Regional observer synthesis for locally Lipschitz non-linear systems

This study discusses a linear matrix inequality (LMI)-based observer design for locally Lipschitz non-linear systems that ensures an ellipsoidal region of stability in the estimation error, via a quadratic Lyapunov function analysis. An ellipsoidal Lipschitz region is defined in which a dynamical non-linearity satisfies the Lipschitz condition locally. First, to provide a state feedback stabilisation scheme, the region of stability in terms of state of a non-linear system is included in the ellipsoidal Lipschitz region, and LMIs are derived to ensure asymptotic stability. This novel state feedback stabilisation technique is broadened to the observer synthesis by defining a new region in the form of state estimation error, satisfying the ellipsoidal Lipschitz region. Further, under a bounded state operation, the region of convergence is included in the new region. Furthermore, a local robust observer synthesis methodology against bounded perturbations, for attaining multiple design objectives, is provided by ensuring the local exponential L₂ stability of the estimation error. The proposed methodologies´ guaranteed regional stability, faster convergence, robustness against disturbances, ease of design, and computational simplicity make them superior from the traditional techniques. The results for the proposed schemes are validated by numerical simulations.