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Parameter identification algorithm for a LTV system with partially unknown state matrix

Olga Kozachek, Nikolay Nikolaev, Olga Slita, Alexey Bobtsov

TL;DR

The paper tackles state estimation and parameter identification for a linear time-varying system where the state matrix contains unknown time-varying components $D(\\theta(t))$. An adaptive, output-based state observer is developed, employing a structured arrangement of $N,G,M,M_c,L$ and a $z=\\hat{x}-Gy$ transformation to ensure exponential convergence of the state error. After obtaining reliable state estimates, a regression-based identification pipeline using LTI filtering and Dynamic Regressor Extension and Mixing (DREM) recovers the unknown frequencies $\\omega_i$ and time-varying components $\\theta_i(t)$; this pipeline iteratively estimates constants $l_i$ and amplitudes, culminating in a full reconstruction of the unknown parameters. Numerical simulations illustrate convergence of the state estimation error, the frequency estimates, and the time-varying parameter trajectories, validating the approach for output-feedback scenarios with non-identity $C$.

Abstract

In this paper an adaptive state observer and parameter identification algorithm for a linear time-varying system are developed under condition that the state matrix of the system contains unknown time-varying parameters of a known form. The state vector is observed using only output and input measurements without identification of the unknown parameters. When the state vector estimate is obtained, the identification algorithm is applied to find unknown parameters of the system.

Parameter identification algorithm for a LTV system with partially unknown state matrix

TL;DR

The paper tackles state estimation and parameter identification for a linear time-varying system where the state matrix contains unknown time-varying components . An adaptive, output-based state observer is developed, employing a structured arrangement of and a transformation to ensure exponential convergence of the state error. After obtaining reliable state estimates, a regression-based identification pipeline using LTI filtering and Dynamic Regressor Extension and Mixing (DREM) recovers the unknown frequencies and time-varying components ; this pipeline iteratively estimates constants and amplitudes, culminating in a full reconstruction of the unknown parameters. Numerical simulations illustrate convergence of the state estimation error, the frequency estimates, and the time-varying parameter trajectories, validating the approach for output-feedback scenarios with non-identity .

Abstract

In this paper an adaptive state observer and parameter identification algorithm for a linear time-varying system are developed under condition that the state matrix of the system contains unknown time-varying parameters of a known form. The state vector is observed using only output and input measurements without identification of the unknown parameters. When the state vector estimate is obtained, the identification algorithm is applied to find unknown parameters of the system.
Paper Structure (10 sections, 1 theorem, 46 equations, 6 figures)

This paper contains 10 sections, 1 theorem, 46 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. MAIN RESULT
  4. STATE OBSERVER DESIGN
  5. TIME-VARYING PARAMETERS IDENTIFICATION
  6. NUMERICAL EXAMPLE AND SIMULATION
  7. State observer
  8. Time-varying parameter estimation
  9. Simulation results
  10. CONCLUSIONS

Key Result

Proposition 1

If for the system sys_new, out_new there exist matrices $N \in \mathbb{R}^{n \times 1}$, $G \in \mathbb{R}^{n \times n}$, $M \in \mathbb{R}^{n \times n}$, $M_c \in \mathbb{R}^{n \times n}$ and $L \in \mathbb{R}^{n \times 1}$ and the following conditions are satisfied with $L$ that ensure an exponential convergence of the $\tilde{x}=x - \hat{x}$ to zero in the system $\dot{\tilde{x}}=M_c\tilde{x}$

Figures (6)

  • Figure 1: The transients of the state vector $x$ and its estimate $\hat{x}$
  • Figure 2: The transient of the state vector estimation error $\tilde{x}$
  • Figure 3: The transient of the estimation error $\tilde{\omega}$
  • Figure 4: The transient of the estimation error $\tilde{a}$
  • Figure 5: The transient of the parameter $\theta(t)$ and its estimate $\hat{\theta}(t)$
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof