def PolyGCD(a: numpy.array, b: numpy.array):
"""Iterative Greatest Common Divisor for polynomial"""
# if len(a) < len(b):
# a, b = b, a
x, xt, y, yt = (1, ), (0, ), (0, ), (1, )
while any(num != 0.0 for num in b):
q = polydiv(a, b)[0]
a, b = b, polydiv(a, b)[1]
x, xt = xt, polysub(x, polymul(xt, q))
y, yt = yt, polysub(y, polymul(yt, q))
return a, x, y
def dbl(x, n, q):
dist = [-1, 0, 1]
prob = [0.25, 0.5, 0.25]
e = np.random.choice(dist, n, p=prob)
dbl_x = (2 * x) % (2 * q)
dbl_x = p.polysub(dbl_x, e) % (2 * q)
return dbl_x
def get_der_rational(num, den):
dnum = poly.polyder(num)
dden = poly.polyder(den)
dnum_den = poly.polymul(dnum, den)
dden_num = poly.polymul(num, dden)
num_new = poly.polysub(dnum_den, dden_num)
den_new = poly.polymul(den, den)
return (num_new, den_new)
def test_polysub(self) :
for i in range(5) :
for j in range(5) :
msg = "At i=%d, j=%d" % (i,j)
tgt = np.zeros(max(i,j) + 1)
tgt[i] += 1
tgt[j] -= 1
res = poly.polysub([0]*i + [1], [0]*j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def compute_polynomials(self, U):
p_unit = (0, 1)
p = []
p.append((1))
p_k_1_1 = P.polymul(p_unit, p[0])
p_k_1_2 = P.polymul(U[0, 0], p[0])
p_new = (1 / U[0, 1]) * P.polysub(p_k_1_1, p_k_1_2)
p.append(p_new)
for k in range(2,self.dim):
#print(k)
p_k_1_1 = P.polymul(p_unit, p[k-1])
p_k_1_2 = P.polymul(U[k-1,k-1], p[k-1])
p_k_2 = P.polymul(U[k-1,k-2], p[k-2])
p_new = (1/U[k-1,k])*P.polysub( P.polysub(p_k_1_1, p_k_1_2), p_k_2 )
p.append(p_new)
self.p = p
self.p_n = P.polyadd(P.polymul(-U[self.dim-2,self.dim-1], p[self.dim-2]), P.polymul(P.polysub(p_unit, U[self.dim-1,self.dim-1]), p[self.dim-1]))
return p
def test_polysub(self):
for i in range(5):
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
tgt = np.zeros(max(i, j) + 1)
tgt[i] += 1
tgt[j] -= 1
res = poly.polysub([0] * i + [1], [0] * j + [1])
assert_equal(trim(res), trim(tgt), err_msg=msg)
def berlekampTest(prime, poly):
m = len(poly) - 1
u = [0, 1]
for i in range(0, m // 2):
u = P.polydiv(P.polypow(u, prime), poly)[1] % prime
d = gcd(P.polysub(u, [0, 1]) % prime, poly, prime)
#print(P.polysub(u, [0, 1]) % prime, poly, d)
if not np.array_equal(np.array([1.]), d):
return False
return True
def PolyGCD_rec(a: numpy.array, b: numpy.array):
"""Recursive Greatest Common Divisor for polynomial"""
if all(num == 0.0 for num in a):
return b, 0, 1
t = polydiv(b, a)
d, x1, y1 = PolyGCD_rec(t[1], a)
x = polysub(y1, polymul(t[0], x1))
y = x1
return d, x, y
def deriv(self):
""" compute first derivative
Returns
-------
dPP : PseudoPolynomial
d(PP)/dx represented as a PseudoPolynomial
"""
dp = pol.polyder(self.p)
dq = pol.polyder(self.q)
p = pol.polysub(pol.polymul((1, 0, -1), dp),
pol.polymul((0, 2 * self.r), self.p))
q = pol.polysub(pol.polymul((1, 0, -1), dq),
pol.polymul((0, 2 * self.r + 1), self.q))
r = self.r - 1
return PseudoPolynomial(p=remove_zeros(p), q=remove_zeros(q), r=r)
def impinvar_causal(*args, **kwargs): """causal version of impinvar, h[0] == 0""" # polynomial.polysub is ordered from lowest to highest from numpy.polynomial.polynomial import polysub bz, az = impinvar(*args, **kwargs) h0 = bz[0] / az[0] bz = polysub(bz, h0 * az) return bz, az
def impinvar_causal(*args, **kwargs):
"""causal version of impinvar, h[0] == 0"""
# polynomial.polysub is ordered from lowest to highest
from numpy.polynomial.polynomial import polysub
bz, az = impinvar(*args, **kwargs)
h0 = bz[0] / az[0]
bz = polysub(bz, h0 * az)
return bz, az
def root_finding_poly(self):
"""
R = p^2 - (1 - x^2) * q^2
Returns
-------
coef : np.ndarray
Coefficients of a root-finding polynomial `R(x)`. Every zero of
the `PseudoPolynomial` is also a zero of `R(x)`.
