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On the solvability of parameter estimation-based observers for nonlinear systems

Bowen Yi, Leyan Fang, Romeo Ortega

Abstract

Parameter estimation-based observer (PEBO) is a recently developed constructive tool to design state observers for nonlinear systems. It reformulates the state estimation problem as one of online parameter identification, effectively addressing many open estimation challenges in practical applications. The feasibility of a PEBO design relies on two fundamental properties: transformability and identifiability. The former pertains to the existence of an injective solution to a suitable partial differential equation, whereas the latter characterizes the uniqueness of the parameterization induced by the resulting nonlinear regression model. In this paper, we analyze the existence of PEBOs for general nonlinear systems by studying these two properties in detail and by providing sufficient conditions under which they hold.

On the solvability of parameter estimation-based observers for nonlinear systems

Abstract

Parameter estimation-based observer (PEBO) is a recently developed constructive tool to design state observers for nonlinear systems. It reformulates the state estimation problem as one of online parameter identification, effectively addressing many open estimation challenges in practical applications. The feasibility of a PEBO design relies on two fundamental properties: transformability and identifiability. The former pertains to the existence of an injective solution to a suitable partial differential equation, whereas the latter characterizes the uniqueness of the parameterization induced by the resulting nonlinear regression model. In this paper, we analyze the existence of PEBOs for general nonlinear systems by studying these two properties in detail and by providing sufficient conditions under which they hold.
Paper Structure (15 sections, 7 theorems, 83 equations, 7 figures)

This paper contains 15 sections, 7 theorems, 83 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Background Material on PEBO
  4. Transformability
  5. Identifiablity
  6. Summary of the problem setting
  7. Transformability to Canonical Form
  8. PDE Solvability
  9. Left Invertibility
  10. Parameter Identifiability
  11. Global Identifiablity Under Differentiable Observability
  12. Local Identifiability
  13. Discussion
  14. Example
  15. Conclusion

Key Result

Proposition 1

Consider the system NLsyst with the $C^1$-smooth vector fields $f$ and $h$ that satisfy Assumption assm:1. For any Hurwitz matrix $A \in \mathbb{R}^{n_z \times n_z}$, any matrix $B \in \mathbb{R}^{n_z\times p}$, and any smooth function $H:\mathbb{R}^p \times \mathbb{R}_{\ge 0} \to \mathbb{R}^p$, the In particular, the function with $\rho > -{\hbox{min}}\{\hbox{Re}(\lambda_i(A))\}$, ensures that t

Figures (7)

  • Figure 1: Signal flow diagram of the PEBO
  • Figure 2: Invariant foliation in PEBO
  • Figure 3: An illustration to Remark \ref{['rem:omega']}
  • Figure 4: Single-time cost function illustrating lack of identifiability on $\mathbb{R}^{n_z}$
  • Figure 5: Simulation results: Using a single batch parameter estimate over the interval $[0.1,1.0]$ s. The parameter vector $\theta$ is estimated only once at the final time $t_f = 1\,\mathrm{s}$ using all the collected data over the interval.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Proposition 1: Solvability of the PDE
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Proposition 2: Injectivity of $\phi(x,t)$
  • proof
  • ...and 18 more