3.5 Model Parametrization
4.1.1 General Estimation Methods
The introductory part of the thesis set out some common approaches for estimating parame-ters of flight vehicles from experimental data. It was pointed out that treating the input-out-put signals has shifted from frequency domain into time domain analysis, and therefore, the time domain methods were considered in the airship identification project.
From the theoretical point of view, all estimation algorithms can be separated into two major classes: deterministic methods and stochastic methods or estimators [11]. The deter-ministic estimators do, in general, solve the problem of the “best-fit” between the model and the real system. In that sense, the fit is treated according to a deterministic measure of error between model output and observed system output, as for example the integral of squared errors do. Opposite to the deterministic methods, the stochastic estimators utilize a statistical approach in interpreting the error. They do not only estimate parameters in the statistical mat-ter, but also provide a quantitative information about the efficiency of estimation [53].
In the field of time domain flight vehicle parameter estimation, both the deterministic and stochastic estimators are widely utilized. According to reference [12], the most common methods are recognized as equation error, output error, filter error and filtering methods. In the following discussion, a short description of the approaches will be presented and their applicability to the airship identification problem is discussed.
Equation Error Method
The equation error (EE) method represents a broad class of methods that are applicable to linear time invariant dynamic systems and based on the least squares regression method [53].
The regression approach requires a direct measurement of all state variables. It constitutes the dynamic equations linear in terms of unknown parameters
, (4.1)
y t( )i = Θ1x1( ) Θti + 2x2( ) … Θti + + nxn( )ti +e t( )i
where denoting the time instance , is the independent variable, the is the dependant variable (state), is the dimensional vector of parame-ters and is the stochastic equation error. Having all independent and all dependant variables measured in discrete points, the equation error can be minimized in one batch iteration using the least squares method:
. (4.2)
This method does not require any initial parameter values and is widely utilized for obtaining primary start-up parameters for other estimation algorithms [55]. An additional benefit of using the equation error method is that it does not require any temporal relation between the measured data points. It is therefore possible to concatenate several data segments in one record. This approach, regarded as data partitioning, is utilized in estimating large amplitude maneuvers by dividing the maneuver into several smaller portions of the flight data [1].
Within simplicity of its realization, the equation error method provides biased esti-mates, if the measurements of the dependant (state) variables are contaminated with the mea-surement noise [66]. Therefore, this method is advisable only if high-quality sensors are used for measuring the system responses.
Output Error Method
The output-error (OE) method has been successfully approved for a variety of flight vehicles using flight test data. The goodness criteria, that is usually employed with this approach is based on the maximum likelihood criteria. The criteria uses a statistical treatment of the error between the model and the system and provides efficient1 parameter estimations.
1. Asymptotically unbiased, minimum covariance of estimations [53].
ti ti, i = 1…N y
xk, k = 1…n Θ n
e y x
N e
Θˆ = (xTx)–1xTy
SYSTEM UNDER TEST
SYSTEM MODEL
LIKELIHOOD PERFORMANCE
CRITERIA CRITERIA
OPTIMIZATION ALGORITHM
_ + + +
Θˆ
u y
v z
e zˆ
Figure 4.1: Output error estimation principle.
The experience obtained by applying the maximum-likelihood identification shows the great adequateness of the models to the real systems, despite the poor initial knowledge of the physical plant during experiments [12]. Furthermore, the output-error method success-fully works on estimation of parameters of nonlinear models [20].
Filter Error Method
This approach is some extension of the output error method. Based on Kalman filter for pro-viding the state estimations of the identified system, the filter error (FE) method has a signif-icant advantage in providing parameter estimates in presence of the process noise. There are several studies available, that utilize the filter error method for estimating aircraft parameters from the flight data in presence of turbulence [15], [19], [64]. The diagram in Figure 4.2 illustrates the principle of the filter error approach, including model dynamics, the presence of additive random process noise and random disturbances corrupting the measurements.
Filtering Method
In the filter error approach the Kalman filter is dedicated to the state estimation only. How-ever, it is possible to use the filter as a set up for the simultaneous estimation of the state and the unknown parameters. This problem is successfully solved if the unknown parameters augmented with the model state vector and Kalman filter estimates in this combination both the state vector and parameters simultaneously. The parameters and states are combined into a composite state vector
, (4.3)
with for the time invariant model parameters.
SYSTEM UNDER TEST
MODEL BASED KALMAN FILTER
ESTIMATOR
LIKELIHOOD PERFORMANCE
CRITERIA CRITERIA
OPTIMIZATION ALGORITHM
_ +
+ +
u z
Θˆ zˆ
y n
e v
Figure 4.2: Diagram of the filter error algorithm
x˜ = x Θ T Θ·
0
=
This method is widely used in the real time parameter estimation applications [2], [61].
However, this approach poses some additional computation difficulties. Even if the system dynamics is approximated by a linear model, the multiplicative nonlinearity appears due to relation . This nonlinearity requires an implementation of the extended Kalman fil-ter, where the system equations are linearized at each successive integration step.