def fourseven(iter_stack):
# load matrices
Ai = iter_stack.A
Bi = iter_stack.B
P1i_roc = iter_stack.P1_roc
P1i_rpinv = iter_stack.P1_rpinv
K2 = roc_hint("B", iter_stack.i, Bi) if mode=="manual" else al.right_ortho_complement(Bi)
K1 = al.regular_completion(K2)
K = st.concat_cols(K1, K2)
assert al.is_regular_matrix(K), "K is not a regular matrix."
Bi_tilde = al.custom_simplify(Bi*K1) # unit matrix
if myStack.calc_G:
Bi_tilde_lpinv = al.left_pseudo_inverse(Bi_tilde)
else:
Bi_tilde_lpinv = None
P1i_tilde_roc = al.custom_simplify( P1i_roc*K1 )
Zi = al.custom_simplify( P1i_roc*K2 )
Zi_lpinv = al.Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi)
assert al.is_unit_matrix( Zi_lpinv*Zi ), "Zi_lpinv seems to be wrong."
# store
iter_stack.store_special_case_matrices( Zi, Zi_lpinv, Bi_tilde, Bi_tilde_lpinv, P1i_tilde_roc )
return Bi_tilde
def Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi):
""" Given a matrix Zi, this function calculates a matrix such that:
Zi_left_pinv * Zi = I
Zi_left_pinv * P1i_tilde_roc = 0
Zi_left_pinv * P1i_rpinv = 0
"""
assert check_row_compatibility(P1i_rpinv, P1i_tilde_roc, Zi),\
"Matrices do not match in row dimension."
C = st.concat_cols(P1i_rpinv, P1i_tilde_roc, Zi)
assert is_regular_matrix(C), "C is not a regular matrix"
C_det = C.berkowitz_det()
C_inv = C.adjugate()/C_det
C_inv = custom_simplify(C_inv)
m, n = Zi.shape
Zi_left_pinv = sp.Matrix([])
for i in xrange(m-n,m):
Zi_left_pinv = st.concat_rows(Zi_left_pinv,C_inv.row(i))
o, p = Zi_left_pinv.shape
assert o==n and p==m, "There must have been a problem with the\
computation of Zi_left_pinv"
assert is_unit_matrix(Zi_left_pinv*Zi), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv*P1i_tilde_roc), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv*P1i_rpinv), "Zi_left_pinv is wrong"
return Zi_left_pinv
def Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi):
""" Given a matrix Zi, this function calculates a matrix such that:
Zi_left_pinv * Zi = I
Zi_left_pinv * P1i_tilde_roc = 0
Zi_left_pinv * P1i_rpinv = 0
"""
assert check_row_compatibility(P1i_rpinv, P1i_tilde_roc, Zi),\
"Matrices do not match in row dimension."
C = st.concat_cols(P1i_rpinv, P1i_tilde_roc, Zi)
assert is_regular_matrix(C), "C is not a regular matrix"
C_det = C.berkowitz_det()
C_inv = C.adjugate() / C_det
C_inv = custom_simplify(C_inv)
m, n = Zi.shape
Zi_left_pinv = sp.Matrix([])
for i in range(m - n, m):
Zi_left_pinv = st.concat_rows(Zi_left_pinv, C_inv.row(i))
o, p = Zi_left_pinv.shape
assert o == n and p == m, "There must have been a problem with the\
computation of Zi_left_pinv"
assert is_unit_matrix(Zi_left_pinv * Zi), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv *
P1i_tilde_roc), "Zi_left_pinv is wrong"
assert is_zero_matrix(Zi_left_pinv * P1i_rpinv), "Zi_left_pinv is wrong"
return Zi_left_pinv
def is_linearly_independent(matrix, column_vector):
m, n = matrix.shape
rank1 = st.rnd_number_rank(matrix)
tmp = st.concat_cols(matrix, column_vector)
rank2 = st.rnd_number_rank(tmp)
assert rank2 >= rank1
return rank2 > rank1
def remove_zero_columns(matrix):
""" this function removes zero columns of a matrix
"""
m, n = matrix.shape
M = sp.Matrix([])
for i in xrange(n):
if not is_zero_matrix(matrix.col(i)):
M = st.concat_cols(M, matrix.col(i))
return M
def is_linearly_independent(matrix, column_vector):
m, n = matrix.shape
rank1 = st.rnd_number_rank(matrix)
tmp = st.concat_cols(matrix, column_vector)
rank2 = st.rnd_number_rank(tmp)
assert rank2 >= rank1
return rank2 > rank1
def remove_zero_columns(matrix):
""" this function removes zero columns of a matrix
"""
m, n = matrix.shape
M = sp.Matrix([])
for i in range(n):
if not is_zero_matrix(matrix.col(i)):
M = st.concat_cols(M, matrix.col(i))
return M
def right_shift_all_in_matrix(self, matrix):
