Page 130 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 119
where (x e ,u e ) is the required “ideal” motion of • integral criterion
the dynamical system.
J(x,k,θ,λ)
Using the solutions of the systems (3.31)and

(3.37), we can define a functional J estimating t f 2 2
= ((x(t) − x e (t)) ,(u(t) − u e (t)) )dt.
the deviation degree of the realized motion (x,u) t 0
from the required one (x e ,u e ).Asshownin[88, (3.41)
93], all possible variants of such estimates can be
It should be emphasized that it is the func-
reduced to one of two possible cases:
tional (3.38), (3.40)or(3.39), (3.41)that“directs”
• guarantee criterion learning the ANN used in the neurocontroller
(3.36), since it is its value that is minimized in
J(x,k,θ,λ) = max (|x(t) − x e (t)|), (3.38) the learning process of the neurocontroller.
t∈[t 0 ,t f ] A discussion of the approach used to get the
μ i (t), μ(t) dependencies and the rule relates to
• integral criterion the decision-making area for the vector-type ef-
ficiency criterion. This approach is based on the
results obtained in [88,94] and it is beyond the
t f
2
J(x,k,θ,λ) = ((x(t)−x e (t)) )dt. (3.39) scope of this book.
t 0
THE ENSEMBLE OF NEURAL CONTROLLERS AS
As we can see from (3.38), with the guar- A TOOL FOR TAKING INTO ACCOUNT THE
anteeing approach the largest of the deviations MULTIMODE NATURE OF DYNAMICAL SYS-
is x i = μ i (t)|x i (t) − x ei (t)|, i = 1,...,n,onthe TEMS
time interval [t 0 ,t f ], as the measure of proximity The dependence k(λ), including its ANN ver-
of the real motion (x,u) to the required (x e ,u e ). sion, may be too complicated to implement on
For the integral case, this measure is the integral board of aircraft due to the limited computing
of the square of the difference between x and resources that can be allocated to such imple-
x e . Here, the coefficients μ i and the rule de- mentation. If λ “does not change too much”
termine the relative importance (significance) of when changing k,wecouldtrytofindsome
the deviations of x i with respect to the corre- “typical” value λ, determine the corresponding
∗
sponding state variables of the dynamical sys- k for it, and then replace k(λ) with this value
∗
tem for different instants of time t ∈[t 0 ,t f ]. k . However, when λ significantly differs from
In the case when, in accordance with the its typical value, the quality of regulation of the
controller obtained in this way may not meet the
specifics of the applied task, it is necessary to
design requirements.
take into account not only the deviations of the
In order to overcome this difficulty, we can
state variables of the controlled object, but also
use a piecewise (piecewise-constant, piecewise-
the required “costs” of control, the indicators linear, piecewise-polynomial, etc.) variant of the
(3.38)and (3.39) will take the following form:
approximation for k(λ). We will clarify consider-
• guarantee criterion ations concerning the assessment of the quality
of regulation in Section 3.4.2.3.
As a tool to implement this kind of approxi-
J(x,k,θ,λ)
mation we introduce the ensemble of neural con-
= max (|x(t) − x e (t)|,μ(t)|u(t) − u e (t)|), trollers (ENC)
t∈[t 0 ,t f ]
(3.40)
= ( 0 , 1 ,..., N ), (3.42)