Demo of the enclosing method to verify robust positive invairance for ellipsoids. This folder is composed of 2 examples, one blank model and the toolbox header. Dependencies: sympy, numpy, scipy, random, math, codac, matplotlib, time.
This repositary is related to the paper Louedec, M., Jaulin, L., & Viel, C. (2023). Computational tractable guaranteed numerical method to study the stability of n-dimensional time-independent nonlinear systems with bounded perturbation. Automatica, 153, 110981.
Frist example, the stability of a simple pendulum with friction. Command to launch:
python3 pendulum_example.py
Three state ellipsoids are tested: Q1 (robust positive invariant), Q2 (not positive invariant) and Q3 (too small to be verified positive invariant). The noise amplitude can be tuned with "Wi_box". Results are visible in pendulum_result/ . If the green ellipsoid (result of the enclosing method) is in the red ellipsoid (Q1, Q2 or Q3), then the red ellipsoid is robust positive invariant. The blue curve simulates state trajectories.
2nd example - the stability of a n-dimensional system. Command to launch:
python3 platooning.py
The cars are controlled to maintain equal distances. Three parameters can be tuned:
- the number of cars "m",
- the disturbance amplitude "w_i",
- the scale factor of the ellipsoid Q "q_i".
Results are visible in platooning_result/ with the simulation of the system.
Follow the instructions in blank_model.py to use the enlosing method on your system.
Tools developped for the enclosing method. The main class is PositiveInvEllipseEnclosureMethod.