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System Structure Identification
Copyright © IFAC Identification and System Parameter E.stimation 1982 , Washington D.C ., USA 1982
SYSTEM STRUCTURE IDENTIFICATION V. Kaminskas and A. Rimidis Institute of Physical and Technical Problems of Energetics, Lithuanw.n A cademy of Sciences, Kaunas, USSR
Abstract. We present here a non-parametric approach to the choice of model structure by estimating the class of the system under identification. We define system class as a logical combination of fundamental system features. Lumped or distributed systems, continuity or discretion of its processes, linearity or non-linearity, static or dynamic behaviour, stationarity or non-stationarity are the basic elements of the system class. We also suggest a sequential solution technique in which each step gives a check to a single property. For certain classes of inputs we give decision rules to detect the above features from input-output measurements. The results were applied to hydraulic behaviour of transportation pipelines. Keywords. Identification; system class; model structure; decision rule; linearity-non-linearity; stationarity-non-stationarity; statics; dynamics.
INTRODUCTION A priori information on the physical nature and relations of variable processes provides approximate postulation of numerous structures of a mathematical operator. Problems of identification deal only occasionaly with the classical "black-box" objects. On the other hand, the only reliable conclusions from a priori information usually refer to the lumped or distributed systems and to the continuous or discrete character of the processes. The absence 0 reliable data on such properties as non-linearity, non-stationarity, dynamic behaviour calls for special techniques of their detection from observation data of system variables.
157
SYS;lh'M CLASS ESTIMATION
Let l4't. be a certain system property, and QJi the opposite property. Then element ~. of set
9
I
=
-
~: (u.: v<0,)t1 ••• N(VmV~m),(i=/'f)}( 1)
will describe a logical combination of system properties, wich we call system class (Kaminskas, 1979). From a priori information and from observation, compile decision rules to classify the system by one of the given classe s ~. (i =1, 'I)' Random errors are a~ost inevitable in observations of signals, and the signals themselves are just realiza-
V. Kaminskas and A. Rimidis
158
tions of random processes, therefore decision rules of zero error rates are actually impossible. A chosen element Q. of set 9 will most probably differ d from the actual class-of the system. Thus our problem is reduced to the estimation of the class of an unknown system, which we call further model class. The notions of model class and model structure differ in that model class denotes just the major properties either of a system or of a model, and one model class may use different mathematical operators. By evaluting model class we restrict the initial set of operators - there exist classes of just one or two operators. The final structure estimation of a mathematical operator may be found in several successive steps, by separating new subclasses from a current class 9. t-:-j) in each step. 1
(i =
DISCRIlHNA"TT ANALYSIS
Our problem of system class evaluation may be actually solved by the techniques of discriminant a:lal:;rsis (Rao, 1965). They employ the so called discriminant functions q.(t1) of property vector 1J' and a non-randomized decision rule
!(tJ): Oifm':l1
cj (tJ);i ~ ~ ,(j=/,f J. (2)
I
Property vector l' is a function of observation signals, and the random nature of the observations leads to discriminant function of the type
IJ
_
{';(tj).-'8~.9(.p(f},y?'J, (t'
/:1/ ~.
!I
I
=/, fj)'
(J)
- a priori probability of class - error losses due to taking class ~' as class ~. ; f(""~.) relative probability denSity of property t!J' for class ~. • For different a priori information on the values of
~.;~.