"""
return remove_zeros(pol.polysub(pol.polymul(self.p, self.p),
pol.polymul(pol.polymul(self.q, self.q),
(1, 0, -1))))
c1[i] = rand.randint(-5, 5)
for i in range(len(c2)):
c2[i] = rand.randint(-5, 5)
# TESTING ADDITION FUNCTION
a = P.polyadd(c1, c2).tolist()
b = q.addPoly(c1, c2)
if (a != b):
print "Failed on Case:"
print "a add b"
print "a", a
print "b", b
break
# TESTING SUBTRACTION FUNCTION
a = P.polysub(c1, c2).tolist()
b = q.subPoly(c1, c2)
if (a != b):
print "Failed on Case:"
print "a subtract b"
print "a", a
print "b", b
break
# TESTING MULTIPLICATION FUNCTION
a = P.polymul(c1, c2).tolist()
b = q.multPoly(c1, c2)
if (a != b):
print "Failed on Case:"
print "a * b"
print "a", a
def polysub(c1, c2):
from numpy.polynomial.polynomial import polysub
return polysub(c1, c2)
def longitudinal(velocity, density, S_gross_w, mac, Cm_q, Cz_alpha, mass, Cm_alpha, Iy, Cm_alpha_dot, Cz_u, Cz_alpha_dot, Cz_q, Cw, Theta, Cx_u, Cx_alpha):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Full_Linearized_Equations.longitudinal(velocity, density, S_gross_w, mac, Cm_q, Cz_alpha, mass, Cm_alpha, Iy, Cm_alpha_dot, Cz_u, Cz_alpha_dot, Cz_q, Cw, Theta, Cx_u, Cx_alpha)
Calculate the natural frequency and damping ratio for the full linearized short period and phugoid modes
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
density - flight density at condition being considered [kg/meters**3]
S_gross_w - area of the wing [meters**2]
mac - mean aerodynamic chord of the wing [meters]
Cm_q - coefficient for the change in pitching moment due to pitch rate [dimensionless] (2 * K * dC_m/di * lt/c where K is approximately 1.1)
Cz_alpha - coefficient for the change in Z force due to the angle of attack [dimensionless] (-C_D - dC_L/dalpha)
mass - mass of the aircraft [kilograms]
Cm_alpha - coefficient for the change in pitching moment due to angle of attack [dimensionless] (dC_m/dC_L * dCL/dalpha)
Iy - moment of interia about the body y axis [kg * meters**2]
Cm_alpha_dot - coefficient for the change in pitching moment due to rate of change of angle of attack [dimensionless] (2 * dC_m/di * depsilon/dalpha * lt/mac)
Cz_u - coefficient for the change in force in the Z direction due to change in forward velocity [dimensionless] (usually -2 C_L or -2C_L - U dC_L/du)
Cz_alpha_dot - coefficient for the change of angle of attack caused by w_dot on the Z force [dimensionless] (2 * dC_m/di * depsilon/dalpha)
Cz_q - coefficient for the change in Z force due to pitching velocity [dimensionless] (2 * K * dC_m/di where K is approximately 1.1)
Cw - coefficient to account for gravity [dimensionless] (-C_L)
Theta - angle between the horizontal axis and the body axis measured in the vertical plane [radians]
Cx_u - coefficient for the change in force in the X direction due to change in the forward velocity [dimensionless] (-2C_D)
Cx_alpha - coefficient for the change in force in the X direction due to the change in angle of attack caused by w [dimensionless] (C_L-dC_L/dalpha)
Outputs:
output - a data dictionary with fields:
short_w_n - natural frequency of the short period mode [radian/second]
short_zeta - damping ratio of the short period mode [dimensionless]
phugoid_w_n - natural frequency of the short period mode [radian/second]
phugoid_zeta - damping ratio of the short period mode [dimensionless]
Assumptions:
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Zero initial conditions
Cm_a = CF_z_a = CF_x_a = 0
Neglect Cx_alpha_dot, Cx_q and Cm_u
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 26-41.
"""
#process
# constructing matrix of coefficients
A = (- Cx_u, mass * velocity / S_gross_w / (0.5*density*velocity**2.)) # X force U term
B = (-Cx_alpha) # X force alpha term
C = (-Cw * np.cos(Theta)) # X force theta term
D = (-Cz_u) # Z force U term
E = (-Cz_alpha, (mass*velocity/S_gross_w/(0.5*density*velocity**2.)-mac*0.5/velocity*Cz_alpha_dot)) # Z force alpha term
F = (- Cw * np.sin(Theta), (-mass*velocity/S_gross_w/(0.5*density*velocity**2.)-mac*0.5/velocity*Cz_q))# Z force theta term
G = (0.) # M moment U term
H = (-Cm_alpha, -mac*0.5/velocity*Cm_alpha_dot) # M moment alpha term
I = (0., - mac*0.5/velocity*Cm_q, Iy/S_gross_w/(0.5*density*velocity**2)/mac) # M moment theta term
# Taking the determinant of the matrix ([A, B, C],[D, E, F],[G, H, I])
EI = P.polymul(E,I)
FH = P.polymul(F,H)
part1 = P.polymul(A,P.polysub(EI,FH))
DI = P.polymul(D,I)
FG = P.polymul(F,G)
part2 = P.polymul(B,P.polysub(FG,DI))
DH = P.polymul(D,H)
GE = P.polymul(G,E)
part3 = P.polymul(C,P.polysub(DH,GE))
total = P.polyadd(part1,P.polyadd(part2,part3))
# Use Synthetic division to split polynomial into two quadratic factors
poly = total / total[4]
poly1 = poly * 1.
