Application of Kalman Filter for Estimation of Aerodynamic Parameters of an Aircraft using Simulated Flight Data

Application of Kalman Filter for Estimation of Aerodynamic Parameters of an Aircraft using Simulated Flight Data

Tusharika

Aeronautical Engineering Department Samalkha Group of Institutions Panipat (Haryana)-132115, India

Abstract An attempt is made to estimate aerodynamic parameters using the simulated flight data of various flight vehicles using Kalman filter technique. The present paper demonstrates application of Kalman filter for the estimation of ballistic coefficient of a falling body by considering the effects of process noise. As compared to rigid aircraft the mathematical model of a flexible aircraft involves larger number of stability derivatives. Furthermore, for a flexible aircraft, additional derivatives need to be included due to aero elastic effects. Applicability of Kalman filter for parameter estimation is validated on simulated flight data generated for a rigid aircraft as well as for a flexible aircraft. It is concluded that Extended Kalman Filter method can be advantageously applied to estimate aerodynamic parameters from flight data of a flexible aircraft.

1. INTRODUCTION

   All modern aerospace vehicles rely upon an understanding of dynamics and control to improve system performance. An understanding of dynamic elements and the trade-off between vehicle dynamic characteristics require for successful system design, control system properties and system performance. Aircraft parameter estimation is one of the most outstanding and illustrated example of the system identification methodologies. The success of the system identification of the flight vehicle has been possible due to better measurement techniques and data processing capabilities provided by digital computers. Other factors that contribute to system identification are the developments in the fields such as estimation and control theory; the design of appropriate flight test and well understood basic principles of aerodynamic modeling1,2,3.. Kalman filter is a linear, optimal estimator of state variables of a linear, time-varying system, operating in a Gaussian stochastic environment. Filtering approach is an extension of Kalman filter. Optimal estimator here is referred to a computational algorithm that processes measurements to deduce a minimum error covariance of the state of a system combining all the information available. Generally, Kalman filter4, 5 is applicable and optimal for linear systems only. When either the system or the measurement equations are non-linear, the same algorithm can still be applied by local linearization of the system about the current state. Such filter applied to nonlinear systems is called Extended Kalman Filter (EKF)6, and it need not be optimal. EKF produces estimates of the parameters that

   approximately minimize the mean square error in the parameter estimates themselves as opposed to ML and least squares which minimize a cost function that is based on matching the output variable behavior given a specific input trajectory. Parameter estimation of an aircraft is done with many measurements like, acceleration (both linear and angular), angular orientation, speed, angle of attack, etc. But from cost effectiveness point of view, it may not be feasible to use many sensors for air borne vehicles that go through many development trials.

   In the present work, EKF method is used for parameter estimation from flight data of air borne vehicles. The motivation here was to conduct a study on the applicability of the method in extracting aerodynamic parameters by processing flight data obtained for different class of flight vehicle/store. The method has been applied to flight data of a one-dimensional ballistic target, flight data of a rigid aircraft in longitudinal short period mode and also to flight data of a flexible aircraft in longitudinal short period mode. Any parameter estimation method requires adequate information about vehicle dynamics to estimate aerodynamic parameters correctly. Flight data of one-dimensional ballistic target contains very limited information regarding its motion/control variables. The flight data obtained through aircraft maneuvers contains more information regarding its motion and control variables. The applicability of the EKF is tested for these three different classes of flight data. It is observed that EKF can be advantageously applied on flight data of flexible aircraft to estimate few aerodynamic parameters with acceptable level of accuracy.

2. KALMAN FILTER

   Kalman filter is an optimal recursive data processing algorithm and it incorporates all information that is available to the filter. It processes all available measurements, regardless of their precision, to estimate the current values of the variables of interest with use of

   • knowledge of the system and measurement device dynamics

   • the statistical description of the system noises, measurement errors, and uncertainty in the dynamic models

   – any available information about initial conditions of

   Q E wwT

   (2)

   the variables of interest

   This filter performs the conditional probability density propagation for problems in which the system can be

   The measurement equation, required for the application of extended Kalman filtering, is considered to be a nonlinear function of the states according to the equation

   described through a linear model and in which system and

   measurement noises are white and Gaussian. The three basic

   z hx v

   (3)

   assumptions in Kalman filter formulation are

   • the model is considered to be linear .

     where v is a zero-mean random process described by the measurement noise matrix [R], which is defined as

   • the measurement noise and system noise are white.