# does nct-package provide this already?
m,n = matrix.shape
matrix_shifted = sp.Matrix([])
t_dep_symbols = [symb for symb in st.atoms(matrix, sp.Symbol) if not symb == s]
for i in xrange(n):
col = matrix.col(i)
col_shifted = sp.Matrix([nct.right_shift_all(expr,s,t, t_dep_symbols) for expr in col])
matrix_shifted = st.concat_cols(matrix_shifted, col_shifted)
return matrix_shifted
def right_shift_all_in_matrix(self, matrix):
# does nct-package provide this already?
m, n = matrix.shape
matrix_shifted = sp.Matrix([])
t_dep_symbols = [
symb for symb in st.atoms(matrix, sp.Symbol) if not symb == s
]
for i in range(n):
col = matrix.col(i)
col_shifted = sp.Matrix([
nct.right_shift_all(expr, s, t, t_dep_symbols) for expr in col
])
matrix_shifted = st.concat_cols(matrix_shifted, col_shifted)
return matrix_shifted
def calculate_Gi_matrix(self, i):
# 1) choose correct matrices for Gi[d/dt]
# 2) change symbols to noncommutative symbols, calculate Gi[d/dt]
# 3) G[d/dt] = G0[d/dt] * G1[d/dt] * ... * Gi[d/dt]
# 4) special case procedure
iteration = self._myStack.get_iteration(i)
if not iteration.is_special_case:
# get matrices
P1_rpinv = iteration.P1_rpinv
P1_roc = iteration.P1_roc
B_lpinv = iteration.B_lpinv
A = iteration.A
P1_rpinv_nc, P1_roc_nc, B_lpinv_nc, A_nc = self.convert_to_nc_matrices(
P1_rpinv, P1_roc, B_lpinv, A)
Gi = P1_rpinv_nc - P1_roc_nc * (B_lpinv_nc * s_ +
B_lpinv_nc * A_nc)
else:
# get matrices
P1_rpinv = iteration.P1_rpinv
P1_tilde_roc = iteration.P1_tilde_roc
B_tilde_lpinv = iteration.B_tilde_lpinv
A = iteration.A
Z = iteration.Z
# convert commutative symbols to non commutative
P1_rpinv_nc = self.make_symbols_non_commutative(P1_rpinv)
P1_tilde_roc_nc = self.make_symbols_non_commutative(P1_tilde_roc)
B_tilde_lpinv_nc = self.make_symbols_non_commutative(B_tilde_lpinv)
A_nc = self.make_symbols_non_commutative(A)
Z_nc = self.make_symbols_non_commutative(Z)
Gi = P1_rpinv_nc - P1_tilde_roc_nc * (B_tilde_lpinv_nc * s_ +
B_tilde_lpinv_nc * A_nc)
Gi = st.concat_cols(Gi, Z_nc)
if False:
# show_Gi_matrices:
pc.print_matrix("G", i, "", Gi)
# store
iteration.Gi = Gi
return Gi
def calculate_Gi_matrix(self, i):
# 1) choose correct matrices for Gi[d/dt]
# 2) change symbols to noncommutative symbols, calculate Gi[d/dt]
# 3) G[d/dt] = G0[d/dt] * G1[d/dt] * ... * Gi[d/dt]
# 4) special case procedure
iteration = self._myStack.get_iteration(i)
if not iteration.is_special_case:
# get matrices
P1_rpinv = iteration.P1_rpinv
P1_roc = iteration.P1_roc
B_lpinv = iteration.B_lpinv
A = iteration.A
P1_rpinv_nc, P1_roc_nc, B_lpinv_nc, A_nc = self.convert_to_nc_matrices(P1_rpinv, P1_roc, B_lpinv, A)
Gi = P1_rpinv_nc - P1_roc_nc*( B_lpinv_nc*s_ + B_lpinv_nc*A_nc )
else:
# get matrices
P1_rpinv = iteration.P1_rpinv
P1_tilde_roc = iteration.P1_tilde_roc
B_tilde_lpinv = iteration.B_tilde_lpinv
A = iteration.A
Z = iteration.Z
# convert commutative symbols to non commutative
P1_rpinv_nc = self.make_symbols_non_commutative(P1_rpinv)
P1_tilde_roc_nc = self.make_symbols_non_commutative(P1_tilde_roc)
B_tilde_lpinv_nc = self.make_symbols_non_commutative(B_tilde_lpinv)
A_nc = self.make_symbols_non_commutative(A)
Z_nc = self.make_symbols_non_commutative(Z)
Gi = P1_rpinv_nc - P1_tilde_roc_nc*( B_tilde_lpinv_nc*s_ + B_tilde_lpinv_nc*A_nc )
Gi = st.concat_cols(Gi, Z_nc)
if False:
# show_Gi_matrices:
pc.print_matrix("G", i, "", Gi)
# store
iteration.Gi = Gi
return Gi
def fourseven(iter_stack):
# load matrices
Ai = iter_stack.A
Bi = iter_stack.B
P1i_roc = iter_stack.P1_roc
P1i_rpinv = iter_stack.P1_rpinv
K2 = roc_hint("B", iter_stack.i,
Bi) if mode == "manual" else al.right_ortho_complement(Bi)
K1 = al.regular_completion(K2)
K = st.concat_cols(K1, K2)
assert al.is_regular_matrix(K), "K is not a regular matrix."
