def left_ortho_complement(matrix):
# commented out for performance
#assert has_left_ortho_complement(matrix), "There is no leftorthocomplement!"
v = custom_simplify( st.nullspaceMatrix(matrix.T).T )
assert is_zero_matrix(v*matrix), "Leftorthocomplement is not correct."
return v
def left_ortho_complement(matrix):
# commented out for performance
#assert has_left_ortho_complement(matrix), "There is no leftorthocomplement!"
v = custom_simplify(st.nullspaceMatrix(matrix.T).T)
assert is_zero_matrix(v * matrix), "Leftorthocomplement is not correct."
return v
def right_ortho_complement(matrix):
assert has_right_ortho_complement(matrix), "There is no rightorthocomplement!"
if roc=="conv":
v = custom_simplify( st.nullspaceMatrix(matrix) )
assert is_zero_matrix(matrix*v), "Leftorthocomplement is not correct."
return v
else:
return alternative_right_ortho_complement(matrix)
def right_ortho_complement(matrix):
assert has_right_ortho_complement(
matrix), "There is no rightorthocomplement!"
if roc == "conv":
v = custom_simplify(st.nullspaceMatrix(matrix))
assert is_zero_matrix(matrix *
v), "Leftorthocomplement is not correct."
return v
else:
return alternative_right_ortho_complement(matrix)
def unimod_inv(M,
s=None,
t=None,
time_dep_symbs=[],
simplify_nsm=True,
max_deg=None):
""" Assumes that M(s) is an unimodular polynomial matrix and calculates its inverse
which is again unimodular
:param M: Matrix to be inverted
:param s: Derivative Symbol
:param time_dep_symbs: sequence of time dependent symbols
:param max_deg: maximum polynomial degree w.r.t. s of the ansatz
:return: Minv
"""
assert isinstance(M, sp.MatrixBase)
assert M.is_square
n = M.shape[0]
degree_m = nc_degree(M, s)
if max_deg is None:
# upper bound according to
# Levine 2011, On necessary and sufficient conditions for differential flatness, p. 73
max_deg = (n - 1) * degree_m
assert int(max_deg) == max_deg
assert max_deg >= 0
C = M * 0
free_params = []
for i in range(max_deg + 1):
prefix = 'c{0}_'.format(i)
c_part = st.symbMatrix(n, n, prefix, commutative=False)
C += nc_mul(c_part, s**i)
free_params.extend(list(c_part))
P = nc_mul(C, M) - sp.eye(n)
P2 = right_shift_all(P, s, t, time_dep_symbs).reshape(n * n, 1)
deg_P = nc_degree(P2, s)
part_eqns = []
for i in range(deg_P + 1):
# omit the highest order (because it behaves like in the commutative case)
res = P2.diff(s, i).subs(s, 0) #/sp.factorial(i)
part_eqns.append(res)
eqns = st.row_stack(*part_eqns) # equations for all degrees of s
# now non-commutativity is inferring
eqns2, st_c_nc = make_all_symbols_commutative(eqns)
free_params_c, st_c_nc_free_params = make_all_symbols_commutative(
free_params)
# find out which of the equations are (in)homogeneous
eqns2_0 = eqns2.subs(st.zip0(free_params_c))
assert eqns2_0.atoms() in ({0, -1}, {-1}, set())
inhom_idcs = st.np.where(st.to_np(eqns2_0) != 0)[0]
hom_idcs = st.np.where(st.to_np(eqns2_0) == 0)[0]
eqns_hom = sp.Matrix(st.np.array(eqns2)[hom_idcs])
eqns_inh = sp.Matrix(st.np.array(eqns2)[inhom_idcs])
assert len(eqns_inh) == n
# find a solution for the homogeneous equations
# if this is not possible, M was not unimodular
Jh = eqns_hom.jacobian(free_params_c).expand()
nsm = st.nullspaceMatrix(Jh, simplify=simplify_nsm, sort_rows=True)
na = nsm.shape[1]
if na < n:
msg = 'Could not determine sufficiently large nullspace. '\
'Either M is not unimodular or the expressions are to complicated.'