:;;i'
P.." I' (~I~' ) we arrive at tayes, minimax, maximum a posterior probability, or maximum likelihood methods. This approach is not simple in its implementation, as relative distribution p(t1I~)iS not only determined by internal features of certain class systems, but also by external factors - input signals. No reliable classification can be expected, if the input signal is not the same, as used in teaching the classifier. Discriminant analysis is only reasonable for systems corrupted by tests noises or special dusturbances (Saridis, Hofstadter, 1974; Kittler, Witchead, 1976). 3EQUElJ'l.'IAL .JECISION
Structure analysis of eleme .lts ~" (i= 1, '1) suggests a POSSibilityJof introducing separate steps into the classification process, so than on each i-th step only a single property "'t' is tested by one or several hypotheses. If ~ - stands for lumped systems, 6)2- ~ontinui ty, aJ~ - stationarity, (i)4 - linearity, VJ5 - statics, then we have a sequential procedure as shown in Fig . 1. The branches end in specific elements -classes 9., (j = J. A priori knowledge of sfstem properties helps in choosing the necessary parts on the branches. Separation of informative features from observations is simplified by the fact that each of the steps has two solutions. The sets of properties can be varied by decisions taken in the preceding steps.
1'1
System Structure Identification
teria according to the decision rule
SEARCH OF MODEL STRUCTURE A definite structure of the model is chosen as a parametric operator from condition
F:
or
R(P) = min ~(F)
,
(4)
FE~
~
I : R(F)~RO'
Fe2.,
R (F)- mean loss function, sible mean loss.
~-
(5)
1.1(t)=r(tJ+Nft) (6 ) fir t) - useful signal, Nft)- stationary error sequence, with autocorrelation function RNN(i) 0 at T ~ ~ Then
=
('-/)r~1 r~ ~
'I'm
(7)
and
"'i=tL(~), fJ=f(~}' and sequence
11 =",(t1 )
(8)
___ fJ :' +!i' (,l ;z), ( 9 ) 11: - is a sequence of independent values with mean zero and a finite dispersion. For a stationary system in steady operation (t "" t'';) :1 (t ) (10) and by testing hypothesis
=/,
%i
r
~· (11)
- statistics of the i-th criterion,~~ - critical value at oC level
of significa~ce. In this case, the probability of decision with the highest power to an unknown alternative is increased. DETECTlll"G NON-LINEARITY
Let us have a one-dimensional lumped system with the input signal X rt) as a periodic function of period 7f • The observed output signal
I-
er"): !71;i71~·~ -1~,(")I}~O~1o.(12)
admis-
DETECTING NON-S'rATIOl'IARITY
~:~
159
t
HJP:I1(~)=ti:,cpnstfor all k (11)
non-stationarity of the system is detected (Kaminskas, Sipienyte, 1976). Zero hypothesis (11) has the advantage of a possibility to test it by nonparametric criterion, which are free from distribution (Kendall, Stuart, 1968). An analysis of numerous non-parametric criteria showed their sufficient sensitivity in detecting non-stationarity of linear systems (Kaminskas, Sipienyte, 1976). It is reasonable to use several non-parametric cri-
Let us have a one-dimensional stationary lumped system. Normalization is an intrinsic feature of linear inertial systems, so that non-linearity of the system cannot be detected by testing the homogeneity of the input and the output. Hypothesis H40 on linearity can be tested by a set (0 (t) oJ of partial hypotheses { H ; H40 ;H40 ; 40 (4) 1 {O . H40 ••• r • If H40 - norma~2/nput ii is true, then a test of H40 - on normality of output LLJ: ' will detect (0 non-linearity of the system. If H40 is not true, then we test mean value Xi is zero. If H~g is true, then by testing Hi~ - mean value of Uk~S zero, we detect non-symmetrical non-linearity.
