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Analysis and analytical solution of incommensurate fuzzy fractional nabla difference systems...
RNN with ν neurons and activation functions q i . However, for commensurate fractional differ-
These RNNs can be expressed as: ential equations, numerous studies 18–20 have es-
∇ x 1 (t) =q 1 (z 1 ), α 1 ∈ (0, 1], tablished a robust framework for fuzzy theory and
α 1
uncertainty analysis.
.
. (1)
. In this paper, we use fuzzy theory to address
∇ x ν (t) =q ν (z ν ), α ν ∈ (0, 1], the uncertainty in IFDSs that arise in the dense
α ν
RNN described by Equation (2).
where z i = w i1 x 1 (t − 1) + · · · + w iν x i (t − 1) + p i .
Typically, for t > 0, x i (t) represents the out- We apply the state-of-the-art method of con-
put at the time step t. Using vector notation, let verting a fuzzy number to an interval using r-
⃗
T
⃗ α = [α 1 , . . . , α ν ] , and define ⃗p, ⃗x, and f in a cuts. In this regard, we carefully define what a
similar fashion. We can rewrite Equation (1) in fuzzy number is in relation to the r-cut represen-
vector form as tation. Then, we obtain the corresponding arith-
metic operations using Zadeh’s extension theory.
⃗ α
∇ ⃗x(t) = ⃗q(W⃗x(t − 1) + ⃗p) (2)
However, we find that for operations such as the
where W = (w ij ) is a ν × ν matrix and ∇ ⃗ α difference, there are inconsistencies. Also, the
is a diagonal matrix with the fractional nabla generalized H-difference suffers from other incon-
operators ∇ α i on the diagonal, i.e., ∇ ⃗ α = sistencies. To overcome such inconsistencies, we
Diag[∇ , . . . , ∇ ]. introduce the concept of H-differenceability. Sim-
α ν
α 1
ilarly, we extend it to the fuzzy nabla fractional
T
Remark 1. Putting ⃗α = [0, . . . , 0] , we obtain difference operation.
⃗x(t) = ⃗q(W⃗x(t − 1) + ⃗p) (3) After obtaining clear definitions, we ana-
which is a layer of classical NN with ν neurons. lyze the fuzzy incommensurate neural network.
T
Putting ⃗α = [1, . . . , 1] , we get This analysis shows that it has a unique H-
differenceable solution. Finally, we introduce an
⃗x(t) = ⃗x(t − 1) + ⃗q(W⃗x(t − 1) + ⃗p) (4)
algorithm to compute the fuzzy solution using a
which is equivalent to an RNN. Therefore, for recursive formula.
α i ∈ (0, 1), it is an RNN with memory. Such The novelty of this paper is highlighted as fol-
NNs are widely used in data classification. 12
lows:
If in System Equation (1) all orders are the
(1) We developed a new RNN model based on
same, i. e., α i = α for all i = 1, . . . , ν, then
it is a commensurate system. In this case ⃗α = fractional nabla operations.
T
[α, . . . , α] . Commensurate systems are much eas- (2) We refine the definition of fuzzy numbers
ier to investigate than incommensurate systems, and present the H-differenceable concept.
since in a commensurate system (3) We prove the existence of a unique H-
differenceable solution for incomensurate
⃗ α
⃗ α
∇ W⃗x(t) = W∇ ⃗x(t), RNNs with fuzzy input.
while this equality does not hold for an incom- (4) We develop a recursive algorithm for cal-
mensurate system. culating fuzzy solutions.
The central problem addressed in this paper is
how fuzzy-valued inputs affect the outputs. This For clarity, we provide tables of mathematical
represents a problem of uncertainty. notations, notation spaces, and abbreviations in
Interval analysis, stochastic analysis, and Tables 1-3, respectively.
fuzzy theory have proven to be valuable tools In Section 2, we review the concept of fuzzy
for analyzing uncertainty. In particular, statis- numbers, along with a minor correction to some
tical analysis has not yet been employed for un- other available definitions that distinguish fuzzy
certainty analysis in incommensurate fractional numbers from fuzzy sets. We present the cor-
differential/difference systems. In contrast, fuzzy responding transforms with r-cats concerning in-
theory and interval analysis have been utilized in terval analysis. In Section 3, fuzzy arithmetic is
a limited number of studies. For example, 13,14 reviewed in connection with shape functions and
harnessed fuzzy theory to investigate state uncer- r-cats, and the nabla fractional difference is ex-
tainties in incommensurate fractional nabla differ- tended to fuzzy numbers. In Section 4, we demon-
ence systems (IFDSs). Refs. 15,16 applied interval strate that fuzzy IFDSs for RNNs admit a unique
analysis to address the uncertainties of the param- solution. In Section 5, we detail the computation
eters. Moreover, Ref. 17 explored a fractional PI of the fuzzy solution for IFDSs. Finally, we pro-
observer for IFDSs with parametric uncertainties. vide an illustrative example in Section 6.
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