def dlsim(system, u, t=None, x0=None):
"""
Simulate output of a discrete-time linear system.
Parameters
----------
system : class instance or tuple
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:
- 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, the
final value in `t` determines the number of steps returned in the
output.
x0 : arry_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.])
"""
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")
if t is None:
out_samples = max(u.shape)
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 cont2discrete(sys, dt, method="zoh", alpha=None):
"""Transform a continuous to a discrete state-space system.
Parameters
-----------
sys : 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)
dt : float
The discretization time step.
method : {"gbt", "bilinear", "euler", "backward_diff", "zoh"}
Which method to use:
* gbt: generalized bilinear transformation
* bilinear: Tustin's approximation ("gbt" with alpha=0.5)
* euler: Euler (or forward differencing) method ("gbt" with
alpha=0)
* backward_diff: Backwards differencing ("gbt" with alpha=1.0)
* zoh: zero-order hold (default).
alpha : float within [0, 1]
The generalized bilinear transformation weighting parameter, which
should only be specified with method="gbt", and is ignored otherwise
Returns
-------
sysd : tuple containing the discrete system
Based on the input type, the output will be of the form
(num, den, dt) for transfer function input
(zeros, poles, gain, dt) for zeros-poles-gain input
(A, B, C, D, dt) for state-space system input
Notes
-----
By default, the routine uses a Zero-Order Hold (zoh) method to perform
the transformation. Alternatively, a generalized bilinear transformation
may be used, which includes the common Tustin's bilinear approximation,
an Euler's method technique, or a backwards differencing technique.
The Zero-Order Hold (zoh) method is based on:
http://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models
Generalize bilinear approximation is based on:
http://techteach.no/publications/discretetime_signals_systems/discrete.pdf
and
G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized bilinear
transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754, 2009.
(http://www.ece.ualberta.ca/~gfzhang/research/ZCC07_preprint.pdf)
"""
if len(sys) == 2:
sysd = cont2discrete(tf2ss(sys[0], sys[1]), dt, method=method,
alpha=alpha)
return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
elif len(sys) == 3:
sysd = cont2discrete(zpk2ss(sys[0], sys[1], sys[2]), dt, method=method,
alpha=alpha)
return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
elif len(sys) == 4:
a, b, c, d = sys
else:
raise ValueError("First argument must either be a tuple of 2 (tf), "
"3 (zpk), or 4 (ss) arrays.")
if method == 'gbt':
if alpha is None:
raise ValueError("Alpha parameter must be specified for the "
"generalized bilinear transform (gbt) method")
elif alpha < 0 or alpha > 1:
raise ValueError("Alpha parameter must be within the interval "
"[0,1] for the gbt method")
if method == 'gbt':
# This parameter is used repeatedly - compute once here
ima = np.eye(a.shape[0]) - alpha*dt*a
ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a)
bd = linalg.solve(ima, dt*b)
# Similarly solve for the output equation matrices
cd = linalg.solve(ima.transpose(), c.transpose())
cd = cd.transpose()
dd = d + alpha*np.dot(c, bd)
elif method == 'bilinear' or method == 'tustin':
return cont2discrete(sys, dt, method="gbt", alpha=0.5)
elif method == 'euler' or method == 'forward_diff':
return cont2discrete(sys, dt, method="gbt", alpha=0.0)
elif method == 'backward_diff':
return cont2discrete(sys, dt, method="gbt", alpha=1.0)
elif method == 'zoh':
# Build an exponential matrix
em_upper = np.hstack((a, b))
# Need to stack zeros under the a and b matrices
em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])),
np.zeros((b.shape[1], b.shape[1])) ))
em = np.vstack((em_upper, em_lower))
ms = linalg.expm(dt * em)
# Dispose of the lower rows
ms = ms[:a.shape[0], :]
ad = ms[:, 0:a.shape[1]]
bd = ms[:, a.shape[1]:]
cd = c
dd = d
else:
raise ValueError("Unknown transformation method '%s'" % method)
return ad, bd, cd, dd, dt
def dlsim(system, u, t=None, x0=None):
"""
Simulate output of a discrete-time linear system.
