Studies on Test System Graying and Its Dynamic Feature Identification

2016 International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2016) ISBN: 978-1-60595-385-4

Studies on Test System Graying and Its Dynamic Feature Identification

Yu-jie ZHANG

and Ling HE

1

Qingshuihe Campus, University of Electronic Science and Technology of China, No. 2006, Xiyuan Ave, West Hi-Tech Zone, 611731, Chengdu, Sichuan, P. R. China

2

Qingshuihe Campus, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi-Tech Zone, 611731, Chengdu, Sichuan, P. R. China

*Corresponding author

Keywords: System graying, Dynamic feature identification, Systematic error, Pseudo-dynamic.

Abstract. At present, some approaches from which the “black box” system transforms to “white” system or “gray” system have gradually been in hot research. Provided that a measurement system is time-varying, dynamic, random and self-correlation, it should be considered as a dynamic system; otherwise, it should be static or “pseudo-dynamic”, we can thus apply static measurement uncertainty theory to process and analyze. In this paper, some theories about system graying are introduced firstly, and dynamic features of measurement are described in detail to fulfill the system identification. Then, the application on an universal hybrid system is shown that based on the modeling analysis mentioned above, the whole processing can be testified its validity.

Introduction

Dynamic measurement is the main trend of development in measurement field. Dynamic measurement have some characteristics, as temporality and spatiality, randomness, dynamism, and self correlation[1]. According to the modern development of dynamic measurement system. Some classic static measurement theories, for example, linearity, stability, and reproducibility, are generally deployed to make sure that the measurement setups are within an acceptable condition to meet manufacturing capabilities in measurement system; however, there are some limitations when they act on the dynamic measurement system due to dynamic measurement’s time-varying characteristic and so on. In broad sense, any static (or pseudo-dynamic) measurement can be considered as special case of dynamic measurement[2]. Whereas, only if some measurement processes are essentially the combination of several static measurement processes, the static measurement methods should be applied instead of dynamic methods. It’s necessary to identify the system feature through an appropriate method to analyze the system as accurately as possible.

This paper focuses on “system graying”. Taylor series expansion is derived and utilized to describe the system; on this basis, systematic feature’s analysis are consequently presented.

Modeling of System Graying

The principle of system graying have proposed in case that the factors are difficult to identify, or the internal relationships are complicate. To study thoroughly on the system, system graying is widely applied.

System Graying

Generally, a measurement system can be treated as a “black box” with input and output. In this “black box”, the transfer function can be defined as F q( i)(i=1, 2 ,, )n , which contains some variables

1, 2, 3 n

q q q q

. Then, the characters will be analyzed base on this model.

The quadratic function in n variables can be represented in this form[4]~[7].

2

0 0 0 0

1 1 1

1

( ) ( ) ( ) ( ) ( ) ( ) ( )

2

n n n

i i i i j j

i i j i i j

F F

F q F q q q q q q q

q q q

= = =

∂ ∂

= + ⋅ − + ⋅ ⋅ − ⋅ −

∂ ∂ ∂

∑ ∑∑

(1) where q _and q0 represent the “black” system’s variable vector and value in a constant state

respectively.

Meanwhile, when the system is fed some external disturbances, it will diverge from the constant state. A truly constant systematic error can be observed. Given y=F q( )_{, we can obtain the systematic} error[4][5][7]

2

1 1 1

1

( ) ( ) ( ) ( ) ( )

2

n n n

i i j

i i j i i j

F F

y q q q

q q q

= = =

∂ ∂

∆ = ⋅ ∆ + ⋅ ⋅ ∆ ⋅ ∆

∂ ∂ ∂

∑ ∑ ∑

(2) Its mathematical expectation according to error propagation theory is[2]

0 0 0 0

1

( ) ( ) [ {( ) ( )}]

2

T

i i i i

E Y =F q + E tr q−q H q−q

(3) The term {( 0)( 0) }

T

E q−q q−q _{is the covariancematrix of the variables vector}(q q− 0) and is also denoted

as ∑q, i.e., {( 0)( 0) } T q=E q−q q−q

∑ _.

