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"""
ltisys -- a collection of classes and functions for modeling linear
time invariant systems.
"""
#
# Author: Travis Oliphant 2001
#
# Feb 2010: Warren Weckesser
# Rewrote lsim2 and added impulse2.
#
from filter_design import tf2zpk, zpk2tf, normalize
import numpy
from numpy import product, zeros, array, dot, transpose, ones, \
nan_to_num, zeros_like, linspace
import scipy.interpolate as interpolate
import scipy.integrate as integrate
import scipy.linalg as linalg
from numpy import r_, eye, real, atleast_1d, atleast_2d, poly, \
squeeze, diag, asarray
__all__ = ['tf2ss', 'ss2tf', 'abcd_normalize', 'zpk2ss', 'ss2zpk', 'lti',
'lsim', 'lsim2', 'impulse', 'impulse2', 'step', 'step2', 'bode']
def tf2ss(num, den):
"""Transfer function to state-space representation.
Parameters
----------
num, den : array_like
Sequences representing the numerator and denominator polynomials.
The denominator needs to be at least as long as the numerator.
Returns
-------
A, B, C, D : ndarray
State space representation of the system.
"""
# Controller canonical state-space representation.
# if M+1 = len(num) and K+1 = len(den) then we must have M <= K
# states are found by asserting that X(s) = U(s) / D(s)
# then Y(s) = N(s) * X(s)
#
# A, B, C, and D follow quite naturally.
#
num, den = normalize(num, den) # Strips zeros, checks arrays
nn = len(num.shape)
if nn == 1:
num = asarray([num], num.dtype)
M = num.shape[1]
K = len(den)
if M > K:
msg = "Improper transfer function. `num` is longer than `den`."
raise ValueError(msg)
if M == 0 or K == 0: # Null system
return array([], float), array([], float), array([], float), \
array([], float)
# pad numerator to have same number of columns has denominator
num = r_['-1', zeros((num.shape[0], K - M), num.dtype), num]
if num.shape[-1] > 0:
D = num[:, 0]
else:
D = array([], float)
if K == 1:
return array([], float), array([], float), array([], float), D
frow = -array([den[1:]])
A = r_[frow, eye(K - 2, K - 1)]
B = eye(K - 1, 1)
C = num[:, 1:] - num[:, 0] * den[1:]
return A, B, C, D
def _none_to_empty(arg):
if arg is None:
return []
else:
return arg
def abcd_normalize(A=None, B=None, C=None, D=None):
"""Check state-space matrices and ensure they are rank-2.
"""
A, B, C, D = map(_none_to_empty, (A, B, C, D))
A, B, C, D = map(atleast_2d, (A, B, C, D))
if ((len(A.shape) > 2) or (len(B.shape) > 2) or \
(len(C.shape) > 2) or (len(D.shape) > 2)):
raise ValueError("A, B, C, D arrays can be no larger than rank-2.")
MA, NA = A.shape
MB, NB = B.shape
MC, NC = C.shape
MD, ND = D.shape
if (MC == 0) and (NC == 0) and (MD != 0) and (NA != 0):
MC, NC = MD, NA
C = zeros((MC, NC))
if (MB == 0) and (NB == 0) and (MA != 0) and (ND != 0):
MB, NB = MA, ND
B = zeros(MB, NB)
if (MD == 0) and (ND == 0) and (MC != 0) and (NB != 0):
MD, ND = MC, NB
D = zeros(MD, ND)
if (MA == 0) and (NA == 0) and (MB != 0) and (NC != 0):
MA, NA = MB, NC
A = zeros(MA, NA)
if MA != NA:
raise ValueError("A must be square.")
if MA != MB:
raise ValueError("A and B must have the same number of rows.")
if NA != NC:
raise ValueError("A and C must have the same number of columns.")
if MD != MC:
raise ValueError("C and D must have the same number of rows.")
if ND != NB:
raise ValueError("B and D must have the same number of columns.")
return A, B, C, D
def ss2tf(A, B, C, D, input=0):
"""State-space to transfer function.
