MATLAB Implementation of Recursive Least Squares (RLS) Algorithm with Code Examples

This article provides a comprehensive guide to implementing the Recursive Least Squares (RLS) algorithm in MATLAB. The RLS algorithm is widely used in practical applications including signal processing, communication systems, and control systems for adaptive filtering tasks. We will demonstrate how to code the RLS algorithm in MATLAB, covering essential implementation aspects such as initializing the forgetting factor, updating the inverse correlation matrix using the Sherman-Morrison formula, and calculating the gain vector. The implementation includes proper handling of the weight vector update equation: w(n) = w(n-1) + K(n)*[d(n) - x(n)'*w(n-1)], where K(n) represents the Kalman gain. The article explores optimization techniques for different datasets and debugging strategies specific to RLS implementations. We'll examine key MATLAB functions like matrix inversion methods and efficient computation of the correlation matrix inverse through recursive updates. Additionally, we analyze the algorithm's advantages, such as fast convergence properties and excellent tracking capabilities for time-varying systems, while also discussing limitations including computational complexity of O(N²) operations per iteration. Through detailed examination of the algorithm's working principles, including the recursive update mechanism and stability considerations, this guide ensures thorough understanding of RLS implementation. The complete MATLAB code structure will be presented with explanations of critical parameters like the forgetting factor (λ) and regularization parameter (δ) for covariance matrix initialization. This serves as a complete practical resource for implementing and understanding RLS algorithms in MATLAB for real-world adaptive filtering applications.