poly2 = poly * 1.
poly3 = poly * 1.
poly4 = poly * 1.
poly1[4] = poly[4] - poly[2]/poly[2]
poly1[3] = poly[3] - poly[1]/poly[2]
poly1[2] = poly[2] - poly[0]/poly[2]
poly2[4] = 0
poly2[3] = poly1[3] - poly[2]*poly1[3]/poly[2]
poly2[2] = poly1[2] - poly[1]*poly1[3]/poly[2]
poly2[1] = poly1[1] - poly[0]*poly1[3]/poly[2]
poly3[4] = poly[4] - poly2[2]/poly2[2]
poly3[3] = poly[3] - poly2[1]/poly2[2]
poly3[2] = poly[2] - poly2[0]/poly2[2]
poly4[3] = poly3[3] - poly2[2]*poly3[3]/poly2[2]
poly4[2] = poly3[2] - poly2[1]*poly3[3]/poly2[2]
poly4[1] = poly3[1] - poly2[0]*poly3[3]/poly2[2]
# Generate natural frequency and damping for Short Period and Phugoid modes
short_w_n = (poly4[2])**0.5
short_zeta = poly3[3]*0.5/short_w_n
phugoid_w_n = (poly2[0]/poly2[2])**0.5
phugoid_zeta = poly2[1]/poly2[2]*0.5/phugoid_w_n
output = Data()
output.short_natural_frequency = short_w_n
output.short_damping_ratio = short_zeta
output.phugoid_natural_frequency = phugoid_w_n
output.phugoid_damping_ratio = phugoid_zeta
return output
def lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r, I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi, Cy_Beta, mass):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Full_Linearized_Equations.lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r, I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi, Cy_Beta)
Calculate the natural frequency and damping ratio for the full linearized dutch roll mode along with the time constants for the roll and spiral modes
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
Cn_Beta - coefficient for change in yawing moment due to sideslip [dimensionless] (no simple relation)
S_gross_w - area of the wing [meters**2]
density - flight density at condition being considered [kg/meters**3]
span - wing span of the aircraft [meters]
I_z - moment of interia about the body z axis [kg * meters**2]
Cn_r - coefficient for change in yawing moment due to yawing velocity [dimensionless] ( - C_D(wing)/4 - 2 * Sv/S * (l_v/b)**2 * (dC_L/dalpha)(vert) * eta(vert))
I_x - moment of interia about the body x axis [kg * meters**2]
Cl_p - change in rolling moment due to the rolling velocity [dimensionless] (no simple relation for calculation)
J_xz - products of inertia in the x-z direction [kg * meters**2] (if X and Z lie in a plane of symmetry then equal to zero)
Cl_r - coefficient for change in rolling moment due to yawing velocity [dimensionless] (Usually equals C_L(wing)/4)
Cl_Beta - coefficient for change in rolling moment due to sideslip [dimensionless]
Cn_p - coefficient for the change in yawing moment due to rolling velocity [dimensionless] (-C_L(wing)/8*(1 - depsilon/dalpha)) (depsilon/dalpha = 2/pi/e/AspectRatio dC_L(wing)/dalpha)
Cy_phi - coefficient for change in sideforce due to aircraft roll [dimensionless] (Usually equals C_L)
Cy_psi - coefficient to account for gravity [dimensionless] (C_L * tan(Theta))
Cy_Beta - coefficient for change in Y force due to sideslip [dimensionless] (no simple relation)
mass - mass of the aircraft [kilograms]
Outputs:
output - a data dictionary with fields:
dutch_w_n - natural frequency of the dutch roll mode [radian/second]
dutch_zeta - damping ratio of the dutch roll mode [dimensionless]
roll_tau - approximation of the time constant of the roll mode of an aircraft [seconds] (positive values are bad)
spiral_tau - time constant for the spiral mode [seconds] (positive values are bad)
Assumptions:
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Zero initial conditions
Neglect Cy_p and Cy_r
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 118-124.
"""
#process
# constructing matrix of coefficients
A = (0, -span * 0.5 / velocity * Cl_p, I_x/S_gross_w/(0.5*density*velocity**2)/span ) # L moment phi term
B = (0, -span * 0.5 / velocity * Cl_r, -J_xz / S_gross_w / (0.5 * density * velocity ** 2.) / span) # L moment psi term
C = (-Cl_Beta) # L moment Beta term
D = (0, - span * 0.5 / velocity * Cn_p, -J_xz / S_gross_w / (0.5 * density * velocity ** 2.) / span ) # N moment phi term
E = (0, - span * 0.5 / velocity * Cn_r, I_z / S_gross_w / (0.5 * density * velocity ** 2.) / span ) # N moment psi term
F = (-Cn_Beta) # N moment Beta term
G = (-Cy_phi) # Y force phi term
H = (-Cy_psi, mass * velocity / S_gross_w / (0.5 * density * velocity ** 2.))