So E Y( ), mathematical expectation of Y, can be derived as[4]

0 1

( ) ( ) { ( ) }

2 q

E Y =F q + tr H q ⋅ Σ

(4) By determining 2

Y _{with squaring Eq.(1), we have}

2 2

0 0

1

( ) ( ( )) ( ) { ( ) } { } { ( ) ( ) } 4

T

q q q q

E Y =F q +F q ⋅tr H t⋅Σ +E gΣ ⋅ +g tr H t⋅Σ ⋅H t⋅Σ

(5) With Eq.(4) and (5), there gives the second order variance of the “block-box”[4],

2 2 2 1

( ) [ ( )] { ( ) ( ) }

2 T

F E Y E Y g q g tr H t q H t q

σ = − = Σ ⋅ + ⋅Σ ⋅ ⋅Σ

(6) The systematic error can be expressed by the statistical estimate of the system. When a system are combination of a few static modules, the variance equation may be expressed as follows[6] [8] [9]:

2 2 2

( )

i n

F x

i i

F x

σ = ∂ ⋅σ

∂

∑

(7)

Dynamic Feature Identification

Differing with the static features, dynamic features, for example, dynamism, time-varying, stability, randomness and relevance, have some important applications in dynamic measurement system. Considering the available limited accuracy, an instrument with measurement error should be shown as Figure1[1].

The multi-input signal {x0}, passing through the “black box” system, correspondingly get the

multi-outputs {y0}={y0(1),y0(2),,y k0( )}_{. Assume that the ideal output are} y t₀( )=F q x t[ ,_{i 0}( )]_,(i=1,2, ) _. Provided with the random error of the input/output of system’s observationnx _and ny_{, the actual} output is y t( )_{, where} nx~ (0,σ12)and

2 2

~ (0, )

y

n σ _{In view of the randomness performance of the system, they}

can both be treated as white noises. Therefore, they are mutually independent and follow normal distribution[3]:

2 1

ˆ( ) ( ) ( ) ~ (0, ) ( )

i x

i

x k x k F q n

F q σ

− =

(8) From Figure1 the output can be written as x kˆ( )=y k( )−ny, so, the Gauss formula’ expectation can be

expressed as

ˆ( ) ( ) ( )

( ) 0

( )

i i

x k x k F q

E

F q −

= .

Denoting the mean value of the input/output are E x k( ( )) _and E y k( ( ))_{, there exists}

0

( ( )) ( ( ) ) ( ( , )) ( ( ))

y i

E y k E y k n

E F q k E x k

= −

= ⋅

(9)

So, the feature of this system can be inferred.

0 ( ( , ))i ( ( )) / ( ( ))

E F q k =E y k E x k ₍₁₀₎

At the same time, since 2 2

1 [( ( ) / ( , )i ( )) ]

E y t F q k −x t =σ _{, the correlation between the output and the system have}

been omitted, the second order variance of the system is expressed as

2 2 2 2 2

2 1 ( ( , ))i y / x

E F q k =σ +σ σ −σ ₍₁₁₎

Systematic feature can be described as a multidimensional stochastic process with a given mean and variance, thus, the system features are randomness, dynamism and time-varying simultaneously appeared in the systematic error and the output data.

The actual output data y t( )_{is expected to equal to} y t0( ), so the error can be written as, e ty( )=y t0( )−y t( )_. ( )

y

e t _{consists of several time variable. Error is usually divided into two parts: the systematic}

component, and the random component [3] [5].

( ) ( )

y t

e t =δ + n t

(12) Where δ _{indicates systematic error; ( )n t (}n ~ (0,σ2)_{) indicates the random item. Suppose there}

are a series of measurement errors ey(1), (2),ey

, namely, e e e1, , ,2 3_{, test statistic is constructed as,}

2 1 2 2 1 ( ) n i i i n i i e e p e − = = − =

∑

∑

(n→∞).The formula can be rewritten as

1 2

2

1

ˆ 2(1 ) 2(1 )

n i i i n i i e e p e ρ − = = ≈ − ≈ − ∑ ∑

,where,

1 2 2 1 ˆ n ii i n i i e e e ρ − = = = ∑ ∑ .

In consideration of Eq.(12), it follows that

2 1

2 1

( ) ( ) 0

n n

i i i

i i

E e e− E e

= =

− ≈

∑

∑

. Obviously, when, ρˆ ≈1 _{, it is} self-correlation in measurement errors, which means the associated system might be dynamic; otherwise, if the system is not self-correlation, it must be a non-dynamic system.

