MODEL AND IDENTIFICATION
OF AERODYNAMIC ONE ROTOR SYSTEM
Przemysław Gorczyca, Maciej Rosół, Andrzej Turnau, Dariusz Marchewka, Krzysztof Kołek
Department of Automatics and Biomedical Engineering, AGH University of Science and Technology
przemgor@agh.edu.pl,mr@agh.edu.pl, atu@agh.edu.pl, dmar@agh.edu.pl, kko@agh.edu.pl
Summary
A real laboratory tethered model of one rotor aero-dynamic system is considered. The studies of aerodynamics are reduced to the measurement of static thrust characteristics. Based on fundamental physical laws the mathematical model has been built. A constructed mathematical model of the one rotor system includes parameters and characteristics obtained by experiment. Based on parametric identification methods the parameters of the model and the thrust force characteristics generated by the rotor are determined. The model is used for the synthesis of control algorithms running in real time.
Keywords: aerodynamic system, identification, modeling, optimization
MODEL I IDENTYFIKACJA AERODYNAMICZNEGO SYSTEMU JEDNOWIRNIKOWEGO
Streszczenie
W pracy omówiono laboratoryjny model jednośmigłowego systemu aerodynamicznego na uwięzi.
Badania aerodynamiki tego systemu ograniczono do pomiaru charakterystyk statycznych siły ciągu śmigła. Korzystając z podstawowych praw fizyki, zbudowano model matematyczny jednośmigłowego systemu aerodynamicznego. Parametry i charakterystyki tego modelu wyznaczono eksperymentalnie, używając metod identyfikacji parametrycznej. Uzyskany model jest stosowany do syntezy algorytmów sterowania działających w czasie rzeczywistym.
Słowa kluczowe: system aerodynamiczny, identyfikacja, modelowania, optymalizacja
1. INTRODUCTION
A number of real flying and not flying systems are considered as far as aerodynamic phenomena are concerned. To demonstrate and show the quantitative relationship between the thrust and counter-rotation contraction it is sufficient to use a simplified not flying model. This does not mean that the modeled dynamics will be expressed in a simple form. On the contrary it will be very complex [1, 2].
The simplified real demonstrative system means also that several technical solutions encountered in a real helicopter are not necessarily present in our laboratory model (for example the blade pitch control). This variety of different models of flying and not flying solutions makes it necessary to explain exactly what constitutes the research aerodynamical laboratory model. The main objective
of the research is to measure the interaction forces and torques sourced by the propeller.
One Rotor Aerodynamic System (ORAS) shown in Fig. 1 is a demonstration and research equipment useful in the generation, testing and understanding control algorithms of aerodynamic rotor systems implemented in the real time [3, 4, 5]. ORAS is an innovative system for its predecessor Two Rotor Aerodynamic System (TRAS) [6, 7]. ORAS is equipped with a single-rotor only. In the TRAS system the aerodynamic force generated by the rotor tail counteracts and balances the reaction torque of the main rotor (carrier) [2]. In the ORAS system a similar effect is achieved by controlling the tilt angle of the rotation axis of the main rotor. An inclination of the rotor plane is achieved by means of a DC motor (which drives the rotor) attached to the motor axis. ORAS as well as TRAS consist of a beam pivoted on its base in such a way that the beam can rotate freely both in the horizontal and vertical planes. The axis of the propeller is tilted by a predetermined angle. It results in splitting a thrust delivered by the rotor into two components:
horizontal and vertical. ORAS is a non-linear system. It is characterized by significant cross- couplings. In a real helicopter the value of the aerodynamic force depends on the adjustment of the rotor blade angle. In the laboratory model the angle of attack is fixed. ORAS in English literature is called NOTAR, which is an abbreviation of the words “no tail rotor”.
Fig. 1. Schematic diagram of ORAS
2. MATHEMATICAL MODEL
When creating a mathematical model of the ORAS dynamics the basic laws of mechanics (primarily the second Newton’s principle) are used.
Although the flow theory says that the thrust force F(ω) generated by the rotor is proportional to the square of the rotor speed ω, however this relationship is more complex. The F(ω) characteristics are obtained by the use of an electronic balance. Due to possibility of tilting of the rotor plane a desired angle αT may be obtained.
Two components of the thrust force F(ω) are generated: Fp=F(ω)cos(αT) related to the αp angle and Fa=F(ω)sin(αT) related to the αa angle.
Formulas (1) and (2) and respectively (3) and (4) are the horizontal and vertical torque equations.
