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Detection of Unsuspected Feedback in Dynamic Systems
©
Copyrig-h t IF !\.( : IdC.'1l1itica lioll alld Systl'1Il Parall1t'l<.:r Estimatioll I ~JH:). York . LI l\. . 1!IH:-)
DETECTION OF UNSUSPECTED FEEDBACK IN DYNAMIC SYSTEMS V. Kaminskas and A. Rimidis IlIslilllll' "r l 'lI.\'si((l/ 111/11 TI'I-iwi((l/ 1'1'0"/1'111.1 0( 1:-'II' lgI'lirs. A((ItlflllY or SriPIICl'J or 1111' I ,illll/lllli'lll SSH. Md"/,, -I . KIIIIIIIIS 2136t!-1. USSR
Abstract_ We p r esent a development o f decision rules on d e t ec ting unsuspected linear feedback in dynamic systems . The technique is based on the appearance o f common roots in the transfer fun c tions of input and noise c hannel s o f open-loop mode l s for c los e d-loop sys tems . We prove t hat the p r esence of such common roots may be established by the statistical behaviour analysis of the prediction e rror, provided the identification employs models with transfer fun ctions , including common fa c tors. The r esu lts of experimental c he cki ng a r e given. Keywords. Sy stem st ructure identification; un s uspec t e d feedback ; decision rule; common polynomial roots; residual er r or analysis.
INTRODUCTION Systems subjec t to iden ti ficatio n may be ope rati ng ei the r as ope n-l oop or as closed-loop , but in numerous occasions no a prio ri information o n the pre sence of f eedback i s known . To ide ntify in such con ditions a specific line must be followed , i n which the stage of system structure estimation on a mathematical model inc ludes t he prob lem of detecting unsuspec t e d feedba c k . To our knowledge there are no publications on this pro bl e m. The o nly known technique that we are aware of (Tsang, Bacon, 1980) requires the internal signal (total linear signal) to be measured . We sugges t in this paper a technique of detect ing un s u spec ted linear feedback from external input-output meas urement s . PROBLEM STATEMENT
r iance
2
a~,
indepe ndent of sequence x ' u - output k k -1 observation sequence, T , TR - time delays, z the backward shift operator. Assume, that the c losed-loo p system is stab l e and the noise channel transfer function is of minimum-phase and stable. The numerator and denominator polynomials of each transfer functions (2) through (4) with parameters a*, B* , h*, which provide actual performance conditions of system (1), have no common factors. Let there be no a priori information on the feedback in this system . The problem then consist s of developing algorithms to detect feedback from input-output o bservation sequenc ies.We consider syste ms including two classes o f input signals (Kamin skas , 1982). The first class contains sequencies of continuous spe c tral densities Sxx(W) that is set
Q ={w:S x
xx
( W»O)
(5)
has a non - zero measure. The second c lass of sequencies have spectral densities S
(w) =
xx
nw x
L
i=l
vX (6(w- w.) +6(w+w. )), ~
~
(6)
l.
where v~ - intensity o f the i -th harmonics , 1
the delta - function. PROPERTIES OF THE LOSS FUNCTION The r e exist ing no a p riori information o n the feedbaCk, we identify system (1) as open-loop -1 -1 (7) u k=W(z ;a)xk+H(Z ;h)~k'
Let u s consider the properties of argument c=argmi n Qs(C), cs)l
cT=(aT,hT)
,
(8 )
c
s
- (c)=M { - 1 Q
s
s-p
L
(9)
k=P+l
where
are the transfer function s of t h e ope n-loop , of the feedback a n d of the noise c hannel, where a , Band h - the parameter ve c t ors . Their coor di nates a re the coefficients of the rel e vant polynomials. x - obsek rved externa l input seque nce , ~ k - an inde pendent
(1 0)
- prediction e r ror sequence , s - number of obser vations, p - n umbe r of preliminary observa tio n s fo r in i ti al condition s , Qc - admissible domain desccribed by
ra ndom sequence with the zero mean and a finite va-
In:1
V. Kaminskas and A. Rimidis
1724 >lc ={c: I zi (a) I
I Zj (p) I
(i=l,n ); a
(9.=l,n ); r
(j=l,n ); p
(t=O,n )} , b
I bt l <
(11)
wIz
-1
;Cl.)Z
-1.