The Criterion of Non-dynamic System

As all known, the test system always involves randomness, dynamism, time-varying due to the inevitable, undesirable measurement disturbances, and should be described as the stochastic process. These features are difficult to distinguish sometimes. As an essential character of dynamic system, however, self- correlation is related with the dynamic feature closely.

Hessian matrix H t( )_{are zero and the covariance matrix} Σf _{is diagonal, which means} Cov x x( ,i j)=0for

i≠ j_{, there must no any correlation ship between the system variables， hence the system must be}

non-dynamic, i.e., it is static or just “pseudo-dynamic”.

Application on the Hybrid System

Modeling Universality Discussion

All measurements should be dynamic, because the change of the measurement feature always caused by the external factors or the internal factors, so there are with the change of the feature，the significant changes of the systematical error that affect the system feature in return; consequently the dynamic feature are presented. However, Static measurements are kinds of measurement under the invariant conditions or that the interference factors are weak. Those system may be the part of the dynamic measurement without including the measurement error, some of the dynamic features are omitted, thereby those static measurement can take as a special case of dynamic measurement, also called “pseudo-dynamic”.

An Universal Hybrid Modeling and Its Analysis

A hybrid representative system be constructed due to the circuit structure are universal, basing on the modeling principle of the whole-system dynamic accuracy theory. So, it’s representative that the graying system principle is applied in order to analyze the hybrid systematic feature[1]. Here follows the Figure2, obviously there are four units.

Figure 2. The hybrid system[1].

From the Figure 2 we can get the formula x t H t( )⋅ ( )=y t( ). Simplify it into y k( )=H x k( ( )).x k( )are the inputs of the system,x(1), (2), (3),x x , ( )x k .y k( ) are the outputs, y(1), (2), (3),y y , ( )y k .

According to the theory of graying-system, the model is described, 4 4 4 2

0 0 0

1 1 1

1

' ( )( ) ( )( )( )

2

s s i i i i j j

i i j i i j

F F

F F f f f f f f

f f f

= = =

∂ ∂

= + − + − −

∂ ∂ ∂

∑ ∑ ∑

x

n ,n_y the random error of input, output, under Gaussian noise with limited energy.

In Eq.(6), where the second order differential vector ofthe system to its variables is

3 2

3 1

2 1

0 1

0 0

( )

0 0 1 0 0 0

f f

f f

H f

f f

     

=_{ }

   _,

the first order differential vector of the system to its variables is T ( _{2 3 4}, _{1 3}, _{1 2}, ₁)

g = f f + f f f f f f , the

covariance matrix of the variables vector is

1 1 2 1 3 1 4

2 1 2 2 3 2 4

3 1 3 2 3 3 4

4 1 4 2 4 3 4

( ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( , ) ( ) f

Var f Cov f f Cov f f Cov f f

Cov f f Var f Cov f f Cov f f

Cov f f Cov f f Var f Cov f f

Cov f f Cov f f Cov f f Var f

 

 

 

Σ =_{ }

 

 

Suppose that the variables f_i with small disturbance are ∆f_i (i=1, 2, 3, 4). So, the variance of f_i is

2 2

1

1 N

i i

n

f N

σ

=

=

∑

1

1 1

i j

N

f f i j

n

C ov f f

N =

= ∆ ∆

−

∑

Substituted these equations into the Eq.(6), then there is clear to get the variance 2

( )

F t

σ .

However, when the covariance equals zero,

2 1

{ ( ) ( ) }

2

F tr H t q H t q

σ = ⋅Σ ⋅ ⋅Σ

for the Hessian matrix

( )

H t

are

zero and the covariance matrix Σ_q is diagonal, namely, 2 4 2 2 1

( )

F i

i i

F f

σ σ

= ∂

= ⋅

∂

∑

system be treated as the combination of some static systems. After getting 2

F

σ , analyze the measurement errors involving errors which caused by the interference of the constituent units within the system, thereby dynamic measurement systematic errors are also presented.

Summary

The paper focuses on converting “black box” system into “graying system” using Taylor series expansion to express the dynamic feature. Some researches are made on the dynamic characteristics of the measurement system to accomplish the system identification. On the condition that lack of any one dynamic feature, the system should be static or “pseudo-dynamic”. By applying this idea on an universal hybrid system which contains several interferences, one of the key dynamic features, self-correlation feature, can be ultimately obtain. As a Consequence, the follow-up identification studies are relatively simple and convenient.

References

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