Formula (5) relates to the aerodynamical force F(ω). Formula (6) describes the rotor dynamics when the control torque u is from an algorithm running on a PC. The u control incorporates the H-
1(ω) static characteristics of the motor/rotor system.
) cos(
) cos(
)
(
22
p a
p a a
a a p
a
k M
dt f d dt
J α d α = − α − ω α + α
(1)) sin(
)
(
Ta a
l F
M = ω α
(2)p p
a p p
p p
p
k M
dt G d dt f d dt
J d + +
+
−
= α α α ω
α
2,
2
(3)
) cos(
)
(
Tp p
l F
M = ω α
(4)2 3
2
1
cos( ) sin( ) sin( ) cos( )
,
+
+
=
dt a d
a dt a
G d α
aα
pα
pα
pα
pα
pα
a(5)
)
1
( ω ω = −
−H dt u
I d
(6)where:
αa is the azimuth angle in the horizontal plane, Ja is the inertia momentum vs. the rotation axis in the horizontal plane,
fa is the coefficient of the viscous friction for the horizontal plane.
la is the distance between the rotor and the axis in the horizontal plane,
ka, kp are the coefficients modeling a precession phenomenon,
Ma is the azimuth rotor thrust torque, αp is the pitch angle in the vertical plane,
Jp is the inertia momentum vs. the rotation axis in the vertical plane,
fp is the coefficient of the viscous friction for the vertical plane.
lp is the distance between the rotor and the axis in the vertical plane,
Mp is the vertical rotor thrust torque,
a1, a2, a3 are the coefficients dependent on dimensions of the rotor blades,
F(ω) is the thrust force dependent on the rotational velocity of the rotor,
G(dαa/dt,αp) is the cross-coupling function, αT is the tilt angle of the rotor axis, I is the inertia momentum of the rotor, u is the rotor control value,
ω is the rotor rotational velocity,
H-1(ω) is the static characteristics of the motor/rotor system.
In Fig. 2 the block diagram of the ORAS model corresponding to equations from (1) to (6) is shown.
For a better diagram clarity certain connections are realized with the use of labels.
Fig. 2. Block diagram of the ORAS model
3. LABORATORY SET-UP
To obtain parameters and characteristics an identification process is conducted by experiments performed at the laboratory rig of ORAS. The block diagram of the rig is shown in Fig. 3. The ORAS system is designed to operate with an external PC- based digital controller. The control computer communicates with the position, speed sensors and motors by a dedicated I/O RT-DAC/USB board and power interface. The I/O board is controlled by the real-time software which operates in the
MATLAB/Simulink RTW and RT-CON
environment. A pre-programmed library of controllers and Simulink models supports the ORAS system. With a uniform platform used as well for the simulation and real-time control it can be effective the ORAS dynamic model identification and verification of the model by comparing the results of the model simulation with real signals measured in the real system.
Fig.3. Laboratory rig for the ORAS system For measuring angles: αT, αp and αa incremental
encoders with the resolution of 4096 pulses per revolution are used. The relevant angular velocities are reconstructed. Additionally the angular velocity
of the rotor is reconstructed from the rotational angle measurements.
DC motors are introduced to control the rotor tilt αT and the rotational velocity ω(u) of the rotor.
These DC motors are steered via the power interface
to achieve the required values of voltages and currents. The power signals are defined as a PWM type. The aforementioned RT-DAC/USB board is equipped with a reconfigurable FPGA chip [8]. Its structure is shown in Fig. 4.
Fig. 4. Structure of the FPGA chip
The primary function of the system is to create a linkage with a USB bridge and to communicate with the computer. Simultaneously ORAS system signals are supported.
The PWM generators block generates control signals for power interfaces. The encoders block working in a quadrature mode, converts the waves generated by the position values to the absolute encoder values. The clock block is a time reference used in particular for reconstruction of velocities.
USB bus transmission is performed in frames, and each frame is a set of actual measurement data and the current time stamp. Speeds are calculated using Simulink algorithms i.e. Simulink models. There are jitters resulting from the characteristics of the operating system and the specific USB transfer. One cannot rely on the measurement of the time realized by a computer and an independent time reference is necessary for the precise measurement of the speed.
4. IDENTIFICATION AND
VERIFICATION OF THE MODEL
The ω(u) function mapping a PWM signal of the control u to the rotational velocity ω of the rotor is obtained on an experimental basis. The ω velocity has been measured for different values of u in the (−1, +1) range. The tilt angle αT has been set to 0.
