def solve_multipliers(self, op_point=None, sol_type='dict'):
"""Solves for the values of the lagrange multipliers symbolically at
the specified operating point
Parameters
----------
op_point : dict or iterable of dicts, optional
Point at which to solve at. The operating point is specified as
a dictionary or iterable of dictionaries of {symbol: value}. The
value may be numeric or symbolic itself.
sol_type : str, optional
Solution return type. Valid options are:
- 'dict': A dict of {symbol : value} (default)
- 'Matrix': An ordered column matrix of the solution
"""
# Determine number of multipliers
k = len(self.lam_vec)
if k == 0:
raise ValueError("System has no lagrange multipliers to solve for.")
# Compose dict of operating conditions
if isinstance(op_point, dict):
op_point_dict = op_point
elif isinstance(op_point, collections.Iterable):
op_point_dict = {}
for op in op_point:
op_point_dict.update(op)
else:
op_point_dict = {}
# Compose the system to be solved
mass_matrix = self.mass_matrix.col_join((-self.lam_coeffs.row_join(
zeros(k, k))))
force_matrix = self.forcing.col_join(self._f_cd)
# Sub in the operating point
mass_matrix = _subs_keep_derivs(mass_matrix, op_point_dict)
force_matrix = _subs_keep_derivs(force_matrix, op_point_dict)
# Solve for the multipliers
sol_list = mass_matrix.LUsolve(-force_matrix)[-k:]
if sol_type == 'dict':
return dict(zip(self.lam_vec, sol_list))
elif sol_type == 'Matrix':
return Matrix(sol_list)
else:
raise ValueError("Unknown sol_type {:}.".format(sol_type))
def to_linearizer(self):
"""Returns an instance of the Linearizer class, initiated from the
data in the KanesMethod class. This may be more desirable than using
the linearize class method, as the Linearizer object will allow more
efficient recalculation (i.e. about varying operating points)."""
if (self._fr is None) or (self._frstar is None):
raise ValueError('Need to compute Fr, Fr* first.')
# Get required equation components. The Kane's method class breaks
# these into pieces. Need to reassemble
f_c = self._f_h
if self._f_nh and self._k_nh:
f_v = self._f_nh + self._k_nh * Matrix(self._u)
else:
f_v = Matrix([])
if self._f_dnh and self._k_dnh:
f_a = self._f_dnh + self._k_dnh * Matrix(self._udot)
else:
f_a = Matrix([])
# Dicts to sub to zero, for splitting up expressions
u_zero = dict((i, 0) for i in self._u)
ud_zero = dict((i, 0) for i in self._udot)
qd_zero = dict((i, 0) for i in self._qdot)
qd_u_zero = dict((i, 0) for i in self._qdot + self._u)
# Break the kinematic differential eqs apart into f_0 and f_1
f_0 = self._f_k.subs(u_zero) + self._k_kqdot * Matrix(self._qdot)
f_1 = self._f_k.subs(qd_zero) + self._k_ku * Matrix(self._u)
# Break the dynamic differential eqs into f_2 and f_3
f_2 = _subs_keep_derivs(self._frstar, qd_u_zero)
f_3 = self._frstar.subs(ud_zero) + self._fr
f_4 = zeros(len(f_2), 1)
# Get the required vector components
q = self._q
u = self._u
if self._qdep:
q_i = q[:-len(self._qdep)]
else:
q_i = q
q_d = self._qdep
if self._udep:
u_i = u[:-len(self._udep)]
else:
u_i = u
u_d = self._udep
# Form dictionary of auxiliary speeds & and their derivatives,
# setting each to 0.
uaux = self._uaux
uauxdot = [diff(i, dynamicsymbols._t) for i in uaux]
uaux_zero = dict((i, 0) for i in uaux + uauxdot)
# Checking for dynamic symbols outside the dynamic differential
# equations; throws error if there is.
insyms = set(q + self._qdot + u + self._udot + uaux + uauxdot)
if any(
_find_dynamicsymbols(i, insyms) for i in [
self._k_kqdot, self._k_ku, self._f_k, self._k_dnh,
self._f_dnh, self._k_d
]):
raise ValueError('Cannot have dynamicsymbols outside dynamic \
forcing vector.')
