Identification of NonStationary Aerodynamic Characteristics of an Aircraft Based on Flight Data

Abstract

The problem of the description of the nonstationary aerodynamic coefficients of an aircraft is considered. An approach based on parametrization and subsequent identification of aerodynamic transient functions by the time-frequency method is proposed. The effectiveness of the approach is shown by the example of identifying the lift coefficient of a short-region aircraft from the flight data.

APPENDIX

The time-frequency identification method is used to calculate the estimates of the parameters of dynamical systems from the observed data [8]. The peculiarities of the method in the problem of identifying mathematical models of the dynamics of aircraft from the flight data, which is considered as a problem of parametric optimization, include the following points:

(a) replacement of the differential equations of a dynamical system with algebraic expressions;

(b) simplicity of calculating the frequency characteristics of the observed variables in the presence of measurement noise;

(c) the ability to independently and randomly select points of the frequency range for each pair of input and output signals;

(d) the possibility of application to identify unstable dynamical systems.

Frequency identification methods make it possible to construct a set of mathematical models of various target orientations, which have their own advantages and disadvantages, as well as their own areas of applicability. The method is based on a transition to the frequency domain using a finite Fourier transform on a specially selected discrete set of frequencies. The discrete set of frequencies at which the finite Fourier transform of the function $x(t),t \in [0,T]$ is calculated has the form

$$\Omega = \left\{ {{{\omega }_{k}}:{{\omega }_{k}} = \frac{{2\pi }}{T}k,\;{\kern 1pt} k = \overline {1,K} } \right\},$$

(A.1)

where $K \leqslant T{{f}_{N}};$ ${{f}_{N}} = 1{\text{/}}2h$ is the Nyquist frequency and h is the measurement step.

It is easy to see that on this set of frequencies the following condition is fulfilled:

$${{e}^{{ - j{{\omega }_{k}}T}}} = {{e}^{{ - j2\pi k}}} = 1,\quad \forall {{\omega }_{k}} \in \Omega .$$

Formulas for the finite Fourier transform of the function x(t) on a discrete set of frequencies (A.1), taking into account the boundary conditions, take the form

$$\begin{gathered} {{X}_{T}}(j\omega ) = {{\mathcal{F}}_{T}}(x(t)) = {{\mathcal{F}}_{x}}(j\omega ) = \int\limits_0^T x(t){{e}^{{ - j\omega t}}}dt; \\ {{\mathcal{F}}_{T}}(C) = 0;\quad {{\mathcal{F}}_{T}}\left( {\frac{{dx(t)}}{{dt}}} \right) = j\omega {{X}_{T}}(j\omega ) + \Delta x; \\ {{\mathcal{F}}_{T}}\left( {\int\limits_0^t x(\tau )d\tau } \right) = \frac{1}{{j\omega }}\left[ {{{X}_{T}}(j\omega ) - \int\limits_0^T x(\tau )d\tau } \right]; \\ {{\mathcal{F}}_{T}}({{e}^{{at}}}) = \frac{{{{e}^{{aT}}} - 1}}{{a - j\omega }};\quad \Delta x = x(T) - x(0). \\ \end{gathered} $$

(A.2)

Here we need to note the following useful properties of the finite Fourier transform on the set of frequencies $\Omega \ni \omega$:

1. for $\forall C \ne 0$ the equality ${{\mathcal{F}}_{T}}(C) = 0$ holds on Ω, which indicates the equivalence of $\forall C \ne 0$ and zero;

2. the influence of the boundary values of transient processes on the finite Fourier transform is determined only by their difference Δx and does not depend on the frequency ${{\omega }_{k}},\forall k \in \overline {1,K}$.

It is recommended to calculate the Fourier integral of a table-defined function $x(t),t \in [0,T]$ by Philon’s formulas [11]. The monograph [8] contains the corresponding program in the language of the mathematical package MATLAB.