Estimating System Transfer Functions from Impulse Response Using Hankel Matrix Method

The Hankel matrix method is a classical system identification technique that estimates transfer functions from impulse response sequences. This approach is particularly effective for linear time-invariant (LTI) systems, enabling the recovery of dynamic system characteristics from finite-length impulse response data.

### Core Methodology Hankel Matrix Construction: Build a Hankel matrix using the acquired impulse response sequence. The matrix structure features rows or columns composed of delayed versions of the original sequence, effectively capturing the system's dynamic properties through its Toeplitz-like arrangement. Matrix Decomposition: Perform Singular Value Decomposition (SVD) on the Hankel matrix to extract dominant singular values and corresponding singular vectors. The number of significant singular values determines the system order, with implementation typically involving thresholding using functions like svd() in MATLAB or numpy.linalg.svd() in Python. State-Space Model Estimation: Utilize the decomposed singular vectors to construct observability and controllability matrices. This involves arranging singular vectors to form approximate state-space matrices (A, B, C, D) through balanced realization algorithms. Transfer Function Conversion: Transform the state-space model into transfer function representation using residue theorem implementation or direct polynomial conversion, resulting in rational fraction form (numerator/denominator coefficients).

### Algorithm Characteristics Effective with finite impulse response data and demonstrates inherent noise robustness through SVD's low-rank approximation capabilities. Automated system order determination via SVD eliminates subjective model complexity selection, with truncation criteria based on singular value magnitude ratios. Computationally efficient for medium-scale datasets, with O(n³) complexity for SVD operations manageable for practical engineering applications.

### Application Scenarios Widely employed in control engineering, signal processing, and system identification, especially when internal system states are unmeasurable and only input-output data is available. Typical applications include mechanical vibration analysis (implemented through vibration response data processing), electrical circuit modeling (using impulse testing results), and chemical process control (from step response experiments). The method's code implementation typically involves Hankel matrix formation using shifting operations, SVD-based model reduction, and state-space to transfer function conversion routines.