System Parameter Identification Using Recursive Least Squares Algorithm

The Recursive Least Squares (RLS) algorithm is an efficient online parameter estimation method particularly suitable for real-time identification of dynamic systems. Its core principle involves continuously updating parameter estimates to minimize the system's output error. In code implementation, the RLS algorithm typically maintains a covariance matrix and parameter vector that get recursively updated with each new data point.

In system identification, selecting appropriate input signals is crucial. The M-sequence (maximum-length linear feedback shift register sequence) possesses excellent pseudo-random properties that can effectively excite the system's dynamic behavior. Using a 6-bit M-sequence means its period length is 63 (2^6-1), providing sufficient complexity to cover the system's dynamic response. Code implementation would involve generating this sequence using shift register operations with proper feedback taps.

Performing 300 recursive iterations indicates the algorithm undergoes 300 update cycles. During each iteration, the RLS algorithm calculates new parameter estimates based on current input-output data and updates the covariance matrix. This process avoids the disadvantage of traditional least squares methods that require storing all historical data, making it particularly suitable for online applications and real-time control systems. The recursive update can be implemented using matrix operations that avoid direct matrix inversion through the Matrix Inversion Lemma.

Key steps of the RLS algorithm include: Initializing parameter estimates and covariance matrix For each sampling instant, computing the gain vector Updating parameter estimates Updating the covariance matrix In MATLAB implementation, these steps would typically involve maintaining P matrix (covariance) and theta vector (parameters) with appropriate initialization values.

Compared to batch least squares methods, the RLS algorithm can adaptively track changes in system parameters, but it's more sensitive to initial conditions and forgetting factor selection. An appropriate forgetting factor balances the influence of new and old data on parameter estimation, making it suitable for time-varying system identification. Code implementation often includes a forgetting factor parameter (usually between 0.95 and 1.0) that controls the memory length of the algorithm.

Typical applications of this method include adaptive filtering, online adjustment of control system parameters, and channel estimation in communication systems. Through sufficient recursive operations, convergent system parameter estimates can be obtained. The algorithm's efficiency makes it implementable in real-time systems with limited computational resources.