reconstruction of the unmeasured signals in a two

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej Nr 49
XLVIII Międzyuczelniana Konferencja Metrologów
MKM 2016
Akademia Górniczo-Hutnicza w Krakowie, 5-7 września 2016
RECONSTRUCTION OF THE UNMEASURED SIGNALS IN A TWO-MASS SYSTEM FILTERING ANALYSIS
Grzegorz SIEKLUCKI1, Adam PRACOWNIK2, Tadeusz ORZECHOWSKI3
1.
2.
3.
AGH w Krakowie, Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej,
tel.: 12 6172888, e-mail: [email protected]
ArcelorMittal, tel.: 666070882, e-mail: [email protected]
AGH w Krakowie, Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej,
tel.: 12 6173171, e-mail: [email protected]
Abstract: The paper describes full-order observer for a two-mass
electric drive with a flexible shaft. The following state variables:
the load machine speed, load torque (disturbance) and torsional
torque are estimated. Optimization of the observer is based on the
LQ problem. The selection of Q matrix in a performance index is
determined by observability condition for LQ problem. This
approach leads to minimal nonzero elements in the weight matrix.
The Bode diagrams for designed observers are shown and analyzed.
The experimental results confirm the usefulness of presented
method in the industrial applications.
Keywords: two-mass system, Bode diagram, observer, LQ
problem.
1. INTRODUCTION
The modeling of drive systems is in most cases made
under a simplifying assumption of absolute stiffness of the
motor and machine connection. The mechanical part of the
drive can be therefore considered as a single equivalent
mass. The obtained mathematical model of a one-mass
system may be not sufficiently correct for some industrial
applications. It may also, due to the simplification, neglect
certain physical phenomena, e.g. the torque and speed
transient oscillations, resulting from the flexible connection.
An example of the drive system with flexible coupling can
be a roll stand, connected with a motor by means of a long
shaft.
Moreover, load machine speed, load torque and
torsional torque are difficult to measured. Thus, the
reconstruction of these signals is usually required.
Many papers discussed observers with complicated
structures, e.g. Kalman Filter, Neural Networks, Fuzzy
Logic, genetic algorithms, etc. The observer presented here
is simpler, without loss of quality of the reconstructed
signals (torsional torque and load machine speed).
In observer-controller (observer-based controller)
system the following rules (principles) should be satisfied:
1. Observer is able to correct for the errors introduced by
"small" model inaccuracies.
2. Observer has good performance with respect to
a mismatch in initial conditions.
3. Observer has good performance with respect to transient
disturbances on the plant.
4. Low-pass filtering should be the natural feature of the
observer (measurement-noise suppression).
The principles 2, 3 are equivalent. If observer is robust
on noisy signals then Kalman Filter does not have to be
used.
The disadvantage of the LQ problem is: the difficulties
in selection of Q and R matrices in the performance index,
hence the observability matrix is used to determine Q.
A novel approach is being considered: observer as
MIMO filter and Bode diagrams are analyzed.
Furthermore, the observer is used with controller so its
dynamics should be taken into account in the parametric
optimization of this controller - instability danger.
2. MATHEMATICAL MODEL AND SETUP
Mathematical description of the two-mass system is
based on a model of an inertialess elastic (fig. 1)[1,2]:
d1 (t )
 ( M e (t )  M s (t ))  D(1 (t )   2 (t ))
dt
d 2 (t )
J2
 ( M s (t )  M m (t ))  D(1 (t )  2 (t ))
dt
dM s ( t )
 k s ( 1 ( t )   2 (t ))
dt
J1
(1)
t

 ( t )   1 ( t )   2 ( t )  ( 1 ( t )   2 ( t ))dt
0
where ks, D, Me(t) ,Mm(t), Ms(t) are: coefficient of shaft
elasticity, damping coefficient of the elastic element, motor
torque, load torque and torsional torque, respectively.
Fig. 1. Two-mass system
where Pc  PcT  0 is solution of ARE which exists if, and
~
only if, the pairs (C , A) and (Q , AT ) are observable.
Model of the two-mass system can be presented as eq. (2),
damping coefficient D is omitted:
Fig. 2. The experimental setup
The diagram of the laboratory plant is presented in fig. 2:
1. Matlab-Simulink and dSPACE software,
2. dSPACE cards: DS1104 and CP1104,
3. power electronics converter PEC,
4. separately excited DC motor S (IN=1,1 A,
MeN=4,07 Nm, J=0,25 kgm2) and generator (load
machine) H (J2=0,25 kgm2) with analog transducers
TG1, TG2,
5. the motor is coupled to the load machine by an
elastic shaft, ks=11,2 Nm/rad,
6. galvanic isolation with LEM converters.
y (t )  Cx (t )
(2)
The error signal of the observer is defined as
e(t )  y (t )  Cxˆ (t )  C ( x(t )  xˆ (t ))
where xˆ (t ) is estimated vector and the full-order observer is
described by the following equation
~
~
~
xˆ (t )  A xˆ (t )  B u (t )  L C ( x(t )  xˆ (t ))
(3)
~
where L is gain matrix of the observer and can be obtained
by duality to LQ regulator problem. The dual system
~
~
x (t )  AT ~
x (t )  C T u~ (t ),
~T ~
~y (t )  B
x (t )


