def test_ode_rk23_simpleCase(self):
# setup parameters and initial point for testing function
x0 = np.array([4.0])
par_a = -0.2
# setup time interval [a,b] for integration
a = 2.0
b = 4.0
# setup algorithm parameters for the chosen ode method to be tested
tol = 1e-8
maxIter = 300
# call integration methos
x, t, failFlag, iter_i = ode.ode_rk23(self.myfun,
a,
b,
x0,
tol,
maxIter,
par_a=par_a)
# compare with expected results
x_expect = math.exp(-0.4) * 4
self.assertTrue(abs(x_expect - x[-1, 0]) < 1e-6, msg=None)
def test_ode_rk23_normalCase(self):
# setup parameters and initial point for testing function
x0 = np.array([0.1, -0.3, 1.0])
matrixA = np.array([[-1.0, 0.1, -0.3], [0.2, -1, 0.1], [0.1, 0.4, -1]])
matrixB = np.array([[0.5, -0.6], [-0.7, 0.6], [0.4, 0.8]])
vectorU = np.array([0.2, 0.3])
vectorF = np.array([-0.1, 0 - 0.02, -0.2])
# setup time interval [a,b] for integration
a = 0.0
b = 5.0
# setup algorithm parameters for the chosen ode method to be tested
tol = 1e-8
maxIter = 300
# call integration methos
x, t, failFlag, iter_i = ode.ode_rk23(self.myfun,
a,
b,
x0,
tol,
maxIter,
matrixA=matrixA,
matrixB=matrixB,
vectorU=vectorU,
vectorF=vectorF)
# compare with expected results
# (note: the values here is not the real expected result, but copied
# from the output of ode_rk23, need a way to get expected result)
x_expect = np.array(
[-3.91136351e-01, -1.45401145e-01, -5.53579890e-01])
self.assertTrue(max(abs(x_expect - x[-1])) < 1e-6, msg=None)
def propagate_u(u_0, qp_vecs, t_start, t_end, sliding_window_instance, q_s_dot,
p_l_dot, p_mf_dot, q_mf_dot, q_mf, u_mf, H_l_D):
'''
Propagate u based on q and p values for one bucket in time.
Inputs:
u_0 (1D np.array of dimension self.control_dim): control values at beginning of bucket
qp_vecs (list of 1D np.arrays each of dimension self.state_dim*3): list of concatenated state, local costate, and mean field costate vectors from q and p propagation.
t_start (np.float): start time of bucket
t_end (np.float): end time of bucket
sliding_window_instance (instance of user-defined class which inherits SlidingWindow): object defining the dynamics, and the initial conditions and parameters for numerical integration/propagation. Same as agent.
q_s_dot (1D np.array of dimension self.state_dim): derivative of local state at end of bucket
p_l_dot (1D np.array of dimension self.state_dim): derivative of local state at end of bucket
p_mf_dot (1D np.array of dimension self.state_dim): derivative of local state at end of bucket
q_mf_dot (1D np.array of dimension total number of states in system): derivative of local state at end of bucket
q_mf (1D np.array of dimension of total number of states in system): mean field state, i.e. state vector holding values for ALL states, not just those pertaiing to the current agen.
u_mf (1D np.array of dimension self.control_dim): mean field control, i.e. control vector holding values for ALL controls, not just those pertaining to the current agent.
