Parameter-Estimation Methods

Several useful techniques are available in the literature for estimation of parameters [1] from flight data, the selection of which is governed by the complexity of the mathematical model, a priori knowledge about the system, and information on the noise characteristics in measured data. In general, the estimation technique provides the estimated values of the parameters along with their accuracies, in the form of standard errors or variances. In this section some widely used methods are briefly discussed.

9.3.2.1 Equation Error Method

In the equation error method (EEM), the measurements of the state variables and their derivatives are assumed to be available and used. The method is computation­ally simple and is a one-shot procedure. Since the measured states and their deriva­tives are used, the method would yield biased estimates, and hence it could be used as a start-up procedure for other methods. It minimizes a quadratic cost function of the error in the state equations to estimate the parameters. The EEM is applicable to linear as well as linear-in-parameter systems. The EEM can be applied to unstable systems because it does not involve any numerical integration of the system equa­tions that would otherwise cause divergence of certain state variables. Thus, the utilization of measured states and state-derivatives for estimation in the algorithm enables estimation of the parameters of unstable systems. If a system is described by the state equation

x = Ax + Bu; x(0) = X0

then the EE can be written as

e(k) xm Axm Bum (9 •11)

НСГС, Aa [A, xam

The cost function is given by

1 ^

J(b) 2 ‘У [Xm(k) Aaxam(k)] [Xm(k) Aaxam(k)]

2 k=1

and the estimator is given as

Example 9.1

The equation error formulation for parameter estimation of an aircraft is illustrated with one such state equation here. Let the moment equation be given

q = Maa + Mqq + MSt Se (9.15)

Then moment coefficients of Equation 9.15 are determined from the system of linear equations given by (Equation 9.15 is multiplied in turn by a, q, and Se)

^qa = M„5>2 + Mq^qa + MSt^ Sea ^qq = Ma^aq + Mq^q2 + Ms^^ Seq (9.16)

]T qSe = M^2 aSe + M^2 qSe + Ms^ S2e

Here, S is the summation over the data points (k = 1,…, N) of a, q, and Se signals. Combining the terms, we get

Eq a		a2	qa	Y, Sea	Ma
Eq q	=	aq	q2	J2seq	Mq	(9.17)
_Eq se_		_J2aSe	J2qse	E«2 J	Ms.

The above formulation can be expressed in a compact form as Y=Xp, and the equation error is formulated as e = Y — X/3 keeping in mind that there will be modeling and estimation errors combined in ‘‘e.’’ It is presumed that measurements of q, a, q, and Se are available. Then the equation error estimates of the parameters can be obtained using Equations 9.7 and 9.14.