# TODO: decide which of the two cases occurs, via substitution of
# random numbers and singular value decomposition
# (or application of st.generic_rank)
raise ValueError(msg)
# parameterize the inhomogenous equations with the solution of the homogeneous equations
# new free parameters:
aa = st.symb_vector('_a1:{0}'.format(na + 1))
nsm_a = nsm * aa
eqns_inh2 = eqns_inh.subs(lzip(free_params_c, nsm_a))
# now solve the remaining equations
# solve the linear system
Jinh = eqns_inh2.jacobian(aa)
rhs_inh = -eqns_inh2.subs(st.zip0(aa))
assert rhs_inh == sp.ones(n, 1)
sol_vect = Jinh.solve(rhs_inh)
sol = lzip(aa, sol_vect)
# get the values for all free_params (now they are not free anymore)
free_params_sol_c = nsm_a.subs(sol)
# replace the commutative symbols with the original non_commutative symbols (of M)
free_params_sol = free_params_sol_c.subs(st_c_nc)
Minv = C.subs(lzip(free_params, free_params_sol))
return Minv
#~ [ 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0],
#~ [W2*x9, W3*x10, W1*x11 - W2*x11, W1*x12 - W3*x12, Rd*W1 + Rl*W2 - Rl*W3, Rl*W2 + Rl*W3, Rd*W1, 0, W2*x1, W3*x2, W1*x3 - W2*x3, W1*x4 - W3*x4]])
P00 = F_eq.jacobian(xx)
#~ P00 = sp.Matrix([
#~ [ x9, 0, -x11, 0, Rl, Rl, 0, 0, x1, 0, -x3, 0],
#~ [ 0, x10, 0, -x12, -Rl, Rl, 0, 0, 0, x2, 0, -x4],
#~ [ 0, 0, 0, 0, -x9, 0, 0, 0, -x5, 0, 0, 0],
#~ [ 0, 0, 0, 0, 0, -x10, 0, 0, 0, -x6, 0, 0],
#~ [ 0, 0, 0, 0, 0, 0, -x11, 0, 0, 0, -x7, 0],
#~ [ 0, 0, 0, 0, 0, 0, 0, -x12, 0, 0, 0, -x8],
#~ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
#~ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
#~ [W2*xdot9, W3*xdot10, W1*xdot11 - W2*xdot11, W1*xdot12 - W3*xdot12, 0, 0, 0, 0, W2*xdot1, W3*xdot2, W1*xdot3 - W2*xdot3, W1*xdot4 - W3*xdot4]])
P10_roc = st.nullspaceMatrix(P10)
#~ P10_roc = sp.Matrix([
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [ 0, 0, 0],
#~ [-W3*x2, x3*(-W1 + W2), x4*(-W1 + W3)],
#~ [ W2*x1, 0, 0],
#~ [ 0, W2*x1, 0],
#~ [ 0, 0, W2*x1]])
# statt P10_rpinv nehme ich jetzt S mit einheitsvektoren sodass
def unimod_inv(M, s=None, t=None, time_dep_symbs=[], simplify_nsm=True, max_deg=None):
""" Assumes that M(s) is an unimodular polynomial matrix and calculates its inverse
which is again unimodular
:param M: Matrix to be inverted
:param s: Derivative Symbol
:param time_dep_symbs: sequence of time dependent symbols
:param max_deg: maximum polynomial degree w.r.t. s of the ansatz
:return: Minv
"""
assert isinstance(M, sp.MatrixBase)
assert M.is_square
n = M.shape[0]
degree_m = nc_degree(M, s)
if max_deg is None:
# upper bound according to
# Levine 2011, On necessary and sufficient conditions for differential flatness, p. 73
max_deg = (n - 1)*degree_m
assert int(max_deg) == max_deg
assert max_deg >= 0
C = M*0
free_params = []
for i in range(max_deg+1):
prefix = 'c{0}_'.format(i)
c_part = st.symbMatrix(n, n, prefix, commutative=False)
C += nc_mul(c_part, s**i)
free_params.extend(list(c_part))
P = nc_mul(C, M) - sp.eye(n)
P2 = right_shift_all(P, s, t, time_dep_symbs).reshape(n*n, 1)
deg_P = nc_degree(P2, s)
part_eqns = []
for i in range(deg_P + 1):
# omit the highest order (because it behaves like in the commutative case)
res = P2.diff(s, i).subs(s, 0)#/sp.factorial(i)
part_eqns.append(res)
eqns = st.row_stack(*part_eqns) # equations for all degrees of s
# now non-commutativity is inferring
eqns2, st_c_nc = make_all_symbols_commutative(eqns)
free_params_c, st_c_nc_free_params = make_all_symbols_commutative(free_params)
# find out which of the equations are (in)homogeneous
eqns2_0 = eqns2.subs(st.zip0(free_params_c))
assert eqns2_0.atoms() in ({0, -1}, {-1}, set())
inhom_idcs = st.np.where(st.to_np(eqns2_0) != 0)[0]
hom_idcs = st.np.where(st.to_np(eqns2_0) == 0)[0]
eqns_hom = sp.Matrix(st.np.array(eqns2)[hom_idcs])
eqns_inh = sp.Matrix(st.np.array(eqns2)[inhom_idcs])
assert len(eqns_inh) == n
# find a solution for the homogeneous equations
# if this is not possible, M was not unimodular
Jh = eqns_hom.jacobian(free_params_c).expand()
nsm = st.nullspaceMatrix(Jh, simplify=simplify_nsm, sort_rows=True)
na = nsm.shape[1]
if na < n:
msg = 'Could not determine sufficiently large nullspace. '\
'Either M is not unimodular or the expressions are to complicated.'
# TODO: decide which of the two cases occurs, via substitution of
# random numbers and singular value decomposition
# (or application of st.generic_rank)
raise ValueError(msg)
# parameterize the inhomogenous equations with the solution of the homogeneous equations
# new free parameters:
aa = st.symb_vector('_a1:{0}'.format(na+1))
nsm_a = nsm*aa
eqns_inh2 = eqns_inh.subs(lzip(free_params_c, nsm_a))
# now solve the remaining equations
# solve the linear system
Jinh = eqns_inh2.jacobian(aa)
rhs_inh = -eqns_inh2.subs(st.zip0(aa))
assert rhs_inh == sp.ones(n, 1)
sol_vect = Jinh.solve(rhs_inh)
sol = lzip(aa, sol_vect)
# get the values for all free_params (now they are not free anymore)
free_params_sol_c = nsm_a.subs(sol)
# replace the commutative symbols with the original non_commutative symbols (of M)
free_params_sol = free_params_sol_c.subs(st_c_nc)
Minv = C.subs(lzip(free_params, free_params_sol))
return Minv