H_l_D (np.float): value of local desired Hamiltonian at beginning of window
Outputs:
u_vecs (list of 1-D numpy arrays): list of propagated control values for entire bucket in time
'''
u_vecs = []
u_vec = u_0
n_s = sliding_window_instance.n_s
steps = np.linspace(t_start, t_end, n_s + 1)
for i in range(n_s):
n_start, n_end = steps[i], steps[i + 1]
qp_vec = qp_vecs[i]
u_vec, t, failFlag, iter_i = ode.ode_rk23(
sliding_window_instance.u_rhs,
n_start,
n_end,
u_vec,
sliding_window_instance.integrateTol,
sliding_window_instance.integrateMaxIter,
state_dim=sliding_window_instance.state_dim,
Gamma=sliding_window_instance.Gamma,
qp_vec=qp_vec,
u_0=u_0,
q_s_dot=q_s_dot,
p_l_dot=p_l_dot,
p_mf_dot=p_mf_dot,
q_mf_dot=q_mf_dot,
q_mf=q_mf,
u_mf=u_mf,
H_l_D=H_l_D)
if len(u_vec) > 1:
u_vec_next = u_vec[-1]
else:
u_vec_next = u_vec
u_vecs.append(
u_vec_next
) # one u_vec for each step, append them and you have all the u_vecs for one bucket
u_vec = u_vec_next
return u_vecs
def test_ode_rk23_backwardCase(self):
'''
test backward integration of 2x2 matrix dynamics (i.e. in the dx/dt function x is 2x2 matrix)
Note: this test case is the reversed test case of
TestCaseMatrix('test_ode_rk23_simpleCase')
i.e. use result of TestCaseMatrix('test_ode_rk23_simpleCase') as
inital point, and integrate from the ending time of
TestCaseMatrix('test_ode_rk23_simpleCase')
and all the way back to the starting time of
TestCaseMatrix('test_ode_rk23_simpleCase').
The expected result of this test case should be very close to the
initial point of TestCaseMatrix('test_ode_rk23_simpleCase')
'''
# setup parameters and initial point for testing function
matrixX0 = np.array([[0.47734137, 0.10362487],
[0.11596301,
0.00871929]]) #np.array([[0.5,-0.2],[1.3,2.4]])
matrixA = np.array([[0.2, -0.1], [-0.4, 0.7]])
matrixB = np.array([[0.006, -0.003], [-0.002, 0.008]])
matrixQ = np.array([[0.006, -0.002], [-0.001, 0.005]])
# setup time interval [a,b] for integration (a>b, backward integration)
a = 5.0
b = 0.0
# setup algorithm parameters for the chosen ode method to be tested
tol = 1e-8
maxIter = 1000
# reshape matrixX0 from 2x2 matrix to 4x1 vector
dim_matrixX0 = matrixX0.shape
vectorX0 = matrixX0.reshape((dim_matrixX0[0] * dim_matrixX0[1], ))
# call integration methos
vectorX, t, failFlag, iter_i = ode.ode_rk23(self.myfun,
a,
b,
vectorX0,
tol,
maxIter,
matrixA=matrixA,
matrixB=matrixB,
matrixQ=matrixQ)
# reshape result from 4x1 to 2x2, and compare with expected result
# (note: the expected result values here is not the reversed values of real expected result, but copied
# from the output of ode_rk23, need a way to get expected result)
matrixX_result = vectorX[-1].reshape(matrixX0.shape)
matrixX_expect = np.array([[0.5, -0.2], [
1.3, 2.4
]]) #np.array([[ 0.47734137, 0.10362487],[0.11596301, 0.00871929]])
self.assertTrue(np.amax(abs(matrixX_expect - matrixX_result)) < 1e-4,
