Print version ISSN 0100-7386
J. Braz. Soc. Mech. Sci. vol.22 n.2 Campinas 2000
Estimation of Aircraft Aerodynamic Derivatives Using Extended Kalman Filter
Centro Técnico Aeroespacial - Instituto de Aeronáutica e Espaço ASA/L, Praça Marechal Eduardo Gomes, no 50, Vila das Acácias - 12228-904, São José dos Campos - SP, Brasil
Design of flight control laws, verification of performance predictions, and the implementation of flight simulations are tasks that require a mathematical model of the aircraft dynamics. The dynamical models are characterized by coefficients (aerodynamic derivatives) whose values must be determined from flight tests. This work outlines the use of the Extended Kalman Filter (EKF) in obtaining the aerodynamic derivatives of an aircraft. The EKF shows several advantages over the more traditional least-square method (LS). Among these the most important are: there are no restrictions on linearity or in the form which the parameters appears in the mathematical model describing the system, and it is not required that these parameters be time invariant. The EKF uses the statistical properties of the process and the observation noise, to produce estimates based on the mean square error of the estimates themselves. Differently, the LS minimizes a cost function based on the plant output behavior. Results for the estimation of some longitudinal aerodynamic derivatives from simulated data are presented.
Keywords: Aerodynamic Coefficient, Aerodynamic Derivative, Kalman Filter, Parameter Estimation, Simulation.
The purpose of this work is to validate, through the use of a simulated case study, the functionality of a computer code, developed by the author, for estimating aerodynamic derivatives of a given aircraft using the Extended Kalman Filter (EKF). The development of such tool is of relevance for the Brazilia n aeronautical industry, which is now developing a new family of transport aircraft that may require the use of fly-by-wire control systems. The EKF is an estimation tool, of optimal stochastic control, that has been developed to attend the demands of the space program. In spite of the fact that it has been in use by other engineering fields (Brinati and Rios Neto 1977), only recently the parameter identification techniques have called the attention of the aeronautical community (Morelli, 1998; Velo and Walker, 1997; Tishler, 1995; Bauer and Andrisani, 1990; Iliff, 1989).
Morelli (1998) presents a method for a system identification cycle consisting of experiment design, and data analysis to be implemented aboard a test aircraft in real time. The technique uses error equation in the frequency domain, and simple design methods employing square wave inputs forms to design the test inputs in flight. Tischler (1995) makes a survey of system identification methods in the frequency domain for the development and integration of aircraft flight control systems. The work illustrates the extraction and analysis of several models of varying complexity. Results are presented for test data of several simulation programs at the Ames Research Center. Velo and Walker (1997) show a comparative study of aerodynamic coefficients obtained from wind tunnel tests and by numerical estimation methods such as: modified stepwise regression (MSR), and the EKF. Bauer and Andrisani (1990) demonstrate de use of the EKF to estimate the parameter of a low-order model, derived from the short period approximation of the longitudinal aircraft dynamics. The parameters obtained from the EKF analysis of flight data are compared to those obtained by frequency response analysis of the flight data, and the results used for determination of the flying qualities of the test aircraft. Lliff (1989) uses the problem of aircraft parameter estimation to illustrate the use of the maximum likelihood (ML) technique. The work discusses basic concepts of minimization and estimation, which are demonstrated for a simple simulated example.
Good estimation of aerodynamic coefficients and stability and control derivatives are of fundamental importance in the design and development process of an aircraft. Generally, these coefficients and derivatives are obtained using theoretical methods and wind tunnel testing. However, due to theirs approximate nature, the results obtained are sometimes far from reality. These coefficients are corrected through some laborious work, using flight test data. The corrected values are then implemented in flight simulators which are used not only for flight crew training, and for flight control systems development, but also for checking compliance with flight quality norms for certification purposes. The availability of an "adequate" filtering technique can be of great advantage in obtaining good estimations of the aerodynamic coefficients and derivatives for an aircraft.
