Robust Moving Horizon Estimation for Nonlinear Systems with Time-Varying Parameters

Core Concepts

The authors propose a moving horizon estimation (MHE) scheme to robustly estimate the states and time-varying parameters of nonlinear systems, where the observability of the parameters may depend on the system's excitation and can be absent during operation.

Abstract

The authors consider a nonlinear discrete-time system with states x and time-varying parameters z. They assume that the states x are uniformly detectable (i-IOSS) while the parameters z are non-uniformly observable, meaning their observability depends on the system's excitation and may be absent during operation.
To address this challenge, the authors propose an MHE scheme that involves a standard quadratic cost function with an adaptive regularization term. This regularization term depends on the current observability of the parameters, which is monitored online. The authors develop robustness guarantees for the overall estimation error that are valid for all times and improve the more often the parameters are detected to be observable during operation.
The key aspects of the proposed approach are:

It does not require a priori guarantees on the observability of the parameters, which is usually impossible for general nonlinear systems.
It relies on online monitoring of parameter observability and selects an appropriate regularization term accordingly.
It provides robustness guarantees for the state and parameter estimation errors that are valid independent of the parameter observability and improve with more frequent parameter observability.
It is illustrated through a simulation example of a modified Chua's circuit system with time-varying parameters.

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Key Insights Distilled From

Moving horizon estimation for nonlinear systems with time-varying parameters

by Juli... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09566.pdf

Deeper Inquiries

How can the online monitoring of parameter observability be implemented efficiently in practice for general nonlinear systems

In practice, online monitoring of parameter observability for general nonlinear systems can be efficiently implemented by utilizing advanced data processing techniques and algorithms. Here are some key steps to implement this monitoring effectively:

Data Collection: Gather input-output data from the system continuously or at regular intervals. This data should include measurements of the system's inputs, outputs, and any disturbances.

Observability Metric Calculation: Develop an observability metric that quantifies the observability of the parameters based on the available data. This metric can be derived from the system dynamics and the relationship between inputs, outputs, and parameters.

Real-Time Analysis: Use real-time data processing and analysis techniques to evaluate the observability metric continuously as new data becomes available. This analysis should consider the current state of the system and the historical data to determine the observability status.

Decision Making: Based on the observability metric and predefined thresholds, make decisions on whether the parameters are observable at any given time. If the observability level falls below a certain threshold, appropriate actions can be taken, such as adjusting the estimation algorithm or introducing regularization terms.

Feedback Loop: Implement a feedback loop mechanism to update the estimation algorithm based on the observability status. This ensures that the estimation process adapts dynamically to changes in parameter observability.

By following these steps and leveraging advanced data processing and real-time analysis techniques, online monitoring of parameter observability can be efficiently implemented in practice for general nonlinear systems.

What are the potential limitations or drawbacks of the proposed approach compared to alternative techniques for joint state and parameter estimation, such as adaptive observers

While the proposed Moving Horizon Estimation (MHE) scheme offers robustness and adaptability in estimating states and time-varying parameters of nonlinear systems, there are potential limitations and drawbacks compared to alternative techniques like adaptive observers:

Computational Complexity: The MHE scheme involves solving optimization problems at each time step, which can be computationally intensive, especially for systems with high-dimensional state and parameter spaces. This complexity can limit real-time applicability in certain scenarios.

Observability Dependency: The effectiveness of the MHE scheme relies on the observability of parameters, which may not always be guaranteed in practical systems. In cases of low observability, the estimation performance may degrade, impacting the overall accuracy.

Tuning Parameters: The performance of the MHE scheme is sensitive to the selection of tuning parameters such as the horizon length, regularization terms, and cost function weights. Finding optimal parameter values can be challenging and may require extensive tuning.

Model Complexity: The MHE scheme assumes specific properties of the system dynamics, such as incremental bounded-energy bounded-state property. Deviations from these assumptions or dealing with highly nonlinear systems can pose challenges for accurate estimation.

Adaptability to Dynamic Changes: Adaptive observers have mechanisms to adjust internal model parameters in real-time based on system behavior. In contrast, the MHE scheme may require manual adjustments or predefined rules for handling dynamic changes in system parameters.

While the MHE scheme offers robustness and stability guarantees, these limitations should be considered when choosing between different estimation techniques for joint state and parameter estimation.

Can the proposed MHE scheme be extended to handle cases where the parameter dynamics do not satisfy the assumed incremental bounded-energy bounded-state property

The proposed Moving Horizon Estimation (MHE) scheme, as described in the context, relies on the assumption that the parameter dynamics satisfy the incremental bounded-energy bounded-state property to ensure boundedness of the estimation error. However, in cases where the parameter dynamics do not meet this specific property, the extension of the MHE scheme may face challenges. Here are some considerations for extending the MHE scheme in such scenarios:

Relaxing Assumptions: One approach could be to relax the strict requirements of the incremental bounded-energy bounded-state property and explore alternative stability criteria that accommodate a broader range of parameter dynamics.

Adaptive Regularization: Introduce adaptive regularization terms or adjustment mechanisms in the MHE scheme to handle variations in parameter dynamics. These adaptations can help mitigate the impact of non-compliant dynamics on the estimation process.

Dynamic Parameter Modeling: Develop dynamic models or adaptive structures within the MHE framework to account for time-varying parameters that do not adhere to the incremental bounded-energy bounded-state property. This can involve updating the parameter dynamics model based on real-time observations.

Hybrid Approaches: Consider hybrid approaches that combine elements of MHE with other estimation techniques, such as adaptive observers or machine learning algorithms, to enhance the robustness and flexibility of parameter estimation in nonlinear systems with diverse dynamics.

By exploring these strategies and potentially integrating them into the MHE scheme, it may be possible to extend its applicability to handle cases where the parameter dynamics deviate from the assumed incremental bounded-energy bounded-state property.