def ab08nd_example():
from numpy import zeros, size
from scipy.linalg import eigvals
A = array([[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0],
[0, 0, 0, -4, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 3]])
B = array([[0, -1], [-1, 0], [1, -1], [0, 0], [0, 1], [-1, -1]])
C = array([[1, 0, 0, 1, 0, 0], [0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 0, 1]])
D = zeros((3, 2))
out = slycot.ab08nd(6, 2, 3, A, B, C, D)
nu = out[0]
print('--- Example for ab08nd ---')
print('The finite invariant zeros are')
print(eigvals(out[8][0:nu, 0:nu], out[9][0:nu, 0:nu]))
def zero(self):
"""Compute the zeros of a state space system."""
if not self.states:
return np.array([])
# Use AB08ND from Slycot if it's available, otherwise use
# scipy.lingalg.eigvals().
try:
from slycot import ab08nd
out = ab08nd(self.A.shape[0], self.B.shape[1], self.C.shape[0],
self.A, self.B, self.C, self.D)
nu = out[0]
if nu == 0:
return np.array([])
else:
return sp.linalg.eigvals(out[8][0:nu, 0:nu], out[9][0:nu,
0:nu])
except ImportError: # Slycot unavailable. Fall back to scipy.
if self.C.shape[0] != self.D.shape[1]:
raise NotImplementedError("StateSpace.zero only supports "
"systems with the same number of "
"inputs as outputs.")
# This implements the QZ algorithm for finding transmission zeros
# from
# https://dspace.mit.edu/bitstream/handle/1721.1/841/P-0802-06587335.pdf.
# The QZ algorithm solves the generalized eigenvalue problem: given
# `L = [A, B; C, D]` and `M = [I_nxn 0]`, find all finite lambda
# for which there exist nontrivial solutions of the equation
# `Lz - lamba Mz`.
#
# The generalized eigenvalue problem is only solvable if its
# arguments are square matrices.
L = concatenate((concatenate(
(self.A, self.B), axis=1), concatenate(
(self.C, self.D), axis=1)),
axis=0)
M = pad(eye(self.A.shape[0]),
((0, self.C.shape[0]), (0, self.B.shape[1])), "constant")
return np.array([
x for x in sp.linalg.eigvals(L, M, overwrite_a=True)
if not isinf(x)
])
def zero(self):
"""Compute the zeros of a state space system."""
if not self.states:
return np.array([])
# Use AB08ND from Slycot if it's available, otherwise use
# scipy.lingalg.eigvals().
try:
from slycot import ab08nd
out = ab08nd(self.A.shape[0], self.B.shape[1], self.C.shape[0],
self.A, self.B, self.C, self.D)
nu = out[0]
if nu == 0:
return np.array([])
else:
return sp.linalg.eigvals(out[8][0:nu, 0:nu], out[9][0:nu, 0:nu])
except ImportError: # Slycot unavailable. Fall back to scipy.
if self.C.shape[0] != self.D.shape[1]:
raise NotImplementedError("StateSpace.zero only supports "
"systems with the same number of "
"inputs as outputs.")
# This implements the QZ algorithm for finding transmission zeros
# from
# https://dspace.mit.edu/bitstream/handle/1721.1/841/P-0802-06587335.pdf.
# The QZ algorithm solves the generalized eigenvalue problem: given
# `L = [A, B; C, D]` and `M = [I_nxn 0]`, find all finite lambda
# for which there exist nontrivial solutions of the equation
# `Lz - lamba Mz`.
#
# The generalized eigenvalue problem is only solvable if its
# arguments are square matrices.
L = concatenate((concatenate((self.A, self.B), axis=1),
concatenate((self.C, self.D), axis=1)), axis=0)
M = pad(eye(self.A.shape[0]), ((0, self.C.shape[0]),
(0, self.B.shape[1])), "constant")
return np.array([x for x in sp.linalg.eigvals(L, M, overwrite_a=True)
if not isinf(x)])
def ab08nd_example():
from numpy import zeros, size
from scipy.linalg import eigvals
A = array([ [1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 3, 0, 0, 0],
[0, 0, 0,-4, 0, 0],
[0, 0, 0, 0,-1, 0],
[0, 0, 0, 0, 0, 3]])
B = array([ [0,-1],
[-1,0],
[1,-1],
[0, 0],
[0, 1],
[-1,-1]])
C = array([ [1, 0, 0, 1, 0, 0],
[0, 1, 0, 1, 0, 1],
[0, 0, 1, 0, 0, 1]])
D = zeros((3,2))
out = slycot.ab08nd(6,2,3,A,B,C,D)
nu = out[0]
print('--- Example for ab08nd ---')
print('The finite invariant zeros are')
print(eigvals(out[8][0:nu,0:nu],out[9][0:nu,0:nu]))
