# Parameter Estimation

## Estimate model parameters

Parameter estimation plays a critical role in accurately describing system behavior through mathematical models such as statistical probability distribution functions, parametric dynamic models, and data-based Simulink® models.

Improving the accuracy of statistical models can involve estimating:

• Parameters of a probability distribution, such as the mean and standard deviation of a normal distribution
• Regression coefficients of a regression model, such as $$y = a'x$$

For more information, see Statistics and Machine Learning Toolbox™, which supports these and similar parameter estimation tasks with more than 40 different probability distributions, including Normal, Weibull, Gamma, Generalized Pareto, and Poisson. The toolbox also supports linear and nonlinear regression.

Creating accurate parametric dynamic models can involve estimating:

• Coefficients of transfer functions, including ARX, ARMAX, Box-Jenkins, and Output-Error
• Entries of state-space matrices
• Coefficients of ODEs or well-structured systems with parameter constraints (grey-box system identification)
• Regression coefficients, saturation levels, or dead-zone limits for nonlinear dynamic systems, including nonlinear ARX and Hammerstein-Wiener

For more information, see System Identification Toolbox™, which supports these tasks with parameter estimation for linear and nonlinear parametric dynamic models.

• Importing and processing input-output test data, such as the voltage input and rotor speed output of a DC motor
• Specifying which model parameters and initial conditions to estimate, such as motor resistance and inertia