Hfci -
(z)
(0
The case of H40 • To test H40 on normaH ty of output we applied the asymmetry-excess criteria based on independent data (Cox, Hinkley,1974) and modified asymmetry-excess criteria accounting for correlation (Gasser, 1975; Kaminskas, Rimidis, 1982). If), and)i2 are estimations of asymmetry and excess coefficients, then statistics
Ll"
~ =p,/If)(~J'j 'z=~t/YO(foJ]"
(13)
have an asymptotically normal distribution with parameters .Ji(U; tJ and Xl:
I: .,.1;
(14)
is distributed as ohi-square with two degrees of freedom. For indepen-
160
V. Kaminskas and A. Rimidis
IJ(/3,J
-or d er errors. Of criteria Si t = t, 2. j) most efficient and most robust are (19) and (20), compiled with the account of correlation of the observed data. A
dent data =
6/1,-'; ~(ft2);:r 24/1,-t (15)
For correlated data if !t1(lii ) and V(lLi) are known, the modified criteria (lJ) and (14) include (Kaminskas, Rimidis, 1982) •
n-t (7.-1 J)(fo,J:O/1,-'D R;(t-lijn)+9n-'C Ri (I-Win), J=-(,,-,)
b-(,,-I) (7.-,
(16)
=
Table 1 illustrates some results on simulation the systems
'r 0,5 z-I
I
D(foz)=24rr'~ Ri (I-lilln).
Lik
~.-(n-I)
For unknown M(£t~) and [)(LLi' we use in (lJ) and (14) the criterion of Gasser (Gasser, 1975)
The (~, 1140 on zero mean value of tlJ; we applied statistics (Kaminskas, Rimidis, 1982)
1
A
S, =i
= '-t,5z- tr o,'r-1 i.t +fi'!i=j(~)·(2l)
The numters in the columns show a percent of successful classification by criteria I. - (13)-(1 5), MC. -(13 ) ( 14 ), ( 16 ), G. - (13), (14), ( 17 ) from 100 experiments ,
TABLE 1 Successful classification percentage for systems J1ll
Nonlineari ty
I(X)=X
(18)
~ G. MC.
J)(ui }n'
For independent data it has an asymptotically normal distribution with parameters ~(O;l). For correlated data
~ ·~/(y!.z f] "i (I-I1Vn )!
(19)
is distributed asymptotically as K(D:!), and the lower bound for Se
~. ~j(11'2
f> . . ,(1-IlI/nJ)(20)
In (16 )-(20) Ri - autocorrelation function of observationsLL". From our earlier experiments (Kaminskas, Rimidis, 1982) we conclude, that orit eria DJ), (14), ( 16) and (lJ), (14), (17), which account for correlation of the observations, are sufficiently efficient and robust. Both first - and second order errors are small. Criteria (13)-(15) based on independent data shows significant first-
(.
f
12<, no
(x) =
0,5x,x~O
1" x >1
rx)= X, IXI~f -I, X<-!
I. G. MC. I. G. MC. I.
1000 96 99 68 99 68 100 99 96 100
5000 97 99 57 100 100 100 100 100 100
iJETEC TING DYNAM ICS Let us have a one-dimensional stationary lumped system. We fix level Xu' and discribe
1;
X (1) = ~,
'7'/1 (~ -'J -) ~ r", .(2 2 )
Then sequence ~ - (8 ), (22) has structure (9). For a static system, either linear or non-linear, apply :& Kx" and t,~:. f (Xo ) and to detect dynamiCS we have j ust to test hypothesis (11). The decision rule
t
expressed by (12) was based on seven
System Structure Identification
non-parametric criteria (Kaminskas,Sipienyte, 1976). Results of experimental analysis of decision rules for detecting dynamics are in (Kaminskas, Rimidis, 1983). Tables 2, 3 illustrate the percentage of successful classification in the simulation of a linear system with impulse response function h(t) e--.tCOS(A)ot, = tT,,,,/"y noise level, fIIft) - white noise, X (0normal process with correlation function Rxx (") :::: e - ",6 IT' ,significance level 0,05, each value is a percent of successful classification from 100 experiments.
r
=
TABLE 2 Successful classification percentage in case =0
r
~ 1
"'u o.
6.28 157
100
250
500
10 100
98 82 62
99 90 67
99 94 67
1 1
95 86
97 91
99 97
TABLE 3 Successful classification percentage in case"c I, (i)q =0
=
0,1 0,5 1.