Bi_tilde = al.custom_simplify(Bi * K1) # unit matrix
if myStack.calc_G:
Bi_tilde_lpinv = al.left_pseudo_inverse(Bi_tilde)
else:
Bi_tilde_lpinv = None
P1i_tilde_roc = al.custom_simplify(P1i_roc * K1)
Zi = al.custom_simplify(P1i_roc * K2)
Zi_lpinv = al.Zi_left_pinv_with_restrictions(P1i_rpinv, P1i_tilde_roc, Zi)
assert al.is_unit_matrix(Zi_lpinv * Zi), "Zi_lpinv seems to be wrong."
# store
iter_stack.store_special_case_matrices(Zi, Zi_lpinv, Bi_tilde,
Bi_tilde_lpinv, P1i_tilde_roc)
return Bi_tilde
def reshape_matrix_columns(P):
m0, n0 = P.shape
# pick m0 of the simplest lin. independent columns of P:
P_new = remove_zero_columns(P)
m1, n1 = P_new.shape
list_of_cols = matrix_to_vectorlist(P_new)
# sort by "complexity"
if pinv_optimization=="free_symbols":
cols_sorted_by_atoms = sorted( list_of_cols, key=lambda x: x.free_symbols, reverse=False)
elif pinv_optimization=="count_ops":
cols_sorted_by_atoms = sorted( list_of_cols, key=lambda x: nr_of_ops(x), reverse=False)
elif pinv_optimization=="none":
cols_sorted_by_atoms = list_of_cols
# sort by number of zero entries
colvecs_sorted = sorted( cols_sorted_by_atoms, key=lambda x: count_zero_entries(x), reverse=True)
# pick m suitable column vectors and add to new matrix A: ----------
A = colvecs_sorted[0]
for j in xrange(len(colvecs_sorted)-1):
column = colvecs_sorted[j+1]
if is_linearly_independent(A,column):
A = st.concat_cols(A,column)
if st.rnd_number_rank(A)==m1:
break
assert A.is_square
# calculate transformation matrix R: -------------------------------
#R_tilde = sp.Matrix([])
used_cols = []
R = sp.Matrix([])
for k in xrange(m1):
new_column_index = k
old_column_index = get_column_index_of_matrix(P,A.col(k))
used_cols.append(old_column_index)
tmp = sp.zeros(n0,1)
tmp[old_column_index] = 1
#R_tilde = st.concat_cols(R_tilde,tmp)
R = st.concat_cols(R,tmp)
#R=R_tilde
# remainder columns of R matrix
#m2,n2 = R_tilde.shape
m2,n2 = R.shape
for l in xrange(n0):
if l not in used_cols:
R_col = sp.zeros(n0,1)
R_col[l] = 1
R = st.concat_cols(R,R_col)
m3, n3 = A.shape
r2 = st.rnd_number_rank(A)
assert m3==r2, "A problem occured in reshaping the matrix."
# calculate B matrix: ----------------------------------------------
B = sp.Matrix([])
tmp = P*R
for i in xrange(n0-m0):
B = st.concat_cols(B,tmp.col(m0+i))
assert is_zero_matrix( (P*R) - st.concat_cols(A,B)), "A problem occured in reshaping the matrix."
return A, B, R
def gen_leqs_for_acc_llmd(self, parameter_values=None):
"""
Create a callable function which returns A, bnum of the linear eqn-system
A*ww = bnum,
where ww := (ttheta_dd, llmnd).
