def plot_trajectory(ppp, x0, u_seq, ssys,
ax=None, color_seed=None):
"""Plots a PropPreservingPartition and the trajectory generated by x0
input sequence u_seq.
See Also
========
C{plot_partition}, plot
@type ppp: L{PropPreservingPartition}
@param x0: initial state
@param u_seq: matrix where each row contains an input
@param ssys: system dynamics
@param color_seed: see C{plot_partition}
@return: axis object
"""
try:
from tulip.graphics import newax
except:
logger.error('failed to import graphics.newax')
return
if ax is None:
ax, fig = newax()
plot_partition(plot_numbers=False, ax=ax, show=False)
A = ssys.A
B = ssys.B
if ssys.K is not None:
K = ssys.K
else:
K = np.zeros(x0.shape)
x = x0.flatten()
x_arr = x0
for i in range(u_seq.shape[0]):
x = (
np.dot(A, x).flatten() +
np.dot(B, u_seq[i, :] ).flatten() +
K.flatten())
x_arr = np.vstack([x_arr, x.flatten()])
ax.plot(x_arr[:,0], x_arr[:,1], 'o-')
return ax
def plot_trajectory(ppp, x0, u_seq, ssys,
ax=None, color_seed=None):
"""Plots a PropPreservingPartition and the trajectory generated by x0
input sequence u_seq.
See Also
========
C{plot_partition}, plot
@type ppp: L{PropPreservingPartition}
@param x0: initial state
@param u_seq: matrix where each row contains an input
@param ssys: system dynamics
@param color_seed: see C{plot_partition}
@return: axis object
"""
try:
from tulip.graphics import newax
except:
logger.error('failed to import graphics.newax')
return
if ax is None:
ax, fig = newax()
plot_partition(plot_numbers=False, ax=ax, show=False)
A = ssys.A
B = ssys.B
if ssys.K is not None:
K = ssys.K
else:
K = np.zeros(x0.shape)
x = x0.flatten()
x_arr = x0
for i in range(u_seq.shape[0]):
x = np.dot(A, x).flatten() +\
np.dot(B, u_seq[i, :] ).flatten() +\
K.flatten()
x_arr = np.vstack([x_arr, x.flatten()])
ax.plot(x_arr[:,0], x_arr[:,1], 'o-')
return ax
def plot(
self, trans=None, ppp2trans=None, only_adjacent=False,
ax=None, plot_numbers=True, color_seed=None
):
"""For details see C{polytope.plot.plot_partition}.
"""
return plot_partition(
self, trans, ppp2trans, only_adjacent,
ax, plot_numbers, color_seed
)
def plot(self,
trans=None,
ppp2trans=None,
only_adjacent=False,
ax=None,
plot_numbers=True,
color_seed=None):
"""For details see C{polytope.plot.plot_partition}.
"""
return plot_partition(self, trans, ppp2trans, only_adjacent, ax,
plot_numbers, color_seed)
def plot_strategy(ab, mealy):
"""Plot strategic transitions on PPP.
Assumes that mealy is feasible for ab.
@type ab: L{AbstractPwa} or L{AbstractSwitched}
@type mealy: L{transys.MealyMachine}
"""
proj_mealy = project_strategy_on_partition(ab.ppp, mealy)
ax = plot_partition(ab.ppp, proj_mealy, color_seed=0)
return ax
def plot_strategy(ab, mealy):
"""Plot strategic transitions on PPP.
Assumes that mealy is feasible for ab.
@type ab: L{AbstractPwa} or L{AbstractSwitched}
@type mealy: L{transys.MealyMachine}
"""
proj_mealy = project_strategy_on_partition(ab.ppp, mealy)
ax = plot_partition(ab.ppp, proj_mealy, color_seed=0)
return ax
def discretize(
part, ssys, N=10, min_cell_volume=0.1,
closed_loop=True, conservative=False,
max_num_poly=5, use_all_horizon=False,
trans_length=1, remove_trans=False,
abs_tol=1e-7,
plotit=False, save_img=False, cont_props=None,
plot_every=1
):
"""Refine the partition and establish transitions
based on reachability analysis.
