def simulate_impulse_response(sys, t=None):
"""
Compute the linear model response to an Dirac delta pulse (or all-zeros
array except the first sample being 1/dt at each channel) sampled at given
time instances.
If the time array is omitted then a time sequence is generated based on
the poles of the model.
Parameters
----------
sys : {State, Transfer}
The system model to be simulated
t : array_like
The real-valued sequence to be used for the evolution of the system.
The values should be equally spaced otherwise an error is raised. For
discrete time models increments different than the sampling period also
raises an error. On the other hand for discrete models this can be
omitted and a time sequence will be generated automatically.
Returns
-------
yout : ndarray
The resulting response array. The array is 1D if sys is SISO and
has p columns if sys has p outputs. If there are also m inputs the
array is 3D array with the shape (<num of samples>, p, m)
tout : ndarray
The time sequence used in the simulation. If the parameter t is not
None then a copy of t is given.
"""
_check_for_state_or_transfer(sys)
# Always works with State Models
try:
_check_for_state(sys)
except ValueError:
sys = transfer_to_state(sys)
if t is None:
tf, ts = _compute_tfinal_and_dt(sys, is_step=False)
t = np.arange(0, tf+ts, ts, dtype=float)
else:
t, ts = _check_custom_time_input(t)
m = sys.shape[1]
u = np.zeros([len(t), m], dtype=float)
u[0] = 1./ts
return simulate_linear_system(sys, u=u, t=t, per_channel=1)
def simulate_step_response(sys, t=None):
"""
Compute the linear model response to an Heaviside function (or all-ones
array) sampled at given time instances.
If the time array is omitted then a time sequence is generated based on
the slowest pole of sys based on the straightforward settling time
estimate
ts = 4/(ζω)
and its discrete equivalent. The sampling period is also approximated by
the fastest stable pole. If the system has all nonnegative poles, -1. is
used for the computations as the threshold for the settling times.
Parameters
----------
sys : {State, Transfer}
The system model to be simulated
t : array_like
The real-valued sequence to be used for the evolution of the system.
The values should be equally spaced otherwise an error is raised. For
discrete time models increments different than the sampling period also
raises an error. On the other hand for discrete models this can be
omitted and a time sequence will be generated automatically.
"""
_check_for_state_or_transfer(sys)
if t is None:
tf, ts = _compute_tfinal_and_dt(sys)
# Massage tf such that it is an integer multiple of ts
mult = int(tf // ts)
if sys._isdiscrete:
t = np.linspace(0, sys._dt * (mult - 1), num=mult)
else:
t = np.linspace(0, ts * (mult - 1), num=mult)
m = sys.shape[1]
u = np.ones([len(t), m])
return simulate_linear_system(sys, u=u, t=t, per_channel=1)
def simulate_linear_system(sys, u, t=None, x0=None, per_channel=False):
"""
Compute the linear model response to an input array sampled at given time
instances.
Parameters
----------
sys : {State, Transfer}
The system model to be simulated
u : array_like
The real-valued input sequence to force the model. 1D arrays for single
input models and 2D arrays that has as many columns as the number of
inputs are valid inputs.
t : array_like, optional
The real-valued sequence to be used for the evolution of the system.
The values should be equally spaced otherwise an error is raised. For
discrete time models increments different than the sampling period also
raises an error. On the other hand for discrete models this can be
omitted and a time sequence will be generated automatically.
x0 : array_like, optional
The initial condition array. If omitted an array of zeros is assumed.
Note that Transfer models by definition assume zero initial conditions
and will raise an error.
per_channel : bool, optional
If this is set to True and if the system has multiple inputs, the
response of each input is returned individually. For example, if a
system has 4 inputs and 3 outputs then the response shape becomes
(num, p, m) instead of (num, p) where k-th slice (:, :, k) is the
response from the k-th input channel. For single input systems, this
keyword has no effect.
Returns
-------
yout : ndarray
The resulting response array. The array is 1D if sys is SISO and
has p columns if sys has p outputs.
tout : ndarray
The time sequence used in the simulation. If the parameter t is not
None then a copy of t is given.
Notes
-----
For Transfer models, first conversion to a state model is performed and
then the resulting model is used for computations.
"""
_check_for_state_or_transfer(sys)
# Quick initial condition checks
if x0 is not None:
if sys._isgain:
raise ValueError('Static system models can\'t have initial '
'conditions set.')
if isinstance(sys, Transfer):
raise ValueError('Transfer models can\'t have initial conditions '
'set.')
x0 = np.asarray(x0, dtype=float).squeeze()
if x0.ndim > 1:
raise ValueError('Initial condition can only be a 1D array.')
else:
x0 = x0[:, None]
if sys.NumberOfStates != x0.size:
raise ValueError('The initial condition size does not match the '
'number of states of the model.')
# Always works with State Models
try:
_check_for_state(sys)
except ValueError:
sys = transfer_to_state(sys)
n, m = sys.NumberOfStates, sys.shape[1]
is_discrete = sys.SamplingSet == 'Z'
u = np.asarray(u, dtype=float).squeeze()
if u.ndim == 1:
u = u[:, None]
t = _check_u_and_t_for_simulation(m, sys._dt, u, t, is_discrete)
# input and time arrays are regular move on
# Static gains are simple matrix multiplications with no x0
if sys._isgain:
if sys._isSISO:
yout = u * sys.d.squeeze()
else:
# don't bother for single inputs
if m == 1:
per_channel = False
if per_channel:
yout = np.einsum('ij,jk->ikj', u, sys.d.T)
else:
yout = u @ sys.d.T
# Dynamic model
else:
# TODO: Add FOH discretization for funky input
# ZOH discretize the continuous system based on the time increment
if not is_discrete:
sys = discretize(sys, t[1] - t[0], method='zoh')
sample_num = len(u)
a, b, c, d = sys.matrices
# Bu and Du are constant matrices so get them ready (transposed)
M_u = np.block([b.T, d.T])
at = a.T
# Explicitly skip single inputs for per_channel
if m == 1:
per_channel = False
# Shape the response as a 3D array
if per_channel:
xout = np.empty([sample_num, n, m], dtype=float)
for col in range(m):
xout[0, :, col] = 0. if x0 is None else x0.T
Bu = u[:, [col]] @ b.T[[col], :]
