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Observability and State Estimation for Smooth and Nonsmooth Differential Algebraic Equation Systems

Hesham Abdelfattah, Sameh A. Eisa, Peter Stechlinski

TL;DR

Addresses local observability and state estimation for smooth and nonsmooth index-1 DAE systems with $n_x$ differential states and $n_w$ algebraic states. Proposes the L-SERC test based on lexicographic sensitivities to assess partial observability and distinguish observable from non-observable states, and introduces a sensitivity-based EKF (S-EKF) that leverages these sensitivities for state estimation. Demonstrates the framework on a wind turbine power system, showing correct identification of observable states and effective state tracking even in partial observability regimes. The results provide a practical methodology for observer design in DAEs with nonsmooth dynamics.

Abstract

In this work, we extend the sensitivity-based rank condition (SERC) test for local observability to another class of systems, namely smooth and nonsmooth differential-algebraic equation (DAE) systems of index-1. The newly introduced test for DAEs, which we call the lexicographic SERC (L-SERC) observability test, utilizes the theory of lexicographic differentiation to compute sensitivity information. Moreover, the newly introduced L-SERC observability test is useful in the context of partial observability as it can judge which states are observable and which are not. Additionally, we introduce a novel sensitivity-based extended Kalman filter (S-EKF) algorithm for state estimation, applicable to both smooth and nonsmooth DAE systems. Finally, we apply the newly developed S-EKF to estimate the states of a wind turbine power system model.

Observability and State Estimation for Smooth and Nonsmooth Differential Algebraic Equation Systems

TL;DR

Addresses local observability and state estimation for smooth and nonsmooth index-1 DAE systems with differential states and algebraic states. Proposes the L-SERC test based on lexicographic sensitivities to assess partial observability and distinguish observable from non-observable states, and introduces a sensitivity-based EKF (S-EKF) that leverages these sensitivities for state estimation. Demonstrates the framework on a wind turbine power system, showing correct identification of observable states and effective state tracking even in partial observability regimes. The results provide a practical methodology for observer design in DAEs with nonsmooth dynamics.

Abstract

In this work, we extend the sensitivity-based rank condition (SERC) test for local observability to another class of systems, namely smooth and nonsmooth differential-algebraic equation (DAE) systems of index-1. The newly introduced test for DAEs, which we call the lexicographic SERC (L-SERC) observability test, utilizes the theory of lexicographic differentiation to compute sensitivity information. Moreover, the newly introduced L-SERC observability test is useful in the context of partial observability as it can judge which states are observable and which are not. Additionally, we introduce a novel sensitivity-based extended Kalman filter (S-EKF) algorithm for state estimation, applicable to both smooth and nonsmooth DAE systems. Finally, we apply the newly developed S-EKF to estimate the states of a wind turbine power system model.

Paper Structure

This paper contains 8 sections, 1 theorem, 25 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sensitivity-Based Observability Method
  4. L-SERC Test for Nonsmooth DAE Observability
  5. Determining Observable and Non-Observable States
  6. Sensitivity-based EKF Method
  7. Applications and Comparison with Literature
  8. Conclusion

Key Result

Theorem III.3

Suppose that $\mathbf{z}^*=(\mathbf{x}^*,\mathbf{w}^*)$ is a regular solution of eq:1 on $[t_0,t_f]$ through $\{(\mathbf{x}_0^*,\mathbf{w}^*_0,\mathbf{u}^*,\mathbf{v}^*)\}$ and, for some $\{t_0,t_1,\ldots,t_N\} \subset [t_0,t_f]$ and $\mathbf{d}\in\mathbb R^{n_x}$, it holds that ${\rm rank}({\pmb \U and the L-sensitivity output functions are and $(\mathbf{X}^*,\mathbf{W}^*)$ uniquely solve the fo

Figures (1)

  • Figure 1: The left panel shows S-EKF with output $y=E"_{q}V+\nu(t)$ and the right panel shows S-EKF with output $y=\textnormal{min}(V+\nu(t),0.98)$. The predicted (green) curve is the solution of \ref{['ex:WTPS DAE']} with $\Omega=0,\nu=0$, the true (light blue) curve is the solution of \ref{['ex:WTPS DAE']}, and the estimated (black) curve represents the filter states from the S-EKF algorithm.

Theorems & Definitions (12)

  • Definition III.1
  • Definition III.2
  • Theorem III.3
  • proof
  • Definition III.4
  • Remark III.5
  • Remark III.6
  • Remark III.7
  • Definition III.8
  • Remark IV.1
  • ...and 2 more