MHE under parametric uncertainty -- Robust state estimation without informative data

TL;DR

This work investigates robust online state estimation for nonlinear systems with unknown constant parameters under process and measurement noise, specifically addressing the absence of persistency of excitation (PE). It shows that standard moving horizon estimation (MHE) for joint state/parameter estimation and classical adaptive observers can yield diverging estimates when excitation is weak. The authors propose a regularized MHE that uses a fixed prior parameter $\bar{\theta}_0$ instead of online updates, and prove that the resulting estimator is practically robustly stable in a $\delta$-IOpS sense under a horizon-based condition, without requiring PE. A mechanical car model with unknown tire parameters demonstrates improved state accuracy and bounded parameter drift, highlighting practical relevance and connections to adaptive-control ideas such as $\sigma$-modification. The results offer a scalable, robust estimation framework for systems where ensuring PE is impractical.

Abstract

In this paper, we study joint state and parameter estimation for general nonlinear systems with uncertain parameters and persistent process and measurement noise. In particular, we are interested in stability properties of the resulting state estimate in the absence of persistency of excitation (PE). With a simple academic example, we show that existing moving horizon estimation (MHE) approaches for joint state and parameter estimation as well as classical adaptive observers can result in diverging state estimates in the absence of PE, even if the noise is small. We propose an MHE formulation involving a regularization based on a constant prior estimate of the unknown system parameters. Only assuming the existence of a stable state estimator, we prove that the proposed MHE approach results in practically robustly stable state estimates irrespective of PE. We discuss the relation of the proposed MHE formulation to state-of-the-art results from MHE and adaptive estimation. The properties of the proposed MHE approach are illustrated with a numerical example of a car with unknown tire friction parameters.

Figures (6)

• Figure 1: Academic example (Section \ref{['sec:motivating_example']}): Absolute state and parameter estimation errors resulting from a standard MHE approach for joint state and parameter estimation \ref{['eq:MHE_state_and_parameter']}.
• Figure 2: Academic example (Section \ref{['sec:motivating_example']}): Absolute state and parameter estimation errors resulting from a standard MHE approach for joint state and parameter estimation with prior on $\hat{\theta}_{t-M_t}$\ref{['eq:MHE_state_and_parameter']} and our proposed MHE approach with prior $\bar{\theta}_0$\ref{['eq:MHE_parametric']}.
• Figure 3: Academic example (Section \ref{['sec:motivating_example']}): Absolute state and parameter estimation errors resulting from the adaptive observer proposed in Dey2022 for both the noise-free case ($w=0$) and the case with measurement noise.
• Figure 4: Miniature race car example (Section \ref{['sec:num']}): The top three plots show the states $x_{\mathrm{p}},\ y_{\mathrm{p}},\ \psi$ with corresponding sensor measurements and estimates resulting from all three considered MHE approaches and an EKF. The bottom three subplots show the estimation error for $v_{\mathrm{x}},\ v_{\mathrm{y}},\ \omega$ resulting from all three MHE approaches and the EKF (dotted line) and a moving average thereof over $40$ time steps (solid line).
• Figure 5: Miniature race car example (Section \ref{['sec:num']}): The top two subplots show the normalized parameter estimation errors resulting from all three considered MHE approaches and an EKF (dotted line) and a moving average thereof over $40$ time steps (solid line). The bottom two subplots show the tire forces $F_{\mathrm{r}}$ and $F_{\mathrm{f}}$ and the corresponding estimates thereof resulting from all three MHE approaches and the EKF.

Theorems & Definitions (13)

• Definition 1: State estimator, adapted from Allan2021
• Definition 2: Incremental input-to-output practical stability
• Definition 3: $\delta$-IOOS Lyapunov function Muntwiler2023
• Proposition 1
• proof
• Remark 1: Persistency of excitation
• Theorem 1: MHE is $\delta$-IOpS
• Corollary 1: FIE is $\delta$-IOS
• proof
• Remark 2: Known parameter