Recursive observer design beyond the uniform observability

We propose a recursive design scheme of state observer for lower triangular nonlinear systems. The design begins from the bottom dynamics and propagates to upper dynamics recalling the backstepping scheme for nonlinear control, The proposed class of systems is fairly general since it includes non-uniformly observable or detectable multi-output systems. The error convergence to zero is proved assuming the boundedness of input a posteriori, which is preferable whereas most results in the literature assume the boundedness a priori. A global observer is proposed with the global Lipschitz condition of the system. However, this condition is removed via the Lipschitz extension technique when the semi-global observer is of interest