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runge_kutta_hw5_1.py
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runge_kutta_hw5_1.py
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import numpy as np
from casadi import *
def collocation_tableau(c):
"""
c, A, b = collocation_tableau(c)
Computes the Butcher tableau corresponding to
a collection of collocation points, c
Input
c - a collection of strictly increasing collocation points in [0,1]
Returns
c, A, b
c - the original collocation points
A - the tableau matrix
b - the final weights
"""
n = len(c) # length of the collocation points (1,2,3)
x_plot = np.linspace(0,1,100)
Lag_polys = []
Lag_ints = []
b = []
for i in range(n):
# Construct the Lagrange Interpolating Polynomial
not_c = np.hstack([c[:i],c[i+1:]]) # find c such that i =! j
L = np.poly1d(not_c,r=True) # construct polynomial such that (x - c_1)(x - c_2)....
L = L / L(c[i]) # Top is the same poly and bottom is substituted with c_i
L_i = L.integ() # Integrate to get a_ij without substitiuting
# The ending weights
b.append(L_i(1.)) # This is L_i evaluated at 1
Lag_polys.append(L)
Lag_ints.append(L_i)
b = np.array(b)
a = []
for i in range(n):
ci = c[i]
row = []
for j in range(n):
L_j = Lag_ints[j]
row.append(L_j(ci)) # evaluate at c_i
a.append(row)
a = np.array(a)
return c,a,b
def runge_kutta_equations(f,t,x,u,h,c,A,b):
"""
G, K, x_next = runge_kutta_equations(f,x,h,c,A,b)
Returns
G - a CasADi symbolic expression such that G == 0
corresponds to the Runge-Kutta equations.
K - CasADi variables for the RK derivative evaluations
x_next - A CasADi symbolic variable for the value of the
state at the end of the RK step. This is consistent with
multiple shooting. For single shooting, the variable x_next
could be eliminated .
Inputs
f - A function of the form x_dot = f(t,x) for the differential equation
t - The initial time for evaluation
x - An initial condition for the Runge-Kutta step
h - A step size
c,A,b - encodes a Butcher Tableau
"""
s = len(c)
n = np.prod(x.shape)
# First make sure that x is an explicit column vector
x0 = x.reshape((n,1))
u0 = u.reshape((1,1))
K = MX.sym('K',(n,s)) # some symbol for K
# RK Evaluation points
X_eval = x0 @ np.ones((1,s)) + h * [email protected]
# The Time evaluation points
t_eval = t + h * np.array(c)
u_eval = u0 + h * np.array(c)
G_list = []
for i in range(s):
G = K[:,i] - f(t_eval[i],X_eval[:,i],u_eval[i])
G_list.append(G)
# The value at the end of the step: This is the last equation
x_next = MX.sym('x_next',(n,1))
u_next = MX.sym('u_next')
G = x_next - (x0 + h * K @ b.reshape((s,1)))
G_list.append(G)
G_list.append(x_next[0]/6 + u_next)
shape_con = np.shape(x_next[0]/6 + u_next)
G = vertcat(*G_list)
return G,K,x_next,u_next,shape_con
flatten = lambda X : X.reshape((np.prod(X.shape),1))
def runge_kutta_sim(f,Time,x0,u0,c,A,b):
"""
Simulate a differential equation via a Runge-Kutta Scheme
"""
N = len(Time)-1
x_c = DM(x0)
u_c = DM(u0)
G_list = []
X_list = []
K_list = []
U_list = []
for i in range(N):
t = Time[i]
h = Time[i+1]-Time[i]
G,K,x_c,u_c,shape_con = runge_kutta_equations(f,t,x_c,u_c,h,c,A,b)
G_list.append(G)
K_list.append(K)
X_list.append(x_c)
U_list.append(u_c)
# Stack the constraints
G_all = vertcat(*G_list)
UB = np.zeros(G_all.shape)
sha = np.array(G_all.shape)
shaa = sha - np.array([1,1])
LB = np.zeros(shaa)
# Stack the variables
X_sym = horzcat(*X_list)
K_sym = horzcat(*K_list)
U_sym = horzcat(*U_list)
X_flat = flatten(X_sym)
K_flat = flatten(K_sym)
U_flat = flatten(U_sym)
Z = vertcat(X_flat,K_flat,U_flat)
nlp = {'x' : Z, 'f' : 0, 'g' : G_all}
opts = {'ipopt' : {'print_level' : 0},
'print_time' : False,
'error_on_fail': True}
solver = nlpsol('solver','ipopt',nlp,opts)
try:
res = solver(ubg = UB,lbg = LB)
X_RK = res['x'][:X_flat.shape[0],0].reshape(X_sym.shape)
X_RK = horzcat(DM(x0),X_RK)
X_RK = np.array(X_RK)
return X_RK
except RuntimeError:
return None