def _dywdy(t, V=None):
cvel = np.load(veldict[t]+'.npy')
delty = ystar(t) - lau.mm_dnssps(cmat, cvel)
if V is None:
return np.dot(delty.T, lau.mm_dnssps(W, delty))
else:
return np.dot(delty.T, lau.mm_dnssps(V, delty))
def _dywdy(t, V=None):
cvel = np.load(veldict[t] + '.npy')
delty = ystar(t) - lau.mm_dnssps(cmat, cvel)
if V is None:
return np.dot(delty.T, lau.mm_dnssps(W, delty))
else:
return np.dot(delty.T, lau.mm_dnssps(V, delty))
def timspanorm(x, mspace=None, mtime=None):
ynorml = []
for col in range(x.shape[1]):
ccol = x[:, col]
ynorml.append(np.dot(ccol.T, lau.mm_dnssps(mspace, ccol)))
ynormvec = np.array(ynorml).reshape((x.shape[1], 1))
return np.sqrt(np.dot(ynormvec.T, lau.mm_dnssps(mtime, ynormvec)))
def burger_fwd_gpresidual(ms=None,
ay=None,
dms=None,
rhs=None,
my=None,
iniv=None,
htittl=None,
uvvdxl=None):
""" function to return the residuals of the burger genpod apprxm
DEPRECATED
"""
import spacetime_pod_utils as spu
if rhs is not None and np.linalg.norm(rhs) > 0:
raise NotImplementedError('rhs not considered in the alg')
# lns = ms.shape[0] # dimension of time disc
lnq = ay.shape[0] # dimension of space disc
dmsz = dms[1:, :1] # the part of the mass matrix related to the ini value
msz = ms[1:, :1]
dmsI = dms[1:, 1:] # mass matrices w/o ini values (test and trial)
msI = ms[1:, 1:]
def _nonlinres(tsvec):
def evabrgquadterm(tvvec):
return eva_burger_quadratic(tvvec=tvvec,
htittl=htittl,
uvvdxl=uvvdxl)
bres = spu.get_spacetimepodres(tvvec=tsvec,
dms=dms,
ms=ms,
my=my,
ared=ay,
nfunc=evabrgquadterm,
rhs=None,
retnorm=False,
iniv=iniv)
return bres[0 * lnq:, ].flatten()
solmat = np.kron(dmsI, my) + np.kron(msI, ay)
linrhs = - np.kron(dmsz, lau.mm_dnssps(my, iniv)) \
- np.kron(msz, lau.mm_dnssps(ay, iniv))
def _linres(tsvec):
tsvecs = tsvec.reshape((tsvec.size, 1))
lres = (np.dot(solmat, tsvecs) - linrhs)
return lres.flatten()
def _lrprime(tsvec):
return solmat
def sollintimspasys(dms=None,
ms=None,
ay=None,
my=None,
iniv=None,
opti=False):
lns = ms.shape[0]
lnq = ay.shape[0]
dmsz = dms[1:, :1]
msz = ms[1:, :1]
dmsI = dms[1:, 1:]
msI = ms[1:, 1:]
solmat = np.kron(dmsI, my) + np.kron(msI, ay)
rhs = -np.kron(dmsz, lau.mm_dnssps(my, iniv)) \
- np.kron(msz, lau.mm_dnssps(ay, iniv))
if opti:
import scipy.optimize as sco
def _linres(tsvec):
tsvecs = tsvec.reshape((tsvec.size, 1))
lres = (np.dot(solmat, tsvecs) - rhs)
return lres.flatten()
def _lrprime(tsvec):
return solmat
sol = sco.fsolve(_linres,
np.zeros(((lns - 1) * lnq, )),
fprime=_lrprime)
else:
sol = np.linalg.solve(solmat, rhs)
fsol = np.hstack([iniv, sol.reshape((lns - 1, lnq)).T])
return fsol
def space_time_norm(errvecsqrd=None,
tmesh=None,
spatimvals=None,
spacemmat=None):
""" compute the space time norm `int_T |v(t)|_M dt` (with the squares
and roots) using the piecewise trapezoidal rule in time """
if errvecsqrd is None:
errvecsql = []
for row in range(spatimvals.shape[0]):
crow = spatimvals[row, :]
errvecsql.append(np.dot(crow.T, lau.mm_dnssps(spacemmat, crow)))
errvecsqrd = np.array(errvecsql)
dtvec = tmesh[1:] - tmesh[:-1]
trapv = 0.5 * (errvecsqrd[:-1] + errvecsqrd[1:])
errvsqrd = (dtvec * trapv).sum()
return np.sqrt(errvsqrd)
def _bbwdrhs(t):
# TODO: -----------------------------> here we need vstar
return (-lau.mm_dnssps(MLV, vfun(te - t)).flatten() +
lau.mm_dnssps(MLV, _vstarfun(te - t)).flatten())
def _contrl_rhs(t):
if t > tE: # the integrator may require values outside [t0, tE]
return rhs(t).flatten() + lau.mm_dnssps(MVU,
ufun(tE)).flatten()
else:
return rhs(t).flatten() + lau.mm_dnssps(MVU, ufun(t)).flatten()
def compare_stepresp(tmesh=None, a_mat=None, c_mat=None, b_mat=None,
m_mat=None, tl=None, tr=None,
iniv=None,
fullresp=None, fsr_soldict=None,
plot=False, jsonstr=None):
""" compare the system's step response to unit inputs in time domain
with reduced system's response.
