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4.3 APPLICATION OF ANN MODELS TO ADAPTIVE CONTROL PROBLEMS UNDER UNCERTAINTY CONDITIONS 145
tation of the Jacobian of this error concerning compensator (additional simple feedback con-
the parameters of the neurocontroller. troller). This simple feedback controller does not
3. Adjustment of parameters on the current seg- depend on the model predictions; thus it is more
ment using the Kalman filter equations. robust to perturbations irrespective of their na-
ture. This kind of compensator integrates very
In real time, the neurocontroller is learned ac-
cording to the same scheme, but with some dif- well into the MRAC system.
ferences: In the simplest case, the compensator (PD
compensator) implements an additional feed-
1. Adjacent segments comprise a sliding win- back control law of the following form [30](see
dow (usually 50 points for 0.5 sec). However, also Figs. 4.6 and 4.14):
the parameters are not updated at each simu-
lation step (0.01 sec), but every 0.1 sec. δ e, add = K p e + K d ˙e, (4.10)
2. The ANN model is trained simultaneously;
therefore the model subnet parameters where e = y rm − y is the tracking error y rm for
change. the reference model. In the control system, the
compensator is used in discrete time; hence ˙e is
It should be noted that the neurocontroller
estimated via finite differences.
learns to control not the object itself, but its
model, so if the ANN model does not have the Despite its simplicity, the compensating cir-
required accuracy, then the control performance cuit reduces the tracking error by about one or-
will be unsatisfactory. der of magnitude. We can compare the effect of
The model cannot be ideally accurate since the compensator using the data given for the
the neural network approach provides only ap- case of hypersonic aircraft in Fig. 4.10 (compen-
proximate solutions. Therefore, with the help sator is used) and in Fig. 4.11 (compensator is
of such a “pure” approach it is impossible to not used). Again, we demonstrate here the per-
achieve precise control (accurate tracking of the formance of the neurocontroller with the real
reference model output). plant and with the ANN model of this plant.
We show this result in Fig. 4.9. For compari- We can also use an integral compensator,
son, the same figure shows the performance of which is a filter of the form [30]
the neurocontroller with the object to which it −1
was trained (ANN model). We can see that the W rm
W comp = , (4.11)
accuracy of the operation of the neurocontroller (τp + 1) n−m − 1
with the real object is somewhat reduced, which
indicates that there is a deviation in the behavior where n is the order of the numerator of the
of the real object from the behavior of its ANN transfer function of the reference model; m is
model. We discuss a way to improve the perfor- the order of its denominator; τ is an arbitrary
mance of the neurocontroller in this situation in constant, a manually adjustable parameter of
the next section. the compensation loop (inversely proportional
to the gain).
4.3.2.3 Compensating Loop in the Model The use of an integral compensator allows us
Reference Adaptive Control to get rid of a steady state error, completely sup-
We can interpret errors introduced by the pressing the constant disturbances.
neural network model as additional perturba- However, in an unsteady mode, the integral
tions leading to a deviation of the trajectory of compensator achieves a similar performance to
the controlled object from the reference trajec- the PD compensator, and since the steady-state
tory. To reduce the tracking error, we can use the regimes for the systems of the considered class