# Find all other dynamic symbols, forming the forcing vector r.
# Sort r to make it canonical.
r = list(_find_dynamicsymbols(self._f_d.subs(uaux_zero), insyms))
r.sort(key=default_sort_key)
# Check for any derivatives of variables in r that are also found in r.
for i in r:
if diff(i, dynamicsymbols._t) in r:
raise ValueError('Cannot have derivatives of specified \
quantities when linearizing forcing terms.')
return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i,
q_d, u_i, u_d, r)
def linearize(self, op_point=None, A_and_B=False, simplify=False):
"""Linearize the system about the operating point. Note that
q_op, u_op, qd_op, ud_op must satisfy the equations of motion.
These may be either symbolic or numeric.
Parameters
----------
op_point : dict or iterable of dicts, optional
Dictionary or iterable of dictionaries containing the operating
point conditions. These will be substituted in to the linearized
system before the linearization is complete. Leave blank if you
want a completely symbolic form. Note that any reduction in
symbols (whether substituted for numbers or expressions with a
common parameter) will result in faster runtime.
A_and_B : bool, optional
If A_and_B=False (default), (M, A, B) is returned for forming
[M]*[q, u]^T = [A]*[q_ind, u_ind]^T + [B]r. If A_and_B=True,
(A, B) is returned for forming dx = [A]x + [B]r, where
x = [q_ind, u_ind]^T.
simplify : bool, optional
Determines if returned values are simplified before return.
For large expressions this may be time consuming. Default is False.
Note that the process of solving with A_and_B=True is computationally
intensive if there are many symbolic parameters. For this reason,
it may be more desirable to use the default A_and_B=False,
returning M, A, and B. More values may then be substituted in to these
matrices later on. The state space form can then be found as
A = P.T*M.LUsolve(A), B = P.T*M.LUsolve(B), where
P = Linearizer.perm_mat.
"""
# Compose dict of operating conditions
if isinstance(op_point, dict):
op_point_dict = op_point
elif isinstance(op_point, collections.Iterable):
op_point_dict = {}
for op in op_point:
op_point_dict.update(op)
else:
op_point_dict = {}
# Extract dimension variables
l, m, n, o, s, k = self._dims
# Rename terms to shorten expressions
M_qq = self._M_qq
M_uqc = self._M_uqc
M_uqd = self._M_uqd
M_uuc = self._M_uuc
M_uud = self._M_uud
M_uld = self._M_uld
A_qq = self._A_qq
A_uqc = self._A_uqc
A_uqd = self._A_uqd
A_qu = self._A_qu
A_uuc = self._A_uuc
A_uud = self._A_uud
B_u = self._B_u
C_0 = self._C_0
C_1 = self._C_1
C_2 = self._C_2
# Build up Mass Matrix
# |M_qq 0_nxo 0_nxk|
# M = |M_uqc M_uuc 0_mxk|
# |M_uqd M_uud M_uld|
if o != 0:
col2 = Matrix([zeros(n, o), M_uuc, M_uud])
if k != 0:
col3 = Matrix([zeros(n + m, k), M_uld])
if n != 0:
col1 = Matrix([M_qq, M_uqc, M_uqd])
if o != 0 and k != 0:
M = col1.row_join(col2).row_join(col3)
elif o != 0:
M = col1.row_join(col2)
else:
M = col1
elif k != 0:
M = col2.row_join(col3)
else:
M = col2
M_eq = _subs_keep_derivs(M, op_point_dict)
# Build up state coefficient matrix A
# |(A_qq + A_qu*C_1)*C_0 A_qu*C_2|
# A = |(A_uqc + A_uuc*C_1)*C_0 A_uuc*C_2|
# |(A_uqd + A_uud*C_1)*C_0 A_uud*C_2|
# Col 1 is only defined if n != 0
if n != 0:
r1c1 = A_qq
if o != 0:
r1c1 += (A_qu * C_1)
r1c1 = r1c1 * C_0
if m != 0:
r2c1 = A_uqc
if o != 0:
r2c1 += (A_uuc * C_1)
r2c1 = r2c1 * C_0
else:
r2c1 = Matrix()
if o - m + k != 0:
r3c1 = A_uqd
if o != 0:
r3c1 += (A_uud * C_1)
r3c1 = r3c1 * C_0
else:
r3c1 = Matrix()
col1 = Matrix([r1c1, r2c1, r3c1])
else:
col1 = Matrix()
# Col 2 is only defined if o != 0
if o != 0:
if n != 0:
r1c2 = A_qu * C_2
else:
r1c2 = Matrix()
if m != 0:
r2c2 = A_uuc * C_2
