Discrete time recurrent neural network sliding mode observer

State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with the state observation problem when the dynamic model of a discrete plant contains uncertainties or is partially unknown. The suggested observer is oriented to solve the state observation problem of discrete time nonlinear systems. Most of the existing results using the neural networks approach have been developed for continuous time systems using. The recurrent neural network (RNN) have shown several advantages to treat many different control and state estimation problems. In this paper, it is presented a new discrete-time observer using the structure of a classical RNN. This observer includes a correction term using the output information and the first order sliding modes. The second method Lyapunov is applied to generate a new learning law, that contains an adaptive adjustment rate. This study proofs the stability condition for the free parameters included in the neural-observer. A numerical example is given using the RNN in the estimation of a mathematical model describing the HIV infection, that includes non-infected cells, infected cells and free virions. This model was used to generated the data using to test the observer.