Page 166 - Neural Network Modeling and Identification of Dynamical Systems

156 4. NEURAL NETWORK BLACK BOX MODELING OF AIRCRAFT CONTROLLED MOTION

4.3.3.2 Neurocontroller Synthesis for the This is done using the method of backprop-
Model Predictive Control agation of the error in time (BPTT – BackProp-
agation Through Time) for a closed-loop neural
In adaptive MPC schemes, only one multi-
network model.
layer neural network is used, which is the ANN
The Gauss–Newton method is very similar
model of the object. The controller is represented
to the Levenberg–Marquardt method; it differs
here by the optimization algorithm. The struc-
only in that the coefficient μ does not change.
tural scheme of the system with the predictive This coefficient we choose experimentally for
model is illustrated in Fig. 4.14. the problem being solved, i.e.,
The criterion of quality for this system is the
T
root-mean-square error in the forecast interval u n+1 = u n + (J J u + λE) −1 T (4.14)
J e.
u
u
(5–7 steps), i.e., it does not take into account the
increase in control. We have With these facts in mind, the calculation of the
control at each step of integration is carried out
in the following sequence:
1 T
E(u) = e e, e(u) = y rm −ˆy(u),
2 1. Build the desired behavior in the forecast in-
T
e =[e k+2 ,e k+3 ,...,e k+T ] , (4.12) terval. To do this, the reference model is cal-
culated on this interval with a constant refer-
T
u =[u k+1 ,u k+2 ,...,u k+T −1 ] ,
ence signal (specifying the signal
r k+1 ...r k+T −1 = r k : for lack of more sophis-
where T is the length of the forecast interval. ticated options, assume the simplest, namely,
last available, value of this signal).
The absence of a control term in the criterion
2. The forecast of the controlled object behav-
makes it possible to do just one iteration at each
ior according to its ANN model for several
step by the Gauss–Newton method.
steps forward. The initial conditions for the
Thus, here the minimization of the deviation
model are the reference trajectory and the
of the behavior of the ANN model from the ref- previous control values obtained using such
erence model is performed not by the parame- a controller.
ters of the neurocontroller (they are not present 3. Determination of the error vector, calculation
here at all), but directly on the control over the of the Jacobian for the error function by con-
forecast interval. To apply effective optimiza- trol for each time instant.
tion methods, we need to calculate the dynamic 4. Adjustment of the control vector by any opti-
control-related Jacobian, i.e., mization method (in this case by the Gauss–
Newton method). Steps 2–4 represent one it-
eration of the optimization procedure per-
⎡ ⎤
∂e k+2 ∂e k+2 ∂e k+2 formed until some reasonable reduction in
...
⎢ ⎥ the forecast error.
∂u k+1 ∂u k+2
⎢ ∂u k+T −1⎥
⎢ ∂e k+3 ∂e k+3 ∂e k+3 ⎥ 5. As the control (in the real object) in the next
...
⎢ ⎥
J u = ⎢ ⎥ . (4.13) step (the optimization procedure takes a part
⎢ ∂u k+1 ∂u k+2 ∂u k+T −1⎥
··· ··· ··· ··· of the step), the first control on the forecast in-
⎢ ⎥
⎢ ⎥
⎣ ∂e k+T ∂e k+T ∂e k+T ⎦ terval (i.e., u k+1 ) is taken. The same value fills
...
∂u k+1 ∂u k+2 ∂u k+T −1 the entire vector of the initial control approx-
imation in the next step.