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 toolbox. Further details can be found in its documentation.

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)