Observer design for inherently nonlinear systems with lower triangular structure

A new observation procedure is proposed for a wide class of single output observable nonlinear systems written in lower triangular form. First, we give the n-th order time-varying differentiator that robustly estimates, in asymptotic manner, the higher derivatives of any model-free continuously differentiable signal. This n-th order differentiator is a generalization of the time-varying differentiator proposed by the author in [1], [2] and [3]. By using an appropriate change of variables, it is shown that the boundedness of the signal to be differentiated is not necessary for the convergence of the differentiator. Based on the fact that systems written in triangular form are algebraically observable then, the system states can be reproduced through a static diffeomorphism that involves the system input, the system output, and their respective higher derivatives. It is shown that the global convergence of the n-th order differentiator implies the asymptotic convergence of the system states without imposing any restrictive condition on the form of nonlinearities.