Nonlinear observers for Lipschitz discrete-time systems. Application to synchronization and input recovery

In this note, a new observer design method for a class of nonlinear Lipschitz discrete-time systems is proposed. The developed method presents significant improvements with respect to the results of [1] and [2]. This is due, firstly, to the use of another structure of the observer introduced recently in [3] and, secondly, to the use of a detailed form of the system by specifying the distribution matrix of the nonlinearities in the system and the dependence matrix of the nonlinearities on the state of the system. The stability analysis is performed using a particular Lyapunov function that leads to the solvability of matrix inequalities which become linear (LMIs) whenever unknown scalar variables are chosen a priori. An illustrative example is given in order to show the efficiency of our method with respect to [1] and [2]. This new design approach is then generalized to systems with unknown inputs. In this paper, we have considered the problem of synchronization and input recovery. This generalization is tested successfully by a numerical application.