RLS Algorithm Implementation for Adaptive Parameter Estimation

In this article, we demonstrate how to implement the RLS (Recursive Least Squares) algorithm for adaptive parameter estimation. Our objective is to estimate the parameters in the system described by the equation: y(n) = 1.5*y(n-1) - 0.7*y(n-2) + 0.3*u(n-2). The RLS algorithm is a recursive least squares estimator that dynamically adjusts model parameters based on incoming data streams. This approach is particularly effective for estimating parameters in dynamic systems, such as the one presented in this example. The implementation typically involves initializing the parameter vector and covariance matrix, then recursively updating them using the Kalman gain calculation. Key algorithmic steps include computing the innovation (prediction error), calculating the Kalman gain vector, updating the parameter estimates, and recursively adjusting the covariance matrix. The forgetting factor parameter allows the algorithm to weight recent data more heavily, making it suitable for tracking time-varying systems. We will provide detailed explanations of the RLS algorithm's implementation process, including code structure considerations and critical mathematical operations. This will help readers better understand how to apply recursive estimation techniques to dynamic system identification problems. The implementation typically requires maintaining a regression vector containing past outputs and inputs, and efficiently updating the inverse covariance matrix using the matrix inversion lemma to avoid direct matrix inversion at each step.