A Fixed-Point State observer with Steffensen-Aitken accelerated convergence

Abstract

In this work, we present an alternative algorithm for the state estimation of discrete-time nonlinear systems, which we have called Fixed-Point state observer with Steffensen-Aitken accelerated convergence. This algorithm decomposes the state estimation task into a set of consecutive fixed point iteration problems. In other words, it considers a system of nonlinear equations for each time instant and uses a fixed point iteration method to solve it, such that the solution given by the method is actually a state estimation of the discrete-time nonlinear system at the current time instant. To increase the convergence speed of the fixed point iteration method, we propose to incorporate the Δ2-Aitken method. Nonetheless, later we show that is possible to increase even more this speed by means of the Steffensen’s method. The main advantage of our algorithm is the lack of complex calculations, such as the Jacobian matrix and its inverse, which are necessary for similar algorithms such as the Newton observer. Therefore, our proposal has a low computational cost, is free of singularities, and is easy to implement. Furthermore, unlike conventional estimators like the Luenberger observer and the Sliding Mode observer, it does not require to calculate gains. To prove the effectiveness of the Fixed-Point state observer, we estimate the unknown states of a modified Chua chaotic attractor and compare the numerical results with those obtained by Luenberger, Sliding Mode, and Newton observers.