We consider the system
.. math::
M\\dot v = Av + Bu, \\quad y = Cv
on the discretized time interval :math:`(t_0, t_E)`
and the reduced system with :math:`\\hat A = t_l^TAt_r`.
Parameters
----------
tmesh : iterable list or ndarray
vector of the time instances
a_mat : (N,N) sparse matrix
system matrix
c_mat : (ny,N) sparse matrix or ndarray
output matrix
b_mat : (N,nu) sparse matrix or ndarray
input operator
m_mat : (N,N) sparse matrix
mass matrix
tl, tr : (N,K) ndarrays
left, right transformation for the reduced system
iniv : (N,1) ndarray
initial value and linearization point of the full system
fullresp : callable f(v, **dict)
returns the response of the full system
fsr_soldict : dictionary
parameters to be passed to `fullresp`
plot : boolean, optional
whether to plot, defaults to `False`
jsonstr: string, optional
if defined, the output is stored in this json file, defaults to `None`
"""
from scipy.integrate import odeint
ahat = np.dot(tl.T, a_mat*tr)
chat = lau.mm_dnssps(c_mat, tr)
inivhat = np.dot(tl.T, m_mat*iniv)
inivout = lau.mm_dnssps(c_mat, iniv).tolist()
red_stp_rsp, ful_stp_rsp = [], []
for ccol in [0]: # , b_mat.shape[1]-1]: # range(2): # b_mat.shape[1]):
bmc = b_mat[:, ccol][:, :]
red_bmc = tl.T * bmc
def dtfunc(v, t):
return (np.dot(ahat, v).flatten() + red_bmc.flatten()) # +\
red_state = odeint(dtfunc, 0*inivhat.flatten(), tmesh)
red_stp_rsp.append(np.dot(chat, red_state.T).T.tolist())
ful_stp_rsp.append(fullresp(bcol=bmc, trange=tmesh, ini_vel=iniv,
cmat=c_mat, soldict=fsr_soldict))
if jsonstr:
try:
tmesh = tmesh.tolist()
except AttributeError:
pass # is a list already
dou.save_output_json(datadict={"tmesh": tmesh,
"ful_stp_rsp": ful_stp_rsp,
"red_stp_rsp": red_stp_rsp,
"inivout": inivout},
fstring=jsonstr,
module='sadptprj_riclyap_adi.bal_trunc_utils',
plotroutine='plot_step_resp')
if plot:
plot_step_resp(tmesh=tmesh, red_stp_rsp=red_stp_rsp,
ful_stp_rsp=ful_stp_rsp, inivout=inivout)
def fv_tmdp_redoutpfb(time=None, curvel=None, memory=None,
linvel=None,
ipsysk_mat_inv=None,
obs_bk=None, cts=None,
b_mat=None, c_mat=None,
xck=None, bk_mat=None,
**kw):
"""realizes a reduced static output feedback as a function
that can be passed to a solution routine for the
unsteady Navier-Stokes equations
For convinience the
Parameters
----------
time : real
current time
curvel : (N,1) nparray
current velocity. For consistency, the full state is taken
as input. However, internally, we only use the observation
`y = c_mat*curvel`
memory : dictionary
contains values from previous call, in particular the
previous state estimate
linvel : (N,1) nparray
linearization point for the linear model
ipsysk_mat_inv : (K,K) nparray
inverse of the system matrix that defines the update
of the state estimate
obs_bk : (K,NU) nparray
input matrix in the observer
cts : real
time step length
b_mat : (N,NU) sparse matrix
input matrix of the full system
c_mat=None,
c_mat : (NY,N) sparse matrix
output matrix of the full system
xck : (K,K) nparray
reduced solution of the CARE
bk_mat : (K,NU) nparray
reduced input matrix
Returns
-------
actua : (N,1) nparray
the current actuation
memory : dictionary
to be passed back in the next timestep
"""
xk_old = memory['xk_old']
buk = cts*np.dot(obs_bk,
lau.mm_dnssps(c_mat, (curvel-linvel)))
xk_old = np.dot(ipsysk_mat_inv, xk_old + buk)
# cts*np.dot(obs_bk,
# lau.mm_dnssps(c_mat, (curvel-linvel))))
memory['xk_old'] = xk_old
actua = -lau.mm_dnssps(b_mat,
np.dot(bk_mat.T, np.dot(xck, xk_old)))
print '\nnorm of deviation', np.linalg.norm(curvel-linvel)
print 'norm of actuation {0}'.format(np.linalg.norm(actua))
return actua, memory
def lqgbt(problemname='drivencavity',
N=10, Re=1e2, plain_bt=False,
use_ric_ini=None, t0=0.0, tE=1.0, Nts=11,
NU=3, NY=3,
bccontrol=True, palpha=1e-5,
paraoutput=True,
trunc_lqgbtcv=1e-6,
nwtn_adi_dict=None,
comp_freqresp=False, comp_stepresp='nonlinear',
closed_loop=False, multiproc=False,
perturbpara=1e-3):
"""Main routine for LQGBT
Parameters
----------
problemname : string, optional
what problem to be solved, 'cylinderwake' or 'drivencavity'
