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Parameter Estimation-Based Extended Observer for Linear Systems with Polynomial Overparameterization

Anton Glushchenko, Konstantin Lastochkin

TL;DR

This paper addresses joint state and parameter estimation for uncertain linear time-invariant systems that are overparametrized and perturbed by bounded disturbances generated by a known exosystem. It introduces an exponentially stable extended adaptive observer that exploits polynomial overparameterization and dynamic regressor extension and mixing to identify observer parameters and reconstruct the original states in an arbitrary state-space form, rather than virtual observer states. Key contributions include regression-based parameterizations that guarantee convergence under a finite excitation condition, an algebraic state-update approach that avoids Luenberger corrections, and simultaneous disturbance reconstruction. Numerical simulations validate the method, showing robust exponential convergence and improved transient behavior compared to prior approaches. The work advances adaptive observer design for overparameterized systems with disturbances and provides a practical framework for reconstructing physical states and disturbances without enforcing strict persistent excitation.

Abstract

We consider a class of uncertain linear time-invariant overparametrized systems affected by bounded disturbances, which are described by a known exosystem with unknown initial conditions. For such systems an exponentially stable extended adaptive observer is proposed, which, unlike existing solutions, simultaneously: (i) allows one to reconstruct original (physical) states of the system represented in arbitrarily chosen state-space form rather than virtual states of the observer canonical form; (ii) ensures convergence of the state observation error to zero under weak requirement of the regressor finite excitation; (iii) does not include Luenberger correction gain and forms states estimate using algebraic rather than differential equation; (iv) additionally reconstructs the unmeasured external disturbance. The proposed solution is based on the new parametrizations to identify the observer parameters obtained with the help of the heterogeneous mappings and the dynamic regressor extension and mixing procedure. Illustrative simulations support obtained theoretical results.

Parameter Estimation-Based Extended Observer for Linear Systems with Polynomial Overparameterization

TL;DR

This paper addresses joint state and parameter estimation for uncertain linear time-invariant systems that are overparametrized and perturbed by bounded disturbances generated by a known exosystem. It introduces an exponentially stable extended adaptive observer that exploits polynomial overparameterization and dynamic regressor extension and mixing to identify observer parameters and reconstruct the original states in an arbitrary state-space form, rather than virtual observer states. Key contributions include regression-based parameterizations that guarantee convergence under a finite excitation condition, an algebraic state-update approach that avoids Luenberger corrections, and simultaneous disturbance reconstruction. Numerical simulations validate the method, showing robust exponential convergence and improved transient behavior compared to prior approaches. The work advances adaptive observer design for overparameterized systems with disturbances and provides a practical framework for reconstructing physical states and disturbances without enforcing strict persistent excitation.

Abstract

We consider a class of uncertain linear time-invariant overparametrized systems affected by bounded disturbances, which are described by a known exosystem with unknown initial conditions. For such systems an exponentially stable extended adaptive observer is proposed, which, unlike existing solutions, simultaneously: (i) allows one to reconstruct original (physical) states of the system represented in arbitrarily chosen state-space form rather than virtual states of the observer canonical form; (ii) ensures convergence of the state observation error to zero under weak requirement of the regressor finite excitation; (iii) does not include Luenberger correction gain and forms states estimate using algebraic rather than differential equation; (iv) additionally reconstructs the unmeasured external disturbance. The proposed solution is based on the new parametrizations to identify the observer parameters obtained with the help of the heterogeneous mappings and the dynamic regressor extension and mixing procedure. Illustrative simulations support obtained theoretical results.
Paper Structure (7 sections, 51 equations, 2 figures)

This paper contains 7 sections, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Main Result
  4. Parametrization of equations w.r.t. $\kappa \left( \theta \right){\rm{,\;}}{T_I}\left( \theta \right){\rm{,\;}}{x_{\delta 0}}$
  5. Adaptive Observer Design
  6. Numerical Experiments
  7. Conclusion

Figures (2)

  • Figure 1: Behavior of state $\tilde{x}\left( t \right)$ and disturbance $\tilde{\delta} \left( t \right)$ observation errors
  • Figure 2: Behavior of $\left\| {\tilde{U}\left( t \right)} \right\|$ and parametric errors $\tilde{\kappa} \left( t \right){\rm{,\;}}{\tilde{x}_{\delta 0}}\left( t \right){\rm{,\;}}{\tilde{T}_I}\left( t \right)$