I have implemented an optimization problem via SOS optimization that output the rho value for a time-varying RoA estimation. It consists in a bilinear alternation of two optimization problems and I am applying it to the dynamics of a torque-limited simple pendulum. The nominal trajectory has been found via Direct Collocation and the control policy comes from a time-varying LQR controller.
My implementation can be found here and can be easily run locally. This code include a verification script and the results can be found in the file "results.pdf". My question regards this, why I am obtaining simulations that fails to achieve the goal region even if I am starting in the estimated RoA? I am not expecting this behaviour from an inner estimation such has the one that I have implemented. Are there other possible reasons for such a result?