N : int, optional
parameter for the dimension of the space discretization
Re : real, optional
Reynolds number, defaults to `1e2`
plain_bt : boolean, optional
whether to try simple *balanced truncation*, defaults to False
use_ric_ini : real, optional
use the solution with this Re number as stabilizing initial guess,
defaults to `None`
t0, tE, Nts : real, real, int, optional
starting and endpoint of the considered time interval, number of
time instancses, default to `0.0, 1.0, 11`
bccontrol : boolean, optional
whether to apply boundary control via penalized robin conditions,
defaults to `False`
NU, NY : int, optional
dimensions of components of in and output space (will double because
there are two components), default to `3, 3`
comp_freqresp : boolean, optional
whether to compute and compare the frequency responses,
defaults to `False`
comp_stepresp : {'nonlinear', False, None}
whether to compute and compare the step responses
| if False -> no step response
| if == 'nonlinear' -> compare linear reduced to nonlinear full model
| else -> linear reduced versus linear full model
defaults to `False`
trunc_lqgbtcv : real, optional
threshold at what the lqgbt characteristiv values are truncated,
defaults to `1e-6`
closed_loop : {'full_state_fb', 'red_output_fb', False, None}
how to do the closed loop simulation:
| if False -> no simulation
| if == 'full_state_fb' -> full state feedback
| if == 'red_output_fb' -> reduced output feedback
| else -> no control is applied
defaults to `False`
"""
typprb = 'BT' if plain_bt else 'LQG-BT'
print '\n ### We solve the {0} problem for the {1} at Re={2} ###\n'.\
format(typprb, problemname, Re)
if nwtn_adi_dict is not None:
nap = nwtn_adi_dict
else:
nap = nwtn_adi_params()['nwtn_adi_dict']
# output
ddir = 'data/'
try:
os.chdir(ddir)
except OSError:
raise Warning('need "' + ddir + '" subdir for storing the data')
os.chdir('..')
# stokesmats = dts.get_stokessysmats(femp['V'], femp['Q'], nu)
# rhsd_vf = dts.setget_rhs(femp['V'], femp['Q'],
# femp['fv'], femp['fp'], t=0)
# # remove the freedom in the pressure
# stokesmats['J'] = stokesmats['J'][:-1, :][:, :]
# stokesmats['JT'] = stokesmats['JT'][:, :-1][:, :]
# rhsd_vf['fp'] = rhsd_vf['fp'][:-1, :]
# # reduce the matrices by resolving the BCs
# (stokesmatsc,
# rhsd_stbc,
# invinds,
# bcinds,
# bcvals) = dts.condense_sysmatsbybcs(stokesmats,
# femp['diribcs'])
# # pressure freedom and dirichlet reduced rhs
# rhsd_vfrc = dict(fpr=rhsd_vf['fp'], fvc=rhsd_vf['fv'][invinds, ])
# # add the info on boundary and inner nodes
# bcdata = {'bcinds': bcinds,
# 'bcvals': bcvals,
# 'invinds': invinds}
# femp.update(bcdata)
femp, stokesmatsc, rhsd_vfrc, rhsd_stbc \
= dnsps.get_sysmats(problem=problemname, N=N, Re=Re,
bccontrol=bccontrol, scheme='TH')
nu = femp['charlen']/Re
# specify in what spatial direction Bu changes. The remaining is constant
uspacedep = femp['uspacedep']
# casting some parameters
invinds, NV = femp['invinds'], len(femp['invinds'])
prbstr = '_bt' if plain_bt else '_lqgbt'
# contsetupstr = 'NV{0}NU{1}NY{2}alphau{3}'.format(NV, NU, NY, alphau)
if bccontrol:
import scipy.sparse as sps
contsetupstr = 'NV{0}_bcc_NY{1}'.format(NV, NY)
stokesmatsc['A'] = stokesmatsc['A'] + 1./palpha*stokesmatsc['Arob']
b_mat = 1./palpha*stokesmatsc['Brob']
u_masmat = sps.eye(b_mat.shape[1], format='csr')
else:
contsetupstr = 'NV{0}NU{1}NY{2}'.format(NV, NU, NY)
def get_fdstr(Re):
return ddir + problemname + '_Re{0}_'.format(Re) + \
contsetupstr + prbstr
fdstr = get_fdstr(Re)
soldict = stokesmatsc # containing A, J, JT
soldict.update(femp) # adding V, Q, invinds, diribcs
# soldict.update(rhsd_vfrc) # adding fvc, fpr
soldict.update(fv=rhsd_stbc['fv']+rhsd_vfrc['fvc'],
fp=rhsd_stbc['fp']+rhsd_vfrc['fpr'],
N=N, nu=nu, data_prfx=fdstr)
#
# Prepare for control
#
# get the control and observation operators
if not bccontrol:
try:
b_mat = dou.load_spa(ddir + contsetupstr + '__b_mat')
u_masmat = dou.load_spa(ddir + contsetupstr + '__u_masmat')
print 'loaded `b_mat`'
except IOError:
print 'computing `b_mat`...'
