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Adaptive State Observers of Linear Time-varying Descriptor Systems: A Parameter Estimation-Based Approach

Romeo Ortega, Alexey Bobtsov, Fernando Castanos, Nikolay Nikolaev

TL;DR

The paper addresses adaptive state estimation for linear time-varying descriptor (DAE) systems with uncertain parameters by deploying generalized parameter estimation-based observers (GPEBO). It reframes state reconstruction as parameter estimation of initial conditions and develops the Standard Canonical Form to separate z_a and z_b dynamics, including strangeness-free and partial-z_b cases. A GPEBO is provided for z_a under both strangeness-free and non-strangeness-free conditions, while a specialized second-order observer handles a subset of z_b for nb = 3 with no uncertainties, plus a benchmark LTV circuit to validate the approach. The work highlights the importance of the strangeness index, the IE/observability interplay, and the potential for extending to delays, disturbances, and port-Hamiltonian models, offering a foundation for robust adaptive observation in complex DAE systems.

Abstract

In this paper, we apply the recently developed generalized parameter estimation-based observer design technique for state-affine systems to the practically important case of linear time-varying descriptor systems with uncertain parameters. We give simulation results of benchmark examples that illustrate the performance of the proposed adaptive observer.

Adaptive State Observers of Linear Time-varying Descriptor Systems: A Parameter Estimation-Based Approach

TL;DR

The paper addresses adaptive state estimation for linear time-varying descriptor (DAE) systems with uncertain parameters by deploying generalized parameter estimation-based observers (GPEBO). It reframes state reconstruction as parameter estimation of initial conditions and develops the Standard Canonical Form to separate z_a and z_b dynamics, including strangeness-free and partial-z_b cases. A GPEBO is provided for z_a under both strangeness-free and non-strangeness-free conditions, while a specialized second-order observer handles a subset of z_b for nb = 3 with no uncertainties, plus a benchmark LTV circuit to validate the approach. The work highlights the importance of the strangeness index, the IE/observability interplay, and the potential for extending to delays, disturbances, and port-Hamiltonian models, offering a foundation for robust adaptive observation in complex DAE systems.

Abstract

In this paper, we apply the recently developed generalized parameter estimation-based observer design technique for state-affine systems to the practically important case of linear time-varying descriptor systems with uncertain parameters. We give simulation results of benchmark examples that illustrate the performance of the proposed adaptive observer.
Paper Structure (14 sections, 5 theorems, 71 equations, 8 figures)

This paper contains 14 sections, 5 theorems, 71 equations, 8 figures.

Table of Contents

  1. Introduction
  2. An Introduction to GPEBO
  3. A motivating example
  4. Discussion
  5. The Standard Canonical Form
  6. Classical representation of the Standard Canonical Form
  7. A suitable representation for the $z_b$ dynamics
  8. Design of GPEBO for the System \ref{['weisys']}
  9. The strangeness-free case
  10. GPEBO-based observer for $z_a$
  11. GPEBO-based observer for (part of) $z_b$
  12. A Benchmark LTV Circuit Example
  13. Concluding Remarks and Future Research
  14. Background Material

Key Result

Proposition 1

Consider the descriptor LTV system oldsys, oldsysc verifying Assumption ass1. There exists an adaptive GPEBO of the form with $\chi(t) \in \mathbb{R}^{n_\chi}$ and the mappings such that, for all initial conditions, we have that exponentially fast, provided some suitable excitation conditions---stated in Assumption ass2 below---are satisfied.

Figures (8)

  • Figure 1: Circuit example from BOB.
  • Figure 2: Transients of the regressor $\psi$
  • Figure 3: Transients of the state vector
  • Figure 4: State $x_a$ and state estimation $\hat{x}_a$
  • Figure 5: State $x_b$ and state estimation $\hat{x}_b$
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Lemma 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • proof
  • Remark 2
  • Proposition 4
  • proof
  • ...and 2 more