Non-Uniform Stability, Detectability, and, Sliding Mode Observer Design for Time Varying Systems with Unknown Inputs

TL;DR

This paper introduces a novel observer design for linear time-varying systems with unknown inputs, ensuring stability and finite-time state reconstruction through non-uniform stability analysis and sliding mode techniques.

Contribution

It proposes a new observer based on Lyapunov exponents and sliding mode differentiators, providing stability conditions and finite-time state estimation for systems with unknown inputs.

Findings

The observer guarantees exponential stability of the error system.

Finite-time exact state reconstruction is achievable despite unknown inputs.

Numerical simulations validate the effectiveness of the proposed method.

Abstract

This paper discusses stability and robustness properties of a recently proposed observer algorithm for linear time varying systems. The observer is based on the approximation and subsequent modification of the non-negative Lyapunov exponents which yields a (non-uniform) exponentially stable error system. Theoretical insights in the construction of the observer are given and the error system is analyzed with respect to bounded unknown inputs. Therefor, new conditions for bounded input bounded state stability for linear time varying systems are presented. It is shown that for a specific class of linear time varying systems, a cascaded observer based on higher order sliding mode differentiators can be designed to achieve finite time exact reconstruction of the system states despite the unknown input. A numerical simulation example shows the applicability of the proposed approach.