Parameters
----------
A, B, C, D : ndarray
State-space representation of linear system.
input : int, optional
For multiple-input systems, the input to use.
Returns
-------
num, den : 1D ndarray
Numerator and denominator polynomials (as sequences)
respectively.
"""
# transfer function is C (sI - A)**(-1) B + D
A, B, C, D = map(asarray, (A, B, C, D))
# Check consistency and
# make them all rank-2 arrays
A, B, C, D = abcd_normalize(A, B, C, D)
nout, nin = D.shape
if input >= nin:
raise ValueError("System does not have the input specified.")
# make MOSI from possibly MOMI system.
if B.shape[-1] != 0:
B = B[:, input]
B.shape = (B.shape[0], 1)
if D.shape[-1] != 0:
D = D[:, input]
try:
den = poly(A)
except ValueError:
den = 1
if (product(B.shape, axis=0) == 0) and (product(C.shape, axis=0) == 0):
num = numpy.ravel(D)
if (product(D.shape, axis=0) == 0) and (product(A.shape, axis=0) == 0):
den = []
return num, den
num_states = A.shape[0]
type_test = A[:, 0] + B[:, 0] + C[0, :] + D
num = numpy.zeros((nout, num_states + 1), type_test.dtype)
for k in range(nout):
Ck = atleast_2d(C[k, :])
num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den
return num, den
def zpk2ss(z, p, k):
"""Zero-pole-gain representation to state-space representation
Parameters
----------
z, p : sequence
Zeros and poles.
k : float
System gain.
Returns
-------
A, B, C, D : ndarray
State-space matrices.
"""
return tf2ss(*zpk2tf(z, p, k))
def ss2zpk(A, B, C, D, input=0):
"""State-space representation to zero-pole-gain representation.
Parameters
----------
A, B, C, D : ndarray
State-space representation of linear system.
input : int, optional
For multiple-input systems, the input to use.
Returns
-------
z, p : sequence
Zeros and poles.
k : float
System gain.
"""
return tf2zpk(*ss2tf(A, B, C, D, input=input))
class lti(object):
"""Linear Time Invariant class which simplifies representation.
Parameters
----------
args : arguments
The `lti` class can be instantiated with either 2, 3 or 4 arguments.
The following gives the number of elements in the tuple and the
interpretation:
* 2: (numerator, denominator)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)
Each argument can be an array or sequence.
Notes
-----
`lti` instances have all types of representations available; for example
after creating an instance s with ``(zeros, poles, gain)`` the state-space
representation (numerator, denominator) can be accessed as ``s.num`` and
``s.den``.
"""
def __init__(self, *args, **kwords):
"""
Initialize the LTI system using either:
- (numerator, denominator)
- (zeros, poles, gain)
- (A, B, C, D) : state-space.
"""
N = len(args)
if N == 2: # Numerator denominator transfer function input
self.__dict__['num'], self.__dict__['den'] = normalize(*args)
self.__dict__['zeros'], self.__dict__['poles'], \
self.__dict__['gain'] = tf2zpk(*args)
self.__dict__['A'], self.__dict__['B'], \
self.__dict__['C'], \
self.__dict__['D'] = tf2ss(*args)
self.inputs = 1
if len(self.num.shape) > 1:
self.outputs = self.num.shape[0]
else:
self.outputs = 1
elif N == 3: # Zero-pole-gain form
self.__dict__['zeros'], self.__dict__['poles'], \
self.__dict__['gain'] = args
self.__dict__['num'], self.__dict__['den'] = zpk2tf(*args)
self.__dict__['A'], self.__dict__['B'], \
self.__dict__['C'], \
self.__dict__['D'] = zpk2ss(*args)
# make sure we have numpy arrays
self.zeros = numpy.asarray(self.zeros)
self.poles = numpy.asarray(self.poles)
self.inputs = 1
if len(self.zeros.shape) > 1:
self.outputs = self.zeros.shape[0]
else:
self.outputs = 1
elif N == 4: # State-space form
self.__dict__['A'], self.__dict__['B'], \
self.__dict__['C'], \
self.__dict__['D'] = abcd_normalize(*args)
self.__dict__['zeros'], self.__dict__['poles'], \
self.__dict__['gain'] = ss2zpk(*args)
self.__dict__['num'], self.__dict__['den'] = ss2tf(*args)
self.inputs = self.B.shape[-1]
self.outputs = self.C.shape[0]
else:
raise ValueError("Needs 2, 3, or 4 arguments.")
