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Численные эксперименты по методу идентификации линейных динамических систем при гармоническом сигнале.

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9/2015
517.98
doi: 10.18097/1994–0866–2015–0–9–76–82
1
©
,
,
-
, 670013,
.
, 40 , e-mail: miarsdu@esstu.ru
, 670013,
.
, 40 , e-mail: elenamadaeva@gmail.com
©
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:
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.
NUMERICAL EXPERIMENTS BY THE METHOD OF IDENTIFICATION OF LINEAR
DYNAMICAL SYSTEMS UNDER HARMONIC SIGNAL
Arsalan D. Mizhidon
DSc, Professor, Applied mathematics Department, East Siberian State University of Technology
and Management
40v Kluchevskaya st., Ulan-Ude 670013, Russia
Elena A. Madaeva
Research Assistant, Applied mathematics Department, East Siberian State University of Technology and Management
40v Kluchevskaya st., Ulan-Ude 670013, Russia
The paper presents the numerical implementation of the method of identification of linear stationary dynamic systems on the input sinusoidal signal. Identification of the system matrix is reduced to
constructing and solving a matrix linear algebraic equation. The construction of the equation is based
on comparison of the representation of Cauchy problem solutions in the form of exponential matrix
series and results of the system phase coordinates measurement under the input sinusoidal signal. For
the implementation of numerical experiments it was compiled a software in Fortran.
Keywords: identification, active identification, linear system, Cauchy problem, fundamental matrix, matrix exponential, interpolation.
,
,
,
–
–
-
.
,
.
,
[1]
-
1
15-41-04020
.
76
. .
, . .
.
-
.
[2],
-
,
.
,
[2]
-
,
.
1.
.
BSin ( t ) ,
B
-
diag b1 , b2 ,..., bn
,
x0
,
( 1 ,....,
n
)T –
( x10 , x20 ,..., xn0 )T
(
).
-
t,
t , x(t ) ( x1 (t ), x2 (t ),..., xn (t ))T ,
.
x(t ) ( x1 (t ), x2 (t ),..., xn (t ))T
:
0
Ax BSin ( t ), x(0) x .
,
x
0
x
A– n
0
1
0
2
0 T
n
( x , x ,..., x ) –
(1)
,
.
A,
-
(1)
2.
(1),
.
[3],
:
t
x(t )
F (t , t0 ) x 0
F (t , )BSin(
)d ,
(2)
t0
F (t , ) -
.
F (t , )
x(t )
x0
ACWb)
Ax 0 t
(1),
Cos( t )
,
( A2 x 0
( A2 m x 0
t3
...
3!
diag
1
,
A2 m 1CW 1b ( 1) m 1 ACW 2 m 1b) 2 m
t
(2m 1)!
2
,...,
n
,
2
-
A CW
C
1
...
(3)
:
C
2
,
t0
,
(1) [2]:
A2CW 1b CWb)t 2
( A3 x 0 A3CW 1b
2!
A2 m CW 1b ( 1) m 1 CW 2 m 1b) 2 m
t
(2m)!
( A2 m 1 x 0
b (b1 ,...., bn )T , W
Sin( t )
E.
(1) x (t )
-
:
x(t )
x ( i ) (t0 )(t t0 )i
x0
i!
i 1
77
.
(4)
9/2015
(3)
t0
(3)
(1) x (t )
-
0 (4).
(4) [2]:
(1)
x (0) Ax0 ,
.........................................
x (2m) (0)
A2 m x0
x (2m 1) (0)
A2 mCW 1b ( 1)m 1CW 2m 1b,
A2 m 1 x0
A2 m 1CW 1b ( 1)m 1 ACW 2 m 1b
Ax (2m) (0).
A:
Ax
0
(1)
x (0),
(1)
Ax (0) x (2) (0) Wb,
.........................
Ax (2 m 1) (0)
x (2 m ) (0) ( 1) m W 2 m 1b,
Ax ( 2 m ) (0)
x (2 m 1) (0).
