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dltisys.py
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dltisys.py
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"""
dltisys - Code related to discrete linear time-invariant systems
"""
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# April 4, 2011
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.interpolate import interp1d
from .ltisys import tf2ss, zpk2ss
__all__ = ['dlsim', 'dstep', 'dimpulse']
def _system_to_statespace(system):
"""
Return a discrete state-space system from a 3, 4, or 5-tuple input
"""
if len(system) == 3:
A, B, C, D = tf2ss(system[0], system[1])
dt = system[2]
elif len(system) == 4:
A, B, C, D = zpk2ss(system[0], system[1], system[2])
dt = system[3]
elif len(system) == 5:
A, B, C, D, dt = system
else:
raise ValueError("System argument should be a discrete transfer " +
"function, zeros-poles-gain specification, or " +
"state-space system")
return A, B, C, D, dt
def dlsim(system, u, t=None, x0=None):
"""
Simulate output of a discrete-time linear system.
Parameters
----------
system : tuple of array_like
A tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
- 3: (num, den, dt)
- 4: (zeros, poles, gain, dt)
- 5: (A, B, C, D, dt)
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, optional
The time steps at which the input is defined. If `t` is given, it
must be the same length as `u`, and the final value in `t` determines
the number of steps returned in the output.
x0 : array_like, optional
The initial conditions on the state vector (zero by default).
Returns
-------
tout : ndarray
Time values for the output, as a 1-D array.
yout : ndarray
System response, as a 1-D array.
xout : ndarray, optional
Time-evolution of the state-vector. Only generated if the input is a
state-space systems.
See Also
--------
lsim, dstep, dimpulse, cont2discrete
Examples
--------
A simple integrator transfer function with a discrete time step of 1.0
could be implemented as:
>>> from scipy import signal
>>> tf = ([1.0,], [1.0, -1.0], 1.0)
>>> t_in = [0.0, 1.0, 2.0, 3.0]
>>> u = np.asarray([0.0, 0.0, 1.0, 1.0])
>>> t_out, y = signal.dlsim(tf, u, t=t_in)
>>> y
array([ 0., 0., 0., 1.])
"""
A, B, C, D, dt = _system_to_statespace(system)
u = np.asarray(u)
if u.ndim == 1:
u = np.atleast_2d(u).T
if t is None:
out_samples = len(u)
stoptime = (out_samples - 1) * dt
else:
stoptime = t[-1]
out_samples = int(np.floor(stoptime / dt)) + 1
# Pre-build output arrays
xout = np.zeros((out_samples, A.shape[0]))
yout = np.zeros((out_samples, C.shape[0]))
tout = np.linspace(0.0, stoptime, num=out_samples)
# Check initial condition
if x0 is None:
xout[0, :] = np.zeros((A.shape[1],))
else:
xout[0, :] = np.asarray(x0)
# Pre-interpolate inputs into the desired time steps
if t is None:
u_dt = u
else:
if len(u.shape) == 1:
u = u[:, np.newaxis]
u_dt_interp = interp1d(t, u.transpose(), copy=False, bounds_error=True)
u_dt = u_dt_interp(tout).transpose()
# Simulate the system
for i in range(0, out_samples - 1):
xout[i+1, :] = np.dot(A, xout[i, :]) + np.dot(B, u_dt[i, :])
yout[i, :] = np.dot(C, xout[i, :]) + np.dot(D, u_dt[i, :])
# Last point
yout[out_samples-1, :] = (np.dot(C, xout[out_samples-1, :]) +
np.dot(D, u_dt[out_samples-1, :]))
if len(system) == 5:
return tout, yout, xout
else:
return tout, yout
def dimpulse(system, x0=None, t=None, n=None):
"""Impulse response of discrete-time system.
Parameters
----------
system : tuple of array_like
A tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 3: (num, den, dt)
* 4: (zeros, poles, gain, dt)
* 5: (A, B, C, D, dt)
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
-------
tout : ndarray
Time values for the output, as a 1-D array.
yout : ndarray
Impulse response of system. Each element of the tuple represents
the output of the system based on an impulse in each input.
See Also
--------
impulse, dstep, dlsim, cont2discrete
"""
# Determine the system type and set number of inputs and time steps
A, B, C, D, dt = _system_to_statespace(system)
n_inputs = B.shape[1]
# Default to 100 samples if unspecified
if n is None:
n = 100
# If time is not specified, use the number of samples
# and system dt
if t is None:
t = np.linspace(0, n * dt, n, endpoint=False)
else:
t = np.asarray(t)
# For each input, implement a step change
yout = None
for i in range(0, n_inputs):
u = np.zeros((t.shape[0], n_inputs))
u[0, i] = 1.0
one_output = dlsim((A, B, C, D, dt), u, t=t, x0=x0)
if yout is None:
yout = (one_output[1],)
else:
yout = yout + (one_output[1],)
tout = one_output[0]
return tout, yout
def dstep(system, x0=None, t=None, n=None):
"""Step response of discrete-time system.
Parameters
----------
system : tuple of array_like
A tuple describing the system.
The following gives the number of elements in the tuple and
the interpretation:
* 3: (num, den, dt)
* 4: (zeros, poles, gain, dt)
* 5: (A, B, C, D, dt)
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
-------
tout : ndarray
Output time points, as a 1-D array.
yout : ndarray
Step response of system. Each element of the tuple represents
the output of the system based on a step response to each input.
See Also
--------
step, dimpulse, dlsim, cont2discrete
"""
# Determine the system type and set number of inputs and time steps
A, B, C, D, dt = _system_to_statespace(system)
n_inputs = B.shape[1]
# Default to 100 samples if unspecified
if n is None:
n = 100
# If time is not specified, use the number of samples
# and system dt
if t is None:
t = np.linspace(0, n * dt, n, endpoint=False)
else:
t = np.asarray(t)
# For each input, implement a step change
yout = None
for i in range(0, n_inputs):
u = np.zeros((t.shape[0], n_inputs))
u[:, i] = np.ones((t.shape[0],))
one_output = dlsim((A, B, C, D, dt), u, t=t, x0=x0)
if yout is None:
yout = (one_output[1],)
else:
yout = yout + (one_output[1],)
tout = one_output[0]
return tout, yout