From 98e71cbcdcfa028dd5c140d8b9c7d873bb68f1dd Mon Sep 17 00:00:00 2001 From: Yuichi NAGAYAMA Date: Sun, 7 Aug 2016 21:17:45 +0900 Subject: [PATCH] Add function to convert a system into observable canonical form --- control/canonical.py | 54 +++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 51 insertions(+), 3 deletions(-) diff --git a/control/canonical.py b/control/canonical.py index 468013e8d..d148ca411 100644 --- a/control/canonical.py +++ b/control/canonical.py @@ -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 @@ -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 @@ -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) @@ -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 pFad - Phonifier reborn

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