# Main loop for xdot eq.
for row in range(1, sample_num):
xout[row, :,
col] = xout[row - 1, :, col] @ at + Bu[row - 1]
# Get the output equation for each slice of inputs
# Cx + Du
yout = np.einsum('ijk,jl->ilk', xout, c.T) + \
np.einsum('ij,jk->ikj', u, d.T)
# Combined output
else:
BDu = u @ M_u
xout = np.empty([sample_num, n], dtype=float)
xout[0] = 0. if x0 is None else x0.T
# Main loop for xdot eq.
for row in range(1, sample_num):
xout[row] = (xout[row - 1] @ at) + BDu[row - 1, :n]
# Now we have all the state evolution get the output equation
yout = xout @ c.T + BDu[:, n:]
return yout, t
def simulate_linear_system(sys, u, t=None, x0=None, per_channel=False):
"""
Compute the linear model response to an input array sampled at given time
instances.
Parameters
----------
sys : {State, Transfer}
The system model to be simulated
u : array_like
The real-valued input sequence to force the model. 1D arrays for single
input models and 2D arrays that has as many columns as the number of
inputs are valid inputs.
t : array_like, optional
The real-valued sequence to be used for the evolution of the system.
The values should be equally spaced otherwise an error is raised. For
discrete time models increments different than the sampling period also
raises an error. On the other hand for discrete models this can be
omitted and a time sequence will be generated automatically.
x0 : array_like, optional
The initial condition array. If omitted an array of zeros is assumed.
Note that Transfer models by definition assume zero initial conditions
and will raise an error.
per_channel : bool, optional
If this is set to True and if the system has multiple inputs, the
response of each input is returned individually. For example, if a
system has 4 inputs and 3 outputs then the response shape becomes
(num, p, m) instead of (num, p) where k-th slice (:, :, k) is the
response from the k-th input channel. For single input systems, this
keyword has no effect.
Returns
-------
yout : ndarray
The resulting response array. The array is 1D if sys is SISO and
has p columns if sys has p outputs.
tout : ndarray
The time sequence used in the simulation. If the parameter t is not
None then a copy of t is given.
Notes
-----
For Transfer models, first conversion to a state model is performed and
then the resulting model is used for computations.
"""
_check_for_state_or_transfer(sys)
# Quick initial condition checks
if x0 is not None:
if sys._isgain:
raise ValueError('Static system models can\'t have initial '
'conditions set.')
if isinstance(sys, Transfer):
raise ValueError('Transfer models can\'t have initial conditions '
'set.')
x0 = np.asarray(x0, dtype=float).squeeze()
if x0.ndim > 1:
raise ValueError('Initial condition can only be a 1D array.')
else:
x0 = x0[:, None]
if sys.NumberOfStates != x0.size:
raise ValueError('The initial condition size does not match the '
'number of states of the model.')
# Always works with State Models
try:
_check_for_state(sys)
except ValueError:
sys = transfer_to_state(sys)
n, m = sys.NumberOfStates, sys.shape[1]
is_discrete = sys.SamplingSet == 'Z'
u = np.asarray(u, dtype=float).squeeze()
if u.ndim == 1:
u = u[:, None]
t = _check_u_and_t_for_simulation(m, sys._dt, u, t, is_discrete)
# input and time arrays are regular move on
# Static gains are simple matrix multiplications with no x0
if sys._isgain:
if sys._isSISO:
yout = u * sys.d.squeeze()
else:
# don't bother for single inputs
if m == 1:
per_channel = False
if per_channel:
yout = np.einsum('ij,jk->ikj', u, sys.d.T)
else:
yout = u @ sys.d.T
# Dynamic model
else:
# TODO: Add FOH discretization for funky input
# ZOH discretize the continuous system based on the time increment
if not is_discrete:
sys = discretize(sys, t[1]-t[0], method='zoh')
sample_num = len(u)
a, b, c, d = sys.matrices
# Bu and Du are constant matrices so get them ready (transposed)
M_u = np.block([b.T, d.T])
at = a.T
# Explicitly skip single inputs for per_channel
if m == 1:
per_channel = False
# Shape the response as a 3D array
if per_channel:
xout = np.empty([sample_num, n, m], dtype=float)
for col in range(m):
xout[0, :, col] = 0. if x0 is None else x0.T
Bu = u[:, [col]] @ b.T[[col], :]
# Main loop for xdot eq.
for row in range(1, sample_num):
xout[row, :, col] = xout[row-1, :, col] @ at + Bu[row-1]
# Get the output equation for each slice of inputs
# Cx + Du
yout = np.einsum('ijk,jl->ilk', xout, c.T) + \
np.einsum('ij,jk->ikj', u, d.T)
# Combined output
else:
BDu = u @ M_u
xout = np.empty([sample_num, n], dtype=float)
xout[0] = 0. if x0 is None else x0.T
# Main loop for xdot eq.
for row in range(1, sample_num):
xout[row] = (xout[row-1] @ at) + BDu[row-1, :n]
# Now we have all the state evolution get the output equation
yout = xout @ c.T + BDu[:, n:]
return yout, t
