Solution Data generation step

A doublet elevator control input (with a pulse width of 2 s) is used in the aircraft model equations (state and measurement model) described in Example 3.3 to generate data for 8 s with a sampling time of 0.03 s. The aircraft data with process noise is simulated for moderate turbulence conditions. In order to have a realistic aircraft response in turbulence, a Dryden model is included in the simulation process (see Section B.14).

State estimation

The parameter vector to be estimated consists of the following unknown elements (see eq. (5.21)):

T = [βT, x₀T, GT]

where β is the vector of aircraft longitudinal stability and control derivatives:

β= [Cx0, Cxα, Cxα2, Cz0, Czα, Czq, Czδe, Cm0, Cmα, Cmα2, Cmq, Cmδe]

x0is the vector of initial values of the states u, w, q and θ :

x0= [u0, w0, q0, θ0]

G is the process noise matrix whose diagonal elements are included in for estimation:

G= [G11, G22, G33, G44]

The procedure for parameter estimation with time varying filter involves the following steps:

a As a first step, Fourier smoothing is applied to the simulated noisy measured data

to estimate the noise characteristics and compute the value of R [9]. This step is executed only once.

Time propagation step

b Predicted response of aircraft states (˜x = [u, w, q, θ]) is obtained by solving eq. (5.40). Assuming the initial values of the parameters defined in vector β to be 50 per cent off from the true parameter values and choosing suitable values for u,

w, q and θ at t = t0, the state model defined in Example 3.3 is integrated using

a fourth order Runge-Kutta method to obtain the time response of the states u,

w, q and θ .

c Using the measurement model defined in Example 3.3, eq. (5.41) is solved to obtain ˜y = [u, w, q, θ, ax, az].

d State matrices A and H are obtained by solving eqs (5.48) and (5.49).

e Next, the transition matrix  is obtained from eq. (5.4).

f With the initial value of the state error covariance matrix P assumed to be zero and assigning starting values of 0.02 to all the elements of matrix G (any set of small values can be used for G to initiate the parameter estimation procedure), eq. (5.42) is used to compute ˜P.

Filter error method 119 Correction step

g With R, ˜P (k)and H computed, the Kalman gain K(k) is obtained from eq. (5.43).

h Updated state error covariance matrix ˆP (k)is computed from eq. (5.45).

i Updated state vector ˆx(k) is computed from eq. (5.44).

Parameter vector update

j Perturbing each element j of the parameter vector one at a time (perturbation

≈ 10−7 _{j), steps (b) to (i) are repeated to compute yci(k), where yci(k)represents}

the changed time history response in each of the components u, w, q, θ , ax, az

due to perturbation in j. The gradient ∂y/∂ can now be computed

from eq. (5.33).

k The covariance matrix S is computed from eq. (5.50).

l Equations (5.36) to (5.39) are used to update the parameter vector .

Steps (b) to (l) are repeated in each iteration and the iterations are continued until the change in the cost function computed from eq. (5.35) is only marginal.

For parameter estimation with output error method, the procedure outlined in Chapter 3 was applied. The approach does not include the estimation of matrix G. For the simulated measurements with process noise considered in the present inves- tigation, the algorithm is found to converge in 20 to 25 iterations. However, the estimated values of the parameters are far from satisfactory (column 4 of Table 5.1).

Table 5.1 Estimated parameters from aircraft data in turbulence [10]

(Example 5.1)

Parameter True values Starting values Estimated values from OEM Estimated values from TVF Cx0 −0.0540 −0.1 −0.0049 −0.533 Cxα 0.2330 0.5 0.2493 0.2260 C_xα2 3.6089 1.0 2.6763 3.6262 Cz0 −0.1200 −0.25 −0.3794 −0.1124 Czα −5.6800 −2.0 −4.0595 −5.6770 Czq −4.3200 −8.0 1.8243 −2.7349 Czδ −0.4070 −1.0 0.7410 −0.3326 Cm0 0.0550 0.1 −0.0216 0.0556 Cmα −0.7290 −1.5 −0.3133 −0.7296 C_mα2 −1.7150 −2.5 −1.5079 −1.7139 Cmq −16.3 −10.0 −10.8531 −16.1744 Cmδ −1.9400 −5.0 −1.6389 −1.9347 G11 – 0.02 – 5.7607 G₂₂ – 0.02 – −6.4014 G33 – 0.02 – 5.3867 G₄₄ – 0.02 – 2.1719 PEEN (%) – – 46.412 9.054

This is in direct contrast to the excellent results obtained with the output error approach (see Table 3.4). This is because the data in Example 3.3 did not have any process noise and as such the output error method gave reliable parameter estimates (see Section B.13) and an excellent match between the measured and model-estimated responses. On the other hand, the response match between the measured and esti- mated time histories of the flight variables in the present case shows significant differences, also reflected in the high value of|R|.

Parameter estimation results with the time varying filter show that the approach converges in about four iterations with adequate agreement between the estimated and measured responses. The estimated parameters from the time varying filter in Table 5.1 compare well with the true parameter values [10]. During the course of investigations with the time varying filter, it was also observed that, for different guesstimates of G, the final estimated values of G were not always the same. How- ever, this had no bearing on the estimated values of the system parameters (vector β), which always converged close to the true parameter values. It is difficult to assign any physical meaning to the estimates of the G matrix, but this is of little signifi- cance considering that we are only interested in the estimated values of derivatives that characterise the aircraft motion. Figure 5.3 shows the longitudinal time history match for the aircraft motion in turbulence, and the estimated derivatives are listed in Table 5.1. 50 40 u , m/s 30 0 2 4 6 8 50 40 u , m/s 30 0 2 4 6 8 20 0 w , m/s –20 0 2 4 6 8 20 10 w , m/s 0 0 2 4 6 8 0.5 0 q , rad/s –0.5 0 2 4 6 8 0.5 0 q , rad/s –0.5 0 2 4 6 8 0.5 0 , rad –0.5 0 2 4

time, s (OEM) time, s (TVF)

6 8 0.5 0 , rad –0.5 0 2 4 6 8 measured estimated

Figure 5.3 Comparison of the measured response in turbulence with the model

Filter error method 121

From the results, it is concluded that the time varying filter is more effective in estimating the parameters from data with turbulence compared with the output error method. Although the time varying filter requires considerably more computational time than the output error method, no convergence problems were encountered during application of this approach to the aircraft data in turbulence.

5.4

Epilogue

The output error method of Chapter 3 accounts for measurement noise only. For parameter estimation from data with appreciable levels of process noise, a filter error method or an extended Kalman filter has to be applied for state estimation. The system parameters and the noise covariances in the filter error method can be estimated by incorporating either a steady state (constant gain) filter or a time varying filter (TVF) in the iterative Gauss-Newton method for optimisation of the cost function. The steady state filter works well for the linear and moderately nonlinear systems, but for a highly nonlinear system, the time varying filter is likely to yield better results. The difficulties arising from complexities in software development and high consumption of CPU time and core (storage/memory) have restricted the use of the time varying filter on a routine basis.

In the field of aircraft parameter estimation, the analysts usually demand the flight manoeuvres to be conducted in calm atmospheric conditions (no process noise). However, in practice, this may not always be possible since some amount of turbu- lence will be present in a seemingly steady atmosphere. The filter error method has been extensively applied to aircraft parameter estimation problems [11,12]. The extended Kalman filter (EKF) is another approach, which can be used to obtain the filtered states from noisy data. EKF is generally used for checking the kinematic consistency of the measured data [13].

5.5

References

1 BALAKRISHNAN, A. V.: ‘Stochastic system identification techniques’, in KARREMAN, H. F. (Ed.): ‘Stochastic optimisation and control’ (Wiley, London, 1968)

2 MEHRA, R. K.: ‘Identification of stochastic linear dynamic systems using Kalman filter representation’, AIAA Journal, 1971, 9, pp. 28–31

3 YAZAWA, K.: ‘Identification of aircraft stability and control derivatives in the presence of turbulence’, AIAA Paper 77-1134, August 1977

4 MAINE, R. E., and ILIFF, K. W.: ‘Formulation and implementation of a practical algorithm for parameter estimation with process and measurement noise’, SIAM

Journal on Applied Mathematics, 1981, 41, pp. 558–579

5 JATEGAONKAR, R. V., and PLAETSCHKE, E.: ‘Maximum likelihood estimation of parameters in linear systems with process and measurement noise’, DFVLR-FB 87-20, June 1987

6 POTTER, J. E.: ‘Matrix quadratic solutions’, SIAM Journal Appl. Math., 1966,

14, pp. 496–501

7 JATEGAONKAR, R. V., and PLAETSCHKE, E.: ‘Algorithms for aircraft parameter estimation accounting for process and measurement noise’, Journal

of Aircraft, 1989, 26, (4), pp. 360–372

8 MAINE, R. E., and ILIFF, K. W.: ‘Identification of dynamic systems’, AGARD AG-300, vol. 2, 1985

9 MORELLI, E. A.: ‘Estimating noise characteristics from flight test data using optimal Fourier smoothing’, Journal of Aircraft, 1995, 32, (4), pp. 689–695 10 SINGH, J.: ‘Application of time varying filter to aircraft data in turbulence’,

Journal of Institution of Engineers (India), Aerospace, AS/1, 1999, 80, pp. 7–17

11 MAINE, R. E., and ILIFF, K. W.: ‘User’s manual for MMLE3 – a general FORTRAN program for maximum likelihood parameter estimation’, NASA TP-1563, 1980

12 JATEGAONKAR, R. V., and PLAETSCHKE, E.: ‘A FORTRAN program for maximum likelihood estimation of parameters in linear systems with process and measurement noise – user’s manual’, DFVLR-IB, 111-87/21, 1987

13 PARAMESWARAN, V., and PLAETSCHKE, E.: ‘Flight path reconstruction using extended Kalman filtering techniques’, DLR-FB 90-41, August 1990

5.6

Exercises