Accurate Identification of Polynomial Nonlinear Systems with Uncontrollable Linearization Using Center-Manifold-Based Approach

Keskeiset käsitteet

A center-manifold-based system identification method can accurately identify polynomial nonlinear systems with uncontrollable linearization, even in the presence of measurement noise.

Tiivistelmä

This paper presents simulation results demonstrating the effectiveness of a center-manifold-based system identification (SID) method for identifying polynomial nonlinear systems (PNS) with uncontrollable linearization.
The key highlights are:

The SID method is able to accurately identify the system order, state-space matrices (A, B, C, D), and the nonlinear coefficients (Fl, Hl) of the PNS, even when the linearized system is uncontrollable.

In the noise-free case, the identification errors are negligible, with the identified models closely matching the true system.

In the presence of measurement noise (around 80dB signal-to-noise ratio), the SID method still provides reasonable identification accuracy, with the average identification errors within an acceptable range.

Two variants of the SID algorithm are compared - Method I and Method II. Both methods exhibit similar performance on the example PNS, indicating the robustness of the center-manifold-based approach.

The simulation results validate the effectiveness of the proposed center-manifold-based SID method for identifying PNS with uncontrollable linearization, even under noisy conditions.

Tilastot

The following key metrics are reported in the paper:
"The identification error is measured by the following norm ratio:
G∆(s) −ˆG∆(s)
∞
∥G∆(s)∥∞
"
The identification errors for the transfer function entries G1,1,1(s), G2,2,1(s), G2,2,2(s), and G2,1,3(s) are provided in Table I for the noise-free case, as well as for the noisy case using Methods I and II.

Lainaukset

None.

Tärkeimmät oivallukset

Simulation Results of Center-Manifold-Based Identification of Polynomial Nonlinear Systems with Uncontrollable Linearization

by Chao Huang, ... klo arxiv.org 10-04-2024

https://arxiv.org/pdf/2410.02169.pdf

Syvällisempiä Kysymyksiä

1. How would the performance of the center-manifold-based SID method compare to other nonlinear system identification techniques, such as Volterra series or neural network-based approaches, on a wider range of polynomial nonlinear systems?

The center-manifold-based system identification (SID) method offers a structured approach to identifying polynomial nonlinear systems, particularly those with uncontrollable linearization. Compared to Volterra series methods, which rely on a series expansion to model nonlinearities, the center-manifold approach can provide a more direct representation of the system dynamics by focusing on the intrinsic behavior of the system near equilibrium points. This can lead to more accurate parameter estimation in cases where the system exhibits polynomial behavior.
In contrast, neural network-based approaches are highly flexible and can approximate a wide variety of nonlinear functions, including those that are not easily captured by polynomial forms. However, they often require extensive training data and can suffer from overfitting, especially in high-dimensional spaces. The center-manifold method, while potentially less flexible, benefits from a more interpretable structure and can leverage prior knowledge about the system's dynamics, which can be advantageous in scenarios where data is limited or noisy.
Overall, the performance of the center-manifold-based SID method may excel in scenarios specifically tailored to polynomial nonlinearities, while Volterra series and neural network approaches may perform better in broader contexts with diverse nonlinear behaviors. The choice of method ultimately depends on the specific characteristics of the system being studied and the available data.

2. What are the potential limitations or challenges in applying the center-manifold-based SID method to high-dimensional or more complex nonlinear systems beyond the second-order example presented?

One significant limitation of the center-manifold-based SID method is its reliance on the assumption of polynomial nonlinearity, which may not hold in high-dimensional or more complex systems. As the dimensionality of the system increases, the complexity of the polynomial terms can grow exponentially, making it challenging to accurately identify all relevant parameters. This can lead to issues such as model overfitting or underfitting, particularly if the system exhibits interactions that are not well-represented by polynomial terms.
Additionally, the method assumes that the linearization of the system is uncontrollable, which can complicate the identification process in higher dimensions where the controllability and observability conditions may not be easily satisfied. The computational burden also increases with dimensionality, as the algorithms may require more extensive numerical computations and data to converge to a reliable solution.
Moreover, the presence of noise, as indicated in the simulation results, can significantly impact the performance of the center-manifold-based SID method. In high-dimensional systems, the noise can obscure the underlying dynamics, making it difficult to extract meaningful information from the data. Therefore, careful consideration of noise reduction techniques and robust estimation methods is essential when applying this approach to more complex systems.

3. Can the center-manifold-based SID method be extended to handle other types of nonlinearities beyond polynomial forms, such as rational, trigonometric, or piecewise-linear functions?

While the center-manifold-based SID method is primarily designed for polynomial nonlinear systems, there is potential for extending its application to other types of nonlinearities, such as rational, trigonometric, or piecewise-linear functions. The key to such extensions lies in the underlying principles of the center manifold theory, which focuses on the behavior of dynamical systems near equilibrium points.
For rational functions, the method could be adapted by considering the center manifold of the system in a transformed state space that captures the rational behavior. This would involve deriving the appropriate center manifold equations that account for the non-polynomial terms, which may require additional mathematical tools and techniques.
In the case of trigonometric functions, the periodic nature of these functions could be incorporated into the center-manifold framework by analyzing the system's dynamics over a periodic interval. This would necessitate a careful formulation of the system equations to ensure that the periodicity is preserved in the identification process.
Piecewise-linear functions present a unique challenge due to their discontinuous nature. However, the center-manifold approach could potentially be adapted by segmenting the state space into regions where linear approximations are valid, allowing for the identification of different linear models within each segment. This would require a robust mechanism for determining the transitions between segments and ensuring continuity in the overall model.
In summary, while the center-manifold-based SID method is inherently suited for polynomial nonlinearities, with appropriate modifications and extensions, it could potentially be applied to a broader class of nonlinear functions, enhancing its versatility in system identification tasks.