Aircraft Mass Estimation
6.3 Mass Estimation based on Least Squares
Since the aircraft mass has been inferred repeatedly by the previous approach, refer-ring to thekthvalue (known at the instanttk+1) asmˆk. These inferences can be used to propose an initial mass that minimizes the error (in a least squares sense) between the real mass, and the inferred mass.
Let the aircraft mass be modeled by the BADA expression
˙
m=−µFthr (6.14)
where the aircraft mass decreases depending on the Thrust force and a thrust specific fuel consumptionµ. This coefficientµcan be extracted from the BADA files, or the ICAO engine databank.
Assumingµ is measured or computed at a sample time tk+1 = tk + ∆t, the mass dynamic is discretized using the Euler method, such that the following equation is obtained:
mk+1 =mk−∆tµkFthrk (6.15)
In this manner, from a general point of view, the measure of the aircraft mass can be assumed to come from the substraction of the weight of the burned fuel (given by equation (6.15)) to an unknown initial mass.
Therefore, assuming that a vector of inferred masses has been computed with the approach based on the longitudinal aircraft model in Section 6.2, the fuel consump-tion model makes possible the knowledge of the initial aircraft mass by adding all the fuel consumed until a determined time. In other words, the initial mass of an aircraft can be estimated using a fuel consumption model and the mass inferences known until the instanttk+1, denoted asmk+10 .
This is possible by minimizing the least-squared error (Els) given by min(Els) =min
wheremˆlis the vector containing all the mass inferences, and the term m0−
l
X
h=0
∆thµhFthrh (6.17)
represents the real mass, given by the substraction of the weight of the burned fuel (given by equation (6.15)) to the initial massm0, considered unknown.
Hence, to solve this minimum problem, the first and second derivatives of (6.16) with respect tom0are obtained. The first derivative is given by
2
and by differentiating one more time (6.18), it is found that the second derivative is positive
6.3. Mass Estimation based on Least Squares 129 Thus, the minimum of (6.16), is found by equalizing to zero equation (6.18) and solving form0, leading to
mk+10 = 1 k
" k X
l=0
ˆ ml+
l
X
h=0
∆thµhFthrh
!#
(6.20) In order to test the approach, the flight simulation of the10minclimb from the pre-vious section was used. Hence, assuming a known fuel consumption, provided by BADA or the ICAO Engine Emission Databank, the initial mass estimation using three different sampling times (2s,5s, and10s) is shown in Figure 6.8. It can be seen how as the available data to solve the least squares problem increases in size, the initial mass estimation becomes more accurate.
Moreover, the first value of the mass inferences (using the longitudinal equations)
FIGURE6.8: Initial mass estimation.
and the initial mass estimation (using the least squares), are compared with respect to the true initial mass. This errors, named initial mass inference error, and initial mass estimation error, are compared in Figure 6.9. The superior performance of the least squares approach at different sampling times is clear.
It must be noted that the presented approach differs completely from a least squares fitting application. If a fitting problem was considered, a polynomial function would be adapted to the mass inferences, such that the polynomial describing the mass evolution would dictate a fuel consumption disregarding any available prior infor-mation such as the BADA or the ICAO Engine Emission Databank model. In our case, the solution of a least squares method is used, but no polynomial is adjusted to fit the data.
At this point, since the initial mass estimation and the fuel consumption are consid-ered known, a recalculation of the mass until the instanttkcan be computed attk+1 (using the mass dyanmic model (6.15)), denoted asmk+1k
ls . This computation, will be increasing its accuracy as every new estimation of the initial mass comes closer to the true value, this is depicted in Figure 6.10. Note that the mass estimation is smoother than the mass inference.
(a) Original plot (b) Zoom
FIGURE6.9: Initial mass estimation and initial mass inference errors at different sampling times.
Let now the error of the mass estimation (usingmk+10 ) and mass inference be com-pared with respect to an accurate mass measure (represented by the simulated real mass without the gaussian noise). These errors are shown at different sampling times in Figure 6.11.
FIGURE6.10: Mass estimation using initial mass estimations.
6.3. Mass Estimation based on Least Squares 131
(a) Original plot (b) Zoom
FIGURE6.11: Mass estimation and mass inference errors at different sampling times.
6.4 Conclusions
A new method for online mass estimation, and initial mass estimation was pro-posed. The mass estimation using the longitudinal equations, despite beginning far away from the true value, converged faster than the mass estimation based on least squares. However, the inclusion of initial guesses into the mass estimation based on least squares approach could significantly improve its convergence time.
The least squares approach provided an accurate initial mass estimation, not only equally accurate, but also smoother, compared to the mass inferences approach (mass estimations using the longitudinal equations). Since the mass estimation process was proposed as a recursive and discrete algorithm, sampling times affected the be-haviour of it.
The approach to estimate the mass relied heavily on previous knowledge of the fuel consumption model, such that the inaccuracy or complete ignorance of this param-eter would have affected severely the method. Nevertheless, an adaptive estimation of the fuel consumption model to solve this issue could be addressed.
Moreover, since the fuel consumption models are well defined for the different flight phases, the method could be easily adjusted to complete flights and not limited to the climb phase.
The implementation of the proposed mass estimation methods could improve air-craft trajectory prediction, and ease the guidance efforts to follow a desired trajec-tory. In addition to this, it completed the simulation of the transport aircraft de-scribed in previous chapters.
133
Chapter 7
Introduction to Control Theory
The first feedback control system ever developed by the human being is though to be the Egyptian water clock of Ktesibios in Alexandria, around the third century b.c., since then, driven by curiosity, need, entertainment or even mistake, mankind has carried on to develop new control systems regardless of the discipline throughout history.
Nowadays, considering the ubiquity of mathematics to describe physical phenom-ena, control theory has gained great interest when a defined system, referred to as plant, need to fulfil some closed-loop behavior. In this manner, the design of a con-troller that allows a system to meet the desired characteristics has been the subject of worldwide research.
Control theory can be roughly divided in two branches, classic and modern. The first one, performs an analysis of physical systems using the Laplace transform as a key tool, and the second, allows the analysis of systems based mainly on Lyapunov theory. System analysis is done using the state-space representation, either for SISO (Single-Input/ Single-Output) or MIMO (Multi-Input/ Multi-Output) systems.
According to (Slotin and Li,1991), physical systems are nonlinear most of the time, and they can be classified in terms of their mathematical properties as continuous or discontinuous. In the case where the operating range to be analyzed is small, and the nonlinearities within this range are smooth, the system may be approximated by a linear one. However, there are "hard" nonlinearities (described by discontinuous functions) that cannot be approximated by linear functions, such as the Coulomb friction, saturation, dead-zones, backlash, and hysteresis.
In this manner, nonlinear and linear control laws are designed depending on the system.
It is a fact that linear control laws are likely to perform poorly, or even lead to unsta-ble behaviours when large range of systems operation are required. Thus, nonlinear control laws are of great interest. Furthermore, many control problems involving model uncertainties can be tolerated using nonlinear control theory. Nevertheless, when a linearized system is able to represent fairly enough a nonlinear one, the com-plexity can be reduced significantly and a linear control law is sufficient.
In this chapter, after providing the essentials about Linear Time-Invariant (LTI) sys-tems and Linear control techniques in Section 7.1, nonlinear syssys-tems theory and linear control techniques are given in Section 7.2. This background of linear / non-linear systems is stated considering that aircraft dynamics are highly nonnon-linear, and that flight control systems are based on control theory, either linear or nonlinear.
Moreover, the flight guidance systems of modern aircraft are briefly described in Section 7.3. Finally, conclusions are given in Section 7.4.