I = (-Cy_Beta, mass * velocity / S_gross_w / (0.5 * density * velocity ** 2.))
# Taking the determinant of the matrix ([A, B, C],[D, E, F],[G, H, I])
EI = P.polymul(E,I)
FH = P.polymul(F,H)
part1 = P.polymul(A,P.polysub(EI,FH))
DI = P.polymul(D,I)
FG = P.polymul(F,G)
part2 = P.polymul(B,P.polysub(FG,DI))
DH = P.polymul(D,H)
GE = P.polymul(G,E)
part3 = P.polymul(C,P.polysub(DH,GE))
total = P.polyadd(part1,P.polyadd(part2,part3))
poly = total / total[5]
# Generate the time constant for the spiral and roll modes along with the damping and natural frequency for the dutch roll mode
root = np.roots(poly)
root = sorted(root,reverse=True)
spiral_tau = 1 * root[0].real
w_n = (root[1].imag**2 + root[1].real**2)**(-0.5)
zeta = -2*root[1].real/w_n
roll_tau = 1 * root [3].real
output = Data()
output.dutch_natural_frequency = w_n
output.dutch_damping_ratio = zeta
output.spiral_tau = spiral_tau
output.roll_tau = roll_tau
return output
print(np.dot(x, y)) #Dot Product
#_______________________________________________________________________________
#Find the roots of the given polynomials
print("Roots of the first polynomial:")
print(np.roots([1, 1.5, 2])) #getting roots of polynomial
print("Roots of the second polynomial:")
print(np.roots([1, -5.09])) #getting roots of polynomial
#_______________________________________________________________________________
# Add, subtract, multiply and divide polynomials
from numpy.polynomial import polynomial as P
x = (10, 20, 30) #declaring polynomials
y = (30, 40, 50)
print("Add one polynomial to another:")
print(P.polyadd(x, y)) #Adding polynomial
print("Subtract one polynomial from another:")
print(P.polysub(x, y)) #Subtracting polynomial
print("Multiply one polynomial by another:")
print(P.polymul(x, y)) #Multiplying polynomial
print("Divide one polynomial by another:")
print(P.polydiv(x, y)) #Dividing polynomial
#_______________________________________________________________________________
#Compute sine, cosine and tangent array of angles given in degrees
print("sine: array of angles given in degrees")
print(np.sin(np.array(
(0., 30., 45., 60., 90.)) * np.pi / 180.)) #Computing sin of angles
print("cosine: array of angles given in degrees")
print(np.cos(np.array(
(0., 30., 45., 60., 90.)) * np.pi / 180.)) #Computing cos of angles
print("tangent: array of angles given in degrees")
print(np.tan(np.array(
(0., 30., 45., 60., 90.)) * np.pi / 180.)) #Computing tan of angles
def polynomialRichardson(f1, f2, degree):
power = 2**(degree + 1)
from numpy.polynomial import polynomial as P
tmp = P.polysub(P.polymul(f2, [power]), f1)
return P.polymul(tmp, [1 / (power - 1)])
def inverse_rational_matrix(my_tf_object):
""" Takes the inverse of a transfer function matrix and returns it.
Parameters:
numArray_pres: a 2D array of the coefficients of the polynomials in the numerator of each entry of the original matrix
denArray: a 2D array of the coefficients of the polynomials in the denominator of each entry of the original matrix
:return:
out: a transfer function object that represents the inverted matrix
"""
# We turn our transfer function into a polynomial coefficient type object for making the algebraic inversion
numArray_reverse = my_tf_object.num
denArray_reverse = my_tf_object.den
# Now we take the coefficients of the polynomial and put them into 2 arrays
# one for numerators and one for denominators
numArray_list = list(numArray_reverse)
denArray_list = list(denArray_reverse)
n = np.shape(denArray_list)[0]
for i in np.arange(n):
for j in np.arange(n):
numArray_list[i][j] = np.flipud(numArray_list[i][j])
denArray_list[i][j] = np.flipud(denArray_list[i][j])
numArray_pres = (numArray_list)
denArray = (denArray_list)
numArray_fut = deepcopy(numArray_pres)
our_ks = np.arange(n - 1, -1, -1)
# We now go through the process of computing the adjoint of the numerator
for k in our_ks:
# One iteration of the pivoting, after pivoting the future become the present and the present becomes the past
# Now we update our past and present and future numerator arrays for the next pivot
numArray_past = deepcopy(numArray_pres)
numArray_pres = deepcopy(numArray_fut)
for i in np.arange(n):
for j in np.arange(n):
# Below are the following cases for the numerator of the adjoint,
# each is based off of the index of the location of the matrix entry:
# Based off of the paper by T. Downs called: On the inversion of a matrix of rational functions (1971)
# Case 1: i,j<k
if i < k and j < k:
if k == n-1:
numArray_fut[i][j] = tuple(poly.polysub(poly.polymul(poly.polymul(numArray_pres[i][j], numArray_pres[k][k]), poly.polymul(denArray[i][k], denArray[k][j])),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][j]), poly.polymul(denArray[i][j], denArray[k][k]))))
else:
numArray_fut[i][j] = tuple(poly.polydiv(poly.polysub(poly.polymul(poly.polymul(numArray_pres[i][j], numArray_pres[k][k]), poly.polymul(denArray[i][k], denArray[k][j])),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][j]), poly.polymul(denArray[i][j], denArray[k][k]))),
numArray_past[k+1][k+1])[0])
# Case 2: i>k>j
elif i > k and k > j:
numArray_fut[i][j] = tuple(poly.polydiv(poly.polysub(poly.polymul(poly.polymul(numArray_pres[i][j], numArray_pres[k][k]), denArray[k][j]),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][j]), denArray[k][k])),
poly.polymul(denArray[k][i], numArray_past[k+1][k+1]))[0])
# Case 2.5: j>k>i
elif j > k and k > i:
numArray_fut[i][j] = tuple(poly.polydiv(poly.polysub(poly.polymul(poly.polymul(numArray_pres[i][j], numArray_pres[k][k]), denArray[i][k]),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][j]), denArray[k][k])),
poly.polymul(denArray[j][k], numArray_past[k+1][k+1]))[0])
# Case 4 from the paper: i=j>k
elif i == j and j > k:
numArray_fut[i][j] = tuple(poly.polydiv(poly.polysub(poly.polymul(numArray_pres[i][i], numArray_pres[k][k]),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][i]), poly.polymul(denArray[k][k], denArray[i][i]))),
poly.polymul(poly.polymul(denArray[k][i], denArray[i][k]), numArray_past[k+1][k+1]))[0])
# Case 3 from the paper: i,j>k
elif i > k and j > k:
numArray_fut[i][j] = tuple(poly.polydiv(poly.polysub(poly.polymul(numArray_pres[i][j], numArray_pres[k][k]),
poly.polymul(poly.polymul(numArray_pres[i][k], numArray_pres[k][j]), denArray[k][k])),
poly.polymul(poly.polymul(denArray[k][i], denArray[j][k]), numArray_past[k+1][k+1]))[0])
# Case 5: i=k!=j
elif i == k and k != j:
numArray_fut[k][j] = tuple(poly.polymul((-1,), numArray_pres[k][j]))
# Case 6: j=k!=i
elif j == k and k != i:
numArray_fut[i][j] = tuple(numArray_pres[i][j])
# Case 7: i=j=k
elif i == j == k:
if k == n-1:
numArray_fut[k][k] = (1,)
else:
numArray_fut[k][k] = tuple(numArray_past[k+1][k+1])
denArray_fut = deepcopy(denArray)
# Below we do the denominator:
for i in np.arange(n):
for j in np.arange(n):
# Each entry of the denominator is the product of all the original denominators except for the ith column
# and the jth row of the original denominator matrix
# First we normalize the initial entry
denArray_fut[i][j] = tuple(poly.polydiv(denArray_fut[i][j], denArray_fut[i][j])[0])
for ii in np.arange(n):
for jj in np.arange(n):
if j != ii and i != jj:
denArray_fut[i][j] = tuple(poly.polymul(denArray_fut[i][j], denArray[ii][jj]))
# Now we collect our results and we return our final inverted matrix
my_adjoint_num = deepcopy(numArray_fut)
my_adjoint_den = deepcopy(denArray_fut)
# We multiply the adjoint by the inverse of the determinant of the matrix, the inverse of the determinant is
# decided to be the past numerator array (or that before preforming the final pivot)
inverse_determinant_den = deepcopy(numArray_pres[0][0])
inverse_determinant_num = (1)
for i in np.arange(n):
for j in np.arange(n):
inverse_determinant_num = tuple(poly.polymul(denArray[i][j], inverse_determinant_num))
newMatrix_num = []
newMatrix_den = []
for i in np.arange(n):
newMatrix_num.append([])
for j in np.arange(n):
newMatrix_num[i].append(tuple(poly.polymul(my_adjoint_num[i][j], inverse_determinant_num)))
# Now we do the denominators
for i in np.arange(n):
newMatrix_den.append([])
for j in np.arange(n):
x = np.shape(my_adjoint_den)
newMatrix_den[i].append(tuple(poly.polymul(my_adjoint_den[i][j], inverse_determinant_den)))
# Ok, we finally have our inverse!
numerators = newMatrix_num
denominators = newMatrix_den
try: # A polynomial solver for smaller degree polynomials to do cancellations and reduce the polynomials
for i in np.arange(n):
for j in np.arange(n):
numerators[i][j] = np.flipud(numerators[i][j])
denominators[i][j] = np.flipud(denominators[i][j])
k = 0
for val in numerators[i][j]:
numerators[i][j][k] = round(val, 4)
k += 1
num_roots = (np.roots(numerators[i][j]))
den_roots = (np.roots(denominators[i][j]))
for r in np.arange(len(num_roots)):
num_roots[r] = round(num_roots[r], 4)*10000.
for dr in np.arange(len(den_roots)):
den_roots[dr] = round(den_roots[dr], 4)*10000.
num_roots = Counter(num_roots)
den_roots = Counter(den_roots)
all_roots_num = num_roots - den_roots
all_roots_den = den_roots - num_roots
if np.any(numerators[i][j]) != 0: # If zero then we want to keep them as they are.