,
(1)
A
AX 0
0
X
0
W
*
(5)
X1 W*,
1
(6)
:
x (0), x (0),..., x ( n ) (0) ,
X
X
(1)
x (0), x (0)..., x ( n 1) (0) , X 1
(1)
0, Wb,...,( 1) m W 2 m 1b ,
(2)
n 2m, m
0, Wb,...,( 1) m W 2 m 1b, 0 ,
;
n 2m 1, m
.
(6),
X1 W*
A
X0
,
X0
1
t
0.
.
X1
-
[1],
:
1.
2.
[1];
X 0 , X1 ,
,
W* ;
BSin ( t ) -
1,
3.
(6).
.
X
0
X
1
,
A,
(6),
A
A, X 0 ,
X1
A X
0
X
0
:
X
1
W
*
X
W*
1
W
*
,
.
[4]
-
X1 , W*
-
.
X 0 , X1 , W*
.
X0,
:
X0 X0
1
X0
1,
X0
1
.
A
:
X1
A
A
.
1
1
X0 X0
A
78
W*
X1 W*
(1)
.
(1),
X0
X0
.
(6)
(5)
. .
, . .
.
-
[2],
X 2m
1
AX 2 m
:
X 2m
X 2m 1 ,
x(0), x (2) (0)..., x(2 n
2)
(0) ,
x (1) (0), x (3) (0),..., x(2 n 1) (0) .
3.
(1),
:
3
4
0
x1 (t )
x2 (t )
x1 (t )
4
5
0
0
2
3
x1 (t )
x2 (t )
2
4
2
x3 (t )
x4 (t )
0 0 2
1
(7)
:
t
1,5 t e
1,25 t e t
1 t e
x2 (t )
t
1 t e
x3 (t )
1,5 t et
x4 (t )
1 t et
Sin(t )
Sin(2t )
,
2Sin(t )
x3 (t )
x4 (t )
4.97
4.32
.
1.86
x (0)
2 Sin(2t )
1.5 3.5
0
4
t
1
0
1
2
.
-
,
(7)
1.56
0
0.6
1.28
1.68
Sin(t )
Cos(t )
0.48
0.08
0.64
1.44
Sin(2t )
Cos (2t )
(8)
(8)
t 0
.
41.
1
(1)
j
(2)
j
j
x j (0)
x (0)
x (0)
1
2
3
4
4.97
4.32
1.86
1.56
0.75
0.8
2.46
2.16
4.37
4.72
5.06
6.76
1
X
1
(3)
j
x (0)
7.75
10.8
1.66
3.36
(4)
j
x (0)
-14.23
-20.88
-3.74
-16.04
X0
(7)
*
W :
X0
(6)
4.97 0.75 4.37 7.75
4.32 0.8 4.72 10.8
, X1 W*
1.86 2.46 5.06 1.66
0.75 3.37 7.75
0.8 2.72 10.8
2.46 3.06 1.66
13.23
12.88
.
1.74
1.56 2.16 6.76 3.36
2.16 2.76 3.36
0.04
A
:
A
3
4
0
3.999999
5
0
0
0
0 1.999999
2
4
.
3
2
2
1
(7),
4
(8) (
79
1).
9/2015
2
t
x1 (t )
4.97
5.216815
5.688296
6.419644
7.439683
8.784953
10.5163
12.73632
15.60607
19.35987
0.0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x2 (t )
4.32
4.587393
5.110366
5.926847
7.050904
8.48708
10.24986
12.38612
14.99818
18.26511
x3 (t )
1.86
2.455224
3.264548
4.302613
5.596578
7.197332
9.190789
11.70842
14.93652
19.12431
x4 (t )
1.56
2.130642
2.984807
4.118829
5.5214
7.189164
9.144857
11.45601
14.25236
17.74054
.
[1]
t
,
0:
(7) (
3).