100
250
500
95 92 82
99 98 99
100 100 100
DETECTING LINEARITY OF PROCESSES HT HYDRAULIC PIPELINES Hydraulic pipelines constitute important parts of different pieces of equipment and aggregates in numerous industrial applicatio n s. They serve for the transport of fluids from storage tanks to consumers. A large part of the cases of failure in pipelines
161
are caused by dynamic loads coming from vibration of pumps and fluctuations of pressure in the fluids. Safe operation may be improved by means of predicting technical dynamics of separate parts in the system. This can only be achieved in the presence of adequate models of dynamic processes for real structural parts of hydraulic systems. A transportation pipeline is a set of standard elements - direct and curvilinear pipes, valves, flow-rate controls, flange pints a.o. The most promising approach to simulation lies in the identification of standard elements, with a subsequent evaluation of properties and states in aggregates of standard representatives. In Fig. 2 we present a part of a hydraulic system. The s y stem was studied in steady - state operation, under static pressure of the fluid, from 1 to 3 MPa, from 4,5 to 10 m per s flowrate. Input-output of a standard element was read as pressure fluctuation in the working fluid. Table 4 gives the results of testing linearity in such a system. TABLE 4 Results of detecting linearity in elements of hydraulic pipelines
Measurement points
1-3 3-4 4-5
G.
MC.
+
+
+
+ +
+
The decision was made at significance level 0,05, and "4o" means decision on linearity.
V. Kaminskas and A. Rimidis
162
dynamic system. Lietuvos TSR
CONCLUSI01'IS Estimation of system class reduces a priori uncertainty on the structure of a mathematical model. The possibilities of discriminant analysis are but limited in this application, because of the difficulties in teaching classifiers on systems with random inputs. The technique can be successfully replaced by sequential decision - based rules of detecting such system properties, as non-li nearity, dynamics, nonstationarity fr om experimental observations. Statistical simulation revealed satisfactory performance of the suggested decision rules. The results were applied to hydraulic behaviour of pipeline elements. REFERENCES j) .1~., Hinkley, D.V. (1 9 74).~ retical Statistics. Chapman and hale, London. Gas s er, T. (1975). Goodlless-of-fit tests for correlated data. Biometrika, §£ill, pp. 563-570. Kaminsias, V. (1979). On the system class estimation in identification problem. Lietuvos TSR Moksly Akademijos Darbai, B serija, 1(110), pp. 117-123 (in russian). Kaminskas, V., Rimidis, A. (1982). Detecting of linearity property of stationary dynamic systems. Lietuvos TSR liloksltt Akademi.jos Darbai, B serija, )(130), p.5967 (in russian). Kaminskas, V., Rimidis, A. (19 83). Detecting a non-stationary and dynamic properties of systems. Lietuvos TSR Moksly Akademijos Darba1, B serija, 1(134), (in print). Kaminskas, V., Sipienyte, D. (1976). Non-parametric method for detecting non-stationarity of linear
Cox,
Moksly Akademijos Darbai, ~ rija, ~, pp. 153-163 (in russian) • Kendall, M.G., Stuart, A. (1968). The advanced Theory of Statis~, vol. 2. Charles Griffin, London. Kittler, J., Witchead, P.G. (1976). Determination of the model structure using pattern recognition techniques. Preprints of 4-th IFAC Symposium on Identification and System Parameter Estimation, ~, pp. 22-30. Rao, C.R. (196 5). Linear Statistical Inference and its Application. J. Weley, New York. Saridis, G. N., Hofs tadter, R.F. (197 4 ). A pattern recognition approach to the cla s sification of nonlinear Systems. IEEE Trans. on Systems, I,ian, and Cybern. vol. SMC-4(4), pp. 362-371.
r
r
t +
Hto - - - -
+H20 - - H21
-1
r+~o-
r
rHso
r ,m 31
,
--H4f
+ H40-
-r r+ l
H"
H50
+
-7
,#
H - ~H
51 r+H50--r51
r+H,o-r
f
Pig. 1. Scheme of seque ntial decisim ( Hio - hyp o t hesis on UJi ).
sv
Fig. 2. Scheme of hydraulic pipeline.
