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ROTORCRAFT LINEAR MODELING
A general state-space form of a model to describe rotorcraft dynamics may be written as
x = f [x(t),u(t),Q] + Fw(t)
y(k) = g[x(k),u(k),Q] + Gv(k) (5.36)
x(0) = X0
The above is the mixed form of continuous-discrete state and observation equations, where x is the state vector, y is the measurement vector, u is the control input vector, Q is the vector of unknown stability and control derivatives, F is the state noise matrix, and G is the measurement noise matrix.
The linear form of the above state-space model can be expressed as
x = Ax(t) + Bu(t) + Fw(t)
y(k) = Cx(k) + Du(k) + Gv(k) (5.37)
x(0) = x0
where A and B are the matrices containing the stability and control derivatives and C and D are the matrices that relate the measured quantities to the rotorcraft states and control variables.
The 6DOF EOMs for modeling rotorcraft dynamics were discussed in Equations 3.44 through 3.47. From these equations and from the state-space model described in Equation 5.36, it is evident that the state, measurement, and control vector will include the following motion variables:
x = [u, v, w, p, q, r, f, U, h]
y [um, vm, wm, pm, qm, rm, fm, Um, hm, axm, aym, azm, pm, qm, rm] (5.38)
u [dlon, dlat, dcol, dped]
The structure of the Equations 3.44 through 3.47, is nonlinear due to the presence of gravity and rotational terms in Equation 3.44 and due to the presence of the product of angular rates in the moment equations given by Equation 3.45.
Assuming small variations in u, v, w, f, and U, the linearized form of Equation 3.44, in the state-space formulation, can be expressed as [1,9]
|
u |
— sin U0 — DU cos U0 |
—W0q + V0r |
|||||
|
v w |
= |
+g |
Df cos U0 cos U0 — DU sin U0 |
+ |
— ЩГ + W0P _ —V0P + u>q _ |
(5.39) |
We have already studied that, for a fixed-wing aircraft, decoupled longitudinal and lateral model equations can be used to model the aircraft dynamics without much loss of accuracy. In the case of helicopters, the degree of coupling between the longitudinal and lateral-directional motion is generally stronger and, therefore, a 6DOF model is preferred [6]. The dynamics of the main rotor in helicopter, however, introduces additional complexities and it becomes necessary to use higher-order models.