Parameters
----------
system : class instance or tuple
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:
- 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, the
final value in `t` determines the number of steps returned in the
output.
x0 : arry_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.])
"""
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")
if t is None:
out_samples = max(u.shape)
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 cont2discrete(sys, dt, method="zoh", alpha=None):
"""Transform a continuous to a discrete state-space system.
Parameters
-----------
sys : 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)
dt : float
The discretization time step.
method : {"gbt", "bilinear", "euler", "backward_diff", "zoh"}
Which method to use:
* gbt: generalized bilinear transformation
* bilinear: Tustin's approximation ("gbt" with alpha=0.5)
* euler: Euler (or forward differencing) method ("gbt" with
alpha=0)
* backward_diff: Backwards differencing ("gbt" with alpha=1.0)
* zoh: zero-order hold (default).
alpha : float within [0, 1]
The generalized bilinear transformation weighting parameter, which
should only be specified with method="gbt", and is ignored otherwise
Returns
-------
sysd : tuple containing the discrete system
Based on the input type, the output will be of the form
(num, den, dt) for transfer function input
(zeros, poles, gain, dt) for zeros-poles-gain input
(A, B, C, D, dt) for state-space system input
Notes
-----
By default, the routine uses a Zero-Order Hold (zoh) method to perform
the transformation. Alternatively, a generalized bilinear transformation
may be used, which includes the common Tustin's bilinear approximation,
an Euler's method technique, or a backwards differencing technique.
The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear
approximation is based on [2]_ and [3].
References
----------
.. [1] http://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models
.. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf
.. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized
bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754,
2009. (http://www.ece.ualberta.ca/~gfzhang/research/ZCC07_preprint.pdf)
"""
if len(sys) == 2:
sysd = cont2discrete(tf2ss(sys[0], sys[1]),
dt,
method=method,
alpha=alpha)
return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt, )
elif len(sys) == 3:
sysd = cont2discrete(zpk2ss(sys[0], sys[1], sys[2]),
dt,
method=method,
alpha=alpha)
return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt, )
elif len(sys) == 4:
a, b, c, d = sys
else:
raise ValueError("First argument must either be a tuple of 2 (tf), "
"3 (zpk), or 4 (ss) arrays.")
if method == 'gbt':
if alpha is None:
raise ValueError("Alpha parameter must be specified for the "
"generalized bilinear transform (gbt) method")
elif alpha < 0 or alpha > 1:
raise ValueError("Alpha parameter must be within the interval "
"[0,1] for the gbt method")
if method == 'gbt':
# This parameter is used repeatedly - compute once here
ima = np.eye(a.shape[0]) - alpha * dt * a
ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0 - alpha) * dt * a)
bd = linalg.solve(ima, dt * b)
# Similarly solve for the output equation matrices
cd = linalg.solve(ima.transpose(), c.transpose())
cd = cd.transpose()
dd = d + alpha * np.dot(c, bd)
elif method == 'bilinear' or method == 'tustin':
return cont2discrete(sys, dt, method="gbt", alpha=0.5)
elif method == 'euler' or method == 'forward_diff':
return cont2discrete(sys, dt, method="gbt", alpha=0.0)
elif method == 'backward_diff':
return cont2discrete(sys, dt, method="gbt", alpha=1.0)
elif method == 'zoh':
# Build an exponential matrix
em_upper = np.hstack((a, b))
# Need to stack zeros under the a and b matrices
em_lower = np.hstack((np.zeros(
(b.shape[1], a.shape[0])), np.zeros((b.shape[1], b.shape[1]))))
em = np.vstack((em_upper, em_lower))
ms = linalg.expm(dt * em)
# Dispose of the lower rows
ms = ms[:a.shape[0], :]
ad = ms[:, 0:a.shape[1]]
bd = ms[:, a.shape[1]:]
cd = c
dd = d
else:
raise ValueError("Unknown transformation method '%s'" % method)
return ad, bd, cd, dd, dt