def __setattr__(self, attr, val):
if attr in ['num', 'den']:
self.__dict__[attr] = val
self.__dict__['zeros'], self.__dict__['poles'], \
self.__dict__['gain'] = \
tf2zpk(self.num, self.den)
self.__dict__['A'], self.__dict__['B'], \
self.__dict__['C'], \
self.__dict__['D'] = \
tf2ss(self.num, self.den)
elif attr in ['zeros', 'poles', 'gain']:
self.__dict__[attr] = val
self.__dict__['num'], self.__dict__['den'] = \
zpk2tf(self.zeros,
self.poles, self.gain)
self.__dict__['A'], self.__dict__['B'], \
self.__dict__['C'], \
self.__dict__['D'] = \
zpk2ss(self.zeros,
self.poles, self.gain)
elif attr in ['A', 'B', 'C', 'D']:
self.__dict__[attr] = val
self.__dict__['zeros'], self.__dict__['poles'], \
self.__dict__['gain'] = \
ss2zpk(self.A, self.B,
self.C, self.D)
self.__dict__['num'], self.__dict__['den'] = \
ss2tf(self.A, self.B,
self.C, self.D)
else:
self.__dict__[attr] = val
def impulse(self, X0=None, T=None, N=None):
return impulse(self, X0=X0, T=T, N=N)
def step(self, X0=None, T=None, N=None):
return step(self, X0=X0, T=T, N=N)
def output(self, U, T, X0=None):
return lsim(self, U, T, X0=X0)
def bode(self, w=None, n=100):
"""Calculate bode magnitude and phase data.
Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude
[dB] and phase [deg]. See scipy.signal.bode for details.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> s1 = signal.lti([1], [1, 1])
>>> w, mag, phase = s1.bode()
>>> plt.figure()
>>> plt.semilogx(w, mag) # bode magnitude plot
>>> plt.figure()
>>> plt.semilogx(w, phase) # bode phase plot
>>> plt.show()
"""
return bode(self, w=w, n=n)
def lsim2(system, U=None, T=None, X0=None, **kwargs):
"""
Simulate output of a continuous-time linear system, by using
the ODE solver `scipy.integrate.odeint`.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2: (num, den)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)
U : array_like (1D or 2D), optional
An input array describing the input at each time T. Linear
interpolation is used between given times. If there are
multiple inputs, then each column of the rank-2 array
represents an input. If U is not given, the input is assumed
to be zero.
T : array_like (1D or 2D), optional
The time steps at which the input is defined and at which the
output is desired. The default is 101 evenly spaced points on
the interval [0,10.0].
X0 : array_like (1D), optional
The initial condition of the state vector. If `X0` is not
given, the initial conditions are assumed to be 0.
kwargs : dict
Additional keyword arguments are passed on to the function
odeint. See the notes below for more details.
Returns
-------
T : 1D ndarray
The time values for the output.
yout : ndarray
The response of the system.
xout : ndarray
The time-evolution of the state-vector.
Notes
-----
This function uses `scipy.integrate.odeint` to solve the
system's differential equations. Additional keyword arguments
given to `lsim2` are passed on to `odeint`. See the documentation
for `scipy.integrate.odeint` for the full list of arguments.