msg=None)
def test_ode_rk23_simpleCase(self):
'''
test forward integration of 2x2 matrix dynamics (i.e. in the dx/dt function x is 2x2 matrix)
'''
# setup parameters and initial point for testing function
matrixX0 = np.array([[0.5, -0.2], [1.3, 2.4]])
matrixA = np.array([[0.2, -0.1], [-0.4, 0.7]])
matrixB = np.array([[0.006, -0.003], [-0.002, 0.008]])
matrixQ = np.array([[0.006, -0.002], [-0.001, 0.005]])
# setup time interval [a,b] for integration
a = 0
b = 5
# setup algorithm parameters for the chosen ode method to be tested
tol = 1e-8
maxIter = 1000
# reshape matrixX0 from 2x2 matrix to 4x1 vector
dim_matrixX0 = matrixX0.shape
vectorX0 = matrixX0.reshape((dim_matrixX0[0] * dim_matrixX0[1], ))
# call integration methos
vectorX, t, failFlag, iter_i = ode.ode_rk23(self.myfun,
a,
b,
vectorX0,
tol,
maxIter,
matrixA=matrixA,
matrixB=matrixB,
matrixQ=matrixQ)
# reshape result from 4x1 to 2x2, and compare with expected result
# (note: the values here is not the real expected result, but copied
# from the output of ode_rk23, need a way to get expected result)
matrixX_result = vectorX[-1].reshape(matrixX0.shape)
matrixX_expect = np.array([[0.47734137, 0.10362487],
[0.11596301, 0.00871929]])
self.assertTrue(np.amax(abs(matrixX_expect - matrixX_result)) < 1e-6,
msg=None)
def test_ode_rk23_normaleCase(self):
'''
test forward integration of 9x9 matrix dynamics (i.e. in the dx/dt function x is 9x9 matrix)
'''
# setup parameters and initial point for testing function
matrixX0=np.array([[ 0.76, 0.22, 0.67, 0.19, 0.71, 0.26, 0.53, 0.14, 0.25],\
[ 0.45, 0.17, 0.74, 0.28, 0.17, 0.25, 0.8 , 0.08, 0.43],\
[ 0.69, 0.02, 0.6 , 0.29, 0.92, 0.32, 0.08, 0.56, 0.32],
[ 0.99, 0.08, 0.84, 0.24, 0.42, 0.25, 0.24, 0.52, 0.95],\
[ 0.31, 0.33, 0.06, 0.13, 0.38, 0.6 , 0.22, 0.83, 0.51],\
[ 0.2 , 0.21, 0.1 , 0.48, 0.89, 0.96, 0.09, 0.48, 0.25],\
[ 0.36, 0.16, 0.34, 0.49, 0.54, 0.32, 0.5 , 0.8 , 0.24],\
[ 0.69, 0.37, 0.69, 0.88, 0.89, 0.26, 0.36, 0.83, 0.72],\
[ 0.8 , 0.64, 0.93, 0.01, 0.59, 0.85, 0.84, 0.13, 0.92]])
matrixX0 = matrixX0 + np.eye(9) * 10
matrixA=np.array([[ 3.54, 0.5 , 0.3 , 0.06, 0.63, 0.68, 0.57, 0.41, 0.13],\
[ 0.04, 3.05, 0.46, 0.42, 0.57, 0.45, 0.49, 0.32, 0.27],\
[ 0.32, 0.79, 3.66, 0.19, 0.11, 0.37, 0.32, 0.63, 0.1 ],\
[ 0.24, 0.35, 0.59, 3.12, 0.93, 0.51, 0.86, 0.76, 0.48],\
[ 0.97, 0.86, 0.4 , 0.92, 3.35, 0.74, 0.62, 0.42, 0.37],\
[ 0.79, 0.02, 0.64, 0.64, 0.17, 3.42, 0.44, 0.26, 0.96],\
[ 0.48, 0.49, 0.4 , 0.45, 0.3 , 0.61, 3.24, 0.26, 0.65],\
[ 0.7 , 0.17, 0.91, 0.5 , 0.65, 0.55, 0.01, 3.69, 0.11],\
[ 0.15, 0.14, 0.39, 0.25, 0.44, 0.53, 0.93, 0.84, 3. ]])
matrixB=np.array([[-9.68, 0.5 , 0.5 , 0.33, 0.8 , 0. , 0.93, 0.91, 0.85],\
[ 0.79, -9.02, 0.55, 0.56, 0.64, 0. , 0.06, 0.46, 0.25],\
[ 0.62, 0.47, -9.82, 0.18, 0.41, 0.7 , 0.03, 0.84, 0.26],\
[ 0.32, 0.05, 0.59, -9.42, 0.63, 0.99, 0.47, 0.37, 0.2 ],\