b_mat, u_masmat = cou.get_inp_opa(cdcoo=femp['cdcoo'], V=femp['V'],
NU=NU, xcomp=uspacedep)
dou.save_spa(b_mat, ddir + contsetupstr + '__b_mat')
dou.save_spa(u_masmat, ddir + contsetupstr + '__u_masmat')
b_mat = b_mat[invinds, :][:, :]
# tb_mat = 1./np.sqrt(alphau)
try:
mc_mat = dou.load_spa(ddir + contsetupstr + '__mc_mat')
y_masmat = dou.load_spa(ddir + contsetupstr + '__y_masmat')
print 'loaded `c_mat`'
except IOError:
print 'computing `c_mat`...'
mc_mat, y_masmat = cou.get_mout_opa(odcoo=femp['odcoo'],
V=femp['V'], NY=NY)
dou.save_spa(mc_mat, ddir + contsetupstr + '__mc_mat')
dou.save_spa(y_masmat, ddir + contsetupstr + '__y_masmat')
c_mat = lau.apply_massinv(y_masmat, mc_mat, output='sparse')
# restrict the operators to the inner nodes
mc_mat = mc_mat[:, invinds][:, :]
c_mat = c_mat[:, invinds][:, :]
c_mat_reg = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=c_mat.T,
transposedprj=True).T
# c_mat_reg = np.array(c_mat.todense())
# TODO: right choice of norms for y
# and necessity of regularization here
# by now, we go on number save
#
# setup the system for the correction
#
#
# compute the uncontrolled steady state Stokes solution
#
v_ss_nse, list_norm_nwtnupd = snu.solve_steadystate_nse(**soldict)
(convc_mat, rhs_con,
rhsv_conbc) = snu.get_v_conv_conts(prev_v=v_ss_nse, invinds=invinds,
V=femp['V'], diribcs=femp['diribcs'])
f_mat = - stokesmatsc['A'] - convc_mat
mmat = stokesmatsc['M']
# ssv_rhs = rhsv_conbc + rhsv_conbc + rhsd_vfrc['fvc'] + rhsd_stbc['fv']
if plain_bt:
get_gramians = pru.solve_proj_lyap_stein
else:
get_gramians = pru.proj_alg_ric_newtonadi
truncstr = '__lqgbtcv{0}'.format(trunc_lqgbtcv)
try:
tl = dou.load_npa(fdstr + '__tl' + truncstr)
tr = dou.load_npa(fdstr + '__tr' + truncstr)
print 'loaded the left and right transformations: \n' + \
fdstr + '__tl/__tr' + truncstr
except IOError:
print 'computing the left and right transformations' + \
' and saving to: \n' + fdstr + '__tl/__tr' + truncstr
try:
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
print 'loaded factor of the Gramians: \n\t' + \
fdstr + '__zwc/__zwo'
except IOError:
zinic, zinio = None, None
if use_ric_ini is not None:
fdstr = get_fdstr(use_ric_ini)
try:
zinic = dou.load_npa(fdstr + '__zwc')
zinio = dou.load_npa(fdstr + '__zwo')
print 'Initialize Newton ADI by zwc/zwo from ' + fdstr
except IOError:
raise UserWarning('No initial guess with Re={0}'.