Reference
=========
U{[NOTM12]
<https://tulip-control.sourceforge.io/doc/bibliography.html#notm12>}
See Also
========
L{prop2partition.pwa_partition}, L{prop2partition.part2convex}
@param part: L{PropPreservingPartition} object
@param ssys: L{LtiSysDyn} or L{PwaSysDyn} object
@param N: horizon length
@param min_cell_volume: the minimum volume of cells in the resulting
partition.
@param closed_loop: boolean indicating whether the `closed loop`
algorithm should be used. default True.
@param conservative: if true, force sequence in reachability analysis
to stay inside starting cell. If false, safety
is ensured by keeping the sequence inside a convexified
version of the original proposition preserving cell.
@param max_num_poly: maximum number of polytopes in a region to use in
reachability analysis.
@param use_all_horizon: in closed loop algorithm: if we should look
for reachability also in less than N steps.
@param trans_length: the number of polytopes allowed to cross in a
transition. a value of 1 checks transitions
only between neighbors, a value of 2 checks
neighbors of neighbors and so on.
@param remove_trans: if True, remove found transitions between
non-neighbors.
@param abs_tol: maximum volume for an "empty" polytope
@param plotit: plot partitioning as it evolves
@type plotit: boolean,
default = False
@param save_img: save snapshots of partitioning to PDF files,
requires plotit=True
@type save_img: boolean,
default = False
@param cont_props: continuous propositions to plot
@type cont_props: list of C{Polytope}
@rtype: L{AbstractPwa}
"""
start_time = os.times()[0]
orig_ppp = part
min_cell_volume = (min_cell_volume /np.finfo(np.double).eps
*np.finfo(np.double).eps)
ispwa = isinstance(ssys, PwaSysDyn)
islti = isinstance(ssys, LtiSysDyn)
if ispwa:
(part, ppp2pwa, part2orig) = pwa_partition(ssys, part)
else:
part2orig = range(len(part))
# Save original polytopes, require them to be convex
if conservative:
orig_list = None
orig = [0]
else:
(part, new2old) = part2convex(part) # convexify
part2orig = [part2orig[i] for i in new2old]
# map new regions to pwa subsystems
if ispwa:
ppp2pwa = [ppp2pwa[i] for i in new2old]
remove_trans = False # already allowed in nonconservative
orig_list = []
for poly in part:
if len(poly) == 0:
orig_list.append(poly.copy())
elif len(poly) == 1:
orig_list.append(poly[0].copy())
else:
raise Exception("discretize: "
"problem in convexification")
orig = list(range(len(orig_list)))
# Cheby radius of disturbance set
# (defined within the loop for pwa systems)
if islti:
if len(ssys.E) > 0:
rd = ssys.Wset.chebR
else:
rd = 0.
# Initialize matrix for pairs to check
IJ = part.adj.copy()
IJ = IJ.todense()
IJ = np.array(IJ)
logger.debug("\n Starting IJ: \n" + str(IJ) )
# next line omitted in discretize_overlap
IJ = reachable_within(trans_length, IJ,
np.array(part.adj.todense()) )
# Initialize output
num_regions = len(part)
transitions = np.zeros(
[num_regions, num_regions],
dtype = int
)
sol = deepcopy(part.regions)
adj = part.adj.copy()
adj = adj.todense()
adj = np.array(adj)
# next 2 lines omitted in discretize_overlap
if ispwa:
subsys_list = list(ppp2pwa)
else:
subsys_list = None
ss = ssys
# init graphics
if plotit:
try:
import matplotlib.pyplot as plt
plt.ion()
fig, (ax1, ax2) = plt.subplots(1, 2)
ax1.axis('scaled')
ax2.axis('scaled')
file_extension = 'pdf'
except:
logger.error('failed to import matplotlib')
plt = None
else:
plt = None
iter_count = 0
# List of how many "new" regions
# have been created for each region
# and a list of original number of neighbors
#num_new_reg = np.zeros(len(orig_list))
#num_orig_neigh = np.sum(adj, axis=1).flatten() - 1
progress = list()
# Do the abstraction
while np.sum(IJ) > 0:
ind = np.nonzero(IJ)
# i,j swapped in discretize_overlap
i = ind[1][0]
j = ind[0][0]
IJ[j, i] = 0
si = sol[i]
sj = sol[j]
si_tmp = deepcopy(si)
sj_tmp = deepcopy(sj)
#num_new_reg[i] += 1
#print(num_new_reg)
if ispwa:
ss = ssys.list_subsys[subsys_list[i]]
if len(ss.E) > 0:
rd, xd = pc.cheby_ball(ss.Wset)
else:
rd = 0.