=W(z
-1 -1 -1;Cl)H (z ;h).
(2 0)
This particular case is indicated by equal denominators of transfer functions W(z-l;ii), H(z-l;h) or by zero multipliers in the denominator H(z-l;h).
Zi (a), Zj(P), z9.(r) - roots of equations Zn a A(z-l)=O, zn p p(z-l)=O, zn r If polynomials B.(z-l) and p.(z-l) have no common
For an autoregression output noise that is 1 H(z-1;h.)=R: (Z-1), the numerator of the transfer function coincides with the denominator of the actual transfer function of the forward part. Therefore,
P
W(z -1 ; a.) z -1. =W (z -1 ; a) -1 (z -1) , H(z-l ;h.)=R- 1 (z-l).
(21 )
The last is the case of a completely known transfer then the minimum value
a~
is achieved at arbitrary ii Rimidis,
of the loss function (9) €
>la and
h€
>lE (Kaminskas,
1983). Domains >l- and >l_ are defined as Cl h
>l_={a:9(z-1)=~(z-1)D.(z-1)B.(z-1)z-'. , Cl -1 -1 [ -1 -1 A(z )=~(z ) A.(z )D.(Z )-1
-1
)G.(z
-B.(Z
)z
-('·+'R.)l J}
function of the noise H(z-l;h.), and then W(z -1 ;a.) z -1 ·=W(z -1 la) H- 1 (z -1 ;h) H (z -1; h.) .
We now come to the development of algorithms of detecting common roots in the corresponding polynomials, which is equivalent to a decision rule of detecting unsuspected feedback. DECISION RULE
(14)
In the identification of real objects parameter estimates are usually found by minimizing Qs(c), i.e. c=arg
-1 -1 -1 [ -1 -1 R(z )=o/(z )R.(z ) A.(z )D.(z )-
-B.(z
-1
)G,.(z
-1
(23 )
min cen
c
+, )]
-(1
(22)
• R. )z},
(15)
where ~(z-l), o/(z-l) - arbitrary polynomials. If the following conditions are satisfied ( 16) (17)
Therefore estimates c in the final sample may differ from terms in domains (14), (15). A direct
c
search of common roots in polynomials
~(z-l)
and
p(z-l) gives then no result. But it follows from (14) and (15) that for the identification of a closed-loop system (1) as openloop, transfer functions w(z-l;a) and H(z-l;h) may be used with polynomials B(z-l)=B' (z-l)B"(z-l), p(z-l)=p' (z-1)B II (z-1),
n * =max{n * +n * a
1
d
,
n * +n * +1.+1 b
g
R
(18)
.} ,
R(z
then domains (14), (15) contain one or more pOints. Here· indicate actual parameter values, polynomials of actual parameter values and actual orders of the relevant polynomials. All this implies the following property describing a feedback system: polynomials B(z-l) and p(z-l) have common roots, if a closed-loop system is identified as open-loop. Therefore the technique of detecting internal feedback reduces to the detection of common roots in the relevant polynomials. After reducing the common roots in polynomials 9(z-1),
A( z -1)
and
P(z - 1), R(z -1); A(z -1) ,R (z -1)
and
B(z -1 ) ,
-1
,-1 -1 )=R (z )A(z ),
where
n' ,-1 b I-i B(z )= l: biz , i=O
(24) nil
11
B (z
-1
)=1+ l:
b,,_i b z i
(25)
i=l
Then sequence A(z) ,-1 -1 - ..----1- u9.- B (z )x9. [ B (z )
J
(26)
becomes a sequence of independent variables ~9. for all
Ci ~ Q= a
and
h€
r2_ where domains I
h
p(z-l) we arrive at transfer function w(z-1;a)=B.(z-1)z-1.
H(z
}
-1 -1 -1 _ A.(z )P.(z ) ;h)= - - - - ; - - - (z -1)
( 19)
(27)
R.
where A.(z-l), R.(z-l) are polynomials, resulting from A.(z-l) and R.(z-l) after the common roots were reduced. But the zero multipliers of the denominators of the transfer function w(z-l;a) is a proof of the presence of a linear feedback, too. It follows from (19), that reconstruction of the actual transfer function W(z-l;Cl.) is possible in specific cases even if polynomials A.(z-l) and R.(z-l) have no common roots. For H(z-l;h.)