(4)
0
~ ~
~ ~
where Q  Q T  0 , R  R T  0 are the weights. The LQ
problem is solved by Algebraic Riccati Equation (ARE)
~
~
~
~
~
~
A Pc  Pc A T  Pc C T R 1CPc  Q  0, L  Pc C T R 1
100
ks ks

 9,46 rad/s
J1 J 2
(7)
The pair (C , A) is observable and the observability matrix
~ ~
Q( A T )2
~ ~
Q( A T )3

is used ( rank (Wo )  4 ) to determine of the weight matrix
~
Q . Thus, two simple weight matrices of (4) are considered:
0
0
~
Q1  
0

0
0 0 0
0 0 0 
,
0 0 0

0 0 q
0
0
~
Q2  
0

0
0
0
0
0
0 q1
0
0
0
0 
0

q
(8)
~
~
and second weight of (4) equals R  1 . The matrix Q1 is the
minimal (simplest) form which satisfies rank (Wo )  4 and
~
matrix Q2 leads to better filtering of the reconstructed
~
signals. Introducing q1 element to the matrix Q was tested
by simulations for several systems.
3. EXPERIMENTAL RESULTS
where ~
x (t ) and ~
y (t ) are the dual signals to x(t ) and y (t ) ,
respectively. Then, the performance index is defined [3]:
~
~
J  [~
x T (t )Q~
x (t )  u~ T (t ) R u~ (t )]dt
on 

A plant is described by the state-space equation:
(6)
where the value of the damping factor of the elastic element
is omitted D  0 . The natural frequency of the eq. (6) equals
~ ~~
Wo  Q QAT
3. OPTIMIZATION OF THE OBSERVER
~
~
x (t )  A x(t )  B u (t ),

0

~
A0

k
 s
 0
ω1(t) 


ω2(t) 
x(t)  
, y(t)  ω1(t), u(t)  M e(t)
 M s(t) 


 M m(t)
1

1
0
0 
J1
J 

1  1 ~  1 
0
, B   0 , C  1 0 0 0
J2 J2 
0
0 
 ks 0
 

 0 
0
0
0 
(5)
The considered Real-time system (embedded system)
can be called quasi-continuous observer:
1. Euler's method for the ordinary differential equations
(observers) is applied.
2. Sampling time (integration time) equals Ts=0.001 s, so
it is about 300 times shorter than the smallest period of
the observers or the model of the two-mass system.
The signals which are measured during the experiments
are shown in fig. 3a, but the 2 is only used to compare
with the estimated signals ̂2 by observer. Input signals of
considered observer are noisy (about 15% of measurement
range), so the filtering properties can be tested.
The transfer-function of the observer (3)
Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki PG, ISSN 2353-1290, Nr 49/2016