if conservative:
# Don't use trans_set
trans_set = None
else:
# Use original cell as trans_set
trans_set = orig_list[orig[i]]
S0 = solve_feasible(
si, sj, ss, N, closed_loop,
use_all_horizon, trans_set, max_num_poly
)
msg = '\n Working with partition cells: ' + str(i) + ', ' + str(j)
logger.info(msg)
msg = '\t' + str(i) +' (#polytopes = ' +str(len(si) ) +'), and:\n'
msg += '\t' + str(j) +' (#polytopes = ' +str(len(sj) ) +')\n'
if ispwa:
msg += '\t with active subsystem: '
msg += str(subsys_list[i]) + '\n'
msg += '\t Computed reachable set S0 with volume: '
msg += str(S0.volume) + '\n'
logger.debug(msg)
#logger.debug('si \cap s0')
isect = si.intersect(S0)
vol1 = isect.volume
risect, xi = pc.cheby_ball(isect)
#logger.debug('si \ s0')
diff = si.diff(S0)
vol2 = diff.volume
rdiff, xd = pc.cheby_ball(diff)
# if pc.is_fulldim(pc.Region([isect]).intersect(diff)):
# logging.getLogger('tulip.polytope').setLevel(logging.DEBUG)
# diff = pc.mldivide(si, S0, save=True)
#
# ax = S0.plot()
# ax.axis([0.0, 1.0, 0.0, 2.0])
# ax.figure.savefig('./img/s0.pdf')
#
# ax = si.plot()
# ax.axis([0.0, 1.0, 0.0, 2.0])
# ax.figure.savefig('./img/si.pdf')
#
# ax = isect.plot()
# ax.axis([0.0, 1.0, 0.0, 2.0])
# ax.figure.savefig('./img/isect.pdf')
#
# ax = diff.plot()
# ax.axis([0.0, 1.0, 0.0, 2.0])
# ax.figure.savefig('./img/diff.pdf')
#
# ax = isect.intersect(diff).plot()
# ax.axis([0.0, 1.0, 0.0, 2.0])
# ax.figure.savefig('./img/diff_cap_isect.pdf')
#
# logger.error('Intersection \cap Difference != \emptyset')
#
# assert(False)
if vol1 <= min_cell_volume:
logger.warning('\t too small: si \cap Pre(sj), '
'so discard intersection')
if vol1 <= min_cell_volume and isect:
logger.warning('\t discarded non-empty intersection: '
'consider reducing min_cell_volume')
if vol2 <= min_cell_volume:
logger.warning('\t too small: si \ Pre(sj), so not reached it')