format(use_ric_ini))
fdstr = get_fdstr(Re)
print 'computing factors of Gramians: \n\t' + \
fdstr + '__zwc/__zwo'
def compobsg():
try:
zwo = dou.load_npa(fdstr + '__zwo')
print 'at least __zwo is there'
except IOError:
zwo = get_gramians(mmat=mmat.T, amat=f_mat.T,
jmat=stokesmatsc['J'],
bmat=c_mat_reg.T, wmat=b_mat,
nwtn_adi_dict=nap,
z0=zinio)['zfac']
dou.save_npa(zwo, fdstr + '__zwo')
def compcong():
try:
zwc = dou.load_npa(fdstr + '__zwc')
print 'at least __zwc is there'
except IOError:
zwc = get_gramians(mmat=mmat, amat=f_mat,
jmat=stokesmatsc['J'],
bmat=b_mat, wmat=c_mat_reg.T,
nwtn_adi_dict=nap,
z0=zinic)['zfac']
dou.save_npa(zwc, fdstr + '__zwc')
if multiproc:
print '\n ### multithread start - ' +\
'output might be intermangled'
p1 = multiprocessing.Process(target=compobsg)
p2 = multiprocessing.Process(target=compcong)
p1.start()
p2.start()
p1.join()
p2.join()
print '### multithread end'
else:
compobsg()
compcong()
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
print 'computing the left and right transformations' + \
' and saving to:\n' + fdstr + '__tr/__tl' + truncstr
tl, tr = btu.\
compute_lrbt_transfos(zfc=zwc, zfo=zwo,
mmat=stokesmatsc['M'],
trunck={'threshh': trunc_lqgbtcv})
dou.save_npa(tl, fdstr + '__tl' + truncstr)
dou.save_npa(tr, fdstr + '__tr' + truncstr)
print 'NV = {0}, NP = {2}, k = {1}'.\
format(tl.shape[0], tl.shape[1], stokesmatsc['J'].shape[0])
if comp_freqresp:
btu.compare_freqresp(mmat=stokesmatsc['M'], amat=f_mat,
jmat=stokesmatsc['J'], bmat=b_mat,
cmat=c_mat, tr=tr, tl=tl,
plot=True, datastr=fdstr + '__tl' + truncstr)
if comp_stepresp is not False:
if comp_stepresp == 'nonlinear':
stp_rsp_nwtn = 3
stp_rsp_dtpr = 'nonl_stepresp_'
else:
stp_rsp_nwtn = 1
stp_rsp_dtpr = 'stepresp_'
def fullstepresp_lnse(bcol=None, trange=None, ini_vel=None,
cmat=None, soldict=None):
soldict.update(fv_stbc=rhsd_stbc['fv']+bcol,
vel_nwtn_stps=stp_rsp_nwtn, trange=trange,
iniv=ini_vel, lin_vel_point=ini_vel,
clearprvdata=True, data_prfx=stp_rsp_dtpr,
return_dictofvelstrs=True)
dictofvelstrs = snu.solve_nse(**soldict)
return cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
c_mat=cmat, load_data=dou.load_npa)
# differences in the initial vector
# print np.dot(c_mat_reg, v_ss_nse)
# print np.dot(np.dot(c_mat_reg, tr),
# np.dot(tl.T, stokesmatsc['M']*v_ss_nse))
jsonstr = fdstr + stp_rsp_dtpr + '_Nred{0}_t0tENts{1}{2}{3}.json'.\
format(tl.shape[1], t0, tE, Nts)
btu.compare_stepresp(tmesh=np.linspace(t0, tE, Nts),
a_mat=f_mat, c_mat=c_mat_reg, b_mat=b_mat,
m_mat=stokesmatsc['M'],
tr=tr, tl=tl, iniv=v_ss_nse,
# ss_rhs=ssv_rhs,
fullresp=fullstepresp_lnse, fsr_soldict=soldict,
plot=True, jsonstr=jsonstr)
# compute the regulated system
trange = np.linspace(t0, tE, Nts)
if closed_loop is False:
return
elif closed_loop == 'full_state_fb':
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
mtxtb = pru.get_mTzzTtb(stokesmatsc['M'].T, zwc, b_mat)
def fv_tmdp_fullstatefb(time=None, curvel=None,
linv=None, tb_mat=None, tbxm_mat=None, **kw):
"""realizes a full state static feedback as a function
that can be passed to a solution routine for the
unsteady Navier-Stokes equations
Parameters
----------
time : real
current time
curvel : (N,1) nparray
current velocity
linv : (N,1) nparray
linearization point for the linear model
tb_mat : (N,K) nparray
input matrix containing the input weighting
tbxm_mat : (N,K) nparray
`tb_mat * gain * mass`
Returns
-------
actua : (N,1) nparray
current contribution to the right-hand side
, : dictionary
dummy `{}` for consistency
"""
actua = -lau.comp_uvz_spdns(tb_mat, tbxm_mat, curvel-linv)
# actua = 0*curvel
print '\nnorm of deviation', np.linalg.norm(curvel-linv)
