12.5.2.   Adaptive Architecture

The proposed architecture involves two adaptive loops. The first adaptive
loop generates approximate model dynamics of the necessary variables, in the
form of Eq (12.11). This is similar to an observer loop and is graphically
shown in Figure 12.5 for the limit parameter y_p. In Figure 12.5, the error
between the measured y_p and the model generated estimate is used to
update an adaptive neural network, representing the approximation Δ₁ of
Eq. (12.11). A proper observer gain matrix K = [k₁k₂ . . . k_l]^T is chosen to
ensure proper convergence of network weights.

The second adaptive loop (see Figure 12.6) is used to calculate the
dynamic trim solution of the model obtained from the first adaptive loop
(Figure 12.5). The condition for dynamic trim of the limit parameter is given
in Eq. (12.12). The initial guess of the dynamic trim is updated by an adaptive
neural network, ANN2, such that Eq. (12.12) is satisfied. The sum of the
approximate linear model and Δ₁ is used to update the adaptive neural
network ANN2. By forcing the error term e₂ to go to zero, the proposed
architecture finds a solution to Eq. (12.12). At each time step, the function

Δ₁ is estimated by ANN1 from the first loop. The weights are ‘‘frozen’’’ for
that instant, and they are used in the dynamic trim solution process in the
second loop.

The second loop in Figure 12.6 can be structured to obtain the limit
control vector estimations of Eq. (12.13) by solving for , given the
limiting parameter boundaries y_{p lim}. The control margins can then be obtained
using

A Lyapunov-based update law can be derived for both networks [24]. Although
it is possible to use the approximate model to obtain initial guesses
for the dynamic trim solutions, faster convergence may be obtained using a
pretrained neural network. A similar approach can also be followed to
calculate the dynamic trim solutions of fast variables [24]. An alternative
method to calculate dynamic trim solutions and control margins from Eq.
(12.11) , without the use of a second network (ANN2) can be found in
reference 36.