SYSTEM STRUCTURE IDENTIFICATION V. Kaminskas and A. Rimidis Institute of Physical and Technical Problems of Energetics, Lithuanw.n A cademy of Sciences, Kaunas, USSR
Abstract. We present here a non-parametric approach to the choice of model structure by estimating the class of the system under identification. We define system class as a logical combination of fundamental system features. Lumped or distributed systems, continuity or discretion of its processes, linearity or non-linearity, static or dynamic behaviour, stationarity or non-stationarity are the basic elements of the system class. We also suggest a sequential solution technique in which each step gives a check to a single property. For certain classes of inputs we give decision rules to detect the above features from input-output measurements. The results were applied to hydraulic behaviour of transportation pipelines. Keywords. Identification; system class; model structure; decision rule; linearity-non-linearity; stationarity-non-stationarity; statics; dynamics.
INTRODUCTION A priori information on the physical nature and relations of variable processes provides approximate postulation of numerous structures of a mathematical operator. Problems of identification deal only occasionaly with the classical "black-box" objects. On the other hand, the only reliable conclusions from a priori information usually refer to the lumped or distributed systems and to the continuous or discrete character of the processes. The absence 0 reliable data on such properties as non-linearity, non-stationarity, dynamic behaviour calls for special techniques of their detection from observation data of system variables.
157
SYS;lh'M CLASS ESTIMATION
Let l4't. be a certain system property, and QJi the opposite property. Then element ~. of set
9
I
=
-
~: (u.: v<0,)t1 ••• N(VmV~m),(i=/'f)}( 1)
will describe a logical combination of system properties, wich we call system class (Kaminskas, 1979). From a priori information and from observation, compile decision rules to classify the system by one of the given classe s ~. (i =1, 'I)' Random errors are a~ost inevitable in observations of signals, and the signals themselves are just realiza-
V. Kaminskas and A. Rimidis
158
tions of random processes, therefore decision rules of zero error rates are actually impossible. A chosen element Q. of set 9 will most probably differ d from the actual class-of the system. Thus our problem is reduced to the estimation of the class of an unknown system, which we call further model class. The notions of model class and model structure differ in that model class denotes just the major properties either of a system or of a model, and one model class may use different mathematical operators. By evaluting model class we restrict the initial set of operators - there exist classes of just one or two operators. The final structure estimation of a mathematical operator may be found in several successive steps, by separating new subclasses from a current class 9. t-:-j) in each step. 1
(i =
DISCRIlHNA"TT ANALYSIS
Our problem of system class evaluation may be actually solved by the techniques of discriminant a:lal:;rsis (Rao, 1965). They employ the so called discriminant functions q.(t1) of property vector 1J' and a non-randomized decision rule
!(tJ): Oifm':l1
cj (tJ);i ~ ~ ,(j=/,f J. (2)
I
Property vector l' is a function of observation signals, and the random nature of the observations leads to discriminant function of the type
IJ
_
{';(tj).-'8~.9(.p(f},y?'J, (t'
/:1/ ~.
!I
I
=/, fj)'
(J)
- a priori probability of class - error losses due to taking class ~' as class ~. ; f(""~.) relative probability denSity of property t!J' for class ~. • For different a priori information on the values of
~.;~.