# We don't want our partitions to be smaller than the disturbance set
# Could be a problem since cheby radius is calculated for smallest
# convex polytope, so if we have a region we might throw away a good
# cell.
if (vol1 > min_cell_volume) and (risect > rd) and \
(vol2 > min_cell_volume) and (rdiff > rd):
# Make sure new areas are Regions and add proposition lists
if len(isect) == 0:
isect = pc.Region([isect], si.props)
else:
isect.props = si.props.copy()
if len(diff) == 0:
diff = pc.Region([diff], si.props)
else:
diff.props = si.props.copy()
# replace si by intersection (single state)
isect_list = pc.separate(isect)
sol[i] = isect_list[0]
# cut difference into connected pieces
difflist = pc.separate(diff)
difflist += isect_list[1:]
n_isect = len(isect_list) -1
num_new = len(difflist)
# add each piece, as a new state
for region in difflist:
sol.append(region)
# keep track of PWA subsystems map to new states
if ispwa:
subsys_list.append(subsys_list[i])
n_cells = len(sol)
new_idx = range(n_cells-1, n_cells-num_new-1, -1)
"""Update transition matrix"""
transitions = np.pad(transitions, (0,num_new), 'constant')
transitions[i, :] = np.zeros(n_cells)
for r in new_idx:
#transitions[:, r] = transitions[:, i]
# All sets reachable from start are reachable from both part's
# except possibly the new part
transitions[i, r] = 0
transitions[j, r] = 0
# sol[j] is reachable from intersection of sol[i] and S0
if i != j:
transitions[j, i] = 1
# sol[j] is reachable from each piece os S0 \cap sol[i]
#for k in range(n_cells-n_isect-2, n_cells):
# transitions[j, k] = 1
"""Update adjacency matrix"""
old_adj = np.nonzero(adj[i, :])[0]
# reset new adjacencies
adj[i, :] = np.zeros([n_cells -num_new])
adj[:, i] = np.zeros([n_cells -num_new])
adj[i, i] = 1
adj = np.pad(adj, (0, num_new), 'constant')
for r in new_idx:
adj[i, r] = 1
adj[r, i] = 1
adj[r, r] = 1
if not conservative:
orig = np.hstack([orig, orig[i]])
# adjacencies between pieces of isect and diff
for r in new_idx:
for k in new_idx:
if r is k:
continue
if pc.is_adjacent(sol[r], sol[k]):
adj[r, k] = 1
adj[k, r] = 1
msg = ''
if logger.getEffectiveLevel() <= logging.DEBUG:
msg += '\t\n Adding states ' + str(i) + ' and '
for r in new_idx:
msg += str(r) + ' and '
msg += '\n'
logger.debug(msg)
for k in np.setdiff1d(old_adj, [i,n_cells-1]):
# Every "old" neighbor must be the neighbor
# of at least one of the new
if pc.is_adjacent(sol[i], sol[k]):
adj[i, k] = 1
adj[k, i] = 1
elif remove_trans and (trans_length == 1):
# Actively remove transitions between non-neighbors
transitions[i, k] = 0
transitions[k, i] = 0
for r in new_idx:
if pc.is_adjacent(sol[r], sol[k]):
adj[r, k] = 1
adj[k, r] = 1
elif remove_trans and (trans_length == 1):
# Actively remove transitions between non-neighbors
transitions[r, k] = 0
transitions[k, r] = 0
"""Update IJ matrix"""
IJ = np.pad(IJ, (0,num_new), 'constant')
adj_k = reachable_within(trans_length, adj, adj)
sym_adj_change(IJ, adj_k, transitions, i)
for r in new_idx:
sym_adj_change(IJ, adj_k, transitions, r)
if logger.getEffectiveLevel() <= logging.DEBUG:
msg = '\n\n Updated adj: \n' + str(adj)
msg += '\n\n Updated trans: \n' + str(transitions)
msg += '\n\n Updated IJ: \n' + str(IJ)
logger.debug(msg)
logger.info('Divided region: ' + str(i) + '\n')
elif vol2 < abs_tol:
logger.info('Found: ' + str(i) + ' ---> ' + str(j) + '\n')
transitions[j,i] = 1
else:
if logger.level <= logging.DEBUG:
msg = '\t Unreachable: ' + str(i) + ' --X--> ' + str(j) + '\n'
msg += '\t\t diff vol: ' + str(vol2) + '\n'
msg += '\t\t intersect vol: ' + str(vol1) + '\n'
logger.debug(msg)
else:
logger.info('\t unreachable\n')
transitions[j,i] = 0
# check to avoid overlapping Regions
if debug:
tmp_part = PropPreservingPartition(
domain=part.domain,
regions=sol, adj=sp.lil_matrix(adj),
prop_regions=part.prop_regions
)
assert(tmp_part.is_partition() )
n_cells = len(sol)
progress_ratio = 1 - float(np.sum(IJ) ) /n_cells**2
progress += [progress_ratio]
msg = '\t total # polytopes: ' + str(n_cells) + '\n'
msg += '\t progress ratio: ' + str(progress_ratio) + '\n'
logger.info(msg)