A good estimation process depends, fundamentally, on the following requisites:
1) quality, quantity, and low collinearity among input and output data;
2) availability of a good dynamical model which closely represents the aircraft motion;
3) introduction of estimation techniques able to deal with non-linear effects, and to reduce losses in the filter learning capacity.
This work demonstrates, through the use of a simulated case study, the possibility to apply the EKF filter in the estimation of aerodynamic derivatives from flight test data, for a given aircraft.
bw = wing span (m)
Cw = mean aerodynamic chord of the wing (m)
Extended Kalman Filter
A relatively simple method for use in parameter estimation is the method of linear regression, based on the least squares (LS) criterion. Of simple implementation, this method has some drawbacks, among which is the fact that it produces highly biased estimates. This undesirable characteristic may be eliminated by the use of an unbiased estimator such as ML. However, the use of such methods in non-linear dynamical system is not always possible, especially for the cases were the system is unstable (relaxed stability aircraft). In these cases, if the estimates are not precise enough the model will diverge during the integration process. This is not the case for the EKF filter, where the estimates are adjusted at each observation (measurement) point such that the system is not destabilized (Kokolios, 1994).
The EKF shows several advantages over the more traditional LS methods. Among these the most important are: there are no restrictions on linearity or in the form which the parameters appears in the mathematical model describing the system, and it is not required that these parameters be time invariant. Finally, the EKF produces estimates of the parameters that approximately minimize the mean square error in the parameters themselves, as opposed to minimizing a cost function that is based in matching the input output behavior of a specific trajectory, which is what the LS and ML techniques do. The EKF method, however, presents some extra challenge. It requires a prior selection of statistical properties of the processes and measurement noises and a model for the parameter dynamics. Some of these informations are typically unknown to the designer, introducing more extra degrees of freedom that can be difficult to resolve (Velo and Walker, 1997).
The estimation problem, using the EKF filter, comprises the three basics steps described bellow:
In this phase, the covariance matrix is initialized with the suspected variance for each state, and the observation error matrix with the precision data of the measurement sensors. The states are initialized using available flight test data, and the parameters with previously estimated values using wind tunnel data, or calculated values using available theoretical methods.
In the propagation phase, the state variable, the parameters to be estimated, and the covariance matrix are integrated between each observation point. The extended state equation is used for the propagation of state variables and parameters, and the Riccati equation for the covariance matrix, Fig. (2).
The correction phase is what distinguishes the EKF from the other methods, Fig. (1). In this phase, the propagated state variables are compared with the observed values and adjusted accordingly, using a gain matrix known as Kalman gain matrix, Eq. (10). The same is true for the propagated covariance matrix This adjustment takes into consideration the variance of local estimates, and information about the precision of the observation at each point in order to minimize the mean square error in the estimated parameters. The Kalman gain matrix is a matrix that carries information about the precision of the observation, the plant disturbances, and the fidelity of the model used in the propagation phase.
A non-linear continuous dynamical system may be modeled in the form of state space variables:
where x(t) is the state, u(t) the control, and w(t) the noise (plant perturbations) vectors, i.e., the effect of atmospheric turbulence over the flight path of the aircraft. The disturbance vector, w(t) , is modeled as being a random process of normal distribution with variance Q.
The initial condition for the state variable vector, and the covariance matrix are then defined as:
The observations are functions of the state variables plus an additional term that represents the precision of the observation.
The observation is a discrete process, where the vector vk represents de measurement precision, modeled as a random process of normal distribution with variance R.
The states are propagated between data points through simple integration of the following equation:
where the control vector, u(t), is a deterministic parameter. The covariance matrix is propagated through the integration of the Riccati equation (Gelb, 1996).