= I,
(28) We note that the values of
a .. n=
and
h ,,>l_
a
also
h
minimize the criterion (9), which is built by using errors (26). For an open-loop system (no feedback) sequence (26) with parameters ~ which minimize (9) a moving average process
becomes
1725
Unsuspected Feedback in Dynamic Systems '" '" -1 Ek(C)=V(z )~k'
a'(z-I)=1-0,25z- l , S"(z-I)=1- 0 ,5z-
(29)
where ~(z-l) ~ 1 a polynomial (Kaminskas, Rimidis, 1983). For example, parameters ~ may be given as values, which satisfy equations
1
}
=. -1 -I = -I -I -2 P (z )=I-O,9z ,A(z )=I-O,4z +0,2z .
(38)
The identification was based on s=1000 observations. The correlation of the residual error seguence was 2
tested by a chisquare test XES(BOX, Jenkins, 1970)
with a significance level a=O,OS (critical value
X~ (31 ) or
~, (z -1) " R* (z - 1),
~'( z -1) "A* ( z -1 ) ,
V(z-l) "p*(z-l)
(32)
and so on. Then from the preceding consideration, we have the following decision rule of detecting unsuspected feedback. An unknown system is identified as open-loop by a conventional method (Kaminskas, 1982), and gives n
a
~ n
*
n
a
p
~ n*
n
p
;;: n *
r
(33)
r
= 119,9).
The estimates of the residual error autocorrelation
function in Fig. lc correspond to a linear open-loop system with transfer functions Wiz
-1
;a*)=
1 1-0.25z-1 - 1 ' H(z ;h*)= -1 . (39) 1-0,9z 1-0,25z
In our case the polynomials satisfying (30) through (32) are
~. (z-I)=1-0,2Sz- l ,
S"(z-I)
=1, ~(z-I)=1-0'9z-I'}
:::, -I -I "' , -I _ '" -1 -I P (z )=1-0,9z ,R (z ) = I, Viz )=1-0,2Sz • (40)
for open-loop system or
It follows from (29) and (40), that for an identification (39) as an open-loop system by using polynomials of the type (24), the residual error E~ (C) (34) for closed-loop operation. If polynomials B(z-l) and p(z-I) have similar roots or other signs of feedback, the identification is repeated with a polynomial of the type (24) in transfer functions w(z-l;a) and H(z-l;h) of an open-loop model with the application of
n' r
n
(3S)
r
Then for an open-loop system, with a successive growth of the order n~ , residual error sequence of the type (26) must behave as a moving average process (29), its order decreasing with n~ EXPERIMENTAL CHECKING Data in Tables 1, 2 and Fig.l present an application example of the technique of detecting unsuspected feedback in a linear system with transfer functions
(36)
of the type (26) must have the properties of a moving average process. This is illustrated by Fig. lc. Thus we conclude from Table I, that for noise-corrupted observations and for short samples, common
roots of polynomials B(z-I) and p(z-I) can hardly be detected directly if feedback exsists. But at the same time Tables 2 and Fig. 1 suggest the efficiency of the technique by using polynomials of the type (24) for the property analysis of the residual error sequence.
CONCLUSIONS For a closed-loop system identified as open-loop common roots appear in the relevant polynomials of the model transfer functions. To detect common roots in the polynomials of input signal and output noise channel, application of transfer functions with common multipliers of the type (24) is recommended. The results of computer simulation, Tables 1, 2, Fig. 1, support the theoretical conclusions on the efficiency of the suggested decision rule of detecting unsuspected feedback in dynamic systems. REFERENCES Box G.E.P. ,Jenkins G.M. Time series analysis.-
1-0,S z
-I
Holden-Day, 1970. Kaminskas V. Dynamic system identification via discrete-time observations. - Vilnius: Mokslas,
and a second-order regression input
1982. - 240 p. Kaminskas V., Rimidis A. Detection of feedback in linear dynamic systems. - Statistical problems of control. Vilnius, 1983, Issue 60, p. 9-26.