:;;i'
P.." I' (~I~' ) we arrive at tayes, minimax, maximum a posterior probability, or maximum likelihood methods. This approach is not simple in its implementation, as relative distribution p(t1I~)iS not only determined by internal features of certain class systems, but also by external factors - input signals. No reliable classification can be expected, if the input signal is not the same, as used in teaching the classifier. Discriminant analysis is only reasonable for systems corrupted by tests noises or special dusturbances (Saridis, Hofstadter, 1974; Kittler, Witchead, 1976). 3EQUElJ'l.'IAL .JECISION
Structure analysis of eleme .lts ~" (i= 1, '1) suggests a POSSibilityJof introducing separate steps into the classification process, so than on each i-th step only a single property "'t' is tested by one or several hypotheses. If ~ - stands for lumped systems, 6)2- ~ontinui ty, aJ~ - stationarity, (i)4 - linearity, VJ5 - statics, then we have a sequential procedure as shown in Fig . 1. The branches end in specific elements -classes 9., (j = J. A priori knowledge of sfstem properties helps in choosing the necessary parts on the branches. Separation of informative features from observations is simplified by the fact that each of the steps has two solutions. The sets of properties can be varied by decisions taken in the preceding steps.
1'1
System Structure Identification
teria according to the decision rule
SEARCH OF MODEL STRUCTURE A definite structure of the model is chosen as a parametric operator from condition
F:
or
R(P) = min ~(F)
,
(4)
FE~
~
I : R(F)~RO'
Fe2.,
R (F)- mean loss function, sible mean loss.
~-
(5)
1.1(t)=r(tJ+Nft) (6 ) fir t) - useful signal, Nft)- stationary error sequence, with autocorrelation function RNN(i) 0 at T ~ ~ Then
=
('-/)r~1 r~ ~
'I'm
(7)
and
"'i=tL(~), fJ=f(~}' and sequence
11 =",(t1 )
(8)
___ fJ :' +!i' (,l ;z), ( 9 ) 11: - is a sequence of independent values with mean zero and a finite dispersion. For a stationary system in steady operation (t "" t'';) :1 (t ) (10) and by testing hypothesis
=/,
%i
r
~· (11)
- statistics of the i-th criterion,~~ - critical value at oC level
of significa~ce. In this case, the probability of decision with the highest power to an unknown alternative is increased. DETECTlll"G NON-LINEARITY
Let us have a one-dimensional lumped system with the input signal X rt) as a periodic function of period 7f • The observed output signal
I-
er"): !71;i71~·~ -1~,(")I}~O~1o.(12)
admis-
DETECTING NON-S'rATIOl'IARITY
~:~
159
t
HJP:I1(~)=ti:,cpnstfor all k (11)
non-stationarity of the system is detected (Kaminskas, Sipienyte, 1976). Zero hypothesis (11) has the advantage of a possibility to test it by nonparametric criterion, which are free from distribution (Kendall, Stuart, 1968). An analysis of numerous non-parametric criteria showed their sufficient sensitivity in detecting non-stationarity of linear systems (Kaminskas, Sipienyte, 1976). It is reasonable to use several non-parametric cri-
Let us have a one-dimensional stationary lumped system. Normalization is an intrinsic feature of linear inertial systems, so that non-linearity of the system cannot be detected by testing the homogeneity of the input and the output. Hypothesis H40 on linearity can be tested by a set (0 (t) oJ of partial hypotheses { H ; H40 ;H40 ; 40 (4) 1 {O . H40 ••• r • If H40 - norma~2/nput ii is true, then a test of H40 - on normality of output LLJ: ' will detect (0 non-linearity of the system. If H40 is not true, then we test mean value Xi is zero. If H~g is true, then by testing Hi~ - mean value of Uk~S zero, we detect non-symmetrical non-linearity.
