Page 128 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 117
We describe the actuator system by the fol- required control efficiency) for all λ ⊂ .Theap-
lowing equations: proach we could apply in this situation is to use
different k for different λ [89–91]. In this case, the
˙ z 1 = z 2 , relationship k = k(λ), ∀λ ⊂ , is realized by the
(3.32)
˙ z 2 = k ϑ ϕ(σ) − T 1 ˙z 1 − T 2 z 1 . control system module, which is called the cor-
rection device or simply the corrector. We will
Here T 1 , T 2 aretimeconstants; k ϑ is the gain; call the combination of the command device and
ϕ(σ) is the desired control law, i.e., some oper- the corrector the controller.
ation algorithm for the controller. The function
ϕ(σ) can take, for example, the following form: NEUROCONTROLLER FOR A SINGLE-MODE
DYNAMICAL SYSTEM AND ITS EFFICIENCY
ϕ(σ) = σ, (3.33)
The formation of the dependence k = k(λ),
3
ϕ(σ) = σ + k n+1 σ , (3.34) ∀λ ⊂ , implemented by the corrector, is a very
3 5 time-consuming task. The traditional approach
ϕ(σ) = σ + k n+1 σ + k n+2 σ , (3.35)
to the realization of dependence k = k(λ) con-
n
sists in its approximation or in interpolation ac-
σ = k j x j .
cording to the table of its values. For large di-
j=1
mensions of the vector λ and the large external
The influence on the control quality for the set , the dependence k(λ) will be, as a rule,
disturbed motion is carried out through vec- very complicated. It obstructs significantly an
tors θ ∈ for parameters of the plant, the com- implementation of this dependence on board of
mand device, and the actuator system, as well as the aircraft. To overcome this situation, we usu-
T n ally try to minimize the dimension of the λ vec-
through the coefficients k = (k 1 ,...,k n ) ⊂ R
included in the control law. tor. In this case, we usually take into account no
The task in this case is to reduce the disturbed more than two or three parameters, and in some
0
0
motion ( x, u) to the reference one ( x , u ) taking cases we use only one parameter, for example,
into account the uncertainty in the parameters the dynamic air pressure in the task of control-
λ. We have to solve this problem in the best, in ling an aircraft motion. This approach, however,
a certain sense, way using a control u added to reduces the control efficiency since it does not
0
the reference signal u . We only know about the take into account a number of factors affecting
λ parameters that they belong to some domain this efficiency.
s
⊂ R . At best, we know the frequency ρ(λ) At the same time, we know from the theory
with which one or another element of λ ⊂ will of ANNs (see, for example, [25–27]) that a feed-
appear. forward neural network with one or two hidden
We term, following [88], the domain as ex- layers can model (approximate) any continuous
ternal set of the dynamical system (3.28). The nonlinear functional relationship between N in-
system that should operate on a subset of the puts and M outputs. Similar results were ob-
Cartesian product X × U × is a multimode dy- tained for RBF networks, as well as for other
namical system (MDS). We can influence the con- types of ANNs (see, for example, [92]). Based
trol efficiency of such MDS by varying the val- on these results, it was proposed in [89]touse
ues of k ∈ K parameters of the command device. ANNs (MLP-type networks with two hidden
If the external set of the considered MDS is layers) to synthesize the required continuous
“large enough,” then, as a rule, there is a situ- nonlinear mapping of the tuning parameters λ
ation when we have no such k ∈ K that would of the controller to the values of the control law
be equally suitable (in the sense of ensuring the coefficients, i.e., to form the dependence k(λ).