     R EvvT

     (4)

   • the measurement noise and system noise are Gaussian.

   z

   The physical implications of these assumptions are

   For systems in which the measurements are discrete, the nonlinear measurement equation is written as

   discussed. A linear model is justifiable because when nonlinearities do exist, the typical engineering approach is

   hx

   vk

   (5)

   k

   k

   to linearize about some nominal point or trajectory, achieving a perturbation model or error model.

   Whiteness implies that the noise value is not correlated in time. The knowledge of the present noise value is no way helpful to predict the noise value of any other time. Whiteness also implies that the noise has equal power at all frequencies. A system will be driven by a wideband noise, noise having power at all frequencies above the system bandpass and essentially constant power at frequencies within the system bandpass. The assumption of noise as white would extend this constant power level out across all frequencies. Within the bandpass of the system of interest, the fictitious white noise looks identical to the real wideband noise. The mathematics involved in the filter is vastly simplified by replacing the real wideband noise with a white noise which from the systems point of view is identical. Gaussianness is related with amplitude whereas whiteness is related with frequency. The probability density of Gaussian noise amplitude takes on the shape of a normal bell -shaped curve. This assumption is justified physically by the fact that a

   system or measurement noise is typically caused by a number of small sources. Mathematically, when a number of

   The discrete measurement noise matrix Rk

   consists of a matrix of variances representing each measurement noise ource. Since, the system equation, Eq.

   (1) and measurement equation, Eq (3) are nonlinear, a first order approximation is used in the continuous Riccati equations for the manipulation of systems dynamics matrix

   F and the measurement matrix H . The matrices are

   related to the nonlinear system and measurement equations according to the relations

   F f x (6)

   x x x

   H hx (7)

   x xx

   The fundamental matrix [ k ], required for the discrete Ricatti equations, can be approximated by the Taylor-series

   expansion for exp( F Ts) and is given by the equation

   independent random variables are added together, the

   F 2 T 2

   F 3 T 3

   summed effect can be described very closely by a Gaussian probability density, regardless of the shape of the individual densities. The first and second order statistics (mean

   k I

   • F Ts

     s s

     2! 3!

     …

     (8)

     and variance or standard deviation) of a noise process can be easily known. In the absence of any higher order statistics, there is no better form to assume than the Gaussian density. The first and second order statistics completely determine a Gaussian density, unlike most densities which require endless number of orders of statistics to specify their shape entirely. The Kalman filter, which propagates the first and second

     order statistics, includes all information contained in the conditional probability density.

     where Ts is the sampling time and I is the identity matrix. In our applications of extended Kalman filtering, the series is approximated by only the first two terms, because [ ]k is only used for the calculation of Kalman gains and the matrix may not necessarily improve the performance of the filter by considering more terms. Therefore [ ]k is given by

     The real world situation is described by a set of

     k

     I F Ts

     (9)

     nonlinear differential equations to apply extended Kalman filter techniques. These equations are expressed in nonlinear

     The matrix Ricatti equations, required for the computation of the Kalman gains, are given by the equations

     state-space form as a set of first-order non linear differential

     equations as

     M

     P T Q

     (10)

     x

     f x w

     (1)

     K

     M H T H M H T R

     1

     k k k 1 k k

     (11)

     where x represents system space, f x is a nonlinear

     k k

     k k

     function of those states and w is a random zero-mean process. The continuous process-noise matrix describing the random process w for the preceding model is given by

     P k

     I Kk H M k

     (12)

     where Pk is the covariance matrix representing errors in

     algorithm for the propagation of states from one time step to the other.

     the state estimates after an update,[Kk] is the Kalman gain and

     M k is the covariance matrix representing errors in the state estimates before an update. The discrete process-noise matrix

     Qk can be found from the continuous process-noise

     matrix according to equation

     Ts

     IV. FLIGHT DATA OF BALLISTIC TARGET (FD-BT)8 Knowledge of target ballistic coefficient is used in

     advance guidance laws such as predictive guidance to relax the interceptor acceleration requirements. In addition, knowledge of the target ballistic coefficient is required for fire control due to the importance of accurate intercept point predictions in launching the interceptor on a collision course.