[ 0.59, 0.38, 0.71, 0.27, -9.75, 0.82, 0.4 , 1. , 0.38],\
[ 0.85, 0.2 , 0.18, 0.46, 0.22, -9.05, 0.61, 0.97, 0.76],\
[ 0.75, 0.93, 0.53, 0.74, 0.52, 0.54, -9.8 , 0.31, 0.63],\
[ 0.32, 0.62, 0. , 0.62, 0.65, 0.9 , 0.74, -9.68, 0.07],\
[ 0.59, 0.4 , 0.57, 0.49, 0.58, 0.5 , 0.86, 0. , -9.94]])
matrixQ=np.array([[ 0.57, 0.08, 0.05, 0. , 0. , 0.06, 0.01, 0.02, 0.06],\
[ 0.07, 0.5 , 0.01, 0.06, 0.02, 0.09, 0.02, 0.09, 0.06],\
[ 0.02, 0.01, 0.57, 0.01, 0.07, 0.07, 0.05, 0.07, 0.06],\
[ 0. , 0.06, 0.02, 0.54, 0.06, 0.03, 0.08, 0.09, 0.02],\
[ 0.03, 0. , 0.05, 0. , 0.54, 0.04, 0.06, 0.03, 0.02],\
[ 0.05, 0.06, 0.04, 0.02, 0.05, 0.6 , 0.07, 0.08, 0.05],\
[ 0. , 0.03, 0.07, 0.09, 0.06, 0.02, 0.52, 0.04, 0.01],\
[ 0.05, 0.06, 0.03, 0.1 , 0.05, 0.03, 0.03, 0.6 , 0.02],\
[ 0.02, 0.02, 0.04, 0.09, 0.05, 0.01, 0.03, 0. , 0.5 ]])
# setup time interval [a,b] for integration
a = 0.0
b = 1.0
# setup algorithm parameters for the chosen ode method to be tested
tol = 1e-8
maxIter = 1000
# reshape matrixX0 from 9x9 matrix to 81x1 vector
dim_matrixX0 = matrixX0.shape
vectorX0 = matrixX0.reshape((dim_matrixX0[0] * dim_matrixX0[1], ))
# call integration methos
vectorX, t, failFlag, iter_i = ode.ode_rk23(self.myfun,
a,
b,
vectorX0,
tol,
maxIter,
matrixA=matrixA,
matrixB=matrixB,
matrixQ=matrixQ)
# reshape result from 81x1 to 9x9, and compare with expected result
# (note: the values here is not the real expected result, but copied
# from the output of ode_rk23, need a way to get expected result)
matrixX_result = vectorX[-1].reshape(matrixX0.shape)
matrixX_expect = np.array([
[-0.05614609, -0.02764642, 0.00442386, -0.03128679, 0.0379626 ,\
-0.00210206, 0.02055731, 0.01708753, -0.01079675],\
[ 0.01076375, -0.075299 , 0.04635011, -0.00160557, 0.02930959,\
-0.01371929, -0.01172549, -0.03845662, 0.00592987],\
[-0.01827535, 0.02592572, -0.06526755, -0.01455321, -0.00696632,\
0.0123018 , 0.00875977, 0.03707331, -0.02007888],\
[-0.01917939, -0.04928098, 0.01923522, -0.07813147, 0.05341208,\
0.0136149 , -0.00085168, 0.00241213, 0.0005026 ],\
[ 0.00643727, 0.06033435, -0.04005195, 0.0275839 , -0.07949216,\
0.00268929, -0.00399012, 0.01671832, -0.00206837],\
[ 0.01743537, 0.00612319, -0.0026365 , 0.01606572, -0.01925 ,\
-0.04091118, -0.00317609, -0.0098151 , 0.00212008],\
[ 0.00996543, 0.00499091, -0.00098197, 0.02704245, -0.01822727,\
-0.00234791, -0.04023058, -0.00917272, -0.00128094],\
[ 0.03103352, 0.002872 , 0.01410333, 0.03743825, -0.0293213 ,\
-0.01646853, -0.01985861, -0.07105586, 0.026175 ],\
[-0.0198831 , -0.00450292, 0.00266541, -0.03444674, 0.01946599,\
0.00968283, 0.01935828, 0.01385639, -0.04228292]])
self.assertTrue(np.amax(abs(matrixX_expect - matrixX_result)) < 1e-6,
msg=None)
def propagate_q_p(qpu_vec, t_start, t_end, sliding_window_instance, q_mf,
u_mf):
'''
This method propagates q and p to the end of one bucket, with a constant control.
