Model linearization function toolbox

A wrapped package to linearize the nonlinear continuous/discrete model. Including numerical and symbolic calculations.

If you have questions, remarks, technical issues etc. feel free to use the issues page of this repository. I am looking forward to your feedback and the discussion.

Github project: link

PyPI: link

Introduction: link

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I. How to use

This package operates within the Python framework.

1. Required packages

• Numpy
• Matplotlib
• Control
• CasADi     <-- 3 <= version <= 4

2. Usage

• Download the modlinear file and save it to your project directory.

• Or install using pip

    pip install modlinear

Then you can use the modlinear in your python project.

II. modlinear toolbox organization

. 
└── modlinear 
    ├── cas_linearize 
    ├── linearize_continuous 
    ├── linearize_c2d
    ├── continuous_to_discrete
    └── plot_matrix

Detailed introduction of each function can be found using help in python.

1. cas_linearize

Symbolic calculation

Obtain the linearized continuous/discrete A, B symbolic functions for the continuous/discrete ODE.

• Continuous/discrete A, B from continuous ODE
• Discrete A, B from discrete ODE

Due to symbolic functions, the A, B at any expand state can be easily obtained by giving the state values.

2. linearize_continuous

Numerical calculation

Obtain the linearized continuous A, B matrices for the continuous ODE.

3. linearize_c2d

Numerical calculation

Obtain the linearized discrete A, B matrices for the continuous ODE.

4. continuous_to_discrete

Numerical calculation

Obtain the discrete model from the continuous model, utilizing control package.

5. plot_matrix

Plot a matrix.

II. Linearization process

1. Indicate the set-point that will be expanded: $x_{ss}, u_{ss}, p_{ss}$.
2. Compute the Jacobian of the system and obtain $A$, $B$, $M$, and $C$ matrix of the continuous linear system.

   $(x_{t+1} - x_{ss}) = A (x_t - x_{ss}) + B (u_{t} - u_{ss}) + M (p_{t} - p_{ss})$

   $y_k = C x_k$

   which equals to: $(x_{k+1} - x_{ss}) = A (x_k - x_{ss}) + [B, M] [u_k - u_{ss}, z_k -z_{ss}]^T$

3. Transform the continuous linear system to discrete linear system and obtain $A_{dis}$, $B_{dis}$, $M_{dis}$, and $C_{dis}$.

   $(x_{k+1} - x_{ss}) = A_{dis} (x_k - x_{ss}) + B_{dis} (u_{k} - u_{ss}) + M_{dis} (p_{k} - p_{ss})$

   $y_k = C_{dis} x_k$

Note: This procedure is applicable to all systems.

III. Tutorial

There is a tutorial example to illustrate how to use the modlinear to linearize nonlinear models.

License

The project is released under the APACHE license. See LICENSE for details.