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Add function to convert a system into observable canonical form #103

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merged 1 commit into from
Nov 18, 2016
Merged

Add function to convert a system into observable canonical form #103

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54 changes: 51 additions & 3 deletions control/canonical.py
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Original file line number Diff line number Diff line change
Expand Up @@ -4,12 +4,12 @@
from .exception import ControlNotImplemented
from .lti import issiso
from .statesp import StateSpace
from .statefbk import ctrb
from .statefbk import ctrb, obsv

from numpy import zeros, shape, poly
from numpy.linalg import inv

__all__ = ['canonical_form', 'reachable_form']
__all__ = ['canonical_form', 'reachable_form', 'observable_form']

def canonical_form(xsys, form='reachable'):
"""Convert a system into canonical form
Expand All @@ -21,7 +21,7 @@ def canonical_form(xsys, form='reachable'):
form : String
Canonical form for transformation. Chosen from:
* 'reachable' - reachable canonical form
* 'observable' - observable canonical form [not implemented]
* 'observable' - observable canonical form
* 'modal' - modal canonical form [not implemented]

Returns
Expand All @@ -35,6 +35,8 @@ def canonical_form(xsys, form='reachable'):
# Call the appropriate tranformation function
if form == 'reachable':
return reachable_form(xsys)
elif form == 'observable':
return observable_form(xsys)
else:
raise ControlNotImplemented(
"Canonical form '%s' not yet implemented" % form)
Expand Down Expand Up @@ -85,3 +87,49 @@ def reachable_form(xsys):
zsys.C = xsys.C * inv(Tzx)

return zsys, Tzx


def observable_form(xsys):
"""Convert a system into observable canonical form

Parameters
----------
xsys : StateSpace object
System to be transformed, with state `x`

Returns
-------
zsys : StateSpace object
System in observable canonical form, with state `z`
T : matrix
Coordinate transformation: z = T * x
"""
# Check to make sure we have a SISO system
if not issiso(xsys):
raise ControlNotImplemented(
"Canonical forms for MIMO systems not yet supported")

# Create a new system, starting with a copy of the old one
zsys = StateSpace(xsys)

# Generate the system matrices for the desired canonical form
zsys.C = zeros(shape(xsys.C))
zsys.C[0, 0] = 1
zsys.A = zeros(shape(xsys.A))
Apoly = poly(xsys.A) # characteristic polynomial
for i in range(0, xsys.states):
zsys.A[i, 0] = -Apoly[i+1] / Apoly[0]
if (i+1 < xsys.states):
zsys.A[i, i+1] = 1

# Compute the observability matrices for each set of states
Wrx = obsv(xsys.A, xsys.C)
Wrz = obsv(zsys.A, zsys.C)

# Transformation from one form to another
Tzx = inv(Wrz) * Wrx

# Finally, compute the output matrix
zsys.B = Tzx * xsys.B

return zsys, Tzx