What are some common methods to compute the Jacobian matrix of a control system model?

2 Numerical differentiation

Another method to compute the Jacobian matrix is to use numerical differentiation. This means that you use numerical approximations to estimate the partial derivatives of your system equations. For example, you can use the finite difference method, which involves perturbing each variable by a small amount and measuring the resulting change in the output. The partial derivative is then calculated as the ratio of the output change to the input change. You can use numerical differentiation functions, such as MATLAB's jacobianest or Python's scipy.optimize.approx_fprime, to implement this method and obtain the Jacobian matrix as a matrix of numerical values.

3 Automatic differentiation

A third method to compute the Jacobian matrix is to use automatic differentiation. This means that you use a software tool that can automatically generate the code for computing the partial derivatives of your system equations. For example, if your system is implemented in a programming language, such as C, C++, or Python, you can use an automatic differentiation tool, such as ADOL-C, CppAD, or JAX, to transform your code into a new code that can evaluate the Jacobian matrix. This method can be more efficient and accurate than numerical differentiation, as it avoids the errors and overhead associated with finite differences.

4 Model-based differentiation

A fourth method to compute the Jacobian matrix is to use model-based differentiation. This means that you use a modeling tool that can automatically generate the system equations and the Jacobian matrix from a graphical or textual representation of your system. For example, if your system is modeled in a tool, such as Simulink, Modelica, or Scilab/Xcos, you can use a model-based differentiation tool, such as Simulink Coder, OpenModelica, or Scilab/Xcos Coder, to export your model as a code that can compute the Jacobian matrix. This method can be more convenient and intuitive than symbolic differentiation, as it avoids the need to write the system equations manually.

5 Sensitivity analysis

A fifth method to compute the Jacobian matrix is to use sensitivity analysis. This means that you use a simulation tool that can compute the sensitivity of the output to the input and state variables. For example, if your system is simulated in a tool, such as MATLAB/Simulink, Python/SciPy, or LabVIEW, you can use a sensitivity analysis tool, such as MATLAB's Simulink Design Optimization, Python's SALib, or LabVIEW's System Identification Toolkit, to run multiple simulations with different input and state values and calculate the Jacobian matrix as the sensitivity matrix. This method can be more flexible and robust than analytical methods, as it can handle complex and uncertain systems.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?