numerators[i][j] = numerators[i][j][0]/denominators[i][j][0]
for k in all_roots_num:
for number in np.arange(all_roots_num[k]):
numerators[i][j] = list(poly.polymul(numerators[i][j], (-k/10000., 1)))
denominators[i][j] = 1
for k in all_roots_den:
for number in np.arange(all_roots_den[k]):
denominators[i][j] = list(poly.polymul(denominators[i][j], (-k/10000., 1)))
# Now we return the values in the form of transfer functions by casting them with control.tf
for i in np.arange(n):
for j in np.arange(n):
if type(numerators[i][j]) == 'npumpy.ndarray' or type(numerators[i][j]) == list:
numerators[i][j] = np.flipud(numerators[i][j])
else:
numerators[i][j] = np.array([float(np.real(numerators[i][j]))])
if type(denominators[i][j]) != int:
denominators[i][j] = np.flipud(denominators[i][j])
else:
denominators[i][j] = np.array([float(denominators[i][j])])
except:
b = "c"
num_list = []
den_list = []
for i in np.arange(n):
num_list.append(list((numerators[i])))
den_list.append(list((denominators[i])))
out = tf.tf(num_list, den_list)
return out
def extrema(self, line, poly):
""" Calculates upper extrema. """
return polyroots(polysub(np.array([line.start.cost]), poly))
def polysub(c1, c2):
from numpy.polynomial.polynomial import polysub
return polysub(c1, c2)
c1[i]=rand.randint(-5,5)
for i in range(len(c2)):
c2[i]=rand.randint(-5,5)
# TESTING ADDITION FUNCTION
a= P.polyadd(c1,c2).tolist()
b= q.addPoly(c1,c2)
if(a!=b):
print "Failed on Case:"
print "a add b"
print "a",a
print "b",b
break
# TESTING SUBTRACTION FUNCTION
a=P.polysub(c1,c2).tolist()
b=q.subPoly(c1,c2)
if(a!=b):
print "Failed on Case:"
print "a subtract b"
print "a",a
print "b",b
break
# TESTING MULTIPLICATION FUNCTION
a=P.polymul(c1,c2).tolist()
b=q.multPoly(c1,c2)
if(a!=b):
print "Failed on Case:"
print "a * b"
print "a",a
def lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r,
I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi,
Cy_Beta, mass):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Full_Linearized_Equations.lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r, I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi, Cy_Beta)
Calculate the natural frequency and damping ratio for the full linearized dutch roll mode along with the time constants for the roll and spiral modes
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
Cn_Beta - coefficient for change in yawing moment due to sideslip [dimensionless] (no simple relation)
S_gross_w - area of the wing [meters**2]
density - flight density at condition being considered [kg/meters**3]
span - wing span of the aircraft [meters]
I_z - moment of interia about the body z axis [kg * meters**2]
Cn_r - coefficient for change in yawing moment due to yawing velocity [dimensionless] ( - C_D(wing)/4 - 2 * Sv/S * (l_v/b)**2 * (dC_L/dalpha)(vert) * eta(vert))
I_x - moment of interia about the body x axis [kg * meters**2]
Cl_p - change in rolling moment due to the rolling velocity [dimensionless] (no simple relation for calculation)
J_xz - products of inertia in the x-z direction [kg * meters**2] (if X and Z lie in a plane of symmetry then equal to zero)
Cl_r - coefficient for change in rolling moment due to yawing velocity [dimensionless] (Usually equals C_L(wing)/4)
Cl_Beta - coefficient for change in rolling moment due to sideslip [dimensionless]
Cn_p - coefficient for the change in yawing moment due to rolling velocity [dimensionless] (-C_L(wing)/8*(1 - depsilon/dalpha)) (depsilon/dalpha = 2/pi/e/AspectRatio dC_L(wing)/dalpha)
Cy_phi - coefficient for change in sideforce due to aircraft roll [dimensionless] (Usually equals C_L)
Cy_psi - coefficient to account for gravity [dimensionless] (C_L * tan(Theta))
Cy_Beta - coefficient for change in Y force due to sideslip [dimensionless] (no simple relation)
mass - mass of the aircraft [kilograms]
Outputs:
output - a data dictionary with fields:
dutch_w_n - natural frequency of the dutch roll mode [radian/second]
dutch_zeta - damping ratio of the dutch roll mode [dimensionless]
roll_tau - approximation of the time constant of the roll mode of an aircraft [seconds] (positive values are bad)
spiral_tau - time constant for the spiral mode [seconds] (positive values are bad)
Assumptions:
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Zero initial conditions
Neglect Cy_p and Cy_r
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 118-124.
"""
#process
# constructing matrix of coefficients
A = (0, -span * 0.5 / velocity * Cl_p,
I_x / S_gross_w / (0.5 * density * velocity**2) / span
) # L moment phi term
B = (0, -span * 0.5 / velocity * Cl_r,
-J_xz / S_gross_w / (0.5 * density * velocity**2.) / span
) # L moment psi term
C = (-Cl_Beta) # L moment Beta term
D = (0, -span * 0.5 / velocity * Cn_p,
-J_xz / S_gross_w / (0.5 * density * velocity**2.) / span
) # N moment phi term
E = (0, -span * 0.5 / velocity * Cn_r,
I_z / S_gross_w / (0.5 * density * velocity**2.) / span
) # N moment psi term
F = (-Cn_Beta) # N moment Beta term
G = (-Cy_phi) # Y force phi term
H = (-Cy_psi, mass * velocity / S_gross_w / (0.5 * density * velocity**2.))
