[Paper Review] Improved Transients in Multiple Frequencies Estimation via Dynamic Regressor Extension and Mixing

Aranovskiy Stanislav, Alexey Bobtsov+2 more|arXiv (Cornell University)|Apr 7, 2016

Fault Detection and Control Systems2 references3 citations

TL;DR

This paper proposes a dynamic regressor extension and mixing (DREM) technique to improve transient performance in multiple frequency estimation for sinusoidal signals. By transforming the vector parameter estimation problem into scalar sub-problems, the method ensures non-strict monotonicity of estimation errors and eliminates oscillations or peaking, significantly enhancing convergence behavior compared to standard gradient-based approaches.

ABSTRACT

A problem of performance enhancement for multiple frequencies estimation is studied. First, we consider a basic gradient-based estimation approach with global exponential convergence. Next, we apply dynamic regressor extension and mixing technique to improve transient performance of the basic approach and ensure non-strict monotonicity of estimation errors. Simulation results illustrate benefits of the proposed solution.

Motivation & Objective

• To address poor transient performance in gradient-based multiple frequency estimation, particularly oscillations and peaking.
• To enhance convergence behavior of existing state-variable filter (SVF)-based frequency estimation methods.
• To ensure non-strict monotonicity of parameter estimation errors, avoiding oscillatory or divergent transients.
• To extend the dynamic regressor extension and mixing (DREM) technique from scalar to multiple frequency estimation problems.
• To validate the proposed method through simulations showing improved transient response and stability.

Proposed method

• The method applies dynamic regressor extension and mixing (DREM) to transform the original vector estimation problem into a set of scalar estimation sub-problems.
• It uses a state-variable filter (SVF) to generate regressor signals that satisfy a linear regression model with exponentially decaying error terms.
• The DREM procedure introduces auxiliary dynamics and mixing matrices to ensure that the estimation error for each parameter component decreases monotonically.
• The approach relies on a time-varying regressor matrix whose determinant is shown to be non-L2-integrable under mild conditions, ensuring persistent excitation.
• A key condition for non-strict monotonicity is that the sampling interval $ d_1 $ is not a half-period or period of any sinusoidal component, preventing singularity in the regressor matrix.
• The method is applied to a gradient-based estimator with a modified gain matrix $ K $, now replaced by individual gains $ \gamma_i $ for each scalar sub-problem.

Experimental results

Research questions

• RQ1 Can DREM be effectively applied to improve transient performance in multiple frequency estimation?
• RQ2 Does the DREM-based approach ensure non-strict monotonicity of estimation errors, avoiding oscillations and peaking?
• RQ3 What conditions on the sampling interval $ d_1 $ are required to maintain persistent excitation and avoid regressor matrix singularity?
• RQ4 How does the DREM-enhanced estimator compare to the basic gradient-based estimator in terms of convergence speed and transient behavior?
• RQ5 Can the method be extended to more than two frequencies while preserving monotonic error convergence?

Key findings

• The DREM-based estimator achieves non-strict monotonic convergence of parameter estimation errors, eliminating oscillatory and peaking behaviors observed in the basic gradient-based method.
• Simulation results show that the DREM-enhanced estimator exhibits significantly improved transient performance, with faster and smoother convergence for both two- and three-frequency signals.
• For the two-frequency case, the transient time was reduced from approximately 10 seconds (basic estimator) to less than 2 seconds (DREM estimator) under identical conditions.
• The condition $ d_1 < \pi \bar{\omega}^{-1} $, where $ \bar{\omega} $ is the maximum frequency, ensures that the determinant of the extended regressor matrix is not in $ \mathcal{L}_2 $, thus guaranteeing persistent excitation.
• The method remains effective even when the signal contains multiple frequencies, as demonstrated by successful estimation for $ N=3 $ with $ \theta = [38, 361, 900]^T $.
• The gains $ \gamma_i $ in the DREM structure can be tuned to further improve transient response, as shown by simulations with different $ \gamma_1, \gamma_2 $ values.