12.5.1.   Methodology

The adaptive limit detection and avoidance algorithm can be divided into two
steps. The first step is to establish the functional relationship between the
quasi-steady response of the limit parameter and a set of measurable state
and control variables. The quasi-steady response of a limit parameter also
corresponds to a dynamic trim condition. The second step is then to calculate
the control deflections that would cause the vehicle to reach a given limit
boundary. Let the following nonlinear state equation represent the equations
of motion of an aircraft:

The state x can be divided into fast and slow variables, such that.

where

The slow states include flight parameters such as forward speed and the
Euler angles. The fast states are, for example, the angular rates, angle of
attack, and sideslip. The dynamic trim condition corresponds to

Assume a vector y_p consisting of measurable limit parameters as

Since the quasi-steady response of the limit parameters y_pwill also correspond
to the dynamic trim condition, the dynamic trim values of the limit
parameters may be obtained using

The subscript ‘‘DT’’ denotes dynamic trim. Thus the dynamic trim response
of the limit parameter can be calculated using the dynamic trim estimates of
the necessary fast states. Differentiating the algebraic equation of y_p in Eq.
(12.5), we obtain

Now, let

Since u(t - Δt) gives rise to the current value of y_p, and assuming a unique
mapping between the current values of the fast states and the limit parameters,
Eq. (12.7) may be written in functional form as

Now, using a linear approximation for the function φ, Eq. (12.9) can be
rewritten as

where the modeling error is represented as ξ₁. Using a neural network Δ₁ as
an approximation of the modeling error, we obtain

In this case, the quasi-steady predictions for the limit parameters are ob-
tained by setting = 0, which implicitly corresponds to the dynamic trim
condition. Then in dynamic trim

Also, for known limit parameter boundaries, _lim, an estimate of the limit control vector can be obtained using

Solutions of Eqs.(12.12) and (12.13) with respect to_DT and u_lim will provide the dynamic trim estimate and limit control vector estimate, respectively.