     Qk

     Q T d

     0

     (13)

     Therefore, accurate estimation of ballistic coefficient of a target re-entering the atmosphere is very important for both guidance and fire control purposes. The flight data for

     The preceding approximations for the

     fundamental and measurement matrices are used in the computation of the Kalman gains. The new state estimate xk

     simulating such a vehicle motion is modeled to investigate the applicability of EKF method8 in extracting parameter (ballistic coefficient) from flight data.

     is the old state estimate

     xk 1

     projected forward to the new

     The one-dimensional example of a ballistic target

     falling on tracking radar is considered. The target was initially

     sampling xk plus a gain times a residual. The residual is the

     at 2, 00,000 ft above the radar and had a velocity of 6000 ft/s

     difference between the actual measurement nonlinear measurement h(xk ) .

     zk and the

     towards the radar, which is located in the surface of a flat Earth. The trajectory of the ballistic target is presented in Fig.

     18. The radar measures the altitude of the target with 25-ft standard deviation measurement accuracy. The radar picks

     xk

     xk

   • Kk

   [zk

   hxk ]

   (14)

   measurements for every 0.1-sec. The simulation is done for 30 sec. An extended Kalman filter is built to estimate the

   The old estimates that have to be propagated

   forward do not have to be done with the fundamental matrix but instead can be propagated directly integrating the actual nonlinear differential equations forward at each sampling interval. Euler integration is applied to the nonlinear system of differential equations and is given by the equation

   altitude, velocity and ballistic coefficient.

   The two forces acting on the object are the drag force and the gravity force. The equation that governs the motion of the object is given by

   Qp g B

   xk xk 1 xk 1Ts

   (15)

   x

   • g B

   (17)

   where the derivative is obtained from

   xk 1 f (xk 1 )

   (16)

   where x is the acceleration acting on the object , g B

   is the acceleration due to gravity and Qp the dynamic

   In the preceding equation the sampling time Ts is

   pressure and is given by the equation

   used as an integration interval. In the problems where the sampling time is large, Ts would have to be replaced by a

   Qp 0.5B

   x 2

   (18)

   small integration interval, or possibly a more accurate method of integration has to be used.

3. GENERATION OF FLIGHT DATA

where x is the velocity of the target and the air density

B in Eq. (18) is an exponential function of altitude and is given by the equation

Due to the non-availability of real flight data, simulated flight data is generated for different class of flight vehicles for the

B

0.0034e x / 22000

(19)

purpose of parameter estimation. Mathematical models used to generate flight data of ballistic target (FD-BT), longitudinal motion of rigid aircraft (FD-RA/C) and longitudinal motion of flexible aircraft (FD-AE A/C1, FD-AE A/C2) are presented.

The term in Eq. (17) is the ballistic coefficient and is expressed as

WB

For the generation of simulated flight data of ballistic target the acceleration equation governing the path of the trajectory was solved using Euler integration. In the case of aircraft,

Sref CD0

(20)

longitudinal equations of motion were solved using fourth- order Runge -Kutta method. Subsequently, the simulated flight-data of the air borne vehicles is used in Kalman filter algorithm and processed for the estimation of aerodynamic parameters. The equations of motion are needed for generating the simulated flight data and also in the Kalman filter

where WB is the weight of the ballistic target, Sref is

the reference area of the object and CD0 is the coefficient of drag. The acceleration acting on the object is expressed as the nonlinear second order differential equation as

0.0034g

e x / 22000

x B g

2 B

(21)

V. SIMULATED FLIGHT DATA OF RIGID AIRCRAFT (FD- RA/C)

In mostcases, longitudinal maneuvers predominantly excite the short period mode and not the phugoid mode. In short period mode the flight velocity is essentially constant during the maneuver. 3-2-1-1 control (elevator) input is considered for the excitation of short period mode. The example aircraft is chosen for the present study. Trim flight conditions correspond to straight and level cruise flight at an altitude of 1500m and at a Mach number of 0.6. The rigid body short period longitudinal response to a given elevator input was simulated for 8 seconds. The short period longitudinal response was simulated using the following equations.

