Page 136 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 125
2
it is possible to implement this decomposition where a = 1/x max , b = 1/u 2 max ,based on thead-
2
in an optimal way. In the general case, such an ditional requirement to keep x below the spec-
2
2
ensemble can be inhomogeneous, containing an ified value x max = const by using the u control
ANN of different architectures, which in prin- not exceeding u 2 max = const.
ciple is an additional source of increasing the In the problem under consideration, the ex-
efficiency of solving complex applied problems. ternal set and the domain K of the values of
It should be noted that, in fact, the ENC is the regulator parameters are one-dimensional,
a neural network implementation and the ex- that is,
tension of the well-known Gain Scheduling ap- 1
=[λ 0 ,λ k ]⊂ R ,
proach [95–97], which is widely used to solve a (3.62)
1
variety of applied problems. K =[k − ,k + ]⊂ R .
3.4.2.4 A Formation Example of an The criterial function f(λ,k) is written, ac-
Ensemble of Neural Controllers for cording to (3.48), in the form
a Simple Multimode Dynamical
∗
System f(λ,k) = J(λ,k) − J(λ,k ). (3.63)
Let us illustrate the application of the main For an arbitrary admissible pair (λ,k), λ ∈ ,
provisions outlined above, on a synthesis exam- k ∈ K, the expression for J(λ,k) applied to the
ple of the optimal ENC for a simple aperiodic system (3.28)–(3.33) takes the form
controlled object (plant) [98], described by
2
2
(a + bk )x 2 0
1 J(λ,k) = . (3.64)
˙ x =− x + u, t ∈[t 0 ,∞). (3.57) 4(1/τ(λ) + k)
τ(λ)
Since the function (3.64)isconvex ∀x ∈ X, we can
Here put x 0 = x max .
According to [98], the value of the functional
2
τ(λ) = c 0 + c 1 λ + c 2 λ ,λ ∈[λ 0 ,λ k ]. (3.58) J(λ,k ) for some arbitrary λ ∈ can be obtained
∗
∗
by knowing the expression for k (λ), i.e.,
As the control law for the plant (3.57), we take

1 a 1
u =−kx, k − k k + . (3.59) k (λ) = + − . (3.65)
∗
2
τ (λ) b τ(λ)
The controller implementing the control law
(3.59) must maintain the state x of the controlled In this problem, the controller realizes the
object in a neighborhood of zero, i.e., as the de- control law (3.59), the neurocorrector repro-
∗
sired (reference) object motion (3.57) we assume duces the dependence (3.65)ofthe k coefficient
adjustment depending on the current value of
x e (t) ≡ 0,u e (t) ≡ 0, ∀t ∈[t 0 ,∞). (3.60) λ ∈ , and collectively, this regulator and neuro-
corrector are neurocontroller 1 , the only one in
The quality criterion (performance index, the ENC .
functional) J for the MDS (3.57)–(3.60) can be As a neurocorrector here we can use an MLP-
written in the form type network with one or two hidden layers or
some RBF network. Due to the triviality of the
∞
formation of the corresponding ANN in the case
1 2 2
J(λ,k) = (ax(λ) + bu(k) )dt, (3.61) under consideration, the details of this process
2 are omitted.
t 0