Inputs:
qpu_vec (1D np.array of dimension self.state_dim*3+control_dim): Most current values for local qpu_vec containing q_s, p_l, p_mf, u_s, concatenated in one array
t_start (np.float): start time of bucket
t_end (np.float): end time of bucket
sliding_window_instance (instance of user-defined class which inherits SlidingWindow): object defining the dynamics, and the initial conditions and parameters for numerical integration/propagation.
q_mf (1D np.array of dimension of total number of states): vector containing mean field state values for states not pertaining to this agent, and containing local values otherwise.
u_mf (1D np.array of dimension of total number of states): vector containing mean field control values for control variables not pertaining to this agent, and local values otherwise.
Outputs:
qp_vecs (list of 1-D numpy arrays): holds qp values for each time interval
qp_dot_vecs (list of 1-D numpy arrays): holds q_s_dot, p_mf_dot, p_l_dot values for each time interval
'''
state_dim = sliding_window_instance.state_dim
state_indices = sliding_window_instance.state_indices
n_s = sliding_window_instance.n_s
q_l_0 = qpu_vec[:state_dim]
p_l_0 = qpu_vec[state_dim:2 * state_dim]
p_mf_0 = qpu_vec[2 * state_dim:3 * state_dim]
u_0 = qpu_vec[3 * state_dim:]
qp_vecs = []
qp_vec = np.concatenate([
q_l_0, p_l_0, p_mf_0
]) # pass in all three: q_0, p_0, u_0, but in the qp_rhs function
qp_dot_vecs = []
steps = np.linspace(t_start, t_end, n_s + 1)
# pass in values from blackboard as kwargs to qp_rhs
for i in range(n_s):
n_start, n_end = steps[i], steps[i + 1]
# update q_mf with the most recent local values in q_s
q_s = qp_vec[:state_dim]
q_mf = update_q_mf(q_mf, q_s, sliding_window_instance)
qp_vec, t, failFlag, iter_i = ode.ode_rk23(
sliding_window_instance.qp_rhs,
n_start,
n_end,
qp_vec,
sliding_window_instance.integrateTol,
sliding_window_instance.integrateMaxIter,
state_dim=sliding_window_instance.state_dim,
Gamma=sliding_window_instance.Gamma,
u_0=u_0,
q_mf=q_mf,
u_mf=u_mf)
qp_vec_orig = qp_vec
# rk23 returns 2 arrays but we remove the first array by doing qp_vec[1] because rk_23 returns the initial value you passed in
if len(np.shape(qp_vec_orig)) >= 2:
qp_vec = qp_vec[-1]
qp_vecs.append(qp_vec)
# get time derivatives
qp_dot_vec = sliding_window_instance.qp_rhs(
0.0,
qp_vec,
state_dim=sliding_window_instance.state_dim,
Gamma=sliding_window_instance.Gamma,
u_0=u_0,
q_mf=q_mf,
u_mf=u_mf)
qp_dot_vecs.append(qp_dot_vec)
return qp_vecs, qp_dot_vecs
# - note: set it to be backward constant, i.e., find rows of the first
# time index > tStart_j, for numerical purpose, add epsilon as tolerance
uTimeIndex = uControl[uControl.index > tbegin + 1e-10].index[0]
uControlData = uControl.loc[uTimeIndex]
tEnd = min(uTimeIndex, tStop_j)
#print ("Var. checking: tbegin={}, tStop_j={} sec. ".format(tbegin, tStop_j, tEnd))
# add noise (varying to each element?)
ns = np.random.normal(mu, sigma, len(vectorStateStart))
vectorStateStart = vectorStateStart + ns
# call the ode integrator ...
fn = system_dyn
vector_out, t, failFlag, iter_i = \
ode.ode_rk23(fn, tbegin, tEnd, vectorStateStart,
integrateTol, integrateMaxIter, C=C, L=L, k=k, V0=V0, Vu= Vu, Rs=np.float(uControlData))
print(
"integration happening from {} - {} sec. with ctrl at {}".