Fig. 2 Simulated q response from flight data FD-RA/C

q uSCL 2m

(22)

m

q u 2 ScC

2IY

(23)

The equations that define coefficient of lift (CL ) and pitching moment coefficient (Cm ) in Eq. (22) and Eq. (23) to describe the aerodynamic model are presented in Eq. (24) and Eq. (25)

qc Fig. 3. 3-2-1-1 Elevator Control Input,

CL CL0 CL CLq 2u CLe e

(24)

max

0.10471 rad

C C C C qc C

m m0 m mq 2u me e

(25)

VI. SIMULATED FLIGHT DATA OF FLEXIBLE AIRCRAFT FD- AE A/C1 & FD-AE A/C2)

The pitching moment coefficient (Cm) is with reference to the center of gravity. The Eq. (4) and Eq. (5) were solved using fourth-order Runge-Kutta algorithm with a time step of 0.001 sec to obtain simulated flight data. The contribution due to

The non linear equations of motion for a flexible aircraft are considered which are obtained from Ref. 7.

aeroelastic effect was neglected for rigid aircraft case. The flight data used for parameter estimation is pictorially

q

• C C

qc C

n C

C

c 2u

presented in Fig. 1 and Fig. 2. This flight data will be referred

L0 L

uS 2m C

Lq 2V

L e e

i 1

Li i

Li i

to as FD-RA/C for further purpose. The elevator input (3-2-1-

1. used to generate this flight data is shown in Fig. 3.

q 2

C

qc 2u C

n C

C

(26)

c 2u

u Sc 2IY Cm mq

m e e

i 1

mi i

mi i

(27)

where angle of attack , pitch rate q and control

input

e represent the perturbations and i

and

i

are the

e

i

generalized displacement coordinates and their

derivatives. CL , CLq ,

CL ,

Cm ,

Cmq ,

Cm e ,

CL ,

CLi ,

Cm

and

Cmi are the aerodynamic coefficients as

i

defined in Ref. 7. The second order differential equation that is satisfied by the generalized displacement coordinates and its

Fig. 1 Simulated response from the flight data FD-RA/C

derivatives is given in Eq. 28. The additional term representing the structural damping is also included

2iii

2 n 2 2

i Ci qc 2u Ci n Ci

• Cii c 2u

i i i i i i

u Sc 2Mi C q

j1

j j

j

j

(28)

where i , i

and Mi are the in-vacuo frequency,

modal damping, and modal generalized mass respectively and

Ci ,

Ci ,

Ci , Ci and

Cii

represent the generalized

q j j

force derivatives due to coupling in elastic and aerodynamic degrees of freedom. Eq. (26), Eq. (27) and Eq. (28) are integrated to generate motion variables and q . The

elevator control input is 3-2-1-1 and is presented in Fig. 3. The aerodynamic model incorporating aero elastic effects is presented in Eq.(29) and Eq.(30)

L L0 L Lq L e e L i L ii i

C C C C qc 2u C 4 C C c 2u

i1

(29)

m m0 m mq m e m i m ie i i

4

C C C C qc 2u C C C c 2u

i1

(30)

The flight data obtained for this case is pictorially presented in Fig. 4 and Fig. 5. To study the effect of flexibility of the structure on parameter estimation, two cases are considered. The two set of modal frequencies for the two different configurations. To start with aircraft with moderate flexibility is considered. The flight data obtained with this case will be referred to as FD-AE A/C. In the second case more flexibility is included and the flight data generated for this case is referred to as FD-AE A/C2.

Fig. 4 Simulated response from the flight data FD-AE AC1 and FD-AE AC2

Fig. 5 Simulated q response from the flight data FD-AE AC1 and FD-AE AC2

In this chapter, estimated parameters obtained through EKF method are presented. In the case of ballistic target the factors that affect the behavior of the filter are discussed. The simulated data corresponding to ballistic target (FD-BT), rigid aircraft (FD-RA) and flexible aircraft (FD-AE AC1 & FD-AE AC2) are used as the measured flight data. The estimated parameters are presented along with their standard deviations to assess the accuracy of the estimates of aerodynamic parameters.

