ERA Program Implementation Using MATLAB for Eigensystem Realization Algorithm

The Eigensystem Realization Algorithm (ERA) is a classical system identification method used to extract modal parameters (such as natural frequencies, damping ratios, and mode shapes) from experimental data for linear dynamic systems. This method finds extensive applications in structural health monitoring, vibration analysis, and control systems.

### Fundamental Approach The ERA algorithm utilizes impulse response data or free decay response data to identify system state-space models through Hankel matrix construction. Key implementation steps include: Data Acquisition: Obtain system impulse response or free decay response data through experiments or simulations. In MATLAB, this can be implemented using data acquisition toolboxes or simulation functions like lsim() for linear systems. Hankel Matrix Construction: Construct the Hankel matrix using response data, which embeds the system's dynamic characteristics. MATLAB's hankel() function can efficiently create this matrix from response sequences. Singular Value Decomposition (SVD): Perform SVD on the Hankel matrix to determine system order and extract dominant modes. The svd() function in MATLAB provides efficient decomposition with automatic thresholding for model order selection. State-Space Model Construction: Build a minimal realization state-space model based on SVD results. This involves matrix operations that MATLAB handles optimally through its built-in matrix manipulation capabilities. Modal Parameter Extraction: Calculate system eigenvalues and eigenvectors from state-space matrices to obtain natural frequencies, damping ratios, and mode shapes. The eig() function and modal parameter conversion algorithms can be implemented for this step.

### MATLAB Implementation Advantages MATLAB's powerful matrix computation capabilities make it ideal for ERA algorithm implementation. Key advantages include: Efficient Matrix Operations: Hankel matrix construction and SVD decomposition can be efficiently performed using built-in functions like hankel() and svd(), which are optimized for large-scale computations. Visualization Support: Easy plotting of modal shapes, frequency responses, and other analysis results using MATLAB's comprehensive plotting functions (plot(), stem(), surf(), etc.). Extensibility: Seamless integration with other toolboxes (Control System Toolbox, Signal Processing Toolbox) for advanced analysis, validation, and controller design.

### Application Scenarios The ERA algorithm is suitable for: Modal parameter identification of mechanical structures (vibration analysis of bridges, aircraft wings, etc.) Reduced-order modeling of control systems Experimental data validation and finite element model updating The MATLAB implementation enables researchers to quickly extract dynamic characteristics from experimental data, providing reliable foundations for structural health monitoring and control system design.