Just a simple visual analysis of the measured trajectories can give an initial confidence about the dynamic behavior of the airship. Studying the measured trajectories (Figure 6.1), one observes that the forward velocity remains nearly unchanged during the whole measurement interval. This behavior agrees with the conclusions made in subsection 3.3.4, where the nom-inal simulation model was considered. The elevator control input effectively excites only vertical and rotational (pitch) variables and has almost no effect on perturbations of the axial velocity. For this reason, the forward velocity damping derivative will be hardly deter-mined from this maneuver.
An additional, but less evident conclusion can be drawn when variations of the axial velocity are analyzed at different pitch angles. Even at large values of there is no apparent change in the trajectory. This behavior indicates that the airship operates at nearly neu-tral buoyancy-gravity condition, i.e. the gravity force is almost compensated by the buoy-ancy force. In this case, the contribution of the and derivatives should be negligibly small.
In order to ensure that these conclusions are valid, a verification test was performed, were the fully parametrized longitudinal model, having 24 unknowns
(6.1) Xu
θ uA CS
Xθ Zθ
Θ Xu Xw Xq Xθ Zw Zq Zθ Mu Mw Mq Mθ XηZηMη…
…bu· bw· bq· bθ· bu bw bq bθ bax baz
T
=
0 20 40 60
Figure 6.1: Filtered time history data of longitudinal F4S1e flight maneuver
0 20 40 60
Figure 6.2: Time histories of measured and estimated (Output-Error) responses, of the full state longitudinal models
and a model with a reduced set of parameters, are estimated. The reduced model neglects the , , , , and derivatives in the dynamic matrix
. (6.2)
The time responses of both models, estimated using the output error algorithm, are illustrated in Figure 6.2. As can be observed, there is a quite small difference between the two models in estimating the forward velocity and no apparent differences in other response variables. Numerical values of the estimated parameters and the supplementary information (Cramer-Rao bounds, TIC coefficients, Eigenvalues) are provided in Table C.2.
Another important indication of the parameter estimation procedure is characterized by the algorithm convergence property. The model with the reduced number of estimation parameters has required 29 iterations to converge, whereas the convergence of the fully parameterized model could not be reached and has been broken after 100 iterations. As will be shown later, this improvement arise due to reduced correlation between estimates and, therefore, a better conditioning of the Hessian matrix (Equation (4.37)).
From this simple verification trial, one concludes that the forward velocity perturbation state , corresponding to the surge mode, can be regarded as decoupled with the remaining states. It follows from the structure of the dynamic matrix (6.2) of the simplified model.
With this knowledge, it is now possible to concentrate efforts on estimating the reduced order model without the state. It should be noticed, that fairly similar flight phenomena was observed not only for this particular flight record, but in all subsequent longitudinal maneu-vers perturbed by elevator deflections. These trajectories are outlined in section C.3.
Low Order Estimation Model
The reduced plant model involves the state-space formulation as described in equation (3.31). Altogether, there are 14 estimation parameters, including 5 stability, 2 control deriva-tives and 7 bias unknowns respectively:
(6.3) As can be concluded from Figure 6.3, in spite of the state reduction, the agreement between the measured and the computed responses is still very good.
So far, only the fit between the measured and the model responses was used as a basic goodness criteria. The trajectory fit does not, however, guarantee the fact that the estimated parameters are reliable. This can be evident when analyzing the correlations between esti-Xw Xq Xθ Zu Zθ Mu
Ared
Xu 0 0 0 0 Zw Zq 0 0 Mw Mq Mθ
0 0 1 0
=
uA CS
δu
Ared δu
Θ = Zw Zq Mw Mq Mθ ZηMηbw· bq· bθ· bw bq bθ baz u0 T.
mated parameters, presented in Figure 6.4(a). In spite of a good trajectory agreement, a nearly linear dependency ( ) exists for several pairs of estimated parameters.
The adopted estimation model with highly correlated parameters should be possibly re-parametrized in order to reduce this interdependence. The reduction of the parameter space of the model can be done based on the following propositions:
• Because the parameters comprising the bias vectors and have only minor impor-tance (e.g. for accounting for the initial condition and the sensor errors), a high correla-tion between them can be allowed unless this does not affect the stability and control estimation or the optimization convergence.
• If a strong dependence between any derivative and the bias parameter exists, the latter should be fixed at some (nominal or zero) value during estimation.