# print 'norm of actuation {0}'.format(np.linalg.norm(actua))
return actua, {}
tmdp_fsfb_dict = dict(linv=v_ss_nse, tb_mat=b_mat, tbxm_mat=mtxtb.T)
fv_tmdp = fv_tmdp_fullstatefb
fv_tmdp_params = tmdp_fsfb_dict
fv_tmdp_memory = None
elif closed_loop == 'red_output_fb':
try:
xok = dou.load_npa(fdstr+truncstr+'__xok')
xck = dou.load_npa(fdstr+truncstr+'__xck')
ak_mat = dou.load_npa(fdstr+truncstr+'__ak_mat')
ck_mat = dou.load_npa(fdstr+truncstr+'__ck_mat')
bk_mat = dou.load_npa(fdstr+truncstr+'__bk_mat')
except IOError:
print 'couldn"t load the red system - compute it'
zwc = dou.load_npa(fdstr + '__zwc')
zwo = dou.load_npa(fdstr + '__zwo')
ak_mat = np.dot(tl.T, f_mat*tr)
ck_mat = lau.mm_dnssps(c_mat_reg, tr)
bk_mat = lau.mm_dnssps(tl.T, b_mat)
tltm, trtm = tl.T*stokesmatsc['M'], tr.T*stokesmatsc['M']
xok = np.dot(np.dot(tltm, zwo), np.dot(zwo.T, tltm.T))
xck = np.dot(np.dot(trtm, zwc), np.dot(zwc.T, trtm.T))
dou.save_npa(xok, fdstr+truncstr+'__xok')
dou.save_npa(xck, fdstr+truncstr+'__xck')
dou.save_npa(ak_mat, fdstr+truncstr+'__ak_mat')
dou.save_npa(ck_mat, fdstr+truncstr+'__ck_mat')
dou.save_npa(bk_mat, fdstr+truncstr+'__bk_mat')
obs_bk = np.dot(xok, ck_mat.T)
DT = (tE - t0)/(Nts-1)
sysmatk_inv = np.linalg.inv(np.eye(ak_mat.shape[1]) - DT*(ak_mat -
np.dot(np.dot(xok, ck_mat.T), ck_mat) -
np.dot(bk_mat, np.dot(bk_mat.T, xck))))
def fv_tmdp_redoutpfb(time=None, curvel=None, memory=None,
linvel=None,
ipsysk_mat_inv=None,
obs_bk=None, cts=None,
b_mat=None, c_mat=None,
xck=None, bk_mat=None,
**kw):
"""realizes a reduced static output feedback as a function
that can be passed to a solution routine for the
unsteady Navier-Stokes equations
For convinience the
Parameters
----------
time : real
current time
curvel : (N,1) nparray
current velocity. For consistency, the full state is taken
as input. However, internally, we only use the observation
`y = c_mat*curvel`
memory : dictionary
contains values from previous call, in particular the
previous state estimate
linvel : (N,1) nparray
linearization point for the linear model
ipsysk_mat_inv : (K,K) nparray
inverse of the system matrix that defines the update
of the state estimate
obs_bk : (K,NU) nparray
input matrix in the observer
cts : real
time step length
b_mat : (N,NU) sparse matrix
input matrix of the full system
c_mat=None,
c_mat : (NY,N) sparse matrix
output matrix of the full system
xck : (K,K) nparray
reduced solution of the CARE
bk_mat : (K,NU) nparray
reduced input matrix
Returns
-------
actua : (N,1) nparray
the current actuation
memory : dictionary
to be passed back in the next timestep
"""
xk_old = memory['xk_old']
buk = cts*np.dot(obs_bk,
lau.mm_dnssps(c_mat, (curvel-linvel)))
xk_old = np.dot(ipsysk_mat_inv, xk_old + buk)
# cts*np.dot(obs_bk,
# lau.mm_dnssps(c_mat, (curvel-linvel))))
memory['xk_old'] = xk_old
actua = -lau.mm_dnssps(b_mat,
np.dot(bk_mat.T, np.dot(xck, xk_old)))
print '\nnorm of deviation', np.linalg.norm(curvel-linvel)
print 'norm of actuation {0}'.format(np.linalg.norm(actua))
return actua, memory
fv_rofb_dict = dict(cts=DT, linvel=v_ss_nse, b_mat=b_mat,
c_mat=c_mat_reg, obs_bk=obs_bk, bk_mat=bk_mat,
ipsysk_mat_inv=sysmatk_inv, xck=xck)
fv_tmdp = fv_tmdp_redoutpfb
fv_tmdp_params = fv_rofb_dict
fv_tmdp_memory = dict(xk_old=np.zeros((tl.shape[1], 1)))
else:
fv_tmdp = None
fv_tmdp_params = {}
fv_tmdp_memory = {}
perturbini = perturbpara*np.ones((NV, 1))
reg_pertubini = lau.app_prj_via_sadpnt(amat=stokesmatsc['M'],
jmat=stokesmatsc['J'],
rhsv=perturbini)
soldict.update(fv_stbc=rhsd_stbc['fv'],
trange=trange,
iniv=v_ss_nse + reg_pertubini,
lin_vel_point=None,
clearprvdata=True, data_prfx=fdstr + truncstr,
fv_tmdp=fv_tmdp,
comp_nonl_semexp=True,
fv_tmdp_params=fv_tmdp_params,