In this case Wiz
H(z
-I
-1
-I ;0.)= ~ A(z-I) ;h)=
p(z-I)
-2 +0,12Sz -I -2 1-0,4z +0,2z
1-0,7Sz
-I
-2
-1
+0,45z -2 -I 1-0,4z +0,2z
1-1,4z
R(z -1)
}
Tsang H.Y., Bacon D.W. Detection of unsuspected
(37)
and the roots of polynomials B(z) and P(z) equal to B
B
P
P
zl=0,2S , z2=0,S and zl=0,9 , z2=0,5 , respectively.
B P Consequently, they have a common root z2=z2=0,S The polynomials of domains (27) and (28) are
feedback in linear dynamic systems. - Technometrics, v. 22, No 4, 1980, p. 509-SI6.
V. Kaminskas and A. Rimidis
In()
Table 1. Parameter estimates for an open-loop
Table 2. Parameter estimates for a open-loop
ide ntification of (36)
identification of (36) by using polynomials (24)
2
2
at/ay
0,01
0,1
1,0
0,01
0,1
1,0
Parameter
1,000
0,989
1,013
b'
1,000
1,000
1,014
-0,754
-0,883
-0,835
°
b;
-0,264
-0,231
-0,230
0,128
0,216
0,182
b~'
0,495
0,505
0,497
-0,403
-0,519
-0,446
a
-0,408
-0,385
-0,346
0,200
0,206
0,221
a
0,201
0,199
0,197
PI
-0,528
-0,154
-1,317
P2
-0,370
-0,759
0,381
0,477
0,821
-0,284
0,212
0,098
0,200
b b b a a
r r
O l 2 l 2
l 2 B
2 2 XEE
0,258
0,447+jO,139
0,412+jO,099
z2
0,496
0,447-jO,139
0,412-jO,099
P zl
0,927
0,951
1,209
P z2
-0,399
-0,798
0,109
zl B
2 XEE
112,02
115,89
l
89,83
89,23
88,04
a
6
[tll c
~&o,g
t~
0.3
rS'
a5 0
0
O,J
~!J1
0,6
- YS'
0
0.3
0
.~
40
- VT
t
0 0
Fig. 1. Estimates of residual error autocorrelation function. Systems: a,b -
(36); c -
(39). Error sequences: a -
(10),
b,c -
(26).
88,86
Copyrig-h t IF !\.( : IdC.'1l1itica lioll alld Systl'1Il Parall1t'l<.:r Estimatioll I ~JH:). York . LI l\. . 1!IH:-)
DETECTION OF UNSUSPECTED FEEDBACK IN DYNAMIC SYSTEMS V. Kaminskas and A. Rimidis IlIslilllll' "r l 'lI.\'si((l/ 111/11 TI'I-iwi((l/ 1'1'0"/1'111.1 0( 1:-'II' lgI'lirs. A((ItlflllY or SriPIICl'J or 1111' I ,illll/lllli'lll SSH. Md"/,, -I . KIIIIIIIIS 2136t!-1. USSR
Abstract_ We p r esent a development o f decision rules on d e t ec ting unsuspected linear feedback in dynamic systems . The technique is based on the appearance o f common roots in the transfer fun c tions of input and noise c hannel s o f open-loop mode l s for c los e d-loop sys tems . We prove t hat the p r esence of such common roots may be established by the statistical behaviour analysis of the prediction e rror, provided the identification employs models with transfer fun ctions , including common fa c tors. The r esu lts of experimental c he cki ng a r e given. Keywords. Sy stem st ructure identification; un s uspec t e d feedback ; decision rule; common polynomial roots; residual er r or analysis.
