def gram(sys, type):
"""Gramian (controllability or observability)
Parameters
----------
sys : StateSpace
System description
type : String
Type of desired computation. `type` is either 'c' (controllability)
or 'o' (observability). To compute the Cholesky factors of Gramians
use 'cf' (controllability) or 'of' (observability)
Returns
-------
gram : 2D array (or matrix)
Gramian of system
Raises
------
ValueError
* if system is not instance of StateSpace class
* if `type` is not 'c', 'o', 'cf' or 'of'
* if system is unstable (sys.A has eigenvalues not in left half plane)
ControlSlycot
if slycot routine sb03md cannot be found
if slycot routine sb03od cannot be found
Notes
-----
The return type for 2D arrays depends on the default class set for
state space operations. See :func:`~control.use_numpy_matrix`.
Examples
--------
>>> Wc = gram(sys, 'c')
>>> Wo = gram(sys, 'o')
>>> Rc = gram(sys, 'cf'), where Wc = Rc' * Rc
>>> Ro = gram(sys, 'of'), where Wo = Ro' * Ro
"""
# Check for ss system object
if not isinstance(sys, statesp.StateSpace):
raise ValueError("System must be StateSpace!")
if type not in ['c', 'o', 'cf', 'of']:
raise ValueError("That type is not supported!")
# TODO: Check for continuous or discrete, only continuous supported for now
# if isCont():
# dico = 'C'
# elif isDisc():
# dico = 'D'
# else:
dico = 'C'
# TODO: Check system is stable, perhaps a utility in ctrlutil.py
# or a method of the StateSpace class?
if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
raise ValueError("Oops, the system is unstable!")
if type == 'c' or type == 'o':
# Compute Gramian by the Slycot routine sb03md
# make sure Slycot is installed
if sb03md is None:
raise ControlSlycot("can't find slycot module 'sb03md'")
if type == 'c':
tra = 'T'
C = -sys.B @ sys.B.T
elif type == 'o':
tra = 'N'
C = -sys.C.T @ sys.C
n = sys.nstates
U = np.zeros((n, n))
A = np.array(sys.A) # convert to NumPy array for slycot
X, scale, sep, ferr, w = sb03md(
n, C, A, U, dico, job='X', fact='N', trana=tra)
gram = X
return _ssmatrix(gram)
elif type == 'cf' or type == 'of':
# Compute cholesky factored gramian from slycot routine sb03od
if sb03od is None:
raise ControlSlycot("can't find slycot module 'sb03od'")
tra = 'N'
n = sys.nstates
Q = np.zeros((n, n))
A = np.array(sys.A) # convert to NumPy array for slycot
if type == 'cf':
m = sys.B.shape[1]
B = np.zeros_like(A)
B[0:m, 0:n] = sys.B.transpose()
X, scale, w = sb03od(
n, m, A.transpose(), Q, B, dico, fact='N', trans=tra)
elif type == 'of':
m = sys.C.shape[0]
C = np.zeros_like(A)
C[0:n, 0:m] = sys.C.transpose()
X, scale, w = sb03od(
n, m, A, Q, C.transpose(), dico, fact='N', trans=tra)
gram = X
return _ssmatrix(gram)
def gram(sys,type):
"""Gramian (controllability or observability)
Parameters
----------
sys: StateSpace
State-space system to compute Gramian for
type: String
Type of desired computation.
`type` is either 'c' (controllability) or 'o' (observability). To compute the
Cholesky factors of gramians use 'cf' (controllability) or 'of' (observability)
Returns
-------
gram: array
Gramian of system
Raises
------
ValueError
* if system is not instance of StateSpace class
* if `type` is not 'c', 'o', 'cf' or 'of'
* if system is unstable (sys.A has eigenvalues not in left half plane)
ImportError
if slycot routine sb03md cannot be found
if slycot routine sb03od cannot be found
Examples
--------
>>> Wc = gram(sys,'c')
>>> Wo = gram(sys,'o')
>>> Rc = gram(sys,'cf'), where Wc=Rc'*Rc
>>> Ro = gram(sys,'of'), where Wo=Ro'*Ro
"""
#Check for ss system object
if not isinstance(sys,statesp.StateSpace):
raise ValueError("System must be StateSpace!")
if type not in ['c', 'o', 'cf', 'of']:
raise ValueError("That type is not supported!")
#TODO: Check for continous or discrete, only continuous supported right now
# if isCont():
# dico = 'C'
# elif isDisc():
# dico = 'D'
# else:
dico = 'C'
#TODO: Check system is stable, perhaps a utility in ctrlutil.py
# or a method of the StateSpace class?
if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
raise ValueError("Oops, the system is unstable!")
if type=='c' or type=='o':
#Compute Gramian by the Slycot routine sb03md
#make sure Slycot is installed
try:
from slycot import sb03md
except ImportError:
raise ControlSlycot("can't find slycot module 'sb03md'")
if type=='c':
tra = 'T'
C = -np.dot(sys.B,sys.B.transpose())
elif type=='o':
tra = 'N'
C = -np.dot(sys.C.transpose(),sys.C)
n = sys.states
U = np.zeros((n,n))
A = np.array(sys.A) # convert to NumPy array for slycot
X,scale,sep,ferr,w = sb03md(n, C, A, U, dico, job='X', fact='N', trana=tra)
gram = X
return gram
elif type=='cf' or type=='of':
#Compute cholesky factored gramian from slycot routine sb03od
try:
from slycot import sb03od
except ImportError:
raise ControlSlycot("can't find slycot module 'sb03od'")
tra='N'
n = sys.states
Q = np.zeros((n,n))
A = np.array(sys.A) # convert to NumPy array for slycot
if type=='cf':
m = sys.B.shape[1]
B = np.zeros_like(A)
B[0:m,0:n] = sys.B.transpose()
X,scale,w = sb03od(n, m, A.transpose(), Q, B, dico, fact='N', trans=tra)
elif type=='of':
m = sys.C.shape[0]
C = np.zeros_like(A)
C[0:n,0:m] = sys.C.transpose()
X,scale,w = sb03od(n, m, A, Q, C.transpose(), dico, fact='N', trans=tra)
gram = X
return gram