Nonlinear Identification Using Polynomial NARMAX Model . . . 105
Table 3 Parameters of the nonlinear stiffness
Parameter Value Identified
k𝛼
0 3 2.99
k𝛼
1 −30 −30.24
k𝛼
2 6600 6592.35
k𝛼
3 −21000 −20325.51
k𝛼
4 48000 35672.42
Fig. 6 A representation of the nonlinear behavior:
moment as function of the angular displacement
where, d=m(I𝛼−mx2𝛼b2) and k4= (−mx𝛼b2𝜌Cl𝛼−m𝜌b2Cm𝛼)∕d. The parametersb2,b3,b4,b5andb6are obtained from the identified discrete-time model parameters followingb2=𝛾3∕(−T),b3=𝛾4∕(−T),b4=𝛾5∕(−T),b5=𝛾6∕(−T)and b6=𝛾7∕(−T), with sample periodT.
The identified parameters of the nonlinear torsional stiffness are also suitable to represent the static nonlinearity present on the system dynamics. Although the parameterk𝛼
4has a very different value in comparison with the true one, the nonlin- earity is in the range of interest, between the operational limits for the pitch angle, around1.5◦. The Fig.6 is obtained with simulated of continuous-time model. The static nonlinear behavior is well captured by the model. The nonlinearity of this sys- tem is assumed as symmetric.
106 R. C. M. G. Barbosa et al.
of this paper is the stability analysis using the identified model. The root locus and the damping factor diagram as function of the free stream velocity and the nonlinear stiffness are presented.
A continuous state-space model may be expressed as follows
̇
x(t) =f𝜇[x(t)] +g[x(t)]𝜇𝛽 (12) where the state-space variables are defined as
{x1(t), x2(t), x3(t), x4(t)}T
={
h(t), 𝜃(t), ̇h(t), ̇𝜃(t)}T
(13) and consequently their derivatives are
{ẋ1(t), ̇x2(t), ̇x3(t), ̇x4(t)}T
={ḣ(t), ̇𝜃(t), ̈h(t), ̈𝜃(t)}T
(14) Therefore the dynamic equation may be written as Eq.15. Note that the termf𝜇(x) depends of the nonlinearity and will compound the Jacobian matrix. The control surface angle is null for the stability analysis.
⎧⎪
⎨⎪
⎩
̇ x1(t)
̇ x2(t)
̇ x3(t)
̇ x4(t)
⎫⎪
⎬⎪
⎭
=
⎡⎢
⎢⎢
⎣
0 0 1 0
0 0 0 1
−k1−[k2𝜇+pkx2(x2(t))] −c1 −c2
−k3−[k4𝜇+qkx2(x2(t))] −c3 −c4
⎤⎥
⎥⎥
⎦
⎧⎪
⎨⎪
⎩ x1(t) x2(t) x3(t) x4(t)
⎫⎪
⎬⎪
⎭ +
⎡⎢
⎢⎢
⎣ 00
g3 g4
⎤⎥
⎥⎥
⎦
𝜇𝛽 (15)
It can be observed that the term kx
2(x2) isk𝛼(𝛼)and due the nonlinearity it is necessary to calculate the Jacobian matrix for evaluating the eigenvalues. Therefore the Jacobian matrix is given by
J=
⎡⎢
⎢⎢
⎣
0 0 1 0
0 0 0 1
−k1−[k2𝜇+p(k𝛼0+k𝛼1𝛼(t) +k𝛼2𝛼(t)2) +k𝛼3𝛼(t)3] −c1−c2
−k3−[k4𝜇+q(k𝛼0+k𝛼1𝛼(t) +k𝛼2𝛼(t)2) +k𝛼3𝛼(t)3] −c3−c4
⎤⎥
⎥⎥
⎦∣𝛼=𝛼
op
(16)
It is possible verify the eigenvalues of the Jacobian matrix depends merely on the state values𝛼. For this system the origin is a equilibrium point, therefore, the Fig.7a shows the root locus forx= {0,0,0,0}T.
From Fig.7a, it can be seen that the instability occurs when the eigenvalue results cross the imaginary axis. This occurs for free stream velocity around15.32m/s, as shown in the Fig.7b. In the instability region starts a LCO and the natural frequency of oscillation varies according to the angular displacement, because of the nonlin- earity effectsk𝛼(𝛼).
Nonlinear Identification Using Polynomial NARMAX Model . . . 107
Fig. 7 aRoot locus for the free stream velocity variation considering the equilibrium point andb damping factor as functions of the free stream velocity
4.1 Influence of the Pitch Angle
According to equations of motion it is possible to verify that the nonlinearity affects the stability when a negative damping factor occurs, resulting in unstable behavior.
The system becomes unstable due to the nonlinearity as shown in Fig.8. The figure is obtained varying the system operation point (namedxop). In this case, the angular displacement sweep occurred because the torsional stiffness is a function of the𝛼 values.
Figure8shows six curves from the left to the right, each one corresponding to a free stream velocity of the range{6; 6.5; 7; 7.5; 8; 8.5}in m/s. The angular displace- ment is varying from0to0.07rad. Note that for reduced values of the free stream velocity, below7m/s, the nonlinearity does not affect the system stability. Above
Fig. 8 aRoot locus for variation in nonlinear stiffness considering six free stream velocities and benlarged region
108 R. C. M. G. Barbosa et al.
Fig. 9 aRoot locus forV∞= 8m/s varying𝛼between0and0.07rad,bdamping factor as function of a nonlinear stiffness
this value, the system is unstable depending on the angular displacement𝛼of the operation point.
The root locus considering the free stream velocity of8 m/s with the stiffness varying approximately from0 to27.07Nm/rad is shown in Fig.9. It is observed in Fig.9b the region of instability for stiffness varying from6.11to8.56Nm/rad.
Through several simulations it is possible to note that the LCO frequency increases as the stiffness rises, considering the stability region.
5 Conclusion
The nonlinear identification applied to an aeroelastic pitch-plunge system is pro- posed using a polynomial NARMAX model estimated from the extended least square estimator (ELS) just to obtain unbiased estimation even with colored noise in mea- sures. The output predictions and an estimation of the static nonlinearity is presented in order to validate the identified model. The estimates converges quite quickly within ten iterations.
The nonlinear identification results in a suitable nonlinear torsional stiffness. The proposed analysis is validated to be applied with the experimental data from the aeroelastic system. Also, the identified system can be applied to the stability analysis and the control system design with nonlinear approach.
From the stability analysis, the mapping of the unstable regions is outlined based on the eigenvalues of the Jacobian matrix. It concludes that the nonlinearity can lead the system to be dynamically unstable. The aeroelastic system is stable for velocity less thanV∞= 15.32m/s even with the nonlinearity.
On the other hand, the system becomes unstable with the increase of the pitch angle and the free stream velocity. It is assumed an excursion of control surface with maximum angle of15◦, positive or negative, for normal operational in flight,
Nonlinear Identification Using Polynomial NARMAX Model . . . 109
but larger angles are tested for the identification process. In this work, the system is excited by the control surface from−30◦to30◦.
The root-locus is outlined for angular displacement between0and0.07rad cor- responding to maximum angular displacement of4.01◦. The system operating in the unstable region can return to stable region by the action of the control surface leading on eigenvalues crossing the imaginary axis and returning to the left plane.
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