Finite-Sample Identification of Continuous-time Parameter-Linear Systems

Conceitos essenciais

A novel method for estimating the parameters of a large class of nonlinear continuous-time systems from discrete-time observations, with finite-sample error guarantees.

Resumo

The content presents a method for parameter estimation of a broad class of nonlinear continuous-time systems, where the state consists of output derivatives and the flow is linear in the parameter. The key ideas are:

Differentiation of the noisy output measurements using a carefully designed finite-difference filter to obtain estimates of the state derivatives. The filter is designed to balance bias and variance, providing quantitative bounds on the mean squared error of the derivative estimates.

Formulating the parameter estimation problem as a linear regression, where the regressors are the estimated state derivatives and the response is the highest-order derivative. This allows the use of regularized least squares to solve for the unknown parameter, while accounting for errors in both the regressors and the response.

The analysis provides the first finite-sample frequentist guarantee for parameter estimation of this broad class of nonlinear continuous-time systems from discrete-time observations. The method is shown to outperform classical nonlinear least squares approaches, which can suffer from multiple local optima and heavy-tailed estimation errors.
The technical contributions include new results on the bias-variance tradeoff in differentiation of noisy signals, as well as a detailed analysis of regularized least squares with errors in both the regressors and the response variable.

Estatísticas

The system dynamics are described by the partial differential equation:
∂^m/∂t^m y(t, θ) = θ^T ϕ(ξ(t, θ))
where y is the output, ξ is the state consisting of output derivatives, and θ is the unknown parameter.
The observations are discrete-time measurements:
z_i = y(i/n * T) + w_i
where w_i is observation noise.

Citações

"We present a method of parameter estimation for large class of nonlinear systems, namely those in which the state consists of output derivatives and the flow is linear in the parameter."
"The method, which solves for the unknown parameter by directly inverting the dynamics using regularized linear regression, is based on new design and analysis ideas for differentiation filtering and regularized least squares."

Principais Insights Extraídos De

Estimation Sample Complexity of a Class of Nonlinear Continuous-time Systems

by Simon Kuang,... às arxiv.org 04-24-2024

https://arxiv.org/pdf/2312.05382.pdf

Perguntas Mais Profundas

How would the estimation performance and guarantees change if the system dynamics were non-autonomous, i.e., the feature map ϕ depended on time t

If the system dynamics were non-autonomous, meaning the feature map ϕ depended on time t, the estimation performance and guarantees would likely be affected. In such cases, the time-varying nature of the feature map would introduce additional complexity into the parameter estimation process. The estimation method would need to account for the time dependence in the feature map ϕ, potentially requiring more sophisticated techniques to handle the varying dynamics accurately. The guarantees of the estimation method may need to be reevaluated to ensure their validity in the presence of time-varying system dynamics.

Can the proposed method be extended to handle systems with multiple unknown parameters, or systems with nonlinear dependence on the parameters

The proposed method could potentially be extended to handle systems with multiple unknown parameters or systems with nonlinear dependence on the parameters. Extending the method to multiple parameters would involve modifying the parameter estimation framework to accommodate the additional parameters and their interactions with the system dynamics. Techniques such as regularization and optimization could be adapted to handle multiple parameters effectively. Similarly, for systems with nonlinear parameter dependencies, the estimation method could be enhanced to capture the nonlinear relationships between the parameters and the system dynamics, possibly through nonlinear regression or optimization approaches.

What are the potential applications of this continuous-time parameter estimation framework beyond the examples discussed, and how could it impact real-world problems in engineering, science, or other domains

The continuous-time parameter estimation framework presented in the context has various potential applications beyond the examples discussed. In engineering, the framework could be applied to fields such as control systems, robotics, and signal processing for accurate parameter identification in complex systems. In science, the framework could be utilized in areas like physics, biology, and environmental science for modeling and analyzing continuous-time systems with unknown parameters. Additionally, the framework could find applications in finance, economics, and social sciences for modeling dynamic systems and making predictions based on parameter estimates. Overall, the framework has the potential to impact real-world problems by providing a robust and reliable method for estimating parameters in nonlinear continuous-time systems, leading to improved modeling, prediction, and control capabilities.