Discrete Tagaki–Sugeno models for control: Where are we?

This work deals with relaxed conditions for stability and stabilization of discrete-time Takagi–Sugeno (TS) models. It recalls classical results found in the literature which use quadratic Lyapunov functions leading to very conservative conditions, and various extensions based on piecewise and non-quadratic Lyapunov functions. Afterwards, a new and powerful way to enhance the previous results is depicted. The basic idea is that waiting long enough a stable model will converge towards its equilibrium and, therefore, the Lyapunov functions under consideration are not necessarily decreasing at every sample, but are allowed to decrease every k samples. Whatever it is k >1, the results are proved to include the standard one-sample case. The potential of this approach is shown through several examples in the paper.