System Identification for Second-Order Systems Using Least Squares Method

This article explores the application of the least squares method for identifying second-order systems and extracting their key parameters. The least squares method serves as a fundamental mathematical tool for finding optimal fitting curves through given data points. For second-order systems, the critical parameters requiring identification include the damping ratio (ζ) and natural frequency (ω_n). We will conduct an in-depth investigation of how to implement the least squares algorithm for parameter estimation, including practical MATLAB code considerations such as constructing the regression matrix using system response data and solving the normal equations through matrix operations like the pseudo-inverse (pinv() function). The implementation typically involves collecting input-output data pairs, formulating the parameter estimation problem as a linear regression model, and computing the parameter vector θ = (XᵀX)⁻¹XᵀY where X contains regressors derived from measured signals. Furthermore, we will analyze the practical significance of these identified parameters in real-world applications such as control system design and vibration analysis, discussing how the accuracy of parameter estimation affects system performance predictions and stability assessments.