State estimation for a class of uncertain nonlinear systems: a finite-time observer approach

ABSTRACT

In this paper, the problem of state estimation for a class of uncertain, and possibly unbounded, nonlinear systems is tackled. In order to deal with such a problem, a nonlinear observer is proposed based on a continuous homogeneous and Levant's nth order differentiators. The proposed nonlinear observer provides finite-time convergence and the observer synthesis is formulated in terms of some linear matrix inequalities (LMIs). Moreover, contrary to what the literature shows, the finite-time observer does not require any type of Bounded-Input Bounded-State (BIBS) condition. Some simulation results illustrate the effectiveness of the proposed finite-time observer applied to an unstable nonlinear system.

Keywords:

• Sliding-mode observer
• finite-time convergence
• nonlinear systems

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 System (1a(1a) $\begin{array}{rl}\dot{x}& =Ax+\varphi (y,u)+D\left(g\right(x)+w(t\left)\right),\end{array}$(1a) ) and (1b(1b) $\begin{array}{rl}y& =Cx,\end{array}$(1b) ) can be obtained from a more general nonlinear system, e.g. $\dot{z}=f(z,u)+g\left(z\right)\overline{w},y=h\left(z\right)$, by means of a nonlinear output injection linearisation (Krener & Isidori, 1983) or through the Fliess's observability canonical form (Fliess, 1987), and a relative degree condition over the output y and the unknown term $\overline{w}$ (Isidori, 1996). Note also that many systems, e.g. mechanical systems as Quad-Rotors, robot manipulators, pendulum-like systems, etc.; or typical bioreactor systems, preserve such a structure.

Additional information

Funding

This work was supported by Consejo Nacional de Ciencia y Tecnología [CVU 270504 project 922].