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 k₁ and k₂, if the condition for k=k₁ is fulfilled, then those corresponding to k=k₂ are also satisfied when k₂ is the divisor of k₁. In this letter, we prove that, if the condition for k=k₂ admits a solution, then those corresponding to any k >; k₂ are also solvable.