fv_tmdp_memory=fv_tmdp_memory,
return_dictofvelstrs=True)
outstr = truncstr + '{0}'.format(closed_loop) \
+ 't0{0}tE{1}Nts{2}N{3}Re{4}'.format(t0, tE, Nts, N, Re)
if paraoutput:
soldict.update(paraviewoutput=True,
vfileprfx='results/vel_'+outstr,
pfileprfx='results/p_'+outstr)
dictofvelstrs = snu.solve_nse(**soldict)
yscomplist = cou.extract_output(strdict=dictofvelstrs, tmesh=trange,
c_mat=c_mat, load_data=dou.load_npa)
dou.save_output_json(dict(tmesh=trange.tolist(), outsig=yscomplist),
fstring=fdstr + truncstr + '{0}'.format(closed_loop) +
't0{0}tE{1}Nts{2}'.format(t0, tE, Nts) +
'inipert{0}'.format(perturbpara))
dou.plot_outp_sig(tmesh=trange, outsig=yscomplist)
def _mm_nonednssps(A, vvec):
if A is None:
return 0 * vvec
else:
return lau.mm_dnssps(A, vvec)
dmsz = rbd['hdms'][1:, :1] # part of the mass matrix related to the ini value
msz = rbd['hms'][1:, :1]
dmsI = rbd['hdms'][1:, 1:] # mass matrices w/o ini values (test and trial)
msI = rbd['hms'][1:, 1:]
solmat = np.kron(dmsI, rbd['hmy']) + np.kron(msI, rbd['hay'])
hshysol = np.dot(Ukslin.T, np.dot(xlin.T, Ukylin))
hshysolvec = hshysol.reshape((hs*hq, 1), order='C')
inivlin = hshysolvec[:hq, :]
insollin = hshysolvec[hq:, :]
linrhs = - np.kron(dmsz, lau.mm_dnssps(rbd['hmy'], inivlin)) \
- np.kron(msz, lau.mm_dnssps(rbd['hay'], inivlin))
print 'residual projsol: ', norm(np.dot(solmat, insollin) - linrhs)
print norm(xlin.T - np.dot(Ukslin, np.dot(hshysol, Ukylin.T)))
linsyssolvec = np.linalg.solve(solmat, linrhs)
diff = linsyssolvec - insollin
print 'diff ', norm(diff)
print 'residual linsyssol: ', norm(np.dot(solmat, linsyssolvec) - linrhs)
print 'cond A ', np.linalg.cond(solmat)
linsyssol = linsyssolvec.reshape((hs-1, hq))
plotmat(vvl, fignum=113)
plotmat(np.dot(Ukslin, np.dot(hshysol, Ukylin.T)), fignum=114)
plotmat(linsyssol, fignum=115)
def proj_alg_ric_newtonadi(mmat=None, amat=None, jmat=None,
bmat=None, wmat=None, z0=None, mtxoldb=None,
transposed=False,
nwtn_adi_dict=dict(adi_max_steps=150,
adi_newZ_reltol=1e-5,
nwtn_max_steps=14,
nwtn_upd_reltol=1e-8),
**kw):
""" solve the projected algebraic ricc via newton adi
`M.T*X*A + A.T*X*M - M.T*X*B*B.T*X*M + J(Y) = -WW.T`
`JXM = 0 and M.TXJ.T = 0`
If `mtxb` is given,
(e.g. as the feedback computed in a previous step of a Newton iteration),
the coefficient matrix with feedback
`A.T <- A.T - mtxb*b`
is considered
"""
if transposed:
mt, at = mmat, amat
else:
mt, at = mmat.T, amat.T
loctransposed = True
if sps.isspmatrix(wmat):
wmat = np.array(wmat.todense())
znc = z0
nwtn_stp, upd_fnorm, upd_fnorm_n = 0, None, None
nwtn_upd_fnorms = []
# import pdb
# pdb.set_trace()
while nwtn_stp < nwtn_adi_dict['nwtn_max_steps']:
if znc is None: # i.e., if z0 was None
rhsadi = wmat
mtxbt = None
else:
mtxb = mt * np.dot(znc, lau.mm_dnssps(znc.T, bmat))
mtxbt = mtxb.T
rhsadi = np.hstack([mtxb, wmat])
# to avoid a dense matrix we use the smw formula
# to compute (A-UV).-T
# for the factorization mTxg.T = tb * mTxtb.T = U*V
# and we add the previous feedback:
if mtxoldb is not None:
mtxbt = mtxbt + mtxoldb.T
znn = solve_proj_lyap_stein(amat=at, mmat=mt, jmat=jmat,
wmat=rhsadi,
umat=bmat, vmat=mtxbt,
transposed=loctransposed,
nwtn_adi_dict=nwtn_adi_dict)['zfac']
if nwtn_adi_dict['full_upd_norm_check']:
if znc is None: # there was no initial guess
znc = 0*znn
upd_fnorm = lau.comp_sqfnrm_factrd_diff(znn, znc)
upd_fnorm = np.sqrt(np.abs(upd_fnorm))
else:
if znc is None: # there was no initial guess
znc = 0*znn
vec = np.random.randn(znn.shape[0], 1)
vecn1 = comp_diff_zzv(znn, znc, vec)
vec = np.random.randn(znn.shape[0], 1)
vecn2 = comp_diff_zzv(znn, znc, vec)
vec = np.random.randn(znn.shape[0], 1)
# to make the estimate relative
vecn3 = np.linalg.norm(np.dot(znn, np.dot(znn.T, vec)))
