THE AERODYNAMIC MODEL

Let x = [/?, p, г, ф]т the state vector, whereas the output vector is у = [ny, p, г, ф]т. и = [Sp, Sr]T denotes the control input vector. The lateral state-space equations of the aircraft are (see appendix A for numerical data):

/3 = Yp(}+ (Yp + sinao)p+ {Yr – cosa0)r + ^Ф + YgvSp + YgrSr p = LpP + Lpp + Lrr + LgpSp + LgrSr r = Np(3 + Npp – I – Nrr + NgTSr

ф = p + tandо r (2.1)

This model is obtained as the linearization of a nonlinear model at the trim value (ao,#o)- The acceleration at the center of gravity is:

V

ny = – — {Ypj3 + Ypp + Yrr + YgpSp + YsrSr)

At a point of coordinates x and z (with respect to the center of gravity), the acceleration is:

n n – (23)

9 9

Uncertainties are introduced in the 14 coefficients which characterize the aerodynamic model, namely the stability derivatives Y@, Yp, Yr, Y&p, Ygr, Lp, Lp, Lr, Lgp, L$r, Np, Np, Nr and Ngr – As an example, the coefficient is rewritten as:

Y0 = Yg{l + xlS1) (2.4)

Yg represents the nominal value of the coefficient. The constant scalar x weights the uncertainty in this coefficient, with respect to uncertainties in the other coefficients. <5i, which is assumed to belong to the interval [­1,1], finally represents the normalized parametric uncertainty in Yp. The scalar xi is called in the following "the weight" in the coefficient У#. The weights in the 14 stability derivatives are chosen in the following as 10 %. A second order actuator is added at the control input Sp:

Ni _ -1.773 + 399 _

Dx ~ s’2 + 48.2.s + 399 ^2’5^

whereas a third order actuator is added at the control input Sr:

N2 2.6.s2 – 1185s + 27350

Th ~ .s3 + 77.7s2 + 3331s + 27350

Remarks:

(i) All quantities /?, p, r and ф are expressed in degree or degree/s. The acceleration output ny is expressed in g,

(ii) For the sake of simplicity, the acceleration ny is measured in the following at the center of gravity.