The obtained measurement data have been approximated (the polyfit function from MATLAB was used) in an analytic form by the following polynomial of the seventh order:
8 . 5 6003 6
. 295 8359 5
. 303
9890 8
. 22 4230 )
(
2 3
4
5 6
7
+ + +
−
−
− +
+
−
=
u u
u u
u u
u
s u
ω (7)
In Fig. 5 the speeds vs. control obtained from the experiment ω(u) and calculated from the polynomial (7) ωs(u) are compared. The total
identification error which is an estimate of the standard deviation of the difference between ω(u) and ωs(u) for successive values of u is up to 204.7 in the examined control interval. The thrust force F(ω) values measured in [N] are obtained by experiments with the electronic balance with the tilt angle αT set to 0 and the velocity ω in the range (−3340, +3240) RPM. Fs(ω) is approximated by the polynomial of the sixth order in the form:
4 5
2 9
3 11 4
15
5 18 6
22
10 94 . 2 10 43 . 3 10
11 . 5
10 14 . 4 10
42 . 4
10 8 . 1 10
29 . 3 ) (
−
−
−
−
−
−
−
⋅
−
⋅ +
⋅ +
+
⋅ +
⋅ +
+
⋅
−
⋅
−
=
ω ω
ω ω
ω ω
s ω F
(8)
-1 -0.5 0 0.5 1
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
PWM [-]
ω [RPM]
ω ωs
Fig. 5. Identified function ω(u)
In Fig.6 experimental thrust force values F are compared to the values Fs calculated from polynomial (8). The total identification error which is an estimate of the standard deviation of the difference between F and Fs for successive values of ω is up to 0.087 in the examined speed interval.
-3000 -2000 -1000 0 1000 2000 3000 -0.5
0 0.5 1
ω [RPM]
F [N]
F Fs
Fig. 6. Identified function F(ω)
The Ja, ka, Jp i kp parameters (see Fig. 7) are identified on the basis of parametric optimization methodology with quality factor (9).
Fig.7. Blok diagram of the parametric optimization method
( ) dt
J
T
∫ −
M=
0
α
α
(9)where: α is the value of rotational angle in the vertical or horizontal planes obtained from a real experiment, αM is as previously defined angle but obtained from a model and T is the experiment period. The difference between α and αM is integrated and creates the penalty function (9) for the optimization procedure. A numerical method implemented in MATLAB looks for the values of ka, Ja, kp and Jp parameters that minimize penalty function (9).
In Figs 8 and 9 the rotational angles αa i αp vs.
time in the real experiment and in the model are compared. Timing charts were recorded for u = 0 and αT = 0. One can notice a high consistency between the model and real system responses.
Vibrations in the azimuth and pitch axes disappear faster for the ORAS system because the model uses simple relationships describing the phenomenon of the kinetic friction.
Fig. 8. Rotational angles in the horizontal plane
Fig. 9. Rotational angles in the vertical plane After a parametric identification the experiment has been performed to verify the simulation model.
In Figs. 10 to 13 the timing charts ω, αa i αp from the experiment and the simulation model are compared. In Fig. 10 in a similar manner rotational velocities of the rotor corresponding to the u(t) = 0.5*1(t) excitation are shown. A high consistency of time diagrams is visible. Fig. 11 illustrates the tilt angle αT of the axis of the rotor (measured in the system), in response to the reference signal (black line). One can recognize the controller operating as a signal follower. In Figs. 12 and 13 the ORAS system responses in the αa i αp forms are shown.
These signals correspond to the input signals shown in Figs. 10 and 11. One can observe highly compatible results obtained from the mathematical model to the measured data of the real system.
Fig. 10. Rotational velocity of the rotor
Fig. 11. Tilt angles of the rotor axis
Fig. 12. Rotational angles in the horizontal plane
Fig. 13. Rotational angles in the vertical plane
5. CONCLUSIONS
The ORAS aerodynamical system equipped with a unique rotor is manufactured to the study of dynamics and control of such systems. At the same MATLAB software platform the simulation of an identified mathematical model and real-time experiments with the ORAS system are performed.
Software and hardware details relevant to the system performance in the real-time are shown. Let us mention at least the I/O USB 2.0 board equipped with the FPGA chip, power interface or PC used as a controller. Simulation and real experiments show the usefulness of the research equipment. In particular, one can tune the parameters of the algorithm to reduce cross-coupling among rotary movements of the rotor and the beam.
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