if (vecn2 + vecn1)/vecn3 < 8e-9:
upd_fnorm, nzn, nzc = lau.\
comp_sqfnrm_factrd_diff(znn, znc, ret_sing_norms=True)
upd_fnorm_n = np.sqrt(np.abs(upd_fnorm) / np.abs(nzn))
nwtn_upd_fnorms.append(upd_fnorm_n)
try:
if np.allclose(upd_fnorm_n, upd_fnorm):
print('no more change in the norm of the update... break')
break
except TypeError:
pass
if nwtn_adi_dict['full_upd_norm_check']:
upd_fnorm = upd_fnorm_n
elif (vecn2 + vecn1)/vecn3 < 8e-9:
upd_fnorm = upd_fnorm_n
try:
if nwtn_adi_dict['verbose']:
print(('Newton ADI step: {1} -- ' +
'rel f norm of update: {0}').format(upd_fnorm,
nwtn_stp + 1))
if not nwtn_adi_dict['full_upd_norm_check']:
print(('btw, we decided whether to compute the actual ' +
'norm on the base of estimates:'))
print('|| upd * vec || / || vec || = {0}'.format(vecn2))
print('||Z*vec|| = {0}'.format(vecn3))
except KeyError:
pass # no verbosity specified - nothing is shown
znc = znn
nwtn_stp += 1
if (upd_fnorm is not None
and upd_fnorm < nwtn_adi_dict['nwtn_upd_reltol']):
break
return dict(zfac=znn, nwtn_upd_fnorms=nwtn_upd_fnorms)
def _bbwdrhs(t):
# TODO: -----------------------------> here we need vstar
return (-lau.mm_dnssps(LVhmy, redvfun(te - t)).flatten() +
lau.mm_dnssps(LVhmy, _redvstarfun(t)).flatten())
def _bbwdnl_vdx(lvec, t):
vdx = vdxoperator(vfun(te - t))
return -(lau.mm_dnssps(vdx, lvec)).flatten()
def redmatfunc(vvec):
return np.dot(lau.mm_dnssps(ULk.T, matfunc(np.dot(UVk, vvec))), ULk)
def compare_stepresp(tmesh=None,
a_mat=None,
c_mat=None,
b_mat=None,
m_mat=None,
tl=None,
tr=None,
iniv=None,
fullresp=None,
fsr_soldict=None,
plot=False,
jsonstr=None):
""" compare the system's step response to unit inputs in time domain
with reduced system's response.
We consider the system
.. math::
M\\dot v = Av + Bu, \\quad y = Cv
on the discretized time interval :math:`(t_0, t_E)`
and the reduced system with :math:`\\hat A = t_l^TAt_r`.
Parameters
----------
tmesh : iterable list or ndarray
vector of the time instances
a_mat : (N,N) sparse matrix
system matrix
c_mat : (ny,N) sparse matrix or ndarray
output matrix
b_mat : (N,nu) sparse matrix or ndarray
input operator
m_mat : (N,N) sparse matrix
mass matrix
tl, tr : (N,K) ndarrays
left, right transformation for the reduced system
iniv : (N,1) ndarray
initial value and linearization point of the full system
fullresp : callable f(v, **dict)
returns the response of the full system
fsr_soldict : dictionary
parameters to be passed to `fullresp`
plot : boolean, optional
whether to plot, defaults to `False`
jsonstr: string, optional
if defined, the output is stored in this json file, defaults to `None`
"""
from scipy.integrate import odeint
ahat = np.dot(tl.T, a_mat * tr)
chat = lau.mm_dnssps(c_mat, tr)
inivhat = np.dot(tl.T, m_mat * iniv)
inivout = lau.mm_dnssps(c_mat, iniv).tolist()
red_stp_rsp, ful_stp_rsp = [], []
for ccol in [0]: # , b_mat.shape[1]-1]: # range(2): # b_mat.shape[1]):
bmc = b_mat[:, ccol][:, :]
red_bmc = tl.T * bmc
def dtfunc(v, t):
return (np.dot(ahat, v).flatten() + red_bmc.flatten()) # +\
red_state = odeint(dtfunc, 0 * inivhat.flatten(), tmesh)
red_stp_rsp.append(np.dot(chat, red_state.T).T.tolist())
ful_stp_rsp.append(
fullresp(bcol=bmc,
trange=tmesh,
ini_vel=iniv,
cmat=c_mat,
soldict=fsr_soldict))
if jsonstr:
try:
tmesh = tmesh.tolist()
except AttributeError:
pass # is a list already
dou.save_output_json(datadict={
"tmesh": tmesh,
"ful_stp_rsp": ful_stp_rsp,
"red_stp_rsp": red_stp_rsp,
"inivout": inivout
},
fstring=jsonstr,
module='sadptprj_riclyap_adi.bal_trunc_utils',
plotroutine='plot_step_resp')
if plot:
plot_step_resp(tmesh=tmesh,
red_stp_rsp=red_stp_rsp,
ful_stp_rsp=ful_stp_rsp,
inivout=inivout)