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
if X0 is None:
X0 = zeros(sys.B.shape[0], sys.A.dtype)
if T is None:
# XXX T should really be a required argument, but U was
# changed from a required positional argument to a keyword,
# and T is after U in the argument list. So we either: change
# the API and move T in front of U; check here for T being
# None and raise an excpetion; or assign a default value to T
# here. This code implements the latter.
T = linspace(0, 10.0, 101)
T = atleast_1d(T)
if len(T.shape) != 1:
raise ValueError("T must be a rank-1 array.")
if U is not None:
U = atleast_1d(U)
if len(U.shape) == 1:
U = U.reshape(-1, 1)
sU = U.shape
if sU[0] != len(T):
raise ValueError("U must have the same number of rows "
"as elements in T.")
if sU[1] != sys.inputs:
raise ValueError("The number of inputs in U (%d) is not "
"compatible with the number of system "
"inputs (%d)" % (sU[1], sys.inputs))
# Create a callable that uses linear interpolation to
# calculate the input at any time.
ufunc = interpolate.interp1d(T, U, kind='linear',
axis=0, bounds_error=False)
def fprime(x, t, sys, ufunc):
"""The vector field of the linear system."""
return dot(sys.A, x) + squeeze(dot(sys.B, nan_to_num(ufunc([t]))))
xout = integrate.odeint(fprime, X0, T, args=(sys, ufunc), **kwargs)
yout = dot(sys.C, transpose(xout)) + dot(sys.D, transpose(U))
else:
def fprime(x, t, sys):
"""The vector field of the linear system."""
return dot(sys.A, x)
xout = integrate.odeint(fprime, X0, T, args=(sys,), **kwargs)
yout = dot(sys.C, transpose(xout))
return T, squeeze(transpose(yout)), xout
def lsim(system, U, T, X0=None, interp=1):
"""
Simulate output of a continuous-time linear system.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2: (num, den)
* 3: (zeros, poles, gain)
* 4: (A, B, C, D)
U : array_like
An input array describing the input at each time `T`
(interpolation is assumed between given times). If there are
multiple inputs, then each column of the rank-2 array
represents an input.
T : array_like
The time steps at which the input is defined and at which the
output is desired.
X0 :
The initial conditions on the state vector (zero by default).
interp : {1, 0}
Whether to use linear (1) or zero-order hold (0) interpolation.
Returns
-------
T : 1D ndarray
Time values for the output.
yout : 1D ndarray
System response.
xout : ndarray
Time-evolution of the state-vector.
"""
# system is an lti system or a sequence
# with 2 (num, den)
# 3 (zeros, poles, gain)
# 4 (A, B, C, D)
# describing the system
# U is an input vector at times T
# if system describes multiple inputs
# then U can be a rank-2 array with the number of columns
# being the number of inputs
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
U = atleast_1d(U)
T = atleast_1d(T)
if len(U.shape) == 1:
U = U.reshape((U.shape[0], 1))
sU = U.shape
if len(T.shape) != 1:
raise ValueError("T must be a rank-1 array.")
if sU[0] != len(T):
raise ValueError("U must have the same number of rows "
"as elements in T.")
if sU[1] != sys.inputs:
raise ValueError("System does not define that many inputs.")
if X0 is None:
X0 = zeros(sys.B.shape[0], sys.A.dtype)
xout = zeros((len(T), sys.B.shape[0]), sys.A.dtype)
xout[0] = X0
A = sys.A
AT, BT = transpose(sys.A), transpose(sys.B)
dt = T[1] - T[0]
lam, v = linalg.eig(A)
vt = transpose(v)
vti = linalg.inv(vt)
GT = dot(dot(vti, diag(numpy.exp(dt * lam))), vt).astype(xout.dtype)
ATm1 = linalg.inv(AT)
ATm2 = dot(ATm1, ATm1)
I = eye(A.shape[0], dtype=A.dtype)
GTmI = GT - I
F1T = dot(dot(BT, GTmI), ATm1)
if interp:
F2T = dot(BT, dot(GTmI, ATm2) / dt - ATm1)
for k in xrange(1, len(T)):
dt1 = T[k] - T[k - 1]
if dt1 != dt:
dt = dt1
GT = dot(dot(vti, diag(numpy.exp(dt * lam))),
vt).astype(xout.dtype)
GTmI = GT - I
F1T = dot(dot(BT, GTmI), ATm1)
if interp:
F2T = dot(BT, dot(GTmI, ATm2) / dt - ATm1)
xout[k] = dot(xout[k - 1], GT) + dot(U[k - 1], F1T)
if interp:
xout[k] = xout[k] + dot((U[k] - U[k - 1]), F2T)
yout = (squeeze(dot(U, transpose(sys.D))) +
squeeze(dot(xout, transpose(sys.C))))
return T, squeeze(yout), squeeze(xout)
def _default_response_times(A, n):
"""Compute a reasonable set of time samples for the response time.