1. ESTIMATION OF BALLISTIC COEFFICIENT FROM FLIGHT DATA OF BALLISTIC TARGET (FD-BT)

   The ballistic target is assumed to fall on a straight line path towards the surface based tracking radar. The ballistic coefficient ( ) is estimated through EKF method. It has

   been observed that there is a negligible change in the estimated value of by the increase of terms in the Taylor

   series expansion for the approximation of fundamental matrix

   k . This is valid because the fundamental matrix is

   actually an infinite Taylor series expansion of the product of sampling time ts and system dynamics matrix F .

   	True Value	Estimated Value by 2 terms	Estimated Value by 3 terms
   Ballistic Coefficient (Lb/Ft/ sec2 )	500	497.8827 (0.290904)*	499.8196 (0.290046)*

   * Standard Deviation

   Table 1 Estimated ballistic coefficient without process noise

   Fig. 6 Estimation of ballistic coefficient without process noise

   Fig. 7 Error in the estimation of ballistic coefficient without process noise

   For the same case the process noise is included assuming that the filters knowledge of the real world is in error. It has been

   e

   observed that addition of process noise to the filter increases the errors in the estimates. Also there is not much difference

   Fig. 8 Estimation of ballistic coefficient with process noise

   Fig. 9 Error in the estimation of ballistic coefficient with process noise

2. ESTIMATION OF AERODYNAMIC PARAMETERS FROM

   FLIGHT DATA OF RIGID AIRCRAFT (FD-RA/C)

   Knowledge of aerodynamic parameters is of paramount importance, to develop accurate mathematical model which represents the longitudinal dynamics of an aircraft. The aerodynamic parameters needed to develop the aerodynamic

   by increase of number of terms in the Taylor series expansion

   model are force derivatives CL ,

   CLq , CL

   and pitching

   e

   for the approximation of fundamental matrix k . From the

   moment derivatives

   Cm ,

   Cmq ,

   Cm . Thus, a study was

   Table. 1 and Table. 2 it is evident that by including process noise the accuracy in the estimates deteriorated. Similar observation can be made by comparing Fig. 6 and Fig. 7 with Fig. 8 and Fig. 9.

   carried out to explore the possibility of extracting the parameters using EKF method. The simulated data generated, FD-RA/C of the example aircraft is added with Gaussian noise and is used as the measured data in the EKF algorithm. The six aerodynamc

   Table 2 Estimation of ballistic coefficient with process noise

   parameters C , C , C

   , C , C and

   C are

   L Lq

   Le

   m mq

   me

   	True Value	Estimated Value by 2 terms	Estimated Value by 3 terms
   Ballistic Coefficient ( Lb/Ft/ sec2 )	500	488.9278 (11.07214)	488.7636 (11.23639)

   considered as state variables along with the flight variables,

   and q . The algorithm is run for an elevator input 3-2-1-

   1 for 8 seconds and the estimated aerodynamic parameters are tabulated along with their standard deviations in Table. 3. The measurement noise variance is 0.016 for both and q.

   In Case A the values of diagonal elements of process noise matrix are same and are randomly chosen to be 0.008 and in Case B the values are differently chosen as an attempt to tune process noise.

   Parameters	True Value	No Process Value	Process Noise Case A	Process Noise Case B
   CL	2.922	2.9096	2.8772	2.8887
   		(0.00764)	(0.06228)	(0.08767)
   CLq	-14.7	-22.7217	-16.6971	-16.8602
   		(0.75708)	(1.78776)	(1.89439)
   CL e	0.435	0.6756 (0.01976)	-0.3212 (0.13688)	-0.2328 (0.14417)
   Cm	-1.66	-1.6078	-1.8501	-1.6099
   		(0.00166)	(0.03303)	(0.0784)
   Cmq	-34.75	-38.3583	-39.0636	-41.527
   		(6.68809)	(6.9463)	(6.94662)
   Cm e	-2.578	-2.25619 (0.00289)	-2.7122 (0.10493)	-2.8064 (0.0588)