The propagated state values are then corrected, at each observation point, using the estimated residues and the Kalman gain matrix:
(-) means estimated value before correction, and (+) estimated value after correction using data point at instant k. The correction for the covariance matrix is given by (Gelb, 1996):
The Kalman gain matrix is a matrix that carries information about the system disturbance (P) and about the observation precision (R), and is given by (Gelb, 1996):
The block diagram illustrated in Fig. (2), depicts the EKF filtering procedure. The upper set of blocks represents the plant dynamics and its sensors. The lower set of blocks represents the filter itself, where the model block represents the propagation phase of the filter, and the estimator block the correction phase.
The aerodynamic model adopted is a classical non-linear model most commonly used in simulations, performance, and stability and control studies.
¨ Lift Force
¨ Drag Force
¨ Pitching Moment
Where q (dynamic pressure) is a function of airspeed and air density:
Some simplifying assumptions, considered for this model are listed bellow:
1) compressibility effects are not taken into account;
2) due to its small magnitude, CDd was neglected;
3) the expression for CD, Eq. (14), is valid only for small perturbations around small values of angle of attack.
The linear model for the drag coefficient is justified by the fact that the aircraft, adopted in this work, has a low aspect ratio wing with zero incidence, and is assumed to be flying in high speed flight regime (in the linear range of the lift curve slope). These conditions lead to a relatively flat drag polar curve for angles of attack in the vicinity of the assumed flight condition.
The dynamical model used for the estimation is a non-linear longitudinal power-off model, described by the following set of equations:
and the state vector by the following set of variables, Fig. (3):
The observation equations are given by:
and the observation vector by:
The measured angle of attack (am) is equal to the angle of attack corrected by a factor that takes into account the distance (d) from the angle of attack vane to the aircraft center of gravity (CG). The last equation of the system represents de vertical load factor (nz).
The Von Karman model for the atmospheric turbulence was included in the simulation, in order to generate plant disturbances. The power spectral density for the vertical component of this turbulence model is given by (Nelson, 1990):
were sw is the standard deviation of the (vertical) gust velocity,Lw is a scaling factor, and W is the spatial frequency. These parameters are dependent on the altitude (h) given in meters, and the type of turbulence, i.e., clear air (high or low altitude) and thunderstorm turbulence. It is assumed that the turbulence field is independent of time (Nelson, 1990).
The following values where adopted:
Turbulence causes variations, of random nature, in the angle of attack (a ), in the airspeed (V), and in all other variables of the dynamic model, Fig. (4).
Estimation of Aerodynamic Derivatives
Simulated data, instead of flight test data, is used to estimate the following aerodynamic derivatives: CLa, CLd e, Cma, Cmq and Cmd e. The use of simulated data does not invalidate the results. Here, the objective is to test a procedure and demonstrate the use of the EKF filter in estimating these coefficients. The technique consists in modifying the original state equation augmenting the state vector with the parameters to be estimated. The augmented vector then becomes:
The propagation of these new state variables may be assumed as constant or as being equal to a normal random process,
This hypothesis is necessary to compensate any unknown disturbance in the dynamics of these parameters. The effect of using the pseudo noise assumption is to speed up the convergence of estimates at the cost of precision (Velo and Walker, 1997). The extended dynamics is then redefined as:
One of the most important aspects of this estimation technique is the modeling of the noises (perturbations) inherent to the processes. Plant noise (Q) was extracted from previous simulations, where disturbances generated by an atmospheric turbulence model were taken into account, Table 4. The measurement noises (R) were simply taken as previously known. The measurement noise was also adopted as initial state variance (P0 = R) for the observable state variables. For the parameters to be estimated the following criterion, as suggested by Bauer and Andrisani (1990), was considered: . These values were subject to some adjustment after the first estimates, using the residuals as guideline (filter tuning). For parameters with initial values equal to or close to zero, the values are established based on engineering judgment (Bauer and Andrisani, 1990). These noise data are shown in Tables 3 and 4.
