Page 159 - Neural Network Modeling and Identification of Dynamical Systems

4.3 APPLICATION OF ANN MODELS TO ADAPTIVE CONTROL PROBLEMS UNDER UNCERTAINTY CONDITIONS 149
The value of the relative damping coefficient takes into account the actuator approximately),
ζ rm in the reference model was experimentally which is adequate until the frequencies of the
chosen to be 0.8 to achieve a compromise be- specified motion (reference model) and the ac-
tween the absence of overshoot and the dura- tuator are far apart and the speed of deflection
tion of the transient process implemented by the of the elevons does not exceed a certain value.
reference model. The prefilter time constant T pf In this formulation, the influence of the actua-
was assumed to be 0.05 sec. tor reduces only to the appearance of a certain
The primary parameter of the reference model delay in the initial part of the transient process.
is the frequency ω rm , which determines the de- We can see also the increasing of elevon deflec-
sired speed of the system as a whole. From the tions if we require some high regulation speed.
physical point of view, it is possible to increase Experiments, the results of which are given in
the reaction rate only using a larger deflection of Figs. 4.12, A.1–A.3, were executed for the hy-
the controls in the transient mode. personic research vehicle. The effectiveness of
It is well known that the desired speed of the elevons in the function of the elevator in this air-
system is selected as a compromise between the craft is low. Besides, we need significant deflec-
speed of the system’s reaction and the speed of tions of the controls for aircraft balancing at dif-
deflection of the control surfaces. It was demon- ferent angles of attack. Thus, it seems reasonable
strated in a series of computational experiments to limit the frequency of the reference model to
applied to the nonlinear system under consider- ω rm = 2 Hz, in order to leave some margin for
ation. the deflection of the control surfaces.
As already noted, the desired control per- The obtained results also show that in some
formance in the considered adaptive control cases the specified actuator speed limitation is
schemes is set using the reference model. We achieved (±60 deg/sec), which affects the tran-
demonstrate the influence of the desired re- sient process (Fig. A.3) and can lead to instability
sponse speed on the required deflection of the in the case of a significant deviation from the de-
control surfaces and the load on the actuator sired motion. To reduce the used actuator speed,
using the MRAC-type adaptive control scheme. it is possible to increase the time constant T pf of
The results of the corresponding computational the reference model, but this can lead to an in-
experiments are shown in Figs. 4.12, A.1–A.3. crease in the delay in the system, while (with a
Based on the results of these experiments, given actuator) the larger the frequency of the
it is evident that the larger the eigenfrequency reference model ω rm , the more part of the tran-
of the reference model, the greater the tracking sient duration will be affected by the actuator.
error during the transient process: it increases
from ±0.1 deg for ω rm = 1.5 /sec to ±2.1 deg for 4.3.2.5 Model Reference Adaptive Control
ω rm = 4 /sec. This is due to the fact that the refer- Applied to the Aircraft Angular
ence model has a third order in accordance with Motion
the above considerations, while the controlled A series of computational experiments was
object (aircraft + actuator) is a fourth-order os- carried out to evaluate the properties of the
cillatory system. It should be noted that the actu- designed MRAC system. Some of the results
ator frequency was set as ω pf = 20 Hz according of these experiments were already presented
to the data for the object under consideration. above. In particular, in Fig. 4.9, as applied to
The neural network model of the controlled ob- the hypersonic research vehicle X-43 (flight at
ject by virtue of the procedure for its formation M = 6), the influence of the accuracy of the ANN
includes an approximate implicit description of model on the performance of the obtained ANN
the actuator (and therefore the controller also mode was shown.