   Table. 3 Estimated Aerodynamic Parameters from FD-RA/C

   identify a simplified model with reduced number of parameters, and to evaluate how the resulting parameters are affected by model simplifications. To start with, a rigid body model was assumed and parameters were estimated from flight data that contain the aero elastic effects. It is expected that the parameter estimates thus obtained would absorb the aero elastic effects. For the convenience of discussion, these parameters are referred by equivalent parameters in Ref. 9. An analytical expression has been proposed to analytically compute the numerical values of the equivalent parameters. It is shown that the numerical value of analytical expression indicates the degree of flexibility of the aircraft, and thereby, a criterion based on it is suggested for deciding adequacy or otherwise of using simpler rigid body models in estimation algorithm. In the EKF algorithm only aerodynamic

   e

   e

   parameters CL , CLq , CL , Cm , Cmq

   estimated and presented in Table. 4.

   and

   Cm

   are

   Fig. 10 Comparison of the estimated value with that of FD-RA/C

   Fig . 11 Comparison of the estimated q value with that of FD-RA/C

   Through the values listed in Table 3 it can be observed that by including process noise the estimated parameters show large variance however we can also infer that by proper choice of noise values the estimate of parameters can be improved.

3. ESTIMATION OF EQUIVALENT AERODYNAMIC PARAMETERS FROM FLIGHT DATA OF FLEXIBLE AIRCRAFT

(FD-AE A/C1 & FD-AE A/C2)

The simulated data from FD-AE A/C1 and FD-AE A/C2 is used in the EKF algorithm by adding Gaussian noise to the data. All the 4 elastic modes of the test aircraft are considered in the generation of the simulated data. A full order model of an aero elastic aircraft has too many parameters to yield satisfactory estimates using any of the conventional parameter estimation methods. In view of this, a study was carried out to

Table 4 Equivalent Aerodynamic Parameters from FD-AE A/C1 and FD-AE A/C2

For the case of FD-AE A/C 1 the measurement noise for and q is 0.016 but for the case of FD-AE A/C 2 the measurement noise is 0.09. For the case of FD-AE A/C 2 the aero elastic effects of more flexible aircraft are got included in the algorithm as measurement noise.

XI. ESTIMATION OF AERODYNAMIC PARAMETERS FROM

FLIGHT DATA OF FLEXIBLE AIRCRAFT (FD-AE A/C1 & FD-AE A/C2)

In the final case both the flight data FD-AE A/C1 and FD- AE A/C2 are processed by the EKF algorithm. The algorithm includes only 15 parameters, taking into account only the first

elastic mode. Along with the flight variables and q the

generalized displacement coordinate 1 and its derivative 1 are also taken as the state variables. The estimated values of aerodynamic parameters are presented in Table. 5. It is seen that the estimation deteriorates as the flexibility increases. Thus for high flexibility aircraft such an approximation may not yield better results Comparison of estimated response of

,q with that of the flight data FD-AE AC1 and FD-AE

AC2 are presented pictorially in Fig.12 to Fig. 15. Based on these Fig. 12 to Fig. 15 it can be easily seen that the estimated response (, q ) closely matches with the simulated response.

However, there is a large difference between these parameters; it can be used for simulators and control law specifications to initiate the analysis.

Fig . 12 Comparison of estimated value of with that of FD-AE A/C1

Fig . 13 Comparison of estimated value of q with that of FD-AE A/C1

Fig . 14 Comparison of estimated value of with that of FD-AE A/C2

Fig . 15 Comparison of estimated value of q with that of FD-AE A/C2

Table 5 Estimated Aerodynamic Parameters from flight data FD-AE AC1 and FD-AE AC2

CONCLUSION

In the present paper, EKF method has been applied to estimate aerodynamic parameters from simulated flight data. The method has been applied starting from flight data of a one- dimensional ballistic target to flight data of a rigid aircraft and then to the flight data of a flexible aircraft. It is observed that EKF method can be applied successfully to estimate the parameters from the flight data of the three

cases. The method also estimates equivalent aerodynamic parameters in which aero elastic effects get absorbed. If appropriate mathematical model of the system is provided then the method can be used advantageously to estimate force and moment derivatives.

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