def system(t, x):
# System parameters
g, mc, m, l = 9.8, 1, 0.1, 0.5
# Controller parameters
r1, r2, r3 = 0.91, 0.09, 2
q1, q2, a1, a2 = 0.5, 0.5, 1, 1.1
i = 4
# Reference
r = np.sin(0.5 * np.pi * t)
dr = 0.5 * np.pi * np.cos(0.5 * np.pi * t)
d2r = -(0.5 * np.pi)**2 * r
# State variables
x1, x2 = x[0], x[1]
# Error variables
e1, e2 = x1 - r, x2 - dr
# Disturbance
Delta = np.sin(10 * x1) + np.cos(x2)
# Sliding variable
s = e2 + sol.odd_pow(
sol.odd_pow(e2, 2) + 2 /
(r1**2) * sol.odd_pow(invdW(i, e1, q1, a1), 2), 0.5)
# Controller
us = -1 / r2 * invdW(i, s, q2, a2) - r3 * np.sign(invdW(
i, s, q2, a2)) - 2 / (r1**2) * proddW(i, e1, q1, a1) * np.sign(
invdW(i, s, q2, a2))
f = (g * np.sin(x1) - m * l * x2**2 * np.cos(x1) * np.sin(x1) /
(mc + m)) / (l * (4 / 3 - m * np.cos(x1)**2 / (mc + m))) - d2r
g = (np.cos(x1) / (mc + m)) / (l * (4 / 3 - m * np.cos(x1)**2 / (mc + m)))
u = (-f + us) / g
return np.array([x2, f + g * u + Delta])
def system(t, x):
# Controller parameters
r1, r2, q = 0.5, 2, 0.4
# Disturbance
Delta = np.sin(2 * np.pi * 5 * t)
# State variables
x1, x2 = x[0], x[1]
# Sliding variable
s = x2 - v(x1, q, r1)
sd = x2 + sol.odd_pow(sol.odd_pow(x2,2) - 2 * sol.odd_pow(v(x1, q, r1), 2), 0.5)
# Controller
u = v(sd, q, r1) - r2 * np.sign(sd) - s + 2 * vdv(x1, q, r1) * np.sign(sd)
return np.array([x2, u+Delta])
def system(t, x):
# Controller parameters
a1, a2, b1, b2 = 128, 64, 128, 64
k = 1.1
# Disturbance
Delta = np.sin(2 * np.pi * 5 * t)
# State variables
x1, x2 = x[0], x[1]
# Sliding variable
s = x2 + sol.odd_pow(
sol.odd_pow(x2, 2) + a1 * x1 + b1 * sol.odd_pow(x1, 3), 0.5)
# Controller
u = -(a1 + 3 * b1 * x1**2 + 2 * k) / 2 * np.sign(s) - sol.odd_pow(
a2 * s + b2 * sol.odd_pow(s, 3), 0.5)
return np.array([x2, u + Delta])
def phi2(x1, x2, r1, r2, r7):
r11, r21, r31, r41, r51, r61 = r1
mr51 = (1-r41*r51)/(r61-r51)
mr61 = (r41*r61-1)/(r61-r51)
g1 = beta(mr51, mr61)/(r21**(mr61)*r31**(mr51)*(r61-r51))
r12, r22, r32, r42, r52, r62 = r2
mr52 = (1-r42*r52)/(r62-r52)
mr62 = (r42*r62-1)/(r62-r52)
g2 = beta(mr52, mr62)/(r22**(mr62)*r32**(mr52)*(r62-r52))
xi = r21*odd_pow(x1, r51)+r31*odd_pow(x1, r61)
s = x2 + odd_pow(odd_pow(x2, 2)+odd_pow(xi, 2*r41), 1/2)
return -((g2/r12)*(r22*np.abs(s)**r52+r32*np.abs(s)**r62)**r42+r7)*np.sign(s)-(2*g1**2*r41/(r11**2))*(r21*r51+r31*r61*np.abs(x1)**(r61-r51))*(r21+r31*np.abs(x1)**(r61-r51))*np.abs(x1)**(2*r41*r51-1)*np.sign(s)
def system(t, x):
import solver as sol
import numpy as np
Delta = np.sin(2 * np.pi * t / 5)
q, Tc, zeta = 0.3, 1, 1
a = 1
return predefined4(x, q, a, Tc) - zeta * sol.odd_pow(x, 0) + Delta
def invdW4(x, q, a):
return gamma(a) / (q) * np.exp(np.abs(x)**q) * sol.odd_pow(x, 1 - a * q)
def invdW3(x, q, a):
return 1 / (a * q) * (np.abs(x)**q + a)**2 * sol.odd_pow(x, 1 - q)
def invdW2(x, q):
return np.pi / (2 * q) * (sol.odd_pow(x, 1 + q) + sol.odd_pow(x, 1 - q))
def invdW1(x, q):
return 1 / q * np.exp(np.abs(x)**q) * sol.odd_pow(x, 1 - q)
x1, x2 = x
# Reference
r = np.sin(0.5 * np.pi * t)
dr = 0.5 * np.pi * np.cos(0.5 * np.pi * t)
d2r = -(0.5 * np.pi)**2 * r
# Error variables
e1, e2 = x1 - r, x2 - dr
# Controller
# System parameters
g, mc, m, l = 9.8, 1, 0.1, 0.5