Hfci -
(z)
(0
The case of H40 • To test H40 on normaH ty of output we applied the asymmetry-excess criteria based on independent data (Cox, Hinkley,1974) and modified asymmetry-excess criteria accounting for correlation (Gasser, 1975; Kaminskas, Rimidis, 1982). If), and)i2 are estimations of asymmetry and excess coefficients, then statistics
Ll"
~ =p,/If)(~J'j 'z=~t/YO(foJ]"
(13)
have an asymptotically normal distribution with parameters .Ji(U; tJ and Xl:
I: .,.1;
(14)
is distributed as ohi-square with two degrees of freedom. For indepen-
160
V. Kaminskas and A. Rimidis
IJ(/3,J
-or d er errors. Of criteria Si t = t, 2. j) most efficient and most robust are (19) and (20), compiled with the account of correlation of the observed data. A
dent data =
6/1,-'; ~(ft2);:r 24/1,-t (15)
For correlated data if !t1(lii ) and V(lLi) are known, the modified criteria (lJ) and (14) include (Kaminskas, Rimidis, 1982) •
n-t (7.-1 J)(fo,J:O/1,-'D R;(t-lijn)+9n-'C Ri (I-Win), J=-(,,-,)
b-(,,-I) (7.-,
(16)
=
Table 1 illustrates some results on simulation the systems
'r 0,5 z-I
I
D(foz)=24rr'~ Ri (I-lilln).
Lik
~.-(n-I)
For unknown M(£t~) and [)(LLi' we use in (lJ) and (14) the criterion of Gasser (Gasser, 1975)
The (~, 1140 on zero mean value of tlJ; we applied statistics (Kaminskas, Rimidis, 1982)
1
A
S, =i
= '-t,5z- tr o,'r-1 i.t +fi'!i=j(~)·(2l)
The numters in the columns show a percent of successful classification by criteria I. - (13)-(1 5), MC. -(13 ) ( 14 ), ( 16 ), G. - (13), (14), ( 17 ) from 100 experiments ,
TABLE 1 Successful classification percentage for systems J1ll
Nonlineari ty
I(X)=X
(18)
~ G. MC.
J)(ui }n'
For independent data it has an asymptotically normal distribution with parameters ~(O;l). For correlated data
~ ·~/(y!.z f] "i (I-I1Vn )!
(19)
is distributed asymptotically as K(D:!), and the lower bound for Se
~. ~j(11'2
f> . . ,(1-IlI/nJ)(20)
In (16 )-(20) Ri - autocorrelation function of observationsLL". From our earlier experiments (Kaminskas, Rimidis, 1982) we conclude, that orit eria DJ), (14), ( 16) and (lJ), (14), (17), which account for correlation of the observations, are sufficiently efficient and robust. Both first - and second order errors are small. Criteria (13)-(15) based on independent data shows significant first-
(.
f
12<, no
(x) =
0,5x,x~O
1" x >1
rx)= X, IXI~f -I, X<-!
I. G. MC. I. G. MC. I.
1000 96 99 68 99 68 100 99 96 100
5000 97 99 57 100 100 100 100 100 100
iJETEC TING DYNAM ICS Let us have a one-dimensional stationary lumped system. We fix level Xu' and discribe
1;
X (1) = ~,
'7'/1 (~ -'J -) ~ r", .(2 2 )
Then sequence ~ - (8 ), (22) has structure (9). For a static system, either linear or non-linear, apply :& Kx" and t,~:. f (Xo ) and to detect dynamiCS we have j ust to test hypothesis (11). The decision rule
t
expressed by (12) was based on seven
System Structure Identification
non-parametric criteria (Kaminskas,Sipienyte, 1976). Results of experimental analysis of decision rules for detecting dynamics are in (Kaminskas, Rimidis, 1983). Tables 2, 3 illustrate the percentage of successful classification in the simulation of a linear system with impulse response function h(t) e--.tCOS(A)ot, = tT,,,,/"y noise level, fIIft) - white noise, X (0normal process with correlation function Rxx (") :::: e - ",6 IT' ,significance level 0,05, each value is a percent of successful classification from 100 experiments.
r
=
TABLE 2 Successful classification percentage in case =0
r
~ 1
"'u o.
6.28 157
100
250
500
10 100
98 82 62
99 90 67
99 94 67
1 1
95 86
97 91
99 97
TABLE 3 Successful classification percentage in case"c I, (i)q =0
=
0,1 0,5 1.