INTRODUCTION Systems subjec t to iden ti ficatio n may be ope rati ng ei the r as ope n-l oop or as closed-loop , but in numerous occasions no a prio ri information o n the pre sence of f eedback i s known . To ide ntify in such con ditions a specific line must be followed , i n which the stage of system structure estimation on a mathematical model inc ludes t he prob lem of detecting unsuspec t e d feedba c k . To our knowledge there are no publications on this pro bl e m. The o nly known technique that we are aware of (Tsang, Bacon, 1980) requires the internal signal (total linear signal) to be measured . We sugges t in this paper a technique of detect ing un s u spec ted linear feedback from external input-output meas urement s . PROBLEM STATEMENT
r iance
2
a~,
indepe ndent of sequence x ' u - output k k -1 observation sequence, T , TR - time delays, z the backward shift operator. Assume, that the c losed-loo p system is stab l e and the noise channel transfer function is of minimum-phase and stable. The numerator and denominator polynomials of each transfer functions (2) through (4) with parameters a*, B* , h*, which provide actual performance conditions of system (1), have no common factors. Let there be no a priori information on the feedback in this system . The problem then consist s of developing algorithms to detect feedback from input-output o bservation sequenc ies.We consider syste ms including two classes o f input signals (Kamin skas , 1982). The first class contains sequencies of continuous spe c tral densities Sxx(W) that is set
Q ={w:S x
xx
( W»O)
(5)
has a non - zero measure. The second c lass of sequencies have spectral densities S
(w) =
xx
nw x
L
i=l
vX (6(w- w.) +6(w+w. )), ~
~
(6)
l.
where v~ - intensity o f the i -th harmonics , 1
the delta - function. PROPERTIES OF THE LOSS FUNCTION The r e exist ing no a p riori information o n the feedbaCk, we identify system (1) as open-loop -1 -1 (7) u k=W(z ;a)xk+H(Z ;h)~k'
Let u s consider the properties of argument c=argmi n Qs(C), cs)l
cT=(aT,hT)
,
(8 )
c
s
- (c)=M { - 1 Q
s
s-p
L
(9)
k=P+l
where
are the transfer function s of t h e ope n-loop , of the feedback a n d of the noise c hannel, where a , Band h - the parameter ve c t ors . Their coor di nates a re the coefficients of the rel e vant polynomials. x - obsek rved externa l input seque nce , ~ k - an inde pendent
(1 0)
- prediction e r ror sequence , s - number of obser vations, p - n umbe r of preliminary observa tio n s fo r in i ti al condition s , Qc - admissible domain desccribed by
ra ndom sequence with the zero mean and a finite va-
In:1
V. Kaminskas and A. Rimidis
1724 >lc ={c: I zi (a) I
I Zj (p) I
(i=l,n ); a
(9.=l,n ); r
(j=l,n ); p
(t=O,n )} , b
I bt l <
(11)
wIz
-1
;Cl.)Z
-1.
=W(z
-1 -1 -1;Cl)H (z ;h).
(2 0)
This particular case is indicated by equal denominators of transfer functions W(z-l;ii), H(z-l;h) or by zero multipliers in the denominator H(z-l;h).
Zi (a), Zj(P), z9.(r) - roots of equations Zn a A(z-l)=O, zn p p(z-l)=O, zn r If polynomials B.(z-l) and p.(z-l) have no common
For an autoregression output noise that is 1 H(z-1;h.)=R: (Z-1), the numerator of the transfer function coincides with the denominator of the actual transfer function of the forward part. Therefore,
P
W(z -1 ; a.) z -1. =W (z -1 ; a) -1 (z -1) , H(z-l ;h.)=R- 1 (z-l).
(21 )
The last is the case of a completely known transfer then the minimum value
a~
is achieved at arbitrary ii Rimidis,
of the loss function (9) €
>la and
h€
>lE (Kaminskas,
1983). Domains >l- and >l_ are defined as Cl h
>l_={a:9(z-1)=~(z-1)D.(z-1)B.(z-1)z-'. , Cl -1 -1 [ -1 -1 A(z )=~(z ) A.(z )D.(Z )-1
-1
)G.(z
-B.(Z
)z
-('·+'R.)l J}
function of the noise H(z-l;h.), and then W(z -1 ;a.) z -1 ·=W(z -1 la) H- 1 (z -1 ;h) H (z -1; h.) .
