Observer-Based Relaxed ${{\cal H}}_{\infty }$ Control for Fuzzy Systems Using a Multiple Lyapunov Function

This short paper proposes a method of designing a fuzzy observer-based ${{\cal H}}_{\infty }$ controller for discrete-time Takagi–Sugeno (T--S) fuzzy systems. To enhance the applicability of the output-feedback controller and improve its performance, this short paper first builds a set of fuzzy control rules with premise variables different from those of the T--S fuzzy system, and sets the overall controller to be dependent on not only the current time but also the one-step-past information on the estimated fuzzy weighting functions. Then, based on the fuzzy control rules, this short paper establishes a less conservative ${{\cal H}}_{\infty }$ stabilization condition incorporated with a multiple Lyapunov function dependent on the estimated fuzzy weighting functions. Through a two-step design procedure, the ${{\cal H}}_{\infty }$ stabilization condition is formulated in terms of parameterized linear matrix equalities (PLMIs), which are reconverted into LMIs with the help of an efficient and effective relaxation scheme.