(
3
,
-
2),
4-
3
(1)
j
(2)
j
(3)
j
j
Pj (0)
P (0)
P (0)
P (0)
1
2
3
4
4.97
4.32
1.86
1.56
0.750057
0.79996
2.460005
2.159965
4.368569
4.720996
5.059994
6.760901
7.77105
10.78604
1.658043
3.347021
Pj( i ) (0)
(4)
j
P (0)
-14.45088
-20.75801
-3.692948
-15.91548
,
i
x j (t ) (
2)
:
4.32 0.799960 4.720996 10.78604
,
1.86 2.460005 5.059994 1.658043
1.56
AP
1
X
X
4.97 0.750057 4.368569 7.771051
X0
A
i , j 1, 4 [1].
0
0
3
X1
t
2.159965 6.760901 3.347021
0.750057 4.368569 7.771051
0.79996 4.720996 10.78604
14.45088
20.75801
2.460005 5.059994 1.658043
3.692948
.
2.159965
(6)
3.08
3.994221
-0.006458
6.760901 3.347021 15.91548
:
-4.096548 -0.030396 2.040963
-4.993414 -1.995434 3.994701
.
-0.007625 3.001408 -2.002228
-0.03089
-0.036105
80
2.012262
-1.016214
. .
, . .
.
-
4
t
x1 (t )
4.970001
5.216816
5.688297
6.419645
7.439682
8.784952
10.5163
12.73632
15.60607
19.35987
0.0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x2 (t )
4.320001
4.587394
5.110367
5.926848
7.050903
8.487079
10.24986
12.38612
14.99818
18.26511
x3 (t )
1.860001
2.455225
3.264549
4.302614
5.596577
7.197331
9.190788
11.70841
14.93652
19.12431
x4 (t )
1.560001
2.130643
2.984808
4.118830
5.521399
7.189163
9.144856
11.45601
14.25236
17.74054
,
(
-
4),
A
3.25413
4.165789
0.16514
4.297418
5.194152
0.193144
0.095888
2.060879
2.935939
0.140825
0.164804
1.946753
5.
A
A
2.128447
4.082123
1.914778
0.928708
:
5
t
x1 (t )
4.970001
5.216794
5.687912
6.417365
7.432735
8.76395
10.47498
12.6654
15.49595
19.20583
0.0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x2 (t )
4.320001
4.587374
5.110089
5.925244
7.046525
8.470778
10.21771
12.33016
14.90804
18.13332
,
x3 (t )
1.860001
2.455211
3.264316
4.301182
5.59192
7.183947
9.163688
11.66056
14.86079
19.01645
(
x4 (t )
1.560001
2.130626
2.9846
4.117638
5.518198
7.177012
9.120893
11.41404
14.18464
17.64201
2)
(
,
5)
-
.
.
.
,
-
,
,
.
1.
. .,
//
2.
. .,
3.
. .
. .
. – 2014. –
. .
//
3. – . 5–12.
. – 2015. –
. – .:
81
1. – . 62–75.
, 1974. – 331 .
9/2015
4.
. .
, 2009. – 457 .
.–
:
References
1. Mizhidon A. D., Madaeva E. A. Ob odnom podkhode k identifikatsii lineinykh dinamicheskikh
system [One approach to the identification of linear dynamic systems]. Vestnik Vostochno-Sibirskogo
gosudarstvennogo universiteta tekhnologii i upravleniya – Bulletin of East Siberian State University of
Technology and Management. 2014. No. 3. Pp. 5–12.
2. Mizhidon A. D., Madaeva E. A. Metod identifikatsii lineinykh dinamicheskikh sistem po
vkhodnomu sinusoidal'nomu vozdeistviyu [The method of linear dynamic system identification by
input sinusoidal action]. Nauchnyi vestnik Novosibirskogo gosudarstvennogo universiteta – Science
Bulletin of Novosibirsk State Technological University. 2015. No. 1. Pp. 62–75.
3. Pontryagin L. S. Obyknovennye differentsial'nye uravneniya [Ordinary differential equations].
Moscow: Nauka, 1974. 331 p.
4. Kabanikhin S.I. Obratnye i nekorrektnye zadachi [Inverse and Ill-Posed Problems].
Novosibirsk: Siberian Scientific publ., 2009, 457 p.
82
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