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Hybrid multi-observer for improving estimation performance

E. Petri, R. Postoyan, D. Astolfi, D. Nesic, V. Andrieu

TL;DR

This work introduces a hybrid multi-observer framework to online-tune the gains of a nominal nonlinear observer while preserving input-to-state stability. By running a bank of modes with diverse gains and scoring them via monitoring variables, the approach adaptively selects the best performing mode through a switching rule, with average dwell-time guarantees to avoid Zeno behavior. Theoretical results establish ISS for the overall hybrid scheme and provide Lyapunov-based proofs of stability and a guaranteed (semiglobal) dwell-time, while also proving potential strict performance improvement over the nominal observer in terms of a quadratic cost. Numerical case studies on a Van der Pol oscillator and a Li-ion battery model illustrate substantial estimation-performance gains, including improved robustness to measurement noise and disturbances. The framework offers flexible gain design options (including optimization-based offline design and null-gain modes) and can be extended to broader observer classes and applications.

Abstract

Various methods are nowadays available to design observers for broad classes of systems, where the primary focus is on establishing the convergence of the estimated states. Nevertheless, the question of the tuning of the observer to achieve satisfactory estimation performance remains largely open. In this context, we present a general design framework for the online tuning of the observer gains. Our starting point is a robust nominal observer designed for a general nonlinear system, for which an input-to-state stability property can be established. Our goal is then to improve the performance of this nominal observer. We present for this purpose a new hybrid multi-observer scheme, whose great flexibility can be exploited to enforce various desirable properties, e.g., fast convergence and good sensitivity to measurement noise. We prove that an input-to-state stability property also holds for the proposed scheme and, importantly, we ensure that the estimation performance in terms of a quadratic cost is (strictly) improved. We illustrate the efficiency of the approach in improving the performance of given nominal observers in two numerical examples (Van der Pol oscillator and Lithium-Ion (Li-Ion) battery model).

Hybrid multi-observer for improving estimation performance

TL;DR

This work introduces a hybrid multi-observer framework to online-tune the gains of a nominal nonlinear observer while preserving input-to-state stability. By running a bank of modes with diverse gains and scoring them via monitoring variables, the approach adaptively selects the best performing mode through a switching rule, with average dwell-time guarantees to avoid Zeno behavior. Theoretical results establish ISS for the overall hybrid scheme and provide Lyapunov-based proofs of stability and a guaranteed (semiglobal) dwell-time, while also proving potential strict performance improvement over the nominal observer in terms of a quadratic cost. Numerical case studies on a Van der Pol oscillator and a Li-ion battery model illustrate substantial estimation-performance gains, including improved robustness to measurement noise and disturbances. The framework offers flexible gain design options (including optimization-based offline design and null-gain modes) and can be extended to broader observer classes and applications.

Abstract

Various methods are nowadays available to design observers for broad classes of systems, where the primary focus is on establishing the convergence of the estimated states. Nevertheless, the question of the tuning of the observer to achieve satisfactory estimation performance remains largely open. In this context, we present a general design framework for the online tuning of the observer gains. Our starting point is a robust nominal observer designed for a general nonlinear system, for which an input-to-state stability property can be established. Our goal is then to improve the performance of this nominal observer. We present for this purpose a new hybrid multi-observer scheme, whose great flexibility can be exploited to enforce various desirable properties, e.g., fast convergence and good sensitivity to measurement noise. We prove that an input-to-state stability property also holds for the proposed scheme and, importantly, we ensure that the estimation performance in terms of a quadratic cost is (strictly) improved. We illustrate the efficiency of the approach in improving the performance of given nominal observers in two numerical examples (Van der Pol oscillator and Lithium-Ion (Li-Ion) battery model).
Paper Structure (27 sections, 8 theorems, 63 equations, 4 figures, 2 tables)

This paper contains 27 sections, 8 theorems, 63 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Hybrid estimation scheme
  4. Additional modes
  5. Monitoring variables
  6. Selection criterion
  7. Reset rule
  8. Hybrid model
  9. Design guidelines
  10. General guidelines for gains selection
  11. Null gain
  12. Optimization-based design
  13. Adjusting $L_1$
  14. Exploiting system and nominal observer structure and/or behaviour
  15. Stability guarantees
  16. ...and 12 more sections

Key Result

Theorem 1

Consider system eq:HybridSystemGeneral-eq:jumpSet and suppose Assumptions NominalAssumption-ASS:ass1 hold. Then there exist $\beta_U \in \mathcal{KL}$ and $\gamma_U \in \mathcal{K}_{\infty}$ such that for any input $u \in \mathcal{L_U}$, disturbance input $v \in \mathcal{L_V}$ and measurement noise for all $(t,j) \in \text{dom}\, q$, with $e:= (e_1, \dots, e_{N+1})$ and $\eta:= (\eta_1, \dots, \e

Figures (4)

  • Figure 1: Block diagram representing the system architecture with $\eta:= (\eta_1, \dots \eta_{N+1})$, $\hat{x}:= (\hat{x}_1, \dots, \hat{x}_{N+1})$.
  • Figure 2: Van der Pol oscillator. Norm of the estimation error $|e|$ (top figure), performance cost $J$ (middle figure) and $\sigma$ (bottom figure). Nominal (yellow), without resets (red), with resets (blue).
  • Figure 3: $f(\text{SOC})$(blue) with its linearization (red) and PHEV current input.
  • Figure 4: Battery example. Norm of the estimation error $|e|$ (top figure), performance cost $J$ (middle figure) and $\sigma$ (bottom figure). Nominal (yellow), without resets (red), with resets (blue).

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • ...and 4 more