def test_solve_proj_sadpnt_smw(self):
"""check the sadpnt as projection"""
n = self.n
# check whether it is a projection
AuvInvZ = lau.solve_sadpnt_smw(amat=self.A,
jmat=self.J,
rhsv=self.Z,
jmatT=self.Jt,
umat=self.U,
vmat=self.V)
auvAUVinv = self.A * AuvInvZ[:n, :] - \
lau.comp_uvz_spdns(self.U, self.V, AuvInvZ[:n, :])
AuvInv2Z = lau.solve_sadpnt_smw(amat=self.A,
jmat=self.J,
rhsv=auvAUVinv,
jmatT=self.Jt,
umat=self.U,
vmat=self.V)
self.assertTrue(np.allclose(AuvInvZ[:n, :], AuvInv2Z[:n, :]),
msg='likely to fail because of ill cond')
prjz = lau.app_prj_via_sadpnt(amat=self.A,
jmat=self.J,
rhsv=self.Z,
jmatT=self.Jt)
prprjz = lau.app_prj_via_sadpnt(amat=self.A,
jmat=self.J,
rhsv=self.Z,
jmatT=self.Jt)
# check projector
self.assertTrue(np.allclose(prprjz, prjz))
# onto kernel J
self.assertTrue(np.linalg.norm(prjz) > 1e-8)
self.assertTrue(
np.linalg.norm(self.J * prjz) / np.linalg.norm(prjz) < 1e-6)
# check transpose
idmat = np.eye(n)
prj = lau.app_prj_via_sadpnt(amat=self.A,
jmat=self.J,
rhsv=idmat,
jmatT=self.Jt)
prjT = lau.app_prj_via_sadpnt(amat=self.A,
jmat=self.J,
rhsv=idmat,
jmatT=self.Jt,
transposedprj=True)
self.assertTrue(np.allclose(prj, prjT.T))
def test_sadpnt_smw_krypy(self):
"""check the sadpnt solver with krypy"""
umat, vmat, k, = self.U, self.V, self.k
# self.Jt = self.J.T
# check the formula
AuvInvZ = lau.solve_sadpnt_smw(amat=self.A, jmat=self.J, rhsv=self.Z,
jmatT=self.Jt, umat=self.U, vmat=self.V,
krylov=True, krpslvprms=self.krpslvprms)
sysm1 = sps.hstack([self.A, self.Jt], format='csr')
sysm2 = sps.hstack([self.J, sps.csr_matrix((k, k))], format='csr')
mata = sps.vstack([sysm1, sysm2], format='csr')
umate = np.vstack([umat, np.zeros((k, umat.shape[1]))])
vmate = np.hstack([vmat, np.zeros((vmat.shape[0], k))])
ze = np.vstack([self.Z, np.zeros((k, self.Z.shape[1]))])
AAinvZ = mata * AuvInvZ - np.dot(umate, np.dot(vmate, AuvInvZ))
# likely to fail because of ill conditioned rand mats
print np.linalg.norm(AAinvZ - ze)
self.assertTrue(np.allclose(AAinvZ, ze),
msg='likely to fail because of ill cond')
def test_sadpnt_smw(self):
"""check the sadpnt solver"""
umat, vmat, k, = self.U, self.V, self.k
# self.Jt = self.J.T
# check the formula
AuvInvZ = lau.solve_sadpnt_smw(amat=self.A,
jmat=self.J,
rhsv=self.Z,
jmatT=self.Jt,
umat=self.U,
vmat=self.V)
sysm1 = sps.hstack([self.A, self.Jt], format='csr')
sysm2 = sps.hstack([self.J, sps.csr_matrix((k, k))], format='csr')
mata = sps.vstack([sysm1, sysm2], format='csr')
umate = np.vstack([umat, np.zeros((k, umat.shape[1]))])
vmate = np.hstack([vmat, np.zeros((vmat.shape[0], k))])
ze = np.vstack([self.Z, np.zeros((k, self.Z.shape[1]))])
AAinvZ = mata * AuvInvZ - np.dot(umate, np.dot(vmate, AuvInvZ))
# likely to fail because of ill conditioned rand mats
self.assertTrue(np.allclose(AAinvZ, ze),
msg='likely to fail because of ill cond')
def test_solve_proj_sadpnt_smw(self):
"""check the sadpnt as projection"""
n = self.n
# check whether it is a projection
AuvInvZ = lau.solve_sadpnt_smw(amat=self.A, jmat=self.J, rhsv=self.Z,
jmatT=self.Jt, umat=self.U, vmat=self.V)
auvAUVinv = self.A * AuvInvZ[:n, :] - \
lau.comp_uvz_spdns(self.U, self.V, AuvInvZ[:n, :])
AuvInv2Z = lau.solve_sadpnt_smw(amat=self.A, jmat=self.J,
rhsv=auvAUVinv,
jmatT=self.Jt,
umat=self.U, vmat=self.V)
self.assertTrue(np.allclose(AuvInvZ[:n, :], AuvInv2Z[:n, :]),
msg='likely to fail because of ill cond')
prjz = lau.app_prj_via_sadpnt(amat=self.A, jmat=self.J,
rhsv=self.Z, jmatT=self.Jt)
prprjz = lau.app_prj_via_sadpnt(amat=self.A, jmat=self.J,
rhsv=self.Z, jmatT=self.Jt)
# check projector
self.assertTrue(np.allclose(prprjz, prjz))
# onto kernel J
self.assertTrue(np.linalg.norm(prjz) > 1e-8)
self.assertTrue(np.linalg.norm(self.J*prjz)/np.linalg.norm(prjz)
< 1e-6)
# check transpose
idmat = np.eye(n)
prj = lau.app_prj_via_sadpnt(amat=self.A, jmat=self.J,
rhsv=idmat, jmatT=self.Jt)
prjT = lau.app_prj_via_sadpnt(amat=self.A, jmat=self.J,
rhsv=idmat, jmatT=self.Jt,
transposedprj=True)
self.assertTrue(np.allclose(prj, prjT.T))
def solve_stst_feedbacknthrough(amat=None, mmat=None, jmat=None,
bmat=None, cmat=None,
fv=None, fl=None, fg=None, fl2=None,
nwtn_adi_dict=dict(adi_max_steps=150,
adi_newZ_reltol=1e-5,
nwtn_max_steps=14,
nwtn_upd_reltol=1e-8)):
"""solve for the stabilizing feedback gain and the feedthrough
for the linear time invariant case"""
Z = proj_alg_ric_newtonadi(mmat=mmat, amat=amat, jmat=jmat,
bmat=bmat, wmat=cmat,
nwtn_adi_dict=nwtn_adi_dict)
mtxb = get_mTzzTtb(mmat.T, Z, bmat)
mtxfv = get_mTzzTtb(mmat.T, Z, fv)
wft = lau.solve_sadpnt_smw(amat=-amat.T, jmat=jmat, rhsv=fl-mtxfv,
umat=-mtxb, vmat=bmat.T)
return Z, wft