def CheckLevelSet(self, prev_x, x0, Vs, Vsdot, rho, multiplier_degree):
prog = MathematicalProgram()
x = prog.NewIndeterminates(len(prev_x),'x')
V = Vs.Substitute(dict(zip(prev_x, x)))
Vdot = Vsdot.Substitute(dict(zip(prev_x, x)))
slack = prog.NewContinuousVariables(1,'a')[0]
#mapping = dict(zip(x, np.ones(len(x))))
#V_norm = 0.0*V
#for i in range(len(x)):
# basis = np.ones(len(x))
# V_norm = V_norm + V.Substitute(dict(zip(x, basis)))
#V = V/V_norm
#Vdot = Vdot/V_norm
#prog.AddConstraint(V_norm == 0)
# in relative state (lambda(xbar)
Lambda = prog.NewSosPolynomial(Variables(x), multiplier_degree)[0].ToExpression()
Lambda = Lambda.Substitute(dict(zip(x, x-x0))) # switch to relative state (lambda(xbar)
prog.AddSosConstraint(-Vdot + Lambda*(V - rho) - slack*V)
prog.AddCost(-slack)
#import pdb; pdb.set_trace()
result = Solve(prog)
if(not result.is_success()):
print('%s, %s' %(result.get_solver_id().name(),result.get_solution_result()) )
print('slack = %f' %(result.GetSolution(slack)) )
print('Rho = %f' %(rho))
#assert result.is_success()
return -1.0
return result.GetSolution(slack)
def is_verified(rho):
# initialize optimization problem
# (with Drake there is no need to specify that
# this is going to be a SOS program!)
prog = MathematicalProgram()
# SOS indeterminates
x = prog.NewIndeterminates(2, 'x')
# Lyapunov function
V = x.dot(P).dot(x)
V_dot = 2*x.dot(P).dot(f(x))
# degree of the polynomial lambda(x)
# no need to change it, but if you really want to,
# keep l_deg even (why?) and do not set l_deg greater than 10
# (otherwise optimizations will take forever)
l_deg = 4
assert l_deg % 2 == 0
# SOS Lagrange multipliers
l = prog.NewSosPolynomial(Variables(x), l_deg)[0].ToExpression()
# main condition above
eps = 1e-3 # do not change
prog.AddSosConstraint(- V_dot - l * (rho - V) - eps*x.dot(x))
# solve SOS program
# no objective function in this formulation
result = Solve(prog)
# return True if feasible, False if infeasible
return result.is_success()
def SOS_compute_1(self, S, rho_prev):
# fix V and rho, search for L and u
prog = MathematicalProgram()
x = prog.NewIndeterminates(2, "x")
# Define u
K = prog.NewContinuousVariables(2, "K")
# Fixed Lyapunov
V = x.dot(np.dot(S, x))
Vdot = Jacobian([V], x).dot(self.dynamics_K(x, K))[0]
# Define the Lagrange multipliers.
(lambda_, constraint) = prog.NewSosPolynomial(Variables(x), 2)
prog.AddLinearConstraint(K[0] * x[0] <= 2.5)
prog.AddSosConstraint(-Vdot - lambda_.ToExpression() * (rho_prev - V))
result = prog.Solve()
# print(lambda_.ToExpression())
# print(lambda_.decision_variables())
lc = [prog.GetSolution(var) for var in lambda_.decision_variables()]
lbda_coeff = np.ones([3, 3])
lbda_coeff[0, 0] = lc[0]
lbda_coeff[0, 1] = lbda_coeff[1, 0] = lc[1]
lbda_coeff[2, 0] = lbda_coeff[0, 2] = lc[2]
lbda_coeff[1, 1] = lc[3]
lbda_coeff[2, 1] = lbda_coeff[1, 2] = lc[4]
lbda_coeff[2, 2] = lc[5]
return lbda_coeff
def CheckLevelSet(prev_x, prev_V, prev_Vdot, rho, multiplier_degree): prog = MathematicalProgram() x = prog.NewIndeterminates(len(prev_x),'x') V = prev_V.Substitute(dict(zip(prev_x, x))) Vdot = prev_Vdot.Substitute(dict(zip(prev_x, x))) slack = prog.NewContinuousVariables(1)[0] Lambda = prog.NewSosPolynomial(Variables(x), multiplier_degree)[0].ToExpression() prog.AddSosConstraint(-Vdot + Lambda*(V - rho) - slack*V) prog.AddCost(-slack) result = Solve(prog) assert result.is_success() return result.GetSolution(slack)
def CheckLevelSet(prev_x, prev_V, prev_Vdot, rho, multiplier_degree):
#import pdb; pdb.set_trace()
prog = MathematicalProgram()
x = prog.NewIndeterminates(len(prev_x),'x')
V = prev_V.Substitute(dict(zip(prev_x, x)))
Vdot = prev_Vdot.Substitute(dict(zip(prev_x, x)))
slack = prog.NewContinuousVariables(1,'a')[0]
Lambda = prog.NewSosPolynomial(Variables(x), multiplier_degree)[0].ToExpression()
#print('V degree: %d' %(Polynomial(V).TotalDegree()))
#print('Vdot degree: %d' %(Polynomial(Vdot).TotalDegree()))
#print('SOS degree: %d' %(Polynomial(-Vdot + Lambda*(V - rho) - slack*V).TotalDegree()))
prog.AddSosConstraint(-Vdot + Lambda*(V - rho) - slack*V)
prog.AddCost(-slack)
result = Solve(prog)
#assert result.is_success()
return result.GetSolution(slack)
def SOS_traj_optim(S, rho_guess):
# S provides the initial V guess
# STEP 1: search for L and u with fixed V and p
mp1 = MathematicalProgram()
x = mp1.NewIndeterminates(3, "x")
V = x.dot(np.dot(S, x))
print(S)
