Estimating Nonlinear Functional Outputs from Noisy Measurements: A Unified Detectability and Estimation Framework

Core Concepts

Functional detectability is a necessary and sufficient condition for the existence of a stable functional estimator that can estimate a nonlinear function of the system state from noisy output measurements.

Abstract

The paper presents a general framework for nonlinear functional estimation, where the goal is to estimate a nonlinear function of the system state from noisy output measurements. The key contributions are:

Introduction of incremental input/output-to-output stability (δ-IOOS) as a notion of functional detectability, and showing that it is necessary and sufficient for the existence of a stable functional estimator.

Design of a full information estimation (FIE) approach for functional estimation, and proof that it is a δ-IOS functional estimator if the system is δ-IOOS.

Demonstration that functional detectability is a necessary and sufficient condition for the existence of a stable functional estimator.

Simplification of the FIE design for the case of exponential functional detectability, allowing the use of a quadratic objective function.

Illustration of the practical applicability of the proposed functional estimation approach on a power system example, where the full system state is not detectable but the total power load can be estimated stably.

The paper provides a unified framework to study functional estimation, establishing necessary and sufficient conditions for the existence of a stable functional estimator, and presenting a corresponding functional estimator design.

Stats

The power system model has 4 buses and 4 transmission lines, resulting in a 16-dimensional state vector. The system is subject to uniformly distributed process noise with a maximum magnitude of 0.005 per state, and uniformly distributed measurement noise with a maximum magnitude of 0.05 per measurement.

Quotes

"Functional detectability is a necessary and sufficient condition for the existence of a stable functional estimator."
"The presented FIE approach is proven to be δ-IOS if the system is δ-IOOS."

Key Insights Distilled From

Nonlinear Functional Estimation: Functional Detectability and Full Information Estimation

by Simo... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2312.13859.pdf

Deeper Inquiries

How can the proposed functional estimation framework be extended to handle time-varying or uncertain functions ϕ(x)

The proposed functional estimation framework can be extended to handle time-varying or uncertain functions ϕ(x) by incorporating adaptive techniques. One approach is to update the functional estimator parameters based on the observed discrepancies between the estimated and actual functional values. This adaptive adjustment can help the estimator adapt to changes in the underlying function ϕ(x) over time. Additionally, incorporating uncertainty quantification methods, such as Bayesian inference or robust optimization, can help account for uncertainties in the function ϕ(x) and improve the robustness of the functional estimator.

What are the implications of the necessary and sufficient conditions for functional detectability on the design of control systems that rely on functional estimates

The necessary and sufficient conditions for functional detectability have significant implications on the design of control systems that rely on functional estimates. By ensuring that the system is functionally detectable, the control system can have a stable and reliable estimate of the desired function of the state, even when the full system state is not directly observable. This allows for the development of control strategies that are robust to uncertainties and disturbances, leading to improved system performance and stability. Additionally, the detectability conditions provide a theoretical foundation for the design and analysis of functional estimators in practical applications, such as fault detection, state estimation, and parameter identification.

Can the FIE approach be further simplified or approximated to reduce the computational complexity for large-scale systems

The FIE approach can be further simplified or approximated to reduce the computational complexity for large-scale systems by implementing techniques such as reduced-order modeling, sparse optimization, or parallel computing. One approach is to use model reduction techniques to reduce the dimensionality of the optimization problem, making it more computationally tractable. Additionally, approximations such as linearization or convex relaxation can be used to simplify the optimization problem while maintaining accuracy. Furthermore, leveraging distributed computing or parallel processing can help speed up the computation of the FIE solution for large-scale systems. These simplifications and approximations can make the FIE approach more scalable and efficient for real-world applications.