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.