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10.3.4 Stability analysis
In this section, the analysis of the dynamic behavior of the proposed model
will be presented. A few definitions, which are used to understand the
analysis, are in order.
Definition 10.6: A FPN is said to have reached an equilibrium state (steadystate)
when N(t * +1) = N(t * ) for some time t= t * , where t * is the minimum
time when the equality of the vectors is first attained. The t * is called the
equilibrium time.
Definition 10.7: A FPN is said to have limit cycles if the fuzzy beliefs nj
of at least one place pj in the network exhibits periodic oscillations, described
by nj (t + k ) = nj (t) for some positive integer k>1 and sufficiently large t,
numerically greater than the number of transitions in the FPN.
The results of stability analysis of the proposed models are presented in the
Theorems 10.2 through 10.4.
Theorem 10.2: The model represented by expression (10.1) is
unconditionally stable and the steady state for the model is attained only
after one belief revision step in the network.
Proof: Proof of the theorem is given in Appendix C.
Theorem 10.3: The model represented by expression (10.8) is
unconditionally stable and the non-zero steady state belief vector N * satisfies
the inequality (10.10).
N* ≥ P' o { R o (Q' o N* c ) c },
when R o (Q o N c (t)) c ≥ Th ,∀ t ≥ 0. (10.10)
Proof: Proof is given in Appendix C.
The following definitions will facilitate the analysis of the model represented
by expression (10.9) .
Definition 10.8: An arc tri ✕ pj is called dominant at time τ if for pjε
(∃k ∩O(trk )), ti (τ ) > tk (τ ); alternatively, an arc px ✕ trv at time τ is
dominant if ∀ w, pw ε I (trv ) , nx (τ ) < nw (τ ) , provided Rv o (∀ w,
∧ nw ) > Th v .