Question about ../src/lyapunov/cubic_polynomial.py about underactuated

[RussTedrake]

There are multiple formulations possible. (I hope that the course notes are informative). The problem with your one line change is that if both lambda and rho are decision variables, then the new expression is bilinear in the decision variables (not a valid SOS constraint).

The code existing code is a clever way to search for both rho and lambda simultaneously. Instead of saying "for all x in which V < rho, Vdot must be < 0" -- which is what you wanted -- we can write "for all x in which Vdot >= 0, V must be > rho". This works everywhere except at the origin (because Vdot=0), so we have to multiply by x^2 to handle that case.

[Jingwen-Zhang-Aaron]

Coool! I just read your online book and found that the code cannot match with what you have written in the chapter so that I got lost a little bit here. Thank you so much for your explanation. What a clever way!

[Jingwen-Zhang-Aaron]

Hello, I wanna use SOS to do the lyapunov analysis of the simple pendulum. There's no bug for my code but I got V = 0 and lambda = 0 which indicates obviously that something goes wrong. Can you help me have a look at my code?
I just followed the following formulation to find the V.

import math
import numpy as np

from pydrake.all import (Jacobian, MathematicalProgram, SolutionResult,
                         Variables)


def dynamics(x):
    return [x[1]*x[2], -x[0]*x[2], -(x[2] + 9.81*x[0])]


prog = MathematicalProgram()
x = prog.NewIndeterminates(3, "x")

# Define the Lyapunov function.
(V, constraint1) = prog.NewSosPolynomial(Variables(x), 2)
Vdot = Jacobian([V.ToExpression()], x).dot(dynamics(x))[0]

# Define the Lagrange multipliers.
(lambda_, constraint2) = prog.NewSosPolynomial(Variables(x), 4)

prog.AddSosConstraint(-Vdot - lambda_.ToExpression()*(pow(x[0],2) + pow(x[1],2) - 1))

result = prog.Solve()

assert(result == SolutionResult.kSolutionFound)

print(prog.SubstituteSolution(V.ToExpression()))
print(prog.SubstituteSolution(lambda_.ToExpression()))

[RussTedrake]

my version of this is here: https://github.com/RussTedrake/underactuated/blob/master/src/pendulum/global_sums_of_squares.py
i didn't check every line, but notice that i was careful to set the scaling by setting V(1) = 1. Otherwise V=0 is perfectly viable.