This article deals with the observability bound problem of a discrete-time nonlinear singularly perturbed systems (SPSs) subject to disturbances and noises. This nonlinear SPS is represented by a coupled state multimodel (MM) with unmeasurable premise variables. A proportional-integral multiobserver ( PI multiobserver), known by its robustness, is designed to accomplish this task. The proposed method is based on the technique to minimize the effect of the disturbances, the noises, and all the unmeasurable premise variables. To design this observer, sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are developed as a first step to ensure the robust stability bound of the considered system represented by a coupled MM for a bound of the singular perturbation parameter based on a quadratic Lyapunov function and the gain. Then, sufficient conditions are developed to ensure the robust stability bound of the state estimation error between the MM and the PI multiobserver for a bound of the singular perturbation parameter that is less than or equal to the considered system’s robust stability bound. Hence, the observability bound of the considered system represented by a coupled state MM is determined. Two simulation examples are then given to validate the proposed strategy.
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