4.4 Robustness Margins
5.1.2 X56A MUTT
The X56A MUTT is another unmanned test platform developed by the U. S. Air Force Research Laboratory and Lockheed Martin Aeronautics Company for the investigation of aeroservoelastic effects and demonstration of active aeroservoelastic control [Beranek et al. 2010]. It is currently used by NASA for research and flight testing [Ryan et al. 2014].
Figure 5.10 shows the aircraft.
Figure 5.10:X56A MUTT aircraft.1
The aircraft is a flying wing with a wing span of 8.4 m and built in a modular fashion such that different sets of wings can be attached. In the considered configuration, the aircraft has ten control surfaces, two at the trailing edge of the body and eight at the trailing edge of the wings. Its empty weight is 86 kg with about 25 kg maximum fuel weight.
Similar to the Body Freedom Flutter Vehicle, the X56A MUTT features six accelerometers, two located at each wing tip and two in the center body, fore and aft, respectively. The aircraft is also equipped with a Pitot tube to measure airspeed and with an IMU to measure rates of rotation and attitude.
A high-fidelity model of the longitudinal dynamics was developed by Schulze et al.
[2016]. It combines rigid-body flight dynamics from first principle modeling, structural dynamics from finite element modeling and unsteady aerodynamics from computational fluid dynamics modeling. The rigid-body states are described in the moving body frame and are represented by angle of attackαand pitch rateq. The flexible modal displacements are represented in terms of assumed mode shapes and generalized coordinatesη. Unsteady aerodynamic states are represented by state variableswand are related to the rigid and flexible degrees of freedom of the system. Specifically, every degree of freedom is coupled to a third-order system that describes the unsteady aerodynamic forces caused by, and acting on, modal displacement. There are 8 structural modes (16 states), the 2 rigid-body states and 30 aerodynamic states, which totals to 48 state variables.
Aiming at a model suitable for longitudinal attitude control and flutter suppression, only a subset of the actual available sensors and actuators is considered for the present model order reduction. As inputs, symmetric deflection of the two outboard wing flap pairs (δ1
andδ2), highlighted in Figure 5.11, are used. The outputs are a pitch rate measurement qmeasand an acceleration signal obtained at the center body (az,center), as well as a wing
1NASA public domain image [https://commons.wikimedia.org/wiki/File:Lockheed_Martin_X-56A_
first_landing.jpg].
tip acceleration signal (az,wing) that averages the measurements from the four sensors shown in Figure 5.11.
Wing Flap (δWing Flap (δ1) 2) Wing Tip Accelerometer
Center Accelerometer
Pitch Rate Gyro
Wing Flap (δ1) Wing Flap (δ2) Wing Tip Accelerometer
Figure 5.11: Schematic of the X56A MUTT unmanned aircraft.
The dynamics of the aircraft depend parametrically on the airspeed V∞and hence the state space model is of the form
˙
x=A(V∞)x+B(V∞)δ
y=C(V∞)x+D(V∞)δ, (5.1)
with state vectorx=
wT |α q|η˙T |ηTT, output vectory= [qmeasaz,centeraz,wing]T, and input vectorδ = [δ1δ2]T. A grid representation with 12 uniformly spaced points is used to cover the domain V∞ ∈ [30.6 68]m/s. The aircraft is naturally stable in this domain but the damping ratio of the lowest-frequency aeroelastic mode decreases dramatically with higher airspeeds. Hence, the dynamics change rapidly.
The available bandwidth of the control system again provides an upper frequency limit on the fidelity requirement and fifth-order Butterworth filters with a cut-off frequency of 100rad/sare selected. The augmented Lyapunov equations (3.29) are solved using the Matlab routinelyapcholat each grid point and a reduced-order model is calculated using Algorithm 2 (p. 50). The calculation takes only seconds and hence the order of the reduced model can be determined by trial and error. A 12th-order model yields satisfactory results.
Figure 5.12 shows the step response of both the full-order and the reduced-order model along a time-varying parameter trajectory. The trajectory covers the complete parameter range from 31 to 68m/sairspeed. The rate of variation in the simulation is further selected to be unreasonably high with a maximum of 40m/s2. The reduced-order model nevertheless approximates the response extremely well in all outputs. This confirms that there are absolutely no rate-dependent errors introduced by the oblique projection method. The most prominent difference in the outputs are the high-frequency transients occurring immediately after the step input is applied. These high-frequency dynamics are not included in the reduced-order model as a consequence of the frequency-weighted approximation.
Figure. 5.13 shows the pole migration of both models over the parameter space with piece-wise linear interpolation between grid points. The plot confirms that the reduced-order
Flapδ2
3 0
−3
−6
−9 q(◦/s)
Flapδ1
0 0.2 0.4 0.6
az,center(g)
0 0.5 1 1.5 2
Time (s)
0 0.5 1 1.5 2
−0.6
−0.4
−0.2 0 0.2
Time (s) az,wing(g)
0 0.5 1 1.5 2
0 0.5 1
Time (s) Input(◦)
0 0.5 1 1.5 2
30 40 50 60 70
Time (s) Airspeed(m/s)
Figure 5.12: LPV simulation of a step response with varying parameter: full-order model ( 48 state variables) and reduced-order model ( 12 state variables).
model obtained by parameter-varying projection indeed retains continuous dependence on the parameter. It further shows that the loci of the lightly damped modes in the reduced-order model almost exactly coincide with those of the original full-order model.
Figures 5.14 and 5.15 finally show frequency response plots of the full-order and the reduced-order model evaluated for frozen parameter values. The reduced-order model agrees very well with the full-order model up the specified frequency of 100rad/s.
10
(a)Full-order model (48 state variables). 10
100
(b) Reduced-order model (12 state variables).
Figure 5.13: Pole migration across flight envelope.
31 40 50 60 68
(a) Full-order model (48 state variables).
31 40 50 60 68
(b)Modal interpolation (12 state variables).
Figure 5.14:Parameterized frequency response from outboard flapsδ1 to wing tip acceleration.
−2020400
(c) Wing flapsδ1 to center acceleration.
−202040600
(d)Wing flapsδ2 to center acceleration.
−202040600
(e)Wing flapsδ1 to wing tip acceleration.
−202040600
(f)Wing flapsδ2 to wing tip acceleration.
Figure 5.15: Frequency response of full-order model (48 state variables) at 33m/s ( ) and 65m/s ( ) airspeed compared to the proposed reduced-order model with 12 state vari-ables ( / ).