else:
r2c2 = Matrix()
if o - m + k != 0:
r3c2 = A_uud * C_2
else:
r3c2 = Matrix()
col2 = Matrix([r1c2, r2c2, r3c2])
else:
col2 = Matrix()
if col1:
if col2:
Amat = col1.row_join(col2)
else:
Amat = col1
else:
Amat = col2
Amat_eq = _subs_keep_derivs(Amat, op_point_dict)
# Build up the B matrix if there are forcing variables
# |0_(n + m)xs|
# B = |B_u |
if s != 0 and o - m + k != 0:
Bmat = zeros(n + m, s).col_join(B_u)
Bmat_eq = _subs_keep_derivs(Bmat, op_point_dict)
else:
Bmat_eq = Matrix()
# kwarg A_and_B indicates to return A, B for forming the equation
# dx = [A]x + [B]r, where x = [q_indnd, u_indnd]^T,
if A_and_B:
A_cont = self.perm_mat.T * M_eq.LUsolve(Amat_eq)
if Bmat_eq:
B_cont = self.perm_mat.T * M_eq.LUsolve(Bmat_eq)
else:
# Bmat = Matrix([]), so no need to sub
B_cont = Bmat_eq
if simplify:
A_cont.simplify()
B_cont.simplify()
return A_cont, B_cont
# Otherwise return M, A, B for forming the equation
# [M]dx = [A]x + [B]r, where x = [q, u]^T
else:
if simplify:
M_eq.simplify()
Amat_eq.simplify()
Bmat_eq.simplify()
return M_eq, Amat_eq, Bmat_eq
def to_linearizer(self):
"""Returns an instance of the Linearizer class, initiated from the
data in the KanesMethod class. This may be more desirable than using
the linearize class method, as the Linearizer object will allow more
efficient recalculation (i.e. about varying operating points)."""
if (self._fr is None) or (self._frstar is None):
raise ValueError('Need to compute Fr, Fr* first.')
# Get required equation components. The Kane's method class breaks
# these into pieces. Need to reassemble
f_c = self._f_h
if self._f_nh and self._k_nh:
f_v = self._f_nh + self._k_nh*Matrix(self._u)
else:
f_v = Matrix([])
if self._f_dnh and self._k_dnh:
f_a = self._f_dnh + self._k_dnh*Matrix(self._udot)
else:
f_a = Matrix([])
# Dicts to sub to zero, for splitting up expressions
u_zero = dict((i, 0) for i in self._u)
ud_zero = dict((i, 0) for i in self._udot)
qd_zero = dict((i, 0) for i in self._qdot)
qd_u_zero = dict((i, 0) for i in self._qdot + self._u)
# Break the kinematic differential eqs apart into f_0 and f_1
f_0 = self._f_k.subs(u_zero) + self._k_kqdot*Matrix(self._qdot)
f_1 = self._f_k.subs(qd_zero) + self._k_ku*Matrix(self._u)
# Break the dynamic differential eqs into f_2 and f_3
f_2 = _subs_keep_derivs(self._frstar, qd_u_zero)
f_3 = self._frstar.subs(ud_zero) + self._fr
f_4 = zeros(len(f_2), 1)
# Get the required vector components
q = self._q
u = self._u
if self._qdep:
q_i = q[:-len(self._qdep)]
else:
q_i = q
q_d = self._qdep
if self._udep:
u_i = u[:-len(self._udep)]
else:
u_i = u
u_d = self._udep
# Form dictionary of auxiliary speeds & and their derivatives,
# setting each to 0.
uaux = self._uaux
uauxdot = [diff(i, dynamicsymbols._t) for i in uaux]
uaux_zero = dict((i, 0) for i in uaux + uauxdot)
# Checking for dynamic symbols outside the dynamic differential
# equations; throws error if there is.
insyms = set(q + self._qdot + u + self._udot + uaux + uauxdot)
if any(_find_dynamicsymbols(i, insyms) for i in [self._k_kqdot,
self._k_ku, self._f_k, self._k_dnh, self._f_dnh, self._k_d]):
raise ValueError('Cannot have dynamicsymbols outside dynamic \
forcing vector.')
# Find all other dynamic symbols, forming the forcing vector r.
# Sort r to make it canonical.
r = list(_find_dynamicsymbols(self._f_d.subs(uaux_zero), insyms))
r.sort(key=default_sort_key)
# Check for any derivatives of variables in r that are also found in r.
for i in r:
if diff(i, dynamicsymbols._t) in r:
raise ValueError('Cannot have derivatives of specified \
quantities when linearizing forcing terms.')
return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i, q_d,
u_i, u_d, r)