# Define the Lagrange multipliers.
(lambda_, constraint) = mp1.NewSosPolynomial(Variables(x), 4)
xd = mp1.NewFreePolynomial(Variables(x), 2)
yd = mp1.NewFreePolynomial(Variables(x), 2)
thetd = mp1.NewFreePolynomial(Variables(x), 2)
u = np.vstack((xd, yd))
u = np.vstack((u, thetd))
Vdot = Jacobian([V], x).dot(plant(x, u))[0]
mp1.AddSosConstraint(-Vdot + lambda_.ToExpression() * (V - rho_guess))
result = mp1.Solve()
# print(type(lambda_).__dict__.keys())
print(type(lambda_.decision_variables()).__dict__.keys())
L = [mp1.GetSolution(var) for var in lambda_.decision_variables()]
# print(lambda_.monomial_to_coefficient_map())
return L, u
# SOS indeterminates
x = prog2.NewIndeterminates(2, 'x')
# Lyapunov function
V = x.dot(P).dot(x)
V_dot = 2*x.dot(P).dot(f(x))
# degree of the polynomial lambda(x)
# no need to change it, but if you really want to,
# keep l_deg even and do not set l_deg greater than 10
l_deg = 4
assert l_deg % 2 == 0
# SOS Lagrange multipliers
l = prog2.NewSosPolynomial(Variables(x), l_deg)[0].ToExpression()
# level set as optimization variable
rho = prog2.NewContinuousVariables(1, 'rho')[0]
# write here the SOS condition described in the "Not quite there yet..." section above
prog2.AddSosConstraint(x.dot(x)*(V-rho) - l*V_dot)
# insert here the objective function (maximize rho)
prog2.AddLinearCost(-rho)
# solve program only if the lines above are filled
if len(prog2.GetAllConstraints()) != 0:
# solve SOS program
result = Solve(prog2)
from pydrake.all import (Jacobian, MathematicalProgram, SolutionResult,
Variables)
def dynamics(x):
return -x + x**3
prog = MathematicalProgram()
x = prog.NewIndeterminates(1, "x")
rho = prog.NewContinuousVariables(1, "rho")[0]
# Define the Lyapunov function.
V = x.dot(x)
Vdot = Jacobian([V], x).dot(dynamics(x))[0]
# Define the Lagrange multipliers.
(lambda_, constraint) = prog.NewSosPolynomial(Variables(x), 4)
prog.AddSosConstraint((V-rho) * x.dot(x) - lambda_.ToExpression() * Vdot)
prog.AddLinearCost(-rho)
result = prog.Solve()
assert(result == SolutionResult.kSolutionFound)
print("Verified that " + str(V) + " < " + str(prog.GetSolution(rho)) +
" is in the region of attraction.")
assert(math.fabs(prog.GetSolution(rho) - 1) < 1e-5)
assert fmin <= 1, "The minimum value is > 1; there is no sub-level set " \ "to plot" # To plot the contour at f = (x-xmin)'H(x-xmin) + fmin = 1, # we make a circle of values y, such that: y'y = 1-fmin, th = np.linspace(0, 2*np.pi, vertices) Y = np.sqrt(1-fmin)*np.vstack([np.sin(th), np.cos(th)]) # then choose L'*(x - xmin) = y, where H = LL'. L = np.linalg.cholesky(H) X = np.tile(xmin, vertices) + np.linalg.inv(np.transpose(L)).dot(Y) return ax.fill(X[0, :]+x0[0], X[1, :]+x0[1], color=color) prog = MathematicalProgram() x = prog.NewIndeterminates(2, 'x') V1 = prog.NewSosPolynomial(Variables(x), 2)[0].ToExpression() V2 = prog.NewSosPolynomial(Variables(x), 2)[0].ToExpression() a = 0.5*Jacobian(V1.Jacobian(x),x) b = 0.5*Jacobian(V2.Jacobian(x),x) pxi = np.array([[-0.5,-0.5], [-0.5, 0.5], [0.5,0.5], [0.5,-0.5]]) for pt in pxi: prog.AddConstraint( pt.T.dot(a).dot(pt) <= 1.0) prog.AddConstraint( pt.T.dot(b).dot(pt) <= 1.0) pxi = np.array([[0.0, 1.0] ]) prog.AddConstraint( pxi[-1].T.dot(b).dot(pxi[-1]) <= 1.0) prog.AddMaximizeLogDeterminantSymmetricMatrixCost(a) prog.AddMaximizeLogDeterminantSymmetricMatrixCost(b)
