Simple Least Squares Approximation Algorithm

The least squares approximation algorithm is a classic and practical parameter estimation method in the field of system identification. This algorithm finds the optimal function match for data by minimizing the sum of squared errors, making it particularly suitable for parameter identification problems in linear systems. In MATLAB implementations, this typically involves constructing a design matrix and solving the normal equations using matrix operations like pinv() or the backslash operator for efficient computation.

In system identification applications, the least squares approximation primarily addresses how to estimate model parameters based on system input-output data. The algorithm constructs observation equations where system outputs are expressed as linear combinations of parameters. When measurement noise is present, the algorithm provides optimal linear unbiased estimates. Code implementations often use vectorized operations to handle batch data processing efficiently, with the parameter estimation formula typically expressed as theta = (X'*X)\X'*Y where X is the input matrix and Y is the output vector.

Noise parameter adjustment constitutes a crucial aspect of algorithm implementation. By modifying noise variance and distribution characteristics, researchers can simulate system identification effects under different environmental conditions. This facilitates the study of algorithm robustness under noise interference. In simulation code, this is commonly implemented using random number generators like randn() for Gaussian noise with adjustable standard deviation parameters.

System parameter configuration is equally critical. Users can freely define system characteristics such as order and time delay parameters. By adjusting these parameters, researchers can investigate their impact on identification accuracy. This flexibility makes the algorithm particularly suitable for teaching demonstrations and simulation experiments. Typical implementations include parameter validation checks to ensure system stability and observability before performing estimation.

In practical applications, the least squares approximation algorithm offers computational simplicity and straightforward implementation. Although it requires certain assumptions about noise statistical properties, it still delivers satisfactory identification results in most engineering practices. For system identification and simulation tasks, this algorithm not only demonstrates fundamental principles but also allows observation of different experimental phenomena through parameter adjustments. The algorithm's efficiency makes it suitable for real-time applications when implemented with recursive least squares variations for online parameter tracking.