1985).

The set of the solutions of inverse problem of

synthesis with fixed operator

p

A

from К

A

contains

elements with unlimited norm (incorrect problem),

therefore size

Z

z+δ can be indefinitely large.

Formally such situation is unacceptable, as it means,

that the error of mathematical modeling is equal to

infinity, if as models to use any function from

h

Q

,δ

.

Hence not all functions from

h

Q

,δ

will be "good"

models of EL.

Further we shall believe, that the size

U

u

δ

exceeds an error of experimental measurements

δ

,

i.e.

δ

<

U

u

δ

. Otherwise the zero element of space

Z belongs to set

h

Q

,δ

with any operator

Ap

KA

∈

,

for which

00

=

p

A

. This case does not represent

practical interest, as the response

u

δ

can be received

with trivial model of EL.

Let's consider the union of sets of the possible

solutions

p

Q

,δ

:

p

Dp

QQ

,

ˆ

δ

∈

∪=

, (∪ – mark of union).

As

the solution of a problem of synthesis for the

class of models

z

midl

we shall accept the element

from

Q

ˆ

(instead of set

h

Q

,δ

)

midl

z ∈ Q

ˆ

which

satisfies the condition:

][inf][

1

ˆ

zz

ZQz

midl

Ω

=

Ω

∩∈

.

For increase of exactness of problem solution of

synthesis for class of models the method of choice of

special MM is used (Menshikov, 1997). For the

realization of such approach it is necessary to choose

IDENTIFICATION OF MODELS OF EXTERNAL LOADS

377

within the vectors Dp ∈ some vector Dp

∈

0

such

that

][][

11

0

xAxA

p

p

−−

Ω≤Ω

for all possible

Xx ∈ and all

Dp ∈

. The operator

0

p

A

with parameter Dp

∈

0

will be called the

special minimal operator.

3 THE UNIFIED

MATHEMATICAL MODEL OF

EXTERNAL LOAD

Let's consider the problem of construction

(synthesis) of EL model

1

Zz

un

∈ which provides the

best results of mathematical modeling uniformly for

all operators

Ap

KA ∈ (Menshikov and Nakonechny,

2005):

2

2

supinf

U

pb

KA

z

U

un

p

uzAuzA

Ab

p

δ

∈

δ

−≤−

for all

Ap

KA ∈ . (4)

Let us name function

un

z as the unified

mathematical model of external load for class

К

А

.

Theorem. The function

un

z

exist and steady to

small variations of initial data if

Ω[z] is stabilizing

functional.

4 IDENTIFICATION OF

EXTERNAL RESISTANCE ON

ROLLING MILLS

One of the important characteristics of rolling

process is the moment of technological resistance

(МТR) arising at the result of plastic deformation of

metal in the center of deformation. Size and

character of change of this moment define loadings

on the main mechanical line of the rolling mill.

However complexity of processes in the center of

deformation do not allow to construct authentic

mathematical model of МТR by usual methods. In

most cases at research of dynamics of the main

mechanical lines of rolling mills МТR is being

created on basis of hypothesis and it is being

imitated as piecewise smooth linear function of time

or corner of turn of the working barrels (Menshikov,

1983,1985,1994). The results of mathematical

modeling of dynamics of the main mechanical lines

of rolling mills with such model МTR are different

among themselves (Menshikov,1994).

In work the problem of construction of models

of technological resistance on the rolling mill is

considered on the basis of experimental

measurements of the responses of the main

mechanical system of the rolling mill under real EL

(Menshikov,1983,1985,1994). Such approach allows

to carry out in a consequence mathematical

modeling of dynamics of the main mechanical lines

of rolling mills with a high degree of reliability and

on this basis to develop optimum technological

modes. The four-mass model with weightless elastic

connections is chosen as MM of dynamic system of

the main mechanical line of the rolling mill

(Menshikov, 1983,1985,1994):

;

; (5)

;

where

112

)(

−−

ϑϑϑ+ϑ=ω

k

ikiik

ik

c ,

ϑ

k

are the

moments of inertia of the concentrated weights, c

ik

are the rigidity of the appropriate elastic connection,

U

rol

M ,

L

rol

M are the moments of technological

resistance put to the upper and lower worker barrel

accordingly, M

eng

(t) is the moment of the engine.

