Grzegorz SIEKLUCKI1, Adam PRACOWNIK2, Tadeusz ORZECHOWSKI3 1. AGH w Krakowie, Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej,
tel.: 12 6172888, e-mail: sieklo@agh.edu.pl
2. ArcelorMittal, tel.: 666070882, e-mail:Adam.Pracownik@arcelormittal.com
3. AGH w Krakowie, Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej, tel.: 12 6173171, e-mail: orzech@agh.edu.pl
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]:
)) elasticity, damping coefficient of the elastic element, motor torque, load torque and torsional torque, respectively.
Fig. 1. Two-mass system
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.
3. OPTIMIZATION OF THE OBSERVER A plant is described by the state-space equation:
)
The error signal of the observer is defined as )) described by the following equation
)) by duality to LQ regulator problem. The dual system
) respectively. Then, the performance index is defined [3]:
problem is solved by Algebraic Riccati Equation (ARE)1
Model of the two-mass system can be presented as eq. (2), damping coefficient Dis omitted:
(t)
where the value of the damping factor of the elastic element is omitted D0. The natural frequency of the eq. (6) equals Q~. Thus, two simple weight matrices of (4) are considered:
matrix Q~2 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
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 is only used to compare 2 with the estimated signals by observer. Input signals of ˆ2 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
101
analysis can be used. The Bode plots of 2-mass system is presented in fig. 3b.3.1. Observer for Q~1 with q=4
The gain matrix and eigenvalues ( ) of the observer are in the following form
the observer, respectively.a)
b)
Fig. 3. The measured signals - (a), Bode plots of the eq. (1) - (b) Moreover, the resonance frequency equals r imaginary part of the eigenvalues and the maximum of the magnitude Bode plot can be seen at this frequency r (fig. 4) and 0nr.
Furthermore, the phase angle varies more than 360 degrees, so the observer is non-minimum-phase system.
Additionally, the magnitude of the transfer-function
)
Fig. 4. Bode diagram of the observer
The reconstruction of ˆ2,Mˆsis 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 transient. ˆ2
Fig. 5. Estimated signals 3.2. Observer for Q~1 with q=100
The following results are obtained:
Fig. 6. Bode diagram of the observer
The Bode plots are presented in fig. 6 and magnitudes are smoother than in the previous subsection. The relationship r 0n is fulfilled.
The signals reconstruction is presented in fig. 7 and the low-pass filtering is shown, too.
Fig. 7. Estimated signals
The obtained results are better, but not perfect. Thus, in further part of the paper the matrix Q~Q~2 is used.
3.3. Observer for Q~2 with q=100, q1=200 The optimization results are:
0,77;0,1
,
4,07;10,41
68 , 2 06 , 3
; 10 88 , 2
10 6 , 17 03 , 5 9 ,
~ 11
0
j j
L
(11)
The Bode plots are presented in fig. 8 and magnitudes are the smoother than in two previous subsections. The relationship r 0n is fulfilled.
Fig. 8. Bode diagram of the observer
The signals reconstruction is presented in fig. 9, low-pass filtering of the observer is shown.
Fig. 9. Estimated signals 4. CONCLUSIONS
Low-pass filtering is the natural feature of the presented observers - measurement-noise suppression. The selection of Q~
matrix based on observability condition 4
) (Wo
rank leads to Q~Q~2 and obtained results confirm the usefulness of presented method in the industrial applications.
The resonance frequency of all observers is: r
n r 0
, but r 0n 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 0n, but ,r selection should be proper. Thus, the quadratic performance index and duality property are right choice.
5. REFERENCES
1. 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.
2. 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.
3. Sieklucki, G., Orzechowski, T., Tondos, M., Sykulski, R.: Optymalizacja obserwatora momentu obciążenia przy kwadratowym wskaźniku jakości. Przegląd Elektrotechniczny, nr 7, 2008, p. 29-35.
Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki PG, ISSN 2353-1290, Nr 49/2016