Sequential Nonlinear Identification

of Stability Derivatives by Gain Adaptation

K. Bousson

Avionics and Control Laboratory

University of Beira Interior; 6201-358 Covilha, Portugal

E-mail: kbousson@mail.telepac.pt; bousson@ubi.pt

The identification of stability derivatives is the backbone of adaptive flight control and airborne simulation systems mainly in the case when these derivatives are subject to changes. The present paper proposes an on-line stability derivative identification method for nonlinear flight models. The identification procedure computes an individual adaptive learning rate for each stability derivative instead of using a single adaptive learning rate for all the stability derivatives, each individual learning rate depending on a meta-learning rate. The method is on the one hand void of probabilistic information about the flight model and the measurement errors contrary to the Kalman filtering, and on the other hand it guarantees fast convergence to optimal values. The convergence properties of the method are stated and proved, and the practical efficiency is successfully demonstrated on an aircraft longitudinal model.

 

Stability derivative identification is required for adaptive flight control and airborne simulation systems. Indeed, the dynamics of an aircraft through its flight envelope is governed by the aerodynamic forces and moments that depend on the aerodynamic coefficients for which the knowledge about the stability derivatives allows one to compute them straightforwardly. It is more appropriate to estimate these parameters along with the acquisition of the necessary data during flight tests or during wind tunnel operations instead of doing that off-line as is often reported in most of existing work about the subject. The interest of estimating the stability derivatives on-line is manifold, for instance, the control strategy may be improved by using the estimated parameters for predicting some state or output variables, by adapting the control parameters according to these predictions as is done in adaptive predictive control, or by estimating the performances of the aircraft so that they can be improved efficiently.

The first approaches for aerodynamic parameter estimation in the past used frequency response data and simple semi-graphical method for the analysis. The methods allowed having the frequency response of the aircraft but not the parameters in the dynamic model expressed as differential equations. Because of the availability of computers, parameter estimation techniques were improved then, and many other techniques have been established. The most used techniques up-to-date are the regression methods based on least-square techniques, Kalman filtering and neural networks. Sri-Jayantha and Stengel proposed a method for identification of nonlinear aerodynamic coefficients based on the Estimation-Before-Modeling method. This consists first in estimating the time histories of the forces and moments from discrete measurement vector by using an extended Kalman-Bucy filter and a smoother, then the optimal estimates and the measured control variables are sorted into a number of subspaces, and finally the aerodynamic modeling is performed using a multiple regression scheme in each subspace. Later on, that method was combined with computational neural network models for improving aerodynamic coefficient identification. Their method makes use of first partial derivative information to estimate weights in individual feedforward neural networks for each aerodynamic coefficient. Due to the time required to train a neural network on sample data, such a method is unlikely to cope efficiently with on-line parameter identification, mainly in case of significant changes in the parameters or when atmospheric disturbances show magnitudes quite different from those known in the sample database. Iliff proposed a method based on the maximum-likelihood concept, which makes use of the Gauss-Newton approximation method for computing the model parameters.  Other significant activities have been conducted about aerodynamic parameter estimation based on statistical and least-square concepts. From the standpoint of modeling, Morelli developed an adequate multivariate polynomial model structure with accurate parameter estimates based on orthogonal modeling function generated from wind tunnel data. This consists in modeling stability derivatives, obtained through wind tunnel experiments, as linear combinations of orthogonal polynomials. The coefficients of the linear expression related to each stability derivative are then estimated through a linear regression technique. Before two on-line stability derivative identification methods are presented, based respectively on the time-domain and the frequency-domain. Both methods are based on the transformation of the problem into a linear regression problem, them classical methods are used for computing the required parameters.

Most of the aforementioned approaches coped  either with off-line identification based on linearized models, or with of on-line identification of nonlinear models that requires probabilistic knowledge about the flight model and the measurement errors. One of the best techniques used for coping with the on-line estimation of nonlinear system parameters is the extended Kalman filtering, however, it is known that the application of the Kalman filtering algorithm may be inefficient if the stochastic behavior of the system is not well known, mainly if the process noise covariance matrix is wrongly chosen. For the sake of accuracy and reliability in the analysis, simulation and control of aerospace systems, it is more suitable to deal with nonlinear models than with linear ones, and without requiring stochastic knowledge about the flight and measurement models.

The purpose of this paper is to investigate the  online identification of stability derivatives from nonlinear flight models. The method is void of prior stochastic knowledge about the flight model and void of linearization of the cost function contrary to the extended Kalman filtering. This work focuses on the suitability of the adaptive learning rate technique for nonlinear parametric identification, and subsumes the work done by Sutton about gain adaptation in that it enables to cope with nonlinear systems whilst the Sutton filtering is linear, and that it gives the answer to an open issue stated by Sutton as to the convergence properties of gain adaptation filtering. The next section states the problem to be solved, then the theoretical development is presented where an efficient adaptive learning rate method is given together with the convergence properties. Following this, the efficiency of the proposed method is demonstrated on the on-line identification of the stability derivatives of a nonlinear flight model.

The method presented in this paper gives some new insights into online stability derivative identification based on nonlinear flight models. It is shown that if the model is twice differentiable in the parameter space, then the learning rate may be adaptively computed from information based on the current learning error so that it achieves a minimum value. The theoretical results have been demonstrated on an aircraft flight model where estimated stability derivatives are shown to converge towards their actual values.

Contrary to the extended Kalman filtering, the method is void of prior probabilistic knowledge about the flight model and measurement errors, and void of the model linearization. Therefore, for nonlinear flight models with high dimensional parameter space, the proposed method may well be an alternative to the extended Kalman filter.

The nonlinear identification technique described and demonstrated here is general for the class of twice-differentiable models, and can well be applied to other physical systems and areas, mainly to adaptive predictive control, optimal control and neural network learning. Future work will investigate the extension of the method in these directions.