:return: None, set self.leqs_acc_lmd_func
"""
if self.leqs_acc_lmd_func is not None and self.acc_of_lmd_func is not None:
return
if parameter_values is None:
parameter_values = []
ntt = self.ntt
nll = self.nll
self.generate_constraints_funcs()
# also respect those values, which have been passed to the constructor
parameter_values = list(self.parameter_values) + list(parameter_values)
# we use mod.eqns here because we do not want ydot-vars inside
eqns = st.concat_rows(self.mod.eqns.subs(parameter_values), self.constraints_dd)
ww = st.concat_rows(self.mod.ttdd, self.mod.llmd)
A = eqns.jacobian(ww)
b = -eqns.subz0(ww) # rhs of the leqs
Ab = st.concat_cols(A, b)
fvars = st.concat_rows(self.mod.tt, self.mod.ttd, self.mod.tau)
actual_symbs = Ab.atoms(sp.Symbol)
expected_symbs = set(fvars)
unexpected_symbs = actual_symbs.difference(expected_symbs)
if unexpected_symbs:
msg = "Equations can only converted to numerical func if all parameters are passed for substitution. " \
"Unexpected symbols: {}".format(unexpected_symbs)
raise ValueError(msg)
A_fnc = st.expr_to_func(fvars, A, keep_shape=True)
b_fnc = st.expr_to_func(fvars, b)
nargs = len(fvars)
# noinspection PyShadowingNames
def leqs_acc_lmd_func(*args):
"""
Calculate the matrices of the linear equation system for ttheta and llmd.
Assume args = (ttheta, theta_d, ttau)
:param args:
:return:
"""
assert len(args) == nargs
Anum = A_fnc(*args)
bnum = b_fnc(*args)
# theese arrays can now be passed to a linear equation solver
return Anum, bnum
self.leqs_acc_lmd_func = leqs_acc_lmd_func
def acc_of_lmd_func(*args):
"""
Calculate ttheta in dependency of args= (yy, ttau) = ((ttheta, ttheta_d, llmd), ttau)
:param args:
:return:
"""
ttheta = args[:ntt]
ttheta_d = args[ntt:2 * ntt]
llmd = args[2 * ntt:2 * ntt + nll]
ttau = args[2 * ntt + nll:]
args1 = np.concatenate((ttheta, ttheta_d, ttau))
Anum = A_fnc(*args1)
A1 = Anum[:ntt, :ntt]
A2 = Anum[:ntt, ntt:]
b1 = b_fnc(*args1)[:ntt]
ttheta_dd_res = np.linalg.solve(A1, b1 - np.dot(A2, llmd))
return ttheta_dd_res
self.acc_of_lmd_func = acc_of_lmd_func
def reshape_matrix_columns(P):
m0, n0 = P.shape
# pick m0 of the simplest lin. independent columns of P:
P_new = remove_zero_columns(P)
m1, n1 = P_new.shape
list_of_cols = matrix_to_vectorlist(P_new)
# sort by "complexity"
if pinv_optimization == "free_symbols":
cols_sorted_by_atoms = sorted(list_of_cols,
key=lambda x: x.free_symbols,
reverse=False)
elif pinv_optimization == "count_ops":
cols_sorted_by_atoms = sorted(list_of_cols,
key=lambda x: nr_of_ops(x),
reverse=False)
elif pinv_optimization == "none":
cols_sorted_by_atoms = list_of_cols
# sort by number of zero entries
colvecs_sorted = sorted(cols_sorted_by_atoms,
key=lambda x: count_zero_entries(x),
reverse=True)
# pick m suitable column vectors and add to new matrix A: ----------
A = colvecs_sorted[0]
for j in range(len(colvecs_sorted) - 1):
column = colvecs_sorted[j + 1]
if is_linearly_independent(A, column):
A = st.concat_cols(A, column)
if st.rnd_number_rank(A) == m1:
break
assert A.is_square
# calculate transformation matrix R: -------------------------------
#R_tilde = sp.Matrix([])
used_cols = []
R = sp.Matrix([])
for k in range(m1):
new_column_index = k
old_column_index = get_column_index_of_matrix(P, A.col(k))
used_cols.append(old_column_index)
tmp = sp.zeros(n0, 1)
tmp[old_column_index] = 1
#R_tilde = st.concat_cols(R_tilde,tmp)
R = st.concat_cols(R, tmp)
#R=R_tilde
# remainder columns of R matrix
#m2,n2 = R_tilde.shape
m2, n2 = R.shape
for l in range(n0):
if l not in used_cols:
R_col = sp.zeros(n0, 1)
R_col[l] = 1
R = st.concat_cols(R, R_col)
m3, n3 = A.shape
r2 = st.rnd_number_rank(A)
assert m3 == r2, "A problem occured in reshaping the matrix."
# calculate B matrix: ----------------------------------------------
B = sp.Matrix([])
tmp = P * R
for i in range(n0 - m0):
B = st.concat_cols(B, tmp.col(m0 + i))
assert is_zero_matrix(
(P * R) -
st.concat_cols(A, B)), "A problem occured in reshaping the matrix."
return A, B, R