Fuzzy theory is widely studied and applied. This article introduces an adaptive control scheme for a class of non-linear systems with Markov jump switching. The introduced scheme supposes that the system is submitted to external disturbances under imperfect premise matching. By using discrete-time Takagi–Sugeno fuzzy models, a sliding mode observer-based control scheme is utilized to estimate unmeasured states of the system. We build two fuzzy switching manifolds for the disturbance and sliding mode observer systems. Then, a linear matrix inequality-based criterion is developed using slack matrices. This criterion proves that the sliding mode dynamics are robustly admissible under an H-infinity performance often used in control theory. Hence, new adaptive sliding mode controllers are synthesized for the disturbance and sliding mode observer systems. This allows the reachability of pre-designed sliding surfaces to be guaranteed. Finally, experimental numerical illustrations on a bio-economic system and a tunnel diode circuit are presented to show potential applications, as well as validating the effectiveness of the scheme proposed in the present investigation.