A Harmonic Approach for Identifying Continuous-Time Linear Time-Periodic Systems

Core Concepts

This paper presents a novel harmonic framework for the identification of continuous-time linear time-periodic (LTP) systems, which converts the LTP system into an equivalent infinite-dimensional linear time-invariant (LTI) system.

Abstract

The paper presents a novel approach for the identification of continuous-time linear time-periodic (LTP) systems. The key ideas are:

LTP systems can be transformed into an equivalent infinite-dimensional LTI system using a harmonic modeling framework. This LTI system is characterized by a block Toeplitz structure formed by the Fourier coefficients (phasors) of the state and input matrices of the original LTP system.

The identification problem is formulated as an infinite-dimensional least-squares problem, which is then approximated by a finite-dimensional problem by truncating the harmonic expansion to a finite number of phasors. It is shown that the solution to the finite-dimensional problem converges to the solution of the infinite-dimensional problem with an arbitrarily small error.

The identification problem is further simplified by focusing only on the "central strip" of the infinite-dimensional LTI system, which contains all the necessary information to identify the original LTP system. This reduces the number of unknowns significantly.

The finite-dimensional least-squares problem can be solved efficiently, and the paper provides theoretical guarantees on the approximation error. The approach is demonstrated on several numerical examples, including a wind turbine system, and is shown to be effective even in the presence of noisy state measurements.

Stats

The paper does not provide specific numerical data to support the key claims. However, it presents the following important figures:

Figure 1: Comparison of true and estimated state trajectories for the finite phasor-order example, with and without noise.
Figure 2: Comparison of true and estimated state trajectories for the infinite phasor-order example, with and without noise.
Figure 3: Moduli of the true and estimated phasors of matrix A for the infinite phasor-order example with noise.
Figure 4: Comparison of true and estimated state trajectories for the wind turbine example, with and without noise.
Figure 5: Identification error convergence for the wind turbine example as a function of the data length.

Quotes

The paper does not contain any striking quotes that support the key logics.

Key Insights Distilled From

A harmonic framework for the identification of linear time-periodic systems

by Flora Verner... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2309.16273.pdf

Deeper Inquiries

How can the proposed identification method be extended to handle nonlinear or time-varying periodic systems

The proposed identification method can be extended to handle nonlinear or time-varying periodic systems by incorporating techniques from nonlinear system identification and time-varying system analysis. For nonlinear systems, the harmonic modeling approach can be adapted to include higher-order terms in the Fourier series expansion to capture nonlinear dynamics. This extension would involve representing the system dynamics using a Volterra series or other nonlinear system representations. Additionally, techniques such as kernel methods or neural networks can be employed to model the nonlinear relationships between the input and output signals in the frequency domain.
For time-varying periodic systems, the identification method can be enhanced by considering time-varying parameters in the harmonic modeling framework. By allowing the system matrices to vary with time, the identification algorithm can adapt to changes in the system dynamics over different periods. This adaptation can be achieved by updating the Fourier coefficients of the state and input matrices at each time interval, enabling the accurate identification of time-varying periodic systems.

What are the practical limitations of the harmonic modeling approach, and how can they be addressed in real-world applications

The practical limitations of the harmonic modeling approach primarily revolve around the assumptions and requirements of the method. One limitation is the need for periodicity in the system dynamics, which may not always hold true in real-world applications. To address this limitation, techniques such as data-driven modeling or adaptive identification algorithms can be integrated to handle non-periodic components in the system behavior.
Another limitation is the sensitivity to noise in the data, which can affect the accuracy of the identified phasors. To mitigate the impact of noise, robust estimation techniques, such as weighted least squares or Kalman filtering, can be employed to enhance the robustness of the identification method in the presence of noise.
Furthermore, the computational complexity of solving the infinite-dimensional identification problem can be a practical limitation. To address this, parallel computing techniques or optimization algorithms tailored for large-scale problems can be utilized to improve the efficiency of the identification process in real-world applications.

Can the identification framework be integrated with control design techniques to improve the performance of LTP systems

The identification framework can be integrated with control design techniques to improve the performance of Linear Time-Periodic (LTP) systems by enabling model-based control strategies. Once the LTP system parameters are accurately identified using the proposed harmonic modeling approach, these parameters can be used to design controllers that stabilize the system and achieve desired performance objectives.
One approach is to design model predictive controllers that utilize the identified LTP system model to predict the system behavior over a finite horizon and optimize control inputs to meet performance criteria. By incorporating the identified system dynamics into the control design, the controller can effectively handle the periodic variations in the system parameters and ensure stability and performance robustness.
Additionally, the identified LTP system model can be used to design adaptive controllers that adjust control strategies based on the real-time estimation of system parameters. This adaptive control approach can enhance the system's ability to adapt to changing operating conditions and disturbances, improving overall control performance in LTP systems.