100
250
500
95 92 82
99 98 99
100 100 100
DETECTING LINEARITY OF PROCESSES HT HYDRAULIC PIPELINES Hydraulic pipelines constitute important parts of different pieces of equipment and aggregates in numerous industrial applicatio n s. They serve for the transport of fluids from storage tanks to consumers. A large part of the cases of failure in pipelines
161
are caused by dynamic loads coming from vibration of pumps and fluctuations of pressure in the fluids. Safe operation may be improved by means of predicting technical dynamics of separate parts in the system. This can only be achieved in the presence of adequate models of dynamic processes for real structural parts of hydraulic systems. A transportation pipeline is a set of standard elements - direct and curvilinear pipes, valves, flow-rate controls, flange pints a.o. The most promising approach to simulation lies in the identification of standard elements, with a subsequent evaluation of properties and states in aggregates of standard representatives. In Fig. 2 we present a part of a hydraulic system. The s y stem was studied in steady - state operation, under static pressure of the fluid, from 1 to 3 MPa, from 4,5 to 10 m per s flowrate. Input-output of a standard element was read as pressure fluctuation in the working fluid. Table 4 gives the results of testing linearity in such a system. TABLE 4 Results of detecting linearity in elements of hydraulic pipelines
Measurement points
1-3 3-4 4-5
G.
MC.
+
+
+
+ +
+
The decision was made at significance level 0,05, and "4o" means decision on linearity.
V. Kaminskas and A. Rimidis
162
dynamic system. Lietuvos TSR
CONCLUSI01'IS Estimation of system class reduces a priori uncertainty on the structure of a mathematical model. The possibilities of discriminant analysis are but limited in this application, because of the difficulties in teaching classifiers on systems with random inputs. The technique can be successfully replaced by sequential decision - based rules of detecting such system properties, as non-li nearity, dynamics, nonstationarity fr om experimental observations. Statistical simulation revealed satisfactory performance of the suggested decision rules. The results were applied to hydraulic behaviour of pipeline elements. REFERENCES j) .1~., Hinkley, D.V. (1 9 74).~ retical Statistics. Chapman and hale, London. Gas s er, T. (1975). Goodlless-of-fit tests for correlated data. Biometrika, §£ill, pp. 563-570. Kaminsias, V. (1979). On the system class estimation in identification problem. Lietuvos TSR Moksly Akademijos Darbai, B serija, 1(110), pp. 117-123 (in russian). Kaminskas, V., Rimidis, A. (1982). Detecting of linearity property of stationary dynamic systems. Lietuvos TSR liloksltt Akademi.jos Darbai, B serija, )(130), p.5967 (in russian). Kaminskas, V., Rimidis, A. (19 83). Detecting a non-stationary and dynamic properties of systems. Lietuvos TSR Moksly Akademijos Darba1, B serija, 1(134), (in print). Kaminskas, V., Sipienyte, D. (1976). Non-parametric method for detecting non-stationarity of linear
Cox,
Moksly Akademijos Darbai, ~ rija, ~, pp. 153-163 (in russian) • Kendall, M.G., Stuart, A. (1968). The advanced Theory of Statis~, vol. 2. Charles Griffin, London. Kittler, J., Witchead, P.G. (1976). Determination of the model structure using pattern recognition techniques. Preprints of 4-th IFAC Symposium on Identification and System Parameter Estimation, ~, pp. 22-30. Rao, C.R. (196 5). Linear Statistical Inference and its Application. J. Weley, New York. Saridis, G. N., Hofs tadter, R.F. (197 4 ). A pattern recognition approach to the cla s sification of nonlinear Systems. IEEE Trans. on Systems, I,ian, and Cybern. vol. SMC-4(4), pp. 362-371.
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Fig. 2. Scheme of hydraulic pipeline.