~ ~
~
G( s )  ( sI  A  L C )1 B
~
L

(8)
can be used to Bode diagrams plotting, where In(1)  M e ,
In(2)  1 , Out (1)  ˆ 2 , Out (2)  Mˆ s . Then the filter
analysis can be used. The Bode plots of 2-mass system is
presented in fig. 3b.
~
3.1. Observer for Q1 with q=4
The gain matrix and eigenvalues (  ) of the observer
are in the following form
~
L  3,31 2,52  1,37  2
   0,21  j 9.49;  1,44  j1,38
  0,72; 0,02, 0  2; 9,48
(9)
where  , 0 are the damping ratio and natural frequency of
the observer, respectively.
a)
Fig. 4. Bode diagram of the observer
The reconstruction of ˆ 2 , M̂ s is presented in fig. 5 and
the performance is not proper (phase angle is about
180 degrees). Moreover, low pass filtering phenomenon for
  r is obtained in smooth ̂2 transient.
b)
Fig. 5. Estimated signals
~
3.2. Observer for Q1 with q=100
The following results are obtained:
~
L  [8,6 6,4  9,3  10],    0,95  j 9.7;  3,4  j 2,8
  0,77; 0,1,
0  4,34; 9,75
(10)
Fig. 3. The measured signals - (a), Bode plots of the eq. (1) - (b)
Moreover, the resonance frequency r equals
imaginary part of the eigenvalues  and the maximum of
the magnitude Bode plot can be seen at this frequency r
(fig. 4) and 0 n  r .
Furthermore, the phase angle varies more than 360
degrees, so the observer is non-minimum-phase system.
Additionally, the magnitude of the transfer-function
ˆ 2 ( s ) / 1 ( s ) varies rapidly near frequency  r .
Fig. 6. Bode diagram of the observer
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101
The Bode plots are presented in fig. 6 and magnitudes
are smoother than in the previous subsection. The
relationship  r  0 n is fulfilled.
The signals reconstruction is presented in fig. 7 and the
low-pass filtering is shown, too.
Fig. 9. Estimated signals
4. CONCLUSIONS
Fig. 7. Estimated signals
The obtained results are better, but not perfect. Thus, in
~ ~
further part of the paper the matrix Q  Q2 is used.
~
3.3. Observer for Q2 with q=100, q1=200
The optimization results are:
~
L  11,9 5,03  17,6  10
   2,88  j10;  3,06  j 2,68
  0,77; 0,1,
0  4,07; 10,41
(11)
The Bode plots are presented in fig. 8 and magnitudes
are the smoother than in two previous subsections. The
relationship  r  0 n is fulfilled.
Low-pass filtering is the natural feature of the
presented observers - measurement-noise suppression. The
~
selection of Q matrix based on observability condition
~ ~
rank (Wo )  4 leads to Q  Q2 and obtained results confirm
the usefulness of presented method in the industrial
applications.
The resonance frequency r of all observers is:
 r  0 n , but  r  0 n and damping ratio   0,707 .
Thus, the presented observers are non-minimum-phase
damped oscillator systems.
The obtained results can be applied in pole placement
which is the most popular method of the observer designing.
Corollary: the right performance of the unmeasured signals
reconstruction does not have to be realized by a observer
with   0,707 and  r  0 n , but  , r selection should
be proper. Thus, the quadratic performance index and
duality property are right choice.
5. REFERENCES
1.
2.
3.
Szabat, K., Orłowska-Kowalska, T.: Vibration
Suppression in a Two-Mass Drive System Using PI
Speed Controller and Additional Feedbacks –
Comparative Study. IEEE Trans. on Industrial
Electronics, vol. 54, No. 2, 2007, p. 1193-1206.
Pracownik A., Sieklucki G., Tondos M., Optimization
of a Digital Controller and Observer in a Two-Mass
System - the LQ Problem. Przegląd Elektrotechniczny,
nr 2, 2012, p. 341-348.
Sieklucki, G., Orzechowski, T., Tondos, M., Sykulski,
R.: Optymalizacja obserwatora momentu obciążenia
przy kwadratowym wskaźniku jakości. Przegląd
Elektrotechniczny, nr 7, 2008, p. 29-35.
Fig. 8. Bode diagram of the observer
The signals reconstruction is presented in fig. 9, lowpass filtering of the observer is shown.
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Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki PG, ISSN 2353-1290, Nr 49/2016
ODTWARZANIE NIEMIERZALNYCH SYGNAŁÓW W UKŁADZIE DWU-MASOWYM
- ANALIZA FILTRACJI
W większości przypadków do modelowania układów napędowych przyjmuje się pewne uproszczenia polegające na
założeniu absolutnej sztywności połączeń pomiędzy silnikiem a maszyną roboczą. Wówczas mechaniczną część układu
napędowego można traktować jako jedną masę zastępczą. Otrzymany model matematyczny układu 1-masowego może być
dla pewnych zastosowań przemysłowych niewystarczająco dokładny. Ponadto, prędkość maszyny roboczej, moment
obciążenia i moment skręcający jest trudno zmierzyć. Oznacza to konieczność odtworzenia tych sygnałów.
W artykule opisano obserwatora pełnego rzędu dla napędu z połączeniem sprężystym. Następujące zmienne stanu:
prędkość maszyny roboczej, moment obciążenia (zakłócenie) oraz moment skręcający są odtwarzane. Optymalizacja
obserwatora jest oparta na problemie LQ. Wybór macierzy Q we wskaźniku jakości jest zrealizowany na podstawie macierzy
obserwowalności. Charakterystyki Bodego prezentowanych obserwatorów zostały przedstawione i przeanalizowane. Wyniki
eksperymentalne potwierdzają przydatność metody w zastosowaniach przemysłowych.
Słowa kluczowe: system dwu-masowy, charakterystyki Bodego, obserwator, problem LQ.
Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki PG, ISSN 2353-1290, Nr 49/2016
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Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki PG, ISSN 2353-1290, Nr 49/2016