recursive least square algorithm

3 min read 19-03-2025

The Recursive Least Squares (RLS) algorithm is a powerful tool for estimating parameters in a linear model. Unlike its batch counterpart, the ordinary least squares (OLS) method, RLS processes data sequentially, updating its estimate with each new observation. This makes it ideal for real-time applications where data streams continuously. This article provides a comprehensive overview of the RLS algorithm, exploring its workings, advantages, and applications.

Understanding the Core Concept

At its heart, the RLS algorithm seeks to minimize the sum of squared errors between predicted and actual values. However, unlike OLS which requires processing the entire dataset at once, RLS performs this minimization iteratively. With each new data point, it updates the parameter estimates efficiently, without needing to reprocess past data. This recursive nature is key to its efficiency in dynamic environments.

Mathematical Formulation

The RLS algorithm is best understood through its mathematical formulation. Let's consider a linear model:

y_k = θ^Tx_k + ε_k

Where:

y_k is the observed output at time k
θ is the vector of unknown parameters we aim to estimate
x_k is the vector of input variables at time k
ε_k is the error term at time k

The goal is to find the optimal θ that minimizes the cost function:

J_k = Σ_i=1^k λ^k-i(y_i - θ^Tx_i)²

The λ (lambda) term, where 0 < λ ≤ 1, is a forgetting factor. This crucial parameter controls the influence of past data. A smaller λ gives more weight to recent observations, making the algorithm more responsive to changes in the underlying system. A λ of 1 gives equal weight to all past observations.

The Recursive Update Equations

The magic of RLS lies in its recursive update equations. Instead of recalculating everything from scratch with each new data point, it updates the parameter estimate using the following equations:

P_k = (1/λ) [P_k-1 - P_k-1x_k(λ + x_k^TP_k-1x_k)^-1x_k^TP_k-1]

θ_k = θ_k-1 + P_kx_k(y_k - x_k^Tθ_k-1)

Where:

P_k is the inverse of the covariance matrix at time k. It represents the uncertainty in the parameter estimates.
θ_k is the parameter estimate at time k.

These equations efficiently update the parameter estimate and its associated uncertainty with each new data point. The algorithm starts with an initial guess for θ and P (often P₀ = αI, where α is a large positive scalar and I is the identity matrix).

Advantages of the RLS Algorithm

Real-time adaptation: Its sequential nature allows for real-time tracking of changes in the system.
Computational efficiency: The recursive updates significantly reduce computational burden compared to re-calculating OLS repeatedly.
Adaptability to changing dynamics: The forgetting factor (λ) allows the algorithm to adapt to non-stationary processes.

Applications of the RLS Algorithm

The RLS algorithm finds widespread use in various fields including:

System identification: Estimating the parameters of dynamic systems from input-output data.
Adaptive control: Adapting control strategies to changing system dynamics.
Signal processing: Estimating parameters in signal models, such as channel equalization.
Robotics: Estimating robot dynamics and controlling robot movements.

Limitations of the RLS Algorithm

Computational complexity: While more efficient than batch methods, RLS still requires matrix inversions which can be computationally demanding, particularly for high-dimensional systems.
Sensitivity to noise: The algorithm can be sensitive to noisy measurements, potentially leading to inaccurate estimates.
Parameter tuning: Proper selection of the forgetting factor (λ) is crucial for performance, and requires careful consideration based on the application.

Conclusion

The Recursive Least Squares algorithm provides a robust and efficient method for estimating parameters in dynamic linear models. Its ability to adapt to changing conditions and process data sequentially makes it a valuable tool in numerous applications. While certain limitations exist, the advantages of its real-time capabilities and relative computational efficiency often outweigh these drawbacks, solidifying its place as a fundamental algorithm in adaptive signal processing and control systems.