Immersion of General Nonlinear Systems Into State-Affine Ones for the Design of Generalized Parameter Estimation-Based Observers: A Simple Algebraic Procedure

TL;DR

This paper introduces a simple algebraic method to embed general nonlinear systems into state-affine form, expanding the applicability of parameter estimation-based observers for complex systems.

Contribution

It proposes an algebraic procedure for immersing nonlinear systems into state-affine form, bypassing complex PDE solutions and restrictive conditions, with practical applications demonstrated.

Findings

The method successfully handles complex benchmark examples.

It simplifies the process of system immersion for physical systems.

The approach is effective for practical nonlinear system analysis.

Abstract

Generalized parameter estimation-based observers have proven very successful to deal with systems described in state-affine form. In this paper, we enlarge the domain of applicability of this method proposing an algebraic procedure to immerse} an $n$-dimensional general nonlinear system into and $n_z$-dimensional system in state affine form, with $n_z>n$. First, we recall the necessary and sufficient condition for the solution of the general problem, which requires the solution of a partial differential equation that, moreover, has to satisfy a restrictive injectivity condition. Given the complexity of this task we propose an alternative simple algebraic method to identify the required dynamic extension and coordinate transformation, a procedure that, as shown in the paper, is rather natural for physical systems. We illustrate the method with some academic benchmark examples from observer theory literature -- that, in spite of their apparent simplicity, are difficult to solve with the existing methods -- as well as several practically relevant physical examples.