System Identification Examples with Code Implementation

System identification is the process of building mathematical models from input-output data, which has wide applications in control engineering and signal processing. This article introduces implementation approaches for several classical identification methods, including recursive least squares algorithm, instrumental variable method, extended least squares method, and bias compensation method, along with corresponding MATLAB programming key points.

### 1. Recursive Least Squares Algorithm The least squares method is the most fundamental system identification approach that estimates model parameters by minimizing the sum of squared errors. The recursive form enables online parameter updates, making it suitable for real-time systems. In MATLAB implementation, you need to initialize the parameter vector and covariance matrix, then continuously refine estimates using recursive formulas. Key programming steps include calculating the gain vector, updating parameter estimates, and adjusting the covariance matrix. The algorithm typically involves matrix operations like P = P - K*phi'*P where K is the gain and phi is the regression vector.

### 2. Instrumental Variable Method When system noise exhibits correlation, ordinary least squares produces biased estimates. The instrumental variable method addresses this by introducing auxiliary variables uncorrelated with noise. MATLAB implementation requires constructing an instrumental variable matrix and ensuring its independence from noise. Common choices for instrumental variables include delayed input signals or model-predicted outputs. The implementation involves solving θ = (Z'*Φ)^{-1}*Z'*y where Z is the instrumental variable matrix.

### 3. Extended Least Squares Method This method incorporates noise model parameters into the estimation process, simultaneously identifying system dynamics and noise characteristics. In MATLAB programming, you need to extend the parameter vector and data matrix to include noise terms, employing solution methods similar to standard least squares. This approach is particularly suitable for systems with colored noise. The implementation typically uses an augmented regression vector containing both input-output terms and noise model terms.

### 4. Bias Compensation Method Targeting the bias issue in least squares estimation, the bias compensation method improves estimation accuracy through compensation terms. The key MATLAB implementation challenge lies in constructing bias terms, usually requiring additional computational steps to estimate noise statistics. This method performs exceptionally well in low signal-to-noise ratio environments. Programming implementation often involves two-stage estimation: first obtaining preliminary estimates, then computing and applying bias corrections.

These methods can be selected based on specific system characteristics in practical applications. MATLAB's powerful matrix computation capabilities make it ideal for implementing these algorithms. Through proper design of data structures and iteration processes, parameter estimation tasks can be efficiently accomplished. For more complex systems, multiple methods can be combined or adaptive mechanisms introduced to enhance identification accuracy.