We now come to the development of algorithms of detecting common roots in the corresponding polynomials, which is equivalent to a decision rule of detecting unsuspected feedback. DECISION RULE
(14)
In the identification of real objects parameter estimates are usually found by minimizing Qs(c), i.e. c=arg
-1 -1 -1 [ -1 -1 R(z )=o/(z )R.(z ) A.(z )D.(z )-
-B.(z
-1
)G,.(z
-1
(23 )
min cen
c
+, )]
-(1
(22)
• R. )z},
(15)
where ~(z-l), o/(z-l) - arbitrary polynomials. If the following conditions are satisfied ( 16) (17)
Therefore estimates c in the final sample may differ from terms in domains (14), (15). A direct
c
search of common roots in polynomials
~(z-l)
and
p(z-l) gives then no result. But it follows from (14) and (15) that for the identification of a closed-loop system (1) as openloop, transfer functions w(z-l;a) and H(z-l;h) may be used with polynomials B(z-l)=B' (z-l)B"(z-l), p(z-l)=p' (z-1)B II (z-1),
n * =max{n * +n * a
1
d
,
n * +n * +1.+1 b
g
R
(18)
.} ,
R(z
then domains (14), (15) contain one or more pOints. Here· indicate actual parameter values, polynomials of actual parameter values and actual orders of the relevant polynomials. All this implies the following property describing a feedback system: polynomials B(z-l) and p(z-l) have common roots, if a closed-loop system is identified as open-loop. Therefore the technique of detecting internal feedback reduces to the detection of common roots in the relevant polynomials. After reducing the common roots in polynomials 9(z-1),
A( z -1)
and
P(z - 1), R(z -1); A(z -1) ,R (z -1)
and
B(z -1 ) ,
-1
,-1 -1 )=R (z )A(z ),
where
n' ,-1 b I-i B(z )= l: biz , i=O
(24) nil
11
B (z
-1
)=1+ l:
b,,_i b z i
(25)
i=l
Then sequence A(z) ,-1 -1 - ..----1- u9.- B (z )x9. [ B (z )
J
(26)
becomes a sequence of independent variables ~9. for all
Ci ~ Q= a
and
h€
r2_ where domains I
h
p(z-l) we arrive at transfer function w(z-1;a)=B.(z-1)z-1.
H(z
}
-1 -1 -1 _ A.(z )P.(z ) ;h)= - - - - ; - - - (z -1)
( 19)
(27)
R.
where A.(z-l), R.(z-l) are polynomials, resulting from A.(z-l) and R.(z-l) after the common roots were reduced. But the zero multipliers of the denominators of the transfer function w(z-l;a) is a proof of the presence of a linear feedback, too. It follows from (19), that reconstruction of the actual transfer function W(z-l;Cl.) is possible in specific cases even if polynomials A.(z-l) and R.(z-l) have no common roots. For H(z-l;h.)
= I,
(28) We note that the values of
a .. n=
and
h ,,>l_
a
also
h
minimize the criterion (9), which is built by using errors (26). For an open-loop system (no feedback) sequence (26) with parameters ~ which minimize (9) a moving average process
becomes
1725
Unsuspected Feedback in Dynamic Systems '" '" -1 Ek(C)=V(z )~k'
a'(z-I)=1-0,25z- l , S"(z-I)=1- 0 ,5z-
(29)
where ~(z-l) ~ 1 a polynomial (Kaminskas, Rimidis, 1983). For example, parameters ~ may be given as values, which satisfy equations
1
}
=. -1 -I = -I -I -2 P (z )=I-O,9z ,A(z )=I-O,4z +0,2z .
(38)
The identification was based on s=1000 observations. The correlation of the residual error seguence was 2
tested by a chisquare test XES(BOX, Jenkins, 1970)
with a significance level a=O,OS (critical value
X~ (31 ) or
~, (z -1) " R* (z - 1),
~'( z -1) "A* ( z -1 ) ,
V(z-l) "p*(z-l)
(32)
and so on. Then from the preceding consideration, we have the following decision rule of detecting unsuspected feedback. An unknown system is identified as open-loop by a conventional method (Kaminskas, 1982), and gives n
a
~ n
*
n
a
p
~ n*
n
p
;;: n *
r
(33)
r
= 119,9).