iter_count += 1
# no plotting ?
if not plotit:
continue
if plt is None or plot_partition is None:
continue
if iter_count % plot_every != 0:
continue
tmp_part = PropPreservingPartition(
domain=part.domain,
regions=sol, adj=sp.lil_matrix(adj),
prop_regions=part.prop_regions
)
# plot pair under reachability check
ax2.clear()
si_tmp.plot(ax=ax2, color='green')
sj_tmp.plot(ax2, color='red', hatch='o', alpha=0.5)
plot_transition_arrow(si_tmp, sj_tmp, ax2)
S0.plot(ax2, color='none', hatch='/', alpha=0.3)
fig.canvas.draw()
# plot partition
ax1.clear()
plot_partition(tmp_part, transitions.T, ax=ax1, color_seed=23)
# plot dynamics
ssys.plot(ax1, show_domain=False)
# plot hatched continuous propositions
part.plot_props(ax1)
fig.canvas.draw()
# scale view based on domain,
# not only the current polytopes si, sj
l,u = part.domain.bounding_box
ax2.set_xlim(l[0,0], u[0,0])
ax2.set_ylim(l[1,0], u[1,0])
if save_img:
fname = 'movie' +str(iter_count).zfill(3)
fname += '.' + file_extension
fig.savefig(fname, dpi=250)
plt.pause(1)
new_part = PropPreservingPartition(
domain=part.domain,
regions=sol, adj=sp.lil_matrix(adj),
prop_regions=part.prop_regions
)
# check completeness of adjacency matrix
if debug:
tmp_part = deepcopy(new_part)
tmp_part.compute_adj()
# Generate transition system and add transitions
ofts = trs.FTS()
adj = sp.lil_matrix(transitions.T)
n = adj.shape[0]
ofts_states = range(n)
ofts.states.add_from(ofts_states)
ofts.transitions.add_adj(adj, ofts_states)
# Decorate TS with state labels
atomic_propositions = set(part.prop_regions)
ofts.atomic_propositions.add_from(atomic_propositions)
for state, region in zip(ofts_states, sol):
state_prop = region.props.copy()
ofts.states.add(state, ap=state_prop)
param = {
'N':N,
'trans_length':trans_length,
'closed_loop':closed_loop,
'conservative':conservative,
'use_all_horizon':use_all_horizon,
'min_cell_volume':min_cell_volume,
'max_num_poly':max_num_poly
}
ppp2orig = [part2orig[x] for x in orig]
end_time = os.times()[0]
msg = 'Total abstraction time: ' +\
str(end_time - start_time) + '[sec]'
print(msg)
logger.info(msg)
if save_img and plt is not None:
fig, ax = plt.subplots(1, 1)
plt.plot(progress)
ax.set_xlabel('iteration')
ax.set_ylabel('progress ratio')
ax.figure.savefig('progress.pdf')
return AbstractPwa(
ppp=new_part,
ts=ofts,
ppp2ts=ofts_states,
pwa=ssys,
pwa_ppp=part,
ppp2pwa=orig,
ppp2sys=subsys_list,
orig_ppp=orig_ppp,
ppp2orig=ppp2orig,
disc_params=param
)