format(tbegin, tEnd, uTimeIndex))
# records results with nicer format,
# note: do not record the starting point since it is given, not computed
countRows = len(t)
sumCountRows = sumCountRows + countRows - 1
if initFlag:
state_j['Time'] = t[1:countRows]
state_j['State'] = vector_out[1:countRows, :]
initFlag = False
else:
state_j['Time'] = np.append(state_j['Time'],
def kfc_propagate_state_and_cov(t0, t1, x0, stateCov0, stateDim, controlDim, observeDim,
uControl, yObserve, dynParA, dynParB, dynParF, observeParH, dynCovW,
observeCovV, outputNum, integrateTol, integrateMaxIter):
'''
# propagate state and state covariance for continuous Kalman Filter by forward integration
# (note: propagate state and covariance of the state simultaneously)
'''
#### define the inital condition of covariance and state
#### note: need to reshape covariance as a vector, and combine covariance vector and state vector as one vector
covVectorDim = stateDim*stateDim
vectorCovStateStart = construct_cov_state_vector(stateCov0, x0, covVectorDim)
### solve integration eqs of state and covariance of the state simultaneously
### note: (1) evenly divide the time interval between two adjacent observation
### and output filtered state values at those time points
### (2) time index of uControl yObserve may not align each other,
### therefore the time interval of two adjacent time points given by (1)
### may be split into multiple segments which each has different uControl data
listFilterResult = [] # it will be a 2D list to record the filtered state and state covariance as well as the time array
tStart_i = t0
for i in range(len(yObserve)-1): # iteration i: the time interval of two adjacent observation
listFilterResult.append([])
tEnd_i = yObserve.index[i+1]
# pull observation data from yObserve for the time interval [tStart_i, tEnd_i],
# note: set it to be backward constant, i.e. observation data = yObserve(tEnd_i)
yObserveData = np.array(yObserve.iloc[i+1])
# evenly divide the time interval between two adjacent observation
# in order to output filtered state values at those time points
stepSize = (tEnd_i-tStart_i)/outputNum
tStart_j = tStart_i
# initiate variable for counting total rows of integration output of the jth iteration
sumCountRows = 0
for j in range(outputNum):
# setup time and initiate dictionary for recording the results of the jth iteration
stateAndCov_j = {"Time" : [], "countRows": 0, "vectorState" : [], "matrixCov" : []}
initFlag = True
tEnd_j = tStart_j + stepSize
tStart = tStart_j
while tStart < tEnd_j - 1e-12: # 1e-12: give some tolerance for numerical purpose
# pull control data from uControl
# - note: set it to be backward constant, i.e., find rows of the first
# time index > tStart_j, for numerical purpose, add epsilon as tolerance
uTimeIndex = uControl[uControl.index > tStart+1e-10].index[0]
uControlData = np.array(uControl.loc[uTimeIndex])
tEnd = min(uTimeIndex, tEnd_j)
# To do: need to raise exception when integration is fail
vectorCovState, t, failFlag, iter_i = \
ode.ode_rk23(continuous_kalman_filter_dyn, tStart, tEnd, vectorCovStateStart,
integrateTol, integrateMaxIter, dynParA = dynParA, dynParB=dynParB,
dynParF = dynParF, observeParH=observeParH, uControlData=uControlData,
yObserveData=yObserveData, dynCovW=dynCovW, observeCovV=observeCovV,
covVectorDim = covVectorDim, stateDim = stateDim)
# records results with nicer format,
# note: do not record the starting point since it is given, not computed
countRows = len(t)
sumCountRows = sumCountRows+countRows-1
if initFlag:
stateAndCov_j['Time'] = t[1:countRows]
stateAndCov_j['vectorState'] = vectorCovState[1:countRows,-stateDim:]
stateAndCov_j['matrixCov'] = vectorCovState[1:countRows,0:covVectorDim].reshape((countRows-1, stateDim,stateDim))