This function is used by `impulse`, `impulse2`, `step` and `step2`
to compute the response time when the `T` argument to the function
is None.
Parameters
----------
A : ndarray
The system matrix, which is square.
n : int
The number of time samples to generate.
Returns
-------
t : ndarray
The 1-D array of length `n` of time samples at which the response
is to be computed.
"""
# Create a reasonable time interval. This could use some more work.
# For example, what is expected when the system is unstable?
vals = linalg.eigvals(A)
r = min(abs(real(vals)))
if r == 0.0:
r = 1.0
tc = 1.0 / r
t = linspace(0.0, 7 * tc, n)
return t
def _default_response_frequencies(A, n):
"""Compute a reasonable set of frequency points for bode plot.
This function is used by `bode` to compute the frequency points (in rad/s)
when the `w` argument to the function is None.
Parameters
----------
A : ndarray
The system matrix, which is square.
n : int
The number of time samples to generate.
Returns
-------
w : ndarray
The 1-D array of length `n` of frequency samples (in rad/s) at which
the response is to be computed.
"""
vals = linalg.eigvals(A)
# Remove poles at 0 because they don't help us determine an interesting
# frequency range. (And if we pass a 0 to log10() below we will crash.)
poles = [pole for pole in vals if pole != 0]
# If there are no non-zero poles, just hardcode something.
if len(poles) == 0:
minpole = 1
maxpole = 1
else:
minpole = min(abs(real(poles)))
maxpole = max(abs(real(poles)))
# A reasonable frequency range is two orders of magnitude before the
# minimum pole (slowest) and two orders of magnitude after the maximum pole
# (fastest).
w = numpy.logspace(numpy.log10(minpole) - 2, numpy.log10(maxpole) + 2, n)
return w
def impulse(system, X0=None, T=None, N=None):
"""Impulse response of continuous-time system.
Parameters
----------
system : LTI class or tuple
If specified as a tuple, the system is described as
``(num, den)``, ``(zero, pole, gain)``, or ``(A, B, C, D)``.
X0 : array_like, optional
Initial state-vector. Defaults to zero.
T : array_like, optional
Time points. Computed if not given.
N : int, optional
The number of time points to compute (if `T` is not given).
Returns
-------
T : ndarray
A 1-D array of time points.
yout : ndarray
A 1-D array containing the impulse response of the system (except for
singularities at zero).
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
if X0 is None:
B = sys.B
else:
B = sys.B + X0
if N is None:
N = 100
if T is None:
T = _default_response_times(sys.A, N)
h = zeros(T.shape, sys.A.dtype)
s, v = linalg.eig(sys.A)
vi = linalg.inv(v)
C = sys.C
for k in range(len(h)):
es = diag(numpy.exp(s * T[k]))
eA = (dot(dot(v, es), vi)).astype(h.dtype)
h[k] = squeeze(dot(dot(C, eA), B))
return T, h
def impulse2(system, X0=None, T=None, N=None, **kwargs):
"""
Impulse response of a single-input, continuous-time linear system.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)
T : 1-D array_like, optional
The time steps at which the input is defined and at which the
output is desired. If `T` is not given, the function will
generate a set of time samples automatically.
X0 : 1-D array_like, optional
The initial condition of the state vector. Default: 0 (the
zero vector).
N : int, optional
Number of time points to compute. Default: 100.
kwargs : various types
Additional keyword arguments are passed on to the function
`scipy.signal.lsim2`, which in turn passes them on to
`scipy.integrate.odeint`; see the latter's documentation for
information about these arguments.
Returns
-------
T : ndarray
The time values for the output.
yout : ndarray
The output response of the system.
