# Synthesizing the Optimal Luenberger-type Observer for Nonlinear Systems

## TL;DR

This paper introduces a novel method for designing Luenberger-type observers for nonlinear systems that maximizes the domain of attraction for state estimation errors, ensuring finite-time convergence regardless of initial conditions.

## Contribution

It formulates the observer design as an infinite-dimensional linear program on measures and solves it using Lasserre's relaxations, enabling optimal observer gain selection.

## Key findings

- Maximized the domain of attraction for nonlinear system observers.
- Demonstrated the approach with two example systems.
- Achieved finite-time convergence for state estimation errors.

## Abstract

Observer design typically requires the observability of the underlying system, which may be hard to verify for nonlinear systems, while guaranteeing asymptotic convergence of errors, which may be insufficient in order to satisfy performance conditions in finite time. This paper develops a method to design Luenberger-type observers for nonlinear systems which guarantee the largest possible domain of attraction for the state estimation error regardless of the initialization of the system. The observer design procedure is posed as a two step problem. In the the first step, the error dynamics are abstractly represented as a linear equation on the space of Radon measures. Thereafter, the problem of identifying the largest set of initial errors that can be driven to within the user-specified error target set in finite-time for all possible initial states, and the corresponding observer gains, is formulated as an infinite-dimensional linear program on measures. This optimization problem is solved, using Lasserre's relaxations via a sequence of semidefinite programs with vanishing conservatism. By post-processing the solution of step one, the set of gains that maximize the size of tolerable initial errors is identified in step two. To demonstrate the feasibility of the presented approach two examples are presented.