  Dynamical models are often corrupted by model uncertainties, external
disturbances, and measurement noise. These factors affect the performance of
model-based observers and as a result, affect the closed-loop performance.
Therefore, it is critical to develop robust model-based estimators that
reconstruct both the states and the model disturbances while mitigating the
effect of measurement noise in order to ensure good system monitoring and
closed-loop performance when designing controllers. In this article, a robust
step by step non-asymptotic observer for triangular nonlinear systems for the
joint estimation of the state and the disturbance is developed. The proposed
approach provides a sequential estimation of the states and the disturbance in
finite time using smooth modulating functions. The robustness of the proposed
observer is both in the sense of model disturbances and measurement noise. In
fact, the structure of triangular systems combined with the modulating
function-based method allows the estimation of the states independently of
model disturbances and the integral operator involved in the modulating
function-based method mitigates the noise. Additionally, the modulating
function method shifts the derivative from the noisy output to the smooth
modulating function which strengthens its robustness properties. The
applicability of the proposed modulating function-based estimator is
illustrated in numerical simulations and compared to a second-order sliding
mode super twisting observer under different measurement noise levels.