See Also
--------
impulse, lsim2, integrate.odeint
Notes
-----
The solution is generated by calling `scipy.signal.lsim2`, which uses
the differential equation solver `scipy.integrate.odeint`.
.. versionadded:: 0.8.0
Examples
--------
Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = u(t)
>>> from scipy import signal
>>> system = ([1.0], [1.0, 2.0, 1.0])
>>> t, y = signal.impulse2(system)
>>> import matplotlib.pyplot as plt
>>> plt.plot(t, y)
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
B = sys.B
if B.shape[-1] != 1:
raise ValueError("impulse2() requires a single-input system.")
B = B.squeeze()
if X0 is None:
X0 = zeros_like(B)
if N is None:
N = 100
if T is None:
T = _default_response_times(sys.A, N)
# Move the impulse in the input to the initial conditions, and then
# solve using lsim2().
U = zeros_like(T)
ic = B + X0
Tr, Yr, Xr = lsim2(sys, U, T, ic, **kwargs)
return Tr, Yr
def step(system, X0=None, T=None, N=None):
"""Step response of continuous-time system.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)
X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
Number of time points to compute if `T` is not given.
Returns
-------
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.
See also
--------
scipy.signal.step2
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
if N is None:
N = 100
if T is None:
T = _default_response_times(sys.A, N)
U = ones(T.shape, sys.A.dtype)
vals = lsim(sys, U, T, X0=X0)
return vals[0], vals[1]
def step2(system, X0=None, T=None, N=None, **kwargs):
"""Step response of continuous-time system.
This function is functionally the same as `scipy.signal.step`, but
it uses the function `scipy.signal.lsim2` to compute the step
response.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)
X0 : array_like, optional
Initial state-vector (default is zero).
T : array_like, optional
Time points (computed if not given).
N : int
Number of time points to compute if `T` is not given.
kwargs :
Additional keyword arguments are passed on the function
`scipy.signal.lsim2`, which in turn passes them on to
`scipy.integrate.odeint`. See the documentation for
`scipy.integrate.odeint` for information about these arguments.
Returns
-------
T : 1D ndarray
Output time points.
yout : 1D ndarray
Step response of system.
See also
--------
scipy.signal.step
Notes
-----
.. versionadded:: 0.8.0
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
if N is None:
N = 100
if T is None:
T = _default_response_times(sys.A, N)
U = ones(T.shape, sys.A.dtype)
vals = lsim2(sys, U, T, X0=X0, **kwargs)
return vals[0], vals[1]
def bode(system, w=None, n=100):
"""Calculate bode magnitude and phase data of a continuous-time system.
Parameters
----------
system : an instance of the LTI class or a tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 2 (num, den)
* 3 (zeros, poles, gain)
* 4 (A, B, C, D)
w : array_like, optional
Array of frequencies (in rad/s). Magnitude and phase data is calculated
for every value in this array. If not given a reasonable set will be
calculated.
n : int, optional
Number of frequency points to compute if `w` is not given. The `n`
frequencies are logarithmically spaced in the range from two orders of
magnitude before the minimum (slowest) pole to two orders of magnitude
after the maximum (fastest) pole.
Returns
-------
w : 1D ndarray
Frequency array [rad/s]
mag : 1D ndarray
Magnitude array [dB]
phase : 1D ndarray
Phase array [deg]
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> s1 = signal.lti([1], [1, 1])
>>> w, mag, phase = signal.bode(s1)
>>> plt.figure()
>>> plt.semilogx(w, mag) # bode magnitude plot
>>> plt.figure()
>>> plt.semilogx(w, phase) # bode phase plot
>>> plt.show()
"""
if isinstance(system, lti):
sys = system
else:
sys = lti(*system)
if w is None:
w = _default_response_frequencies(sys.A, n)
else:
w = numpy.asarray(w)
jw = w * 1j
y = numpy.polyval(sys.num, jw) / numpy.polyval(sys.den, jw)
mag = 20.0 * numpy.log10(abs(y))
phase = numpy.arctan2(y.imag, y.real) * 180.0 / numpy.pi
return w, mag, phase
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