The estimates of the residual error autocorrelation
function in Fig. lc correspond to a linear open-loop system with transfer functions Wiz
-1
;a*)=
1 1-0.25z-1 - 1 ' H(z ;h*)= -1 . (39) 1-0,9z 1-0,25z
In our case the polynomials satisfying (30) through (32) are
~. (z-I)=1-0,2Sz- l ,
S"(z-I)
=1, ~(z-I)=1-0'9z-I'}
:::, -I -I "' , -I _ '" -1 -I P (z )=1-0,9z ,R (z ) = I, Viz )=1-0,2Sz • (40)
for open-loop system or
It follows from (29) and (40), that for an identification (39) as an open-loop system by using polynomials of the type (24), the residual error E~ (C) (34) for closed-loop operation. If polynomials B(z-l) and p(z-I) have similar roots or other signs of feedback, the identification is repeated with a polynomial of the type (24) in transfer functions w(z-l;a) and H(z-l;h) of an open-loop model with the application of
n' r
n
(3S)
r
Then for an open-loop system, with a successive growth of the order n~ , residual error sequence of the type (26) must behave as a moving average process (29), its order decreasing with n~ EXPERIMENTAL CHECKING Data in Tables 1, 2 and Fig.l present an application example of the technique of detecting unsuspected feedback in a linear system with transfer functions
(36)
of the type (26) must have the properties of a moving average process. This is illustrated by Fig. lc. Thus we conclude from Table I, that for noise-corrupted observations and for short samples, common
roots of polynomials B(z-I) and p(z-I) can hardly be detected directly if feedback exsists. But at the same time Tables 2 and Fig. 1 suggest the efficiency of the technique by using polynomials of the type (24) for the property analysis of the residual error sequence.
CONCLUSIONS For a closed-loop system identified as open-loop common roots appear in the relevant polynomials of the model transfer functions. To detect common roots in the polynomials of input signal and output noise channel, application of transfer functions with common multipliers of the type (24) is recommended. The results of computer simulation, Tables 1, 2, Fig. 1, support the theoretical conclusions on the efficiency of the suggested decision rule of detecting unsuspected feedback in dynamic systems. REFERENCES Box G.E.P. ,Jenkins G.M. Time series analysis.-
1-0,S z
-I
Holden-Day, 1970. Kaminskas V. Dynamic system identification via discrete-time observations. - Vilnius: Mokslas,
and a second-order regression input
1982. - 240 p. Kaminskas V., Rimidis A. Detection of feedback in linear dynamic systems. - Statistical problems of control. Vilnius, 1983, Issue 60, p. 9-26.
In this case Wiz
H(z
-I
-1
-I ;0.)= ~ A(z-I) ;h)=
p(z-I)
-2 +0,12Sz -I -2 1-0,4z +0,2z
1-0,7Sz
-I
-2
-1
+0,45z -2 -I 1-0,4z +0,2z
1-1,4z
R(z -1)
}
Tsang H.Y., Bacon D.W. Detection of unsuspected
(37)
and the roots of polynomials B(z) and P(z) equal to B
B
P
P
zl=0,2S , z2=0,S and zl=0,9 , z2=0,5 , respectively.
B P Consequently, they have a common root z2=z2=0,S The polynomials of domains (27) and (28) are
feedback in linear dynamic systems. - Technometrics, v. 22, No 4, 1980, p. 509-SI6.
V. Kaminskas and A. Rimidis
In()
Table 1. Parameter estimates for an open-loop
Table 2. Parameter estimates for a open-loop
ide ntification of (36)
identification of (36) by using polynomials (24)
2
2
at/ay
0,01
0,1
1,0
0,01
0,1
1,0
Parameter
1,000
0,989
1,013
b'
1,000
1,000
1,014
-0,754
-0,883
-0,835
°
b;
-0,264
-0,231
-0,230
0,128
0,216
0,182
b~'
0,495
0,505
0,497
-0,403
-0,519
-0,446
a
-0,408
-0,385
-0,346
0,200
0,206
0,221
a
0,201
0,199
0,197
PI
-0,528
-0,154
-1,317
P2
-0,370
-0,759
0,381
0,477
0,821
-0,284
0,212
0,098
0,200
b b b a a
r r
O l 2 l 2
l 2 B
2 2 XEE
0,258
0,447+jO,139
0,412+jO,099
z2
0,496
0,447-jO,139
0,412-jO,099
P zl
0,927
0,951
1,209
P z2
-0,399
-0,798
0,109
zl B
2 XEE
112,02
115,89
l
89,83
89,23
88,04
a
6
[tll c
~&o,g
t~
0.3
rS'
a5 0
0
O,J
~!J1
0,6
- YS'
0
0.3
0
.~
40
- VT
t
0 0
Fig. 1. Estimates of residual error autocorrelation function. Systems: a,b -
(36); c -
(39). Error sequences: a -
(10),
b,c -
(26).
88,86