initFlag = False
else:
stateAndCov_j['Time'] = np.append(stateAndCov_j['Time'],t[1:countRows])
stateAndCov_j['vectorState'] = np.append(stateAndCov_j['vectorState'],
vectorCovState[1:countRows,-stateDim:], axis=0)
stateAndCov_j['matrixCov'] = np.append(stateAndCov_j['matrixCov'],
vectorCovState[1:countRows,0:covVectorDim].reshape((countRows-1, stateDim,stateDim)), axis=0)
# update tStart and the start point, i.e. vectorCovStateStart
tStart = tEnd
vectorCovStateStart = vectorCovState[-1]
# add results of jth iteration to the list, and update start time of j loop
listFilterResult[i].append(stateAndCov_j)
tStart_j = tEnd_j
# update start time of i loop
tStart_i = tEnd_i
return listFilterResult
def lqt_propogate_state(x0, t0, t1, stateDim, controlDim, listRiccati,
riccatiInvRB, dynParA, dynParB, dynParF, integrateTol, integrateMaxIter):
'''
#forward integration of state dynamics with results of riccati equations as input
# note: form a given time interval time to be processed,
# set results of riccati equations to be constants equal the
# values at the end of the time interval
'''
vectorStateStart = x0
tStart = t0
listStateAndControl = [] # it will be a 2D list to record the results of state
startFlag = True
iterNum = len(listRiccati)
for i in range(iterNum): # iteration i: work in the ith time interval in which zTrack is a constant
listStateAndControl.append([])
listRiccatiRow = iterNum-i-1
if i+1 < iterNum:
tEnd = listRiccati[listRiccatiRow-1][-1]['Time'][-1] #zTrack.index[i+1]
else:
tEnd = t1
# evenly divided time interval [tStart_i, tEnd_i) by outputNum
#tStep = (tEnd-tStart)/outputNum
### outputNum = len(listRiccati[listRiccatiRow-1])
# forward integration for each tStep
riccatiSteps_j = len(listRiccati[listRiccatiRow-1])
tStart_j = tStart
for j in range(riccatiSteps_j): # iteration j: work in the jth sub-time-interval within the zTrack constant interval
listRiccatiCol = riccatiSteps_j-j-1
if listRiccatiCol > 0:
tEnd_j = listRiccati[listRiccatiRow][listRiccatiCol-1]['Time'][-1] # tStart_j+tStep
else:
tEnd_j = tEnd
# initiate dictionary for recording the results of the jth iteration
stateAndControl_j = {"Time" : [], "countRows": 0, "vectorState" : [], "vectorControl" : []}
initFlag = True
sumCountRows = 0
tStart_k = tStart_j
for k in range(listRiccati[listRiccatiRow][listRiccatiCol]['countRows']-1,-1,-1):
# iteration k: the smallest time interval in which ricatti numerical results are constant
if k > 0:
tEnd_k = listRiccati[listRiccatiRow][listRiccatiCol]['Time'][k-1]
else:
tEnd_k = tEnd_j
# construct parameters from results of riccati equations for forward integration
riccatiKlq = -riccatiInvRB.dot(listRiccati[listRiccatiRow][listRiccatiCol]['matrixSigma'][k])
riccatiPsi = -riccatiInvRB.dot(listRiccati[listRiccatiRow][listRiccatiCol]['vectorPhi'][k])
stateAPlusBKlq = dynParA+dynParB.dot(riccatiKlq)
stateBPsiPlusF = dynParB.dot(riccatiPsi)+dynParF
# TO DO: need raise failture message when integration fail
# propogate state dynamics with feedback policy
vectorState, t, failFlag, iter_i = \
ode.ode_rk23(feedback_policy_state_dot, tStart_k, tEnd_k, vectorStateStart,
integrateTol, integrateMaxIter, stateAPlusBKlq=stateAPlusBKlq,
stateBPsiPlusF=stateBPsiPlusF)
# compute optimal feedback control
listControl = []
for tmp_i in range(len(t)):
listControl.append(riccatiKlq.dot(vectorState[tmp_i])+riccatiPsi)
vectorControl = np.array(listControl)
# records results with nicer format for forward integration,
# note: if at time tStart_k=t0, record the state and control values at starting time point
# if at tStart_k>t0, no need to record the values at tStart_k,
# since it has been recorded at the previous iteration
countRows = len(t)
