# NLcontrollability
MATLAB toolbox for analysing controllability and accessibility of nonlinear systems.
# To use NLcontrollability you need to:
1. Download the code.
2. Open a MATLAB session.
3. Edit the `run.m` file and run it.
You can find an example in the `run.m` file and more in `ctrl_reach_unit_test_script.m`
# Options
- **opts.LC** --> 1 if you want to check the Linearization Condition, 0 if not
- **opts.ARC** --> 1 if you want to check the Accessibility Rank Condition, 0 if not
- **opts.LARC** --> 1 if you want to check the Lie Algebraic Rank Condition, 0 if not
- **opts.LARC** --> 1 if you want to check the Lie Algebraic Rank Condition, 0 if not
- **opts.GSC** --> 1 if you want to check Sussmann's General Sufficient Condition, 0 if not
- **'model_name'** --> Name of a .mat file placed in the 'models' folder
- **opts.numericLC** --> 0 if you want to check the Linearization Condition symbolically, 1 for numeric computation
- **opts.maxtime** --> max time for each test in seconds
- **x0** --> Specific initial point. If no point is given it would try to compute equilibrium points.
# Entering the model
NLcontrollability reads models stored as MATLAB MAT-files (.mat). Models are defined as follows:
First, all parameters, states, and any other entities (such as inputs or known constants) appearing in the
model must be defined as symbolic variables. Example:
syms x1 x2 x3 x4 lambda u1
Then we define the state variables, by creating a column vector named x. Example:
x = [x1; x2; x3; x4];
Similarly, we define the known input vector, u. If there are no inputs, enter a blank vector. Example:
u = u1; % Or, for more than one input --> u = [u1; u2];
The model ODEs (dx/dt=...) must also be entered as a column vector, called f, which must have the same
length as the state vector x. Example:
f = [u1;
x1;
x1^3;
x3^2-lambda*x2^2*x1^4];
Finally, save all the variables in a MAT-file. Example:
save(‘MAPK’,‘x’,‘u’,‘f’);
This model description format is the same used by the [STRIKE-GOLDD](https://github.com/afvillaverde/strike-goldd) toolbox.
Further details can be found in its [documentation](https://github.com/afvillaverde/strike-goldd/blob/master/STRIKE-GOLDD/doc/STRIKE-GOLDD_manual.pdf).
# Software contents
- **`run.m`**: file where the user enters the model and options. Running it executes the code.
- **`ctrl_analysis_MAIN.m`:** main file; it calls the scripts with the accessibility and controllability tests.
- **`ctrl_LC.m`**: implementation of the Linearization Condition.
- **`ctrl_ARC.m`**: implementation of the Accessibility Rank Condition.
- **`ctrl_LARC_GSC.m`**: implementation of the Lie Algebraic Rank Condition and Sussmann's General Sufficient Condition.
- **`ctrl_reach_unit_tests_script.m`**: file with some examples ready to run.
- **`Lie_bracket.m`**: auxiliary function that computes the Lie bracket of two vectors w.r.t. a variable.
- **`frac_elem_sym.m`**: auxiliary function that changes a rational equation into its numerator and denominator.
- **`make_affine_known_u.m`**: function that converts a model into affine in inputs form
# Publications
Publication of the methodology (pending)
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