I = (-Cy_Beta,
mass * velocity / S_gross_w / (0.5 * density * velocity**2.))
# Taking the determinant of the matrix ([A, B, C],[D, E, F],[G, H, I])
EI = P.polymul(E, I)
FH = P.polymul(F, H)
part1 = P.polymul(A, P.polysub(EI, FH))
DI = P.polymul(D, I)
FG = P.polymul(F, G)
part2 = P.polymul(B, P.polysub(FG, DI))
DH = P.polymul(D, H)
GE = P.polymul(G, E)
part3 = P.polymul(C, P.polysub(DH, GE))
total = P.polyadd(part1, P.polyadd(part2, part3))
poly = total / total[5]
# Generate the time constant for the spiral and roll modes along with the damping and natural frequency for the dutch roll mode
root = np.roots(poly)
root = sorted(root, reverse=True)
spiral_tau = 1 * root[0].real
w_n = (root[1].imag**2 + root[1].real**2)**(-0.5)
zeta = -2 * root[1].real / w_n
roll_tau = 1 * root[3].real
output = Data()
output.dutch_natural_frequency = w_n
output.dutch_damping_ratio = zeta
output.spiral_tau = spiral_tau
output.roll_tau = roll_tau
return output
def subpoly(f, g, m):
return to_ring(poly.polysub(f, g), m)
pto = input("Digite el punto a evaluar: ")
print "El resultado es: "+str(obj.evalPto(p1,pto))
elif op == 2:
print "Suma"
gr1 = input("Grado del polinomio 1")
p1 = obj.insertPol(gr1,p1)
gr1 = input("Grado del polinomio 2")
p2 = obj.insertPol(gr1,p2)
print "La suma es:"+str(npy.polyadd(p1,p2))
elif op == 3:
print "Resta"
gr1 = input("Grado del polinomio 1")
p1 = obj.insertPol(gr1,p1)
gr1 = input("Grado del polinomio 2")
p2 = obj.insertPol(gr1,p2)
print "La resta es:"+str(npy.polysub(p1,p2))
elif op == 4:
print "Multiplicacion"
gr1 = input("Grado del polinomio 1")
p1 = obj.insertPol(gr1,p1)
gr1 = input("Grado del polinomio 2")
p2 = obj.insertPol(gr1,p2)
print "La multiplicacion es:"+str(npy.polymul(p1,p2))
elif op == 5:
print "Igualdad"
gr1 = input("Grado del polinomio 1")
p1 = obj.insertPol(gr1,p1)
gr1 = input("Grado del polinomio 2")
p2 = obj.insertPol(gr1,p2)
elif op == 6:
print "Polinomio Opuesto"
def extrema(self, line, poly):
""" Calculates upper extrema. """
return np.concatenate((polyroots(polysub(np.array([line.start.cost, np.inner(self.inexact["c_u"]*line.start.grad, line.dire)]), poly)),
polyroots(polysub(np.array([line.start.cost, np.inner(self.inexact["c_l"]*line.start.grad, line.dire)]), poly))))
def longitudinal(velocity, density, S_gross_w, mac, Cm_q, Cz_alpha, mass,
Cm_alpha, Iy, Cm_alpha_dot, Cz_u, Cz_alpha_dot, Cz_q, Cw,
Theta, Cx_u, Cx_alpha):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Full_Linearized_Equations.longitudinal(velocity, density, S_gross_w, mac, Cm_q, Cz_alpha, mass, Cm_alpha, Iy, Cm_alpha_dot, Cz_u, Cz_alpha_dot, Cz_q, Cw, Theta, Cx_u, Cx_alpha)
Calculate the natural frequency and damping ratio for the full linearized short period and phugoid modes
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
density - flight density at condition being considered [kg/meters**3]
S_gross_w - area of the wing [meters**2]
mac - mean aerodynamic chord of the wing [meters]
Cm_q - coefficient for the change in pitching moment due to pitch rate [dimensionless] (2 * K * dC_m/di * lt/c where K is approximately 1.1)
Cz_alpha - coefficient for the change in Z force due to the angle of attack [dimensionless] (-C_D - dC_L/dalpha)
mass - mass of the aircraft [kilograms]
Cm_alpha - coefficient for the change in pitching moment due to angle of attack [dimensionless] (dC_m/dC_L * dCL/dalpha)
Iy - moment of interia about the body y axis [kg * meters**2]
Cm_alpha_dot - coefficient for the change in pitching moment due to rate of change of angle of attack [dimensionless] (2 * dC_m/di * depsilon/dalpha * lt/mac)
Cz_u - coefficient for the change in force in the Z direction due to change in forward velocity [dimensionless] (usually -2 C_L or -2C_L - U dC_L/du)
Cz_alpha_dot - coefficient for the change of angle of attack caused by w_dot on the Z force [dimensionless] (2 * dC_m/di * depsilon/dalpha)
Cz_q - coefficient for the change in Z force due to pitching velocity [dimensionless] (2 * K * dC_m/di where K is approximately 1.1)
Cw - coefficient to account for gravity [dimensionless] (-C_L)
Theta - angle between the horizontal axis and the body axis measured in the vertical plane [radians]
Cx_u - coefficient for the change in force in the X direction due to change in the forward velocity [dimensionless] (-2C_D)
Cx_alpha - coefficient for the change in force in the X direction due to the change in angle of attack caused by w [dimensionless] (C_L-dC_L/dalpha)
Outputs:
output - a data dictionary with fields:
short_w_n - natural frequency of the short period mode [radian/second]
short_zeta - damping ratio of the short period mode [dimensionless]
phugoid_w_n - natural frequency of the short period mode [radian/second]
phugoid_zeta - damping ratio of the short period mode [dimensionless]
Assumptions:
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Zero initial conditions
Cm_a = CF_z_a = CF_x_a = 0
Neglect Cx_alpha_dot, Cx_q and Cm_u
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 26-41.