if startFlag:
startRow = 0
sumCountRows = sumCountRows+countRows
startFlag = False
else:
startRow = 1
sumCountRows = sumCountRows+countRows-1
if initFlag:
stateAndControl_j['Time'] = t[startRow:countRows]
stateAndControl_j['vectorState'] = vectorState[startRow:countRows]
stateAndControl_j['vectorControl'] = vectorControl[startRow:countRows]
initFlag = False
else:
stateAndControl_j['Time'] = np.append(stateAndControl_j['Time'],t[startRow:countRows])
stateAndControl_j['vectorState'] = np.append(stateAndControl_j['vectorState'],
vectorState[startRow:countRows], axis=0)
stateAndControl_j['vectorControl'] = np.append(stateAndControl_j['vectorControl'],
vectorControl[startRow:countRows], axis=0)
# update tStart_k and the start point, i.e. vectorStateStart
tStart_k = tEnd_k
vectorStateStart = vectorState[-1]
# update tStart_j for outer loop j
stateAndControl_j['countRows']=sumCountRows
listStateAndControl[i].append(stateAndControl_j)
tStart_j = tEnd_j
# update tStart for outer loop i
tStart = tEnd
return listStateAndControl
def lqt_solve_riccati(t1, stateDim, zTrack, dynParA, dynParF, trackParF,
trackParQ, riccatiBInvRB, outputNum, integrateTol, integrateMaxIter):
'''
# solve riccati equations
# i.e. backward integrate sigma and phi dynamics together:
# sigma_dot = -sigma*dynParA-Transpose(dynParA)*sigma+sigma*dynParB*Inv(trackParR)*Transpose(dynParB)-trackParQ
# phi_dot = (-Transpose(dynParA)+sigma*dynParB*Inv(trackParR)*Transpose(dynParB))*phi-sigma*dynParF+trackParQ*zTrack
# with terminal condition sigma_end=trackParF, phi_end=(nx1 zeros)
# Note: sigma is a symmetric 2D matrix
'''
#### define the terminal condition of sigma and phi
#### note: need to reshape sigma as a vector, and make sigma vector and phi as one vector
sigmaVectorDim = stateDim*stateDim
vectorSigmaPhiBackStart = construct_sigma_phi_vector(trackParF, -trackParF.dot(np.array(zTrack.iloc[-1])), sigmaVectorDim)
### solve riccati equations by backward integration
#### i.e. for each piecewise constant of zTrack time interval,
#### call ode method to barckward integrate riccati equations
iterNum = len(zTrack)-1
tStart = t1
listRiccati = [] # it will be a 2D list to record the results of riccati equations
for i in range(iterNum-1,-1,-1): # iteration i: work in the ith time interval in which zTrack is a constant
listRiccati.append([])
listRiccatiRow = iterNum-i-1
tEnd = zTrack.index[i]
zTrackData = np.array(zTrack.iloc[i])
# evenly divided time interval [tStart_i, tEnd_i) by outputNum
tStep = (tEnd-tStart)/outputNum
# backward integration for each tStep
tStart_j = tStart
for j in range(outputNum):
# iteration j: work in the jth sub-time-interval within the zTrack constant interval
tEnd_j = tStart_j+tStep
# TO DO: need raise failure message when integration fail
# backward propagate riccati equations
vectorSigmaPhiBack, t, failFlag, iter_i = \
ode.ode_rk23(riccati_eqs, tStart_j, tEnd_j, vectorSigmaPhiBackStart,
integrateTol, integrateMaxIter, dynParA = dynParA, dynParF = dynParF,
trackParQ = trackParQ, riccatiBInvRB = riccatiBInvRB,
zTrackData = zTrackData, sigmaVectorDim = sigmaVectorDim,
stateDim = stateDim)
# records results with nicer format for forward integration,
# note: do not record the starting point since it is given, not computed
countRows = len(vectorSigmaPhiBack)
riccatiResults = {"Time" : t[1:countRows], "countRows":countRows-1,
"matrixSigma" : vectorSigmaPhiBack[1:countRows,0:sigmaVectorDim].reshape((countRows-1,stateDim,stateDim)),
"vectorPhi" : vectorSigmaPhiBack[1:countRows,-stateDim:]}
listRiccati[listRiccatiRow].append(riccatiResults)
# update tStart_j and the start point, i.e. vectorSigmaPhiBackStart
tStart_j = tEnd_j
vectorSigmaPhiBackStart = vectorSigmaPhiBack[