Finding a region of attraction using sum of squares optimization

lieskjur
Fri, 25 Jul 2025 21:05:25 +0000
Optimization (Mathematical)

Out of interest I decided to explore sum of squares optimization in the context of Lyapunov functions. Right now I’d like to use it to find the region of attraction for the value function (viewed as a Lyapunov function) of the LQR. As an example I have a single pendulum which I am keeping in its unstable equilibrium using the regulator. As for the approach I am trying to use the “The equality-constrained formulation” in section 9.2.3 of the Underactuated MIT course.

I’ve written the problem using SumOfSquares.jl but even though the simulation shows that the controlled system is stable the region found is practically zero. The script is not too long (83 lines) and hopefully commented thoroughly enough. Any help would be appreciated.

It is quite possible that there is some theoretical mistake based on the fact that I am searching over a region that is a cylinder but to my understanding that shouldn’t be the case.

11 posts - 4 participants