# Parameter Estimation

## Compute parameter values to improve model accuracy

Parameter estimation is the process of computing a model’s parameter values from measured data. You can apply parameter estimation to different types of mathematical models, including statistical models, parametric dynamic models, and data-based Simulink® models.

Statistical Models

Engineers and scientists apply parameter estimation to statistical models to estimate:

• Parameters of a probability distribution, such as the mean and standard deviation of a normal distribution
• Regression coefficients of a regression model, such as $$Y = a′X$$

The Statistics and Machine Learning Toolbox™ 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.

Dynamic Models

Engineers apply parameter estimation to dynamic models to compute:

• Coefficients of transfer functions, including ARX, ARMAX, Box-Jenkins, and output-error models
• Entries in 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

The System Identification Toolbox™ supports parameter estimation for linear and nonlinear parametric dynamic models.

Engineers developing Simulink models can apply parameter estimation to develop an accurate plant model for control system design, or to create a digital twin.

To create an accurate plant model for control design using Simulink:

• Collect input-output test data from the plant
• Use optimization to estimate the model’s parameter values, so the simulated model output matches the measured plant output

You can use Simulink Design Optimization™ to interactively preprocess test data, automatically estimate model parameters, and validate estimation results.

To create a digital twin of a current hardware asset:

• Model the machine’s dynamics in Simulink or Simscape™
• Automatically update parameters of a model when new data from the physical asset becomes available

You can develop algorithms to estimate digital twin parameters with Simulink Design Optimization. You can also deploy the algorithms on premises, cloud, or edge systems using Simulink Compiler™.