# Controller parameters
r1, r2, r3 = 0.91, 0.09, 2
q1, q2, a1, a2 = 0.5, 0.5, 1, 1.1
i = 4
s = e2 + sol.odd_pow(
sol.odd_pow(e2, 2) + 2 /
(r1**2) * sol.odd_pow(invdW(i, e1, q1, a1), 2), 0.5)
us = -1 / r2 * invdW(i, s, q2, a2) - r3 * np.sign(invdW(i, s, q2, a2)) - 2 / (
r1**2) * proddW(i, e1, q1, a1) * np.sign(invdW(i, s, q2, a2))
f = (g * np.sin(x1) - m * l * x2**2 * np.cos(x1) * np.sin(x1) /
(mc + m)) / (l * (4 / 3 - m * np.cos(x1)**2 / (mc + m))) - d2r
g = (np.cos(x1) / (mc + m)) / (l * (4 / 3 - m * np.cos(x1)**2 / (mc + m)))
u = (-f + us) / g
## Trajectories
#plt.figure(num=1)
#plt.plot(t, r, color=0.5*np.ones(3), lw=3, label='$r(t)$')
#plt.plot(t, dr, color=0.7*np.ones(3), lw=3, label='$\dot{r}(t)$')
#plt.plot(t, x1, '--', color=0*np.ones(3), lw=2, label='$x_1(t)$')
#plt.plot(t, x2, '--', color=0.3*np.ones(3), lw=2, label='$x_2(t)$')
#plt.ylim(-2,5)
x1, x2 = x
# Plot of the trajectories
ax1.plot(t, x1, color=0 * np.ones(3), lw=2, label='$x_1(t)$')
ax1.plot(t, x2, '--', color=0.5 * np.ones(3), lw=2, label='$x_2(t)$')
ax1.set_xlim(0, 1.2)
ax1.set_ylim(-5, 5)
ax1.set_xlabel('$t$', fontsize=14)
ax1.axvline(x=1, ymin=-1, ymax=2, linestyle='dashed', color=0.6 * np.ones(3))
ax1.legend(loc='best')
ax1.grid()
# Controller parameters
a1, a2, b1, b2 = 128, 64, 128, 64
k = 1.1
# Sliding variable
s = x2 + sol.odd_pow(
sol.odd_pow(x2, 2) + a1 * x1 + b1 * sol.odd_pow(x1, 3), 0.5)
# Controller
u = -(a1 + 3 * b1 * x1**2 + 2 * k) / 2 * np.sign(s) - sol.odd_pow(
a2 * s + b2 * sol.odd_pow(s, 3), 0.5)
# Plot of the control
ax2.plot(t, u, color=0 * np.ones(3), lw=2)
ax2.set_xlim(0, 1.2)
ax2.set_ylim(-100, 100)
ax2.set_xlabel('$t$', fontsize=14)
ax2.grid()
fig.savefig('figures/poly_controller.eps',
bbox_inches='tight',
format='eps',
dpi=1500)
# Simulation
t, x = sol.ode1(system, np.array([1000, 0]), t0, tf, h)
# States
x1, x2 = x
# Plot of the trajectories
ax1.plot(t, x1, color=0*np.ones(3), lw=2, label='$x_1(t)$')
ax1.plot(t, x2, '--', color=0.5*np.ones(3), lw=2, label='$x_2(t)$')
ax1.set_xlim(0, 1.2)
ax1.set_ylim(-5, 5)
ax1.set_xlabel('$t$', fontsize = 14)
ax1.axvline(x = 1, ymin = -1, ymax = 2, linestyle='dashed', color = 0.6*np.ones(3))
ax1.legend(loc='best')
ax1.grid()
# Controller parameters
r1, r2, q = 0.5, 2, 0.4
# Sliding variable
s = x2 - v(x1, q, r1)
sd = x2 + sol.odd_pow(sol.odd_pow(x2,2) - 2 * sol.odd_pow(v(x1, q, r1), 2), 0.5)
# Controller
u = v(sd, q, r1) - r2 * np.sign(sd) - s + 2 * vdv(x1, q, r1) * np.sign(sd)
# Plot of the control
ax2.plot(t, u, color=0*np.ones(3), lw=2)
ax2.set_xlim(0, 1.2)
ax2.set_ylim(-100, 100)
ax2.set_xlabel('$t$', fontsize = 14)
ax2.grid()
fig.savefig('figures/my_controller.eps', bbox_inches='tight', format='eps', dpi=1500)
def predefined4(x, q, a, Tc):
from scipy.special import gamma
return -gamma(a) / (q * Tc) * np.exp(np.abs(x)**q) * sol.odd_pow(
x, 1 - a * q)
def predefined3(x, q, a, Tc):
return -1 / (a * q * Tc) * (np.abs(x)**q + a)**2 * sol.odd_pow(x, 1 - q)
def predefined2(x, q, Tc):
return -np.pi / (2 * q * Tc) * (sol.odd_pow(x, 1 + q) +
sol.odd_pow(x, 1 - q))
def predefined1(x, q, Tc):
return -1 / (q * Tc) * np.exp(np.abs(x)**q) * sol.odd_pow(x, 1 - q)