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

Romeo Ortega, Alexey Bobtsov, Jose Guadalupe Romero, Leyan Fang

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.

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

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 -dimensional general nonlinear system into and -dimensional system in state affine form, with . 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.

Paper Structure

This paper contains 18 sections, 3 theorems, 64 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Mathematically Intractable General Problem Formulation
  3. A change of coordinates and systems immersion problem
  4. A necessary and sufficient condition
  5. A Simple Algebraic Immersion Procedure
  6. Main result
  7. Discussion
  8. Benchmark Examples
  9. An academic example LEVMAR
  10. Two academic examplesBERbook
  11. Magnetic levitation system
  12. Surface-mount permanent magnet synchronous motor
  13. Mechanical systems
  14. Simulations
  15. Magnetic levitation system
  16. ...and 3 more sections

Key Result

proposition 1

There exists a change of coordinates $z_w=T_w(x,w)$ that transform the augmented system augsys into augsysaffif and only if the mapping $T_w(x,w)$ satisfies the PDE $\Box \Box \Box$

Figures (8)

  • Figure 1: Two DoF prismatic robot.
  • Figure 2: Robotic leg
  • Figure 3: Transient behavior of the state $x(t)$
  • Figure 4: Transient behavior of the parameter estimation errors $\tilde{\theta}(t)$
  • Figure 5: Transient behavior of the state estimation errors $\tilde{x}(t)$
  • ...and 3 more figures

Theorems & Definitions (12)

  • remark 1
  • proposition 1
  • remark 2
  • proposition 2
  • proof
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • remark 7
  • ...and 2 more