Results and Discussions
The results obtained are encouraging as shown by Fig. (8) through Fig. (16), and Tables 1 and 2. The estimated values were coherent with the real values, and the residuals of the state variables are small. An interesting fact to notice, is some level of correlation among these coefficients. Especially between CLde, CLa , and Cmq as shown in Table 2. This interdependence has some important implications in the estimation processes. If a good estimation of one of these parameters cannot be obtained, the estimation of other may be adversely affected. Choosing a different dynamic model may minimize the correlation.
The simulations reveled that the terms: Cmdede, and Cmaa are very influential in the pitching moment equation, indicating that there is a good possibility for the accurate estimation of Cmde, and Cma . On the other hand the significance of terms like CLdede and Cmq . (qcw / 2Vare small so that the chance for an accurate estimation of CLde and Cmq is small. New studies with the purpose of implementing advanced procedures such as: factorization, to avoid possible numerical problems (Bierman, 1977; Kuga at all, 1990) or else adaptive schemes to get away from modeling related problems, are highly recommended (Rios Neto and Kuga, 1985).
In the following example, the coefficients used to initialize the estimation processes were equal to the values used in the simulation, called true values, corrupted by ± 15%, and called theoretical values.
Simulated State Variables:
The following set of graphics, Fig. (7a) through Fig. (7d), shows the simulated measurements of the observed variables. Comparisons between the simulated state variables and the simulated measurements are shown, whenever the measured variable corresponds to a state variable. This is not the case for the load factor (nz), which is a measured variable but not a state variable.
Estimated State Variables
The EKF estimates of the longitudinal state variables (V, a , q and q ), shown by Fig. (8a) through Fig. (11a), seems to agree reasonably well with the simulated longitudinal state variables. The residuals time histories are plotted against a ±s error bond, based on the values of the matrix, and are shown by Fig. (8b) through Fig (11b). The residuals for the angle of attack (a) and the pitch rate (q) are subject to large variations, that exceeds by much the error limits, during the maneuver. This may be due to the fact that these variables are subject to proportionally large oscillations relative to their steady state values, which is not the case for the airspeed (V) and the angle of attitude (q).
Estimated Parameters (Aerodynamic Derivatives)
Figures (12-16) shows the time histories for de aerodynamic derivative estimates. In all cases, the derivative values settle almost immediately after the maneuver. These results are encouraging. If the time histories of these parameters kept varying even after the maneuver, through the end of the estimation time history, then little confidence could be held in the estimation of these coefficients. These estimates behavior, however, were mixed in character. Some parameters are easier to estimate than others; this behavior seems to be directly linked to observation parameter.
For the purpose of comparison, the estimated aerodynamic coefficients data shown in Figs. 12 through 16 are all summarized in Table 1. At the same table, the values used for simulations are presented as well as the initial values for the estimation problem. The final values for the elements of the covariance matrix are displayed in Table 2.
The noise data used for estimation are shown in the following tables. Table 3 contains the noise data related to the state variables initial variance (Po) and the noise due to plant disturbance (Q). Table 4 contains the noise due to measurement precision (R).
A procedure was tested, and proved capable of providing good parameter estimations. The results obtained are encouraging and points towards future research aiming the implementation of advanced numerical methods and modeling techniques.
The EKF filter seems to be well suited to applications in adaptive control systems. It can provide good parameter estimates to a set of control laws, which uses these estimates to adapt to changing flight conditions. For this type of application, the control should be designed to rely on parameters associated with the pitching rate, which is reasonable since the pitching rate derivatives are better observed.
The algorithm is simple enough to be packed as a subroutine and included in the flight control computer, so that the estimation could be conducted in near real time during flight tests. This capability has obvious attraction in real time monitoring of the flight envelope expansion, during flight test campaigns.
All the geometric, mass and aerodynamic data of the airplane used in the calculations are presented in the APPENDIX.
This work was done as part of the requirement for post graduation in Engineering and Space Technology, at the department of Space Mechanics and Control of INPE (Instituto Nacional de Pesquisas Espaciais).
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Aircraft Data Used in the Simulations:
Manuscript received: April 1999, Technical Editor: Angela Ourívio Nieckele.