Further Theoretical Justification of the $k$ -Samples Variation Approach for Discrete-Time Takagi–Sugeno Fuzzy Systems

The recently developed
k
-samples variation approach is known as a powerful way to reduce the conservativeness of existing stability and stabilization conditions for discrete-time Takagi-Sugeno (T-S) fuzzy systems. In this approach, the Lyapunov functions under consideration are not necessarily decreasing at every sample but are allowed to decrease every
k
samples, which is evidently less restrictive than classical approaches. Consequently, less-conservative linear-matrix-inequality (LMI) conditions were derived. In addition, it was proved that, for two positive integers
k
₁
and
k
₂
, if the condition for
k
=
k
₁
is fulfilled, then those corresponding to
k
=
k
₂
are also satisfied when
k
₂
is the divisor of
k
₁
. In this letter, we prove that, if the condition for
k
=
k
₂
admits a solution, then those corresponding to any
k
>;
k
₂
are also solvable.