Core Concepts

This work derives efficient batch and recursive least squares algorithms for identifying matrix parameters that minimize the same cost function as the standard vec-permutation approach, while requiring significantly less computational and storage complexity.

Abstract

This paper presents efficient batch and recursive least squares (BLS and RLS) algorithms for identifying matrix parameters in a linear measurement process. The key insights are:
The traditional vec-permutation approach, which transforms the matrix parameter estimation problem into a standard vector form, introduces extraneous zero terms in the regressor matrix, leading to increased computational and storage requirements.
The authors derive BLS and RLS formulations that, under mild assumptions, minimize the same cost function as the vec-permutation approach, but with significantly lower computational and storage complexity.
Two variants are presented:
Independent column weighting, which computes a separate covariance matrix for each column of the parameter matrix, resulting in a O(m^2) improvement in complexity over vec-permutation.
Independent identical column weighting, which uses the same covariance matrix for all columns, resulting in a O(m^3) improvement in complexity.
It is shown that persistent excitation guarantees convergence of the matrix RLS algorithm to the true matrix parameters, extending established results for vector parameter estimation.
The proposed matrix RLS algorithm is applied to improve the online identification step in an indirect adaptive model predictive control scheme, demonstrating significant reductions in computation time.

Stats

The computation time per step for the truss example was reduced by 97.6% using the matrix RLS algorithm compared to the vec-permutation approach.

Quotes

"This work derives batch and recursive least squares algorithms for the identification of matrix parameters. Under the assumption of independent, identical column weighting, these methods minimize the same cost function as the vec-permutation approach."
"It is also shown how, under persistent excitation, convergence guarantees can be extended from the vector case to the matrix case."

Key Insights Distilled From

by Brian Lai,De... at **arxiv.org** 04-18-2024

Deeper Inquiries

The proposed matrix BLS and RLS algorithms can be extended to handle time-varying or nonlinear measurement processes by incorporating adaptive techniques. For time-varying processes, the algorithm parameters can be updated recursively at each time step to adapt to the changing system dynamics. This adaptation can involve updating the covariance matrices, regularization terms, or forgetting factors based on the evolving system behavior. Additionally, for nonlinear measurement processes, the regressor matrices can be modified to account for the nonlinear relationships between the measurements and the parameters. Techniques such as extended Kalman filtering or particle filtering can be integrated into the algorithm to handle nonlinearity and improve estimation accuracy in such scenarios.

The implications of the matrix parameter estimation approach on the stability and performance of the adaptive model predictive control (MPC) scheme are significant. By accurately estimating the matrix parameters in real-time, the adaptive MPC controller can effectively adjust its predictive model to match the actual system behavior. This leads to improved tracking performance, disturbance rejection, and robustness of the control system. The stability of the adaptive MPC scheme is ensured by the convergence properties of the matrix parameter estimation algorithms. If the estimated parameters converge to the true values, the control system will exhibit stable behavior and achieve the desired control objectives. Moreover, the reduced computational complexity and improved efficiency of the matrix parameter estimation approach contribute to enhancing the overall performance of the adaptive MPC scheme.

The matrix parameter estimation techniques proposed in the context of adaptive MPC can be applied to various areas of system identification and control beyond adaptive MPC, such as robust control design and state estimation. In robust control design, accurate estimation of system parameters is crucial for designing controllers that can handle uncertainties and disturbances effectively. By utilizing matrix BLS and RLS algorithms, system parameters can be estimated with improved accuracy, leading to the design of robust controllers that can maintain stability and performance under varying operating conditions. Additionally, in state estimation applications, the matrix parameter estimation techniques can be used to estimate the state variables of a dynamic system based on noisy measurements, enabling accurate state reconstruction for feedback control and monitoring purposes. The versatility and efficiency of the matrix parameter estimation approach make it applicable to a wide range of system identification and control tasks, contributing to enhanced system performance and robustness.

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