"""
velocity = np.mean(velocity)
density = np.mean(density)
S_gross_w = np.mean(S_gross_w)
mac = np.mean(mac)
Cm_q = np.mean(Cm_q)
Cz_alpha = np.mean(Cz_alpha)
mass = np.mean(mass)
Cm_alpha = np.mean(Cm_alpha)
Iy = np.mean(Iy)
Cm_alpha_dot = np.mean(Cm_alpha_dot)
Cz_u = np.mean(Cz_u)
Cz_alpha_dot = np.mean(Cz_alpha_dot)
Cz_q = np.mean(Cz_q)
Cw = np.mean(Cw)
Theta = np.mean(Theta)
Cx_u = np.mean(Cx_u)
Cx_alpha = np.mean(Cx_alpha)
# constructing matrix of coefficients
A = (-Cx_u, mass * velocity / S_gross_w / (0.5 * density * velocity**2.)
) # X force U term
B = (-Cx_alpha) # X force alpha term
C = (-Cw * np.cos(Theta)) # X force theta term
D = (-Cz_u) # Z force U term
E = (-Cz_alpha,
(mass * velocity / S_gross_w / (0.5 * density * velocity**2.) -
mac * 0.5 / velocity * Cz_alpha_dot)) # Z force alpha term
F = (-Cw * np.sin(Theta),
(-mass * velocity / S_gross_w / (0.5 * density * velocity**2.) -
mac * 0.5 / velocity * Cz_q)) # Z force theta term
G = 0. # M moment U term
H = (-Cm_alpha, -mac * 0.5 / velocity * Cm_alpha_dot
) # M moment alpha term
I = (0., -mac * 0.5 / velocity * Cm_q,
Iy / S_gross_w / (0.5 * density * velocity**2) / mac
) # M moment theta term
# Taking the determinant of the matrix ([A, B, C],[D, E, F],[G, H, I])
EI = P.polymul(E, I)
FH = P.polymul(F, H)
part1 = P.polymul(A, P.polysub(EI, FH))
DI = P.polymul(D, I)
FG = P.polymul(F, G)
part2 = P.polymul(B, P.polysub(FG, DI))
DH = P.polymul(D, H)
GE = P.polymul(G, E)
part3 = P.polymul(C, P.polysub(DH, GE))
total = P.polyadd(part1, P.polyadd(part2, part3))
# Use Synthetic division to split polynomial into two quadratic factors
poly = total / total[4]
poly1 = poly * 1.
poly2 = poly * 1.
poly3 = poly * 1.
poly4 = poly * 1.
poly1[4] = poly[4] - poly[2] / poly[2]
poly1[3] = poly[3] - poly[1] / poly[2]
poly1[2] = poly[2] - poly[0] / poly[2]
poly2[4] = 0
poly2[3] = poly1[3] - poly[2] * poly1[3] / poly[2]
poly2[2] = poly1[2] - poly[1] * poly1[3] / poly[2]
poly2[1] = poly1[1] - poly[0] * poly1[3] / poly[2]
poly3[4] = poly[4] - poly2[2] / poly2[2]
poly3[3] = poly[3] - poly2[1] / poly2[2]
poly3[2] = poly[2] - poly2[0] / poly2[2]
poly4[3] = poly3[3] - poly2[2] * poly3[3] / poly2[2]
poly4[2] = poly3[2] - poly2[1] * poly3[3] / poly2[2]
poly4[1] = poly3[1] - poly2[0] * poly3[3] / poly2[2]
# Generate natural frequency and damping for Short Period and Phugoid modes
short_w_n = (poly4[2])**0.5
short_zeta = poly3[3] * 0.5 / short_w_n
phugoid_w_n = (poly2[0] / poly2[2])**0.5
phugoid_zeta = poly2[1] / poly2[2] * 0.5 / phugoid_w_n
output = Data()
output.short_natural_frequency = short_w_n
output.short_damping_ratio = short_zeta
output.phugoid_natural_frequency = phugoid_w_n
output.phugoid_damping_ratio = phugoid_zeta
return output
def __sub__(self, other):
if isinstance(other, MyPolynomial):
return MyPolynomial(
polynomial.polysub(self.polynomial, other.polynomial)[0])
else:
return MyPolynomial(self.polynomial - other)