The problem of synthesis of MM of EI can be

formulated so: it is necessary to define such external

models of technological resistance on the part of

metal which would cause in elastic connections of

model of fluctuations identical experimental (in

points of measurements) taking into account of an

error of measurements and error of MM of the main

mechanical line of rolling mill. Such type of

problems and the methods of their solutions can find

applications at construction of MM of EI in other

similar situations.

The information on the real motion of the main

mechanical line of rolling mill is received by an

experimental way (Menshikov, 1983,1985,1994).

Such information is being understood as presence of

functions M

12

(t), M

23

(t), M

24

(t). Let's consider a

problem of construction of models of EL to the

upper working barrel. On the lower working barrel

all calculations will be carried out similarly. From

system (5) the equation concerning required model

U

rol

M

can be received.

)(

1

12

24

2

12

23

2

12

12

2

1212

tM

c

M

c

M

c

MM

eng

ϑϑϑ

ω

=−−+

&&

)(

3

23

24

2

23

12

2

23

23

2

2323

tM

c

M

c

M

c

MM

U

rol

ϑϑϑ

ω

=+−+

&&

)(

4

24

23

4

24

12

2

24

24

2

2424

tM

c

M

c

M

c

MM

L

rol

ϑϑϑ

ω

=+−+

&&

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

378

∫

δ

=τττ−ω

t

rol

tudMt

0

)()()(sin .

The size of the maximal deviation of the

operators

AT

KA ∈ was defined by numerical

methods and it equal h = 0.12. An error initial data

for a case Z = U = C [0,T] is equal δ = 0.066 Мнм.

In figure 1 the diagrams of functions

un

midl

zz ,

for a typical case of rolling on the smooth working

barrel are submitted as solution of last equation

(Menshikov, 1983,1985,1994).

Figure 1: The diagrams of change of models of the

moment of technological resistance on the rolling mill.

The results of calculations are showing that the

rating from above of accuracy of mathematical

modeling with model

un

z for all

AT

KA ∈ does not

exceed 11 % in the uniform metrics with error of

MM parameters of the main mechanical line of

rolling mill in average 10 % and errors of

experimental measurements 7 % in the uniform

metrics.

The calculations of model of EL

z

~

for a class of

models К

A

on set of the possible solutions

h

Q

,δ

was

executed for comparison. This function has the

maximal deviation from zero as 0.01 Мнм.

In work [4] the comparative analysis of

mathematical modeling with various known models

of EL was executed. The model of load

un

z turn out

to be correspond to experimental observations in the

greater degree [4].

4 CONCLUSIONS

The offered approach to synthesis of mathematical

models of external loads on dynamical system can

find application in cases when the information about

external loads is absent or poor and also for check of

hypotheses on the basis of which were constructed

the known models of external loads.

REFERENCES

Gelfandbein Ju.K., Кolosov L.V., 1972. Retrospective

identification of impacts and hindrances, Мoscow,

Science.

Ikeda S.I., Migamoto S., Sawaragi Y., 1976.

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parameters of systems. IY a Symposium IFAC, Tbilisi,

USSR, 1976, vol.3, Preprint. Мoscow.

Menshikov Yu.L., 1983. About a problem of identification

of external loads on dynamical objects as a problem of

synthesis. In J. Dynamics and strength of heavy

machines, Dnepropetrovsk University, vol.7,

Dnepropetrovsk, Ukraine.

Menshikov Yu.L., 1994. The Models of External Actions

for Mathematical Simulation. In: System Analysis and

Mathematical Simulation (SAMS), New-York, vol.14,

n.2.

Menshikov Yu.L., 1985. The synthesis of external load for

the class of models of mechanical objects. In J. of

Differential equations and their applications in

Physics. Dnepropetrovsk University, Dniepropetrovsk,

Ukraine.

Тikhonov A.N., Аrsenin V.J., 1979. Methods of solution of

ill-posed problems, Мoscow, Science.

Menshikov Yu.L., 1997. The Reduction of initial Date

Inaccuracy in ill-posed problems. In Proc. of 15th

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Menshikov Yu.L., Nakonechny A.G.: Constraction of the

Model of an External Action on Controlled Objects. In

Journal of Automation and Information Sciences ,

vol.37, is. 7, (2005) 20-29.

0

0,1

0,2

0,3

0,4

0,5

0,6

0246810

time [0.05s]

The models of technological resistance moment [

MHM

]

Zun

Zmidl

IDENTIFICATION OF MODELS OF EXTERNAL LOADS

379