Mass Estimations based on the longitudinal equations of motion

The mass estimations based on the longitudinal aircraft model will be referred to as mass inferences, while the ones based on least squares (described later) asmass estima-tions, even if both are mass estimation processes. Also, for the reader’s convenience throughout the chapter, a non-exhaustive scheme of the mass estimation processes is provided in Figure 6.3.

According to the aircraft performance model proposed by BADA, the

computa-FIGURE6.3: Mass estimation scheme.

tion of the lift coefficient is directly dependent on the aircraft mass, and the drag coefficient is directly dependent on the lift (see (6.6)).

C_L= 2mg

ρV_a²Scos(φ) (6.6a)

C_D =C_D0+C_DiC_L² (6.6b)

Therefore, when these coefficients are used analytically to obtain mass inferences, a certain inaccuracy is self-induced when standard mass values are considered.

In spite of the fact that this inaccuracy is corrected by taking into account observa-tions of the flown trajectory, this section describes an algorithm capable of providing mass and thrust inferences without the use of mass-dependent lift and drag coeffi-cients. Thus, the proposed approach relies on machine learning techniques, more specifically, in the neural networks proposed in Section 4.1.3, to first estimate the lift and drag coefficients, and then infer the aircraft mass using a reduced longitudinal aircraft model. In this scenario, the training part of the neural networks becomes crucial, as it is determinant for the neural network performance.

Assuming that a well-trained neural network is available for the lift and drag coeffi-cients, which could be feasible thanks to wind tunnel testing, experimental-aircraft measures, or any other, a mass estimation can be performed as follows.

The assumption of a zero bank angle (φ≈ 0) as well as no yawing motion (ψ ≈ 0), produce a reduced three degrees of freedom longitudinal aircraft model, expressed in the wind frame by

V˙_a= F_thrc_α−D

m −gs_γ (6.7a)

˙

γ= F_thrsα+L mV_a − g

V_acγ (6.7b)

Assuming that V_a, γ and α are measured or computed at a sample time t_k+1 = tk+∆t, the system (6.7) is discretized using the Euler method, such that the following

equations are obtained: Besides, it is assumed that the aerodynamic forcesD andL are known using ap-proximated models. On the other hand, the thrust force and mass are assumed to be unknown and unmeasurable. Thus, an on-line inference of these variables can be proposed.

For this purpose, (6.8a) and (6.8b) are rearranged as Va_k+1−Va_k

Then, dividing (6.9a) by (6.9b), an auxiliary variableσ_kis defined as σ_k= Fthr_kcαk−Dk Hence, after manipulating (6.10a), an estimate of the thrust force at the instantt_k, but computed at the instanttk+1, can be expressed as

Fˆ_thrk = D_k+σ_kL_k cαk−σksαk

(6.11) whereσ_k can be computed from (6.10b). Finally, introducing (6.11) into (6.8a) and (6.8b), two possible expressions for mass inference are obtained:

ˆ wheremˆ1k is feasible at any flight condition with the exception of constant speed and zero flight path angle. In this case, themˆ2_k value can be used. Therefore, the inferred mass at the instantt_k, referred to it asmˆ_kis known at the instantt_k+1. In order to corroborate the feasibility of this mass inference, data corresponding to the climb phase of a 10min simulated flight using the model described in Chapter 4 was obtained. The aircraft climbs from 1,000m to approximately 4,000m (FL30 to FL130) (see Figure 6.4), starting with an airspeed (TAS) of100m/s (194knot) and finishing at122m/s (237.5kn). During the whole simulation, a pitch angle of10^◦ is maintained. The variables used to infer the aircraft mass and the Thrust are shown in Figure 6.5. The quality of these data is assumed to be reliable (accurate, inte-gral, continuous and available), thanks to the use of up-to-date navaids+GNSS and

6.2. Mass Estimations based on the longitudinal equations of motion 125

FIGURE6.4: Altitude and Rate of Climb for mass inference.

FIGURE6.5: Variables used to infer aircraft mass and Thrust profile.

ADIRS systems.

Regarding the aircraft mass, it is assumed that a measure of this parameter is subject to uncertainty. Therefore an initial mass of50,000kg, decreasing under the

FIGURE6.6: Fuel consumption and Thrust profile inference.

effects of a gaussian noise (with standard deviation of100kgto avoid the simulation of an ideal measure), provoked by a fuel consumption dependent on the thrust and airspeed, is considered as the "real" mass of the aircraft during the climb. Concern-ing the "real" thrust profile, it is dependent on the altitude, speed, and drag. The fuel consumption associated to the real mass behaviour and the inferred thrust under a 12ssampling are shown in Figure 6.6. Although the generation of the thrust profile is out of the scope of this section, it must be noted that as the altitude increases, the fuel consumption and the thrust decrease, always maintaining a growing airspeed, which fits the general behaviour of aircraft during the climb phase.

Regarding the mass inference, since different values of ∆t affect the inference, in Figure 6.7 are shown the results of mass inferences at different sampling times with respect to the real mass. For a sampling at1s, the inferred mass converges within a bounded error of±400kg, and for2,5, and12s, bounds of±300,±300, and±200kg are sufficient. This error reduction is thought to be related to the low-pass filter effect that the increase in the sampling time has on the variables affected by the gaussian noise added to avoid an ideal measure of the mass. The error of these inferences with respect to to an average value of the real mass (an accurate mass measure), is analyzed later.

Hence, it could be assumed that the mass is now available for further treatment, but in addition to this, thanks to a method based on least squares and the computed mass inference, the initial mass of the aircraft can be estimated.

6.2. Mass Estimations based on the longitudinal equations of motion 127

(a) Sampling time of1s. (b) Sampling time of2s.

(c) Sampling time of5s. (d) Sampling time of12s.

FIGURE6.7: Mass inference at different sampling times.