These steps should be regarded as a trade off between the performance of the model and esti-mation confidence of important derivatives.
A serious difficulty arises from the fact that the estimated derivatives , and are highly correlated. Neglecting one of these parameters is undesired, because otherwise it will change the essential properties of the model (such as removing of the second order dynamics of the longitudinal pendulum motion). One advisable solution is to fix the static pitch stability coefficient at some nominal value during estimation. Fixing the has a particular advantage, because its value basically characterizes the metacentric position
ρij>0.98
Figure 6.3: Time histories of measured and estimated (Output-Error) response, of the reduced order longitudinal models
Mw Mq Mθ
Mθ Mθ
zCG
1
Figure 6.4: Correlation map of estimated parameters (numerical values provided for ρij>0.8)
Zw ZqMwMq ZηMηbw· bq· b bθ
Figure 6.5: Time histories of measured and estimated (Output-Error) responses with fixed derivatie
Mθ
and is quite insensitive to variations of the trimmed velocity. This derivative can be alterna-tively calculated using approximation [23]:
. (6.4)
Therefore, the same value of can be used for several flight segments, under supposition that environmental conditions and configuration of the airship remain unchanged. Because the static pitch stability derivative was not known exactly, was initially decided to use an averaged value of the estimations from several maneuvers.
The model with the reduced parameter space has been evaluated and compared with the response of the model that uses the estimated derivative for the same flight maneu-ver. Because the bias parameters show (Figure 6.4(a)) a little correlation with important derivatives, they have not been discarded. Figure 6.5 shows that there is almost no qualitative difference in the time plots provided by the nominal model and the model with fixed . Fixing this parameter also results in reduction of interdependency between and , as illustrated by the correlation map in Figure 6.4(b). As a consequence, estimation of the Cramer-Rao bounds of these two derivatives , has been significantly reduced (see Table C.3).
Accounting for Large Pitch Deviations
One interesting result is that the purely linear model fits the measurements even at the cases, where nonlinear effects are evident. For example, examining the system responses shown in Figure 6.6, one observes that the pitch angle reaches very large values . The trigo-nometric nonlinearities, associated with these large deviations, invalidate the modelling assumptions, given in section 3.2.1. On the other hand, even at large pitch deviations, the aerodynamic linearity assumption is acceptable due to relatively small perturbations of the angle of attack .
Mθ mgzCG Iyy –M,q·
( )
---≈ –
Mθ
Mθ
Mθ
Mθ Mw Mq Mw Mq
θmax>35°
α <8°
0 10 20 30 40 50 60
-20 -10 0 10 20 30 40
-30
Figure 6.6: Measured pitch deviations and angle of attack during 3-2-1-1 identification maneuver
time [s]
αCRθ [°],
αCR θ
In order to keep the physical validity of the estimation model, its basic structure estab-lished in equation (3.31) has been slightly changed. Because the measured Euler angles are normally subjected to a relatively small sensor noise, a directly measured value of can be taken for precomputing the trigonometric nonlinearity. The resulted signal is then applied to the model as a deterministic pseudo-input. In the state-space representation, the enhanced dynamic model yields
. (6.5)
Because the measured pitch angle was used as the pseudo control variable, the system output was reduced to
.
In this case the observation model appears as:
. (6.6)
In this formulation, the model has been estimated using the output error method. The esti-mated time histories are presented in Figure 6.7.
In addition to the proposed parametrization, Figure 6.8 illustrates the trajectory estima-tions of the model that assumes the zero static pitching moment . This “worst-case”
parametrization, corresponding to the case when , , is used here to ensure the significance of the derivative in the system dynamics.
The estimated response of the enhanced model provides no visible improvement in tra-jectory. However, the overall performance of the extended model is more confident with regard to the performance of the pure linear model. The responses of the model that discards the static pitching moment , has lead to inacceptable large model deviations from the flight trajectories.
Application of the enhanced model, however, did not reduce the problem of a high cor-relation between estimates. A nearly linear dependency (not shown) between the derivatives , and remains unchanged. Therefore, the demand on fixing the is actual for this model formulation as well.
θ
0 20 40 60
Figure 6.7: Estimated trajectories of the enhanced model for estimating the large pitch deviations
0 20 40 60
Figure 6.8: Estimated trajectories of the model with Mθ = 0.