Direct Estimation of Nonlinear Aerodynamic Coefficients

Abstract

(Abstract) In this paper a parameter estimation technique is proposed which allows for direct updates to the aerodynamic coefficients represented in the form of table lookup. The proposed method is capable of capturing the nonlinearities in the coefficients. It is also possible to capture joint dependence of the coefficient on one or more independent variables. The estimation is cast in the form of linear least-squares problem. The recursive form of this algorithm is also proposed. The results are demonstrated using simulated data as well as flight data. I. Introduction Aerodynamic data of an aircraft is updated by Flight-testing. The response of the aircraft to control inputs is used to compute the dynamic model of the aircraft. Estimation of nonlinear dependence of the aerodynamic coefficients with respect to the variables like Mach number, angle of attack, sideslip, angular rates and control effectors is a particularly important aspect of the update process. The mathematical model is validated by comparing outputs obtained by simulation with flight data. There are many techniques developed over the years for estimation of flight characteristics (1,9). Most of these the techniques result in estimation of the linear equivalent model of the aircraft. This model is valid only in a restricted region of the flight envelope. The linear model can prove useful in quickly establishing the level of stability and for handling qualities evaluations (2). The linear model assumes particular importance for the calculation of the gain and phase margins for an unstable aircraft (3). Linear least squares is an efficient tool for the estimation of flight derivatives. The recursive version of least squares has been shown to be computationally efficient and can be used in real time (4). Many aircraft do not possess linear characteristics. This exposes two limitations of the linear model approach for such aircraft. Firstly, the estimated derivatives are capable of giving only the general trend of the coefficient without capturing the variations. Secondly, since linearity of response is restricted to small signal amplitudes only, this presents a problem in choosing the proper level of input excitation. At higher input levels, the nonlinearities dominate the response, while at very low levels estimation accuracy is compromised due to low response levels. This is undesirable particularly when the stability margins or loss of control effectiveness are to be estimated from flight data. The wind tunnel model of an aircraft can be quite comprehensive. This is achieved by representing the nonlinear variations in the form of Table lookup. Even with linear interpolation, it is possible to capture the characteristics with sufficient fidelity by choosing break points at proper locations. The linear estimation techniques do not permit direct update of these tables because they capture the average trend rather than local variations. Techniques for determination of airplane model structure from flight data using splines and modified stepwise regression are available in literature (5). Wind tunnel predictions through incremental coefficients obtained from flight data analysis are presented in (6,7). Also, postulating suitable derivative models in an analytical form can carry out identification of an aerodynamic database (8). In this paper we propose an estimation technique, which directly updates the aerodynamic coefficient dependency on one or more independent variables represented in table lookup form. This paper is organized as follows. In Section 2, a brief survey of existing methods is presented. In Section 3, a nonlinear model of the aircraft short period dynamics is formulated. The simulation response of this system is used to generate data for the estimation problem. Section 4 reformulates the multi-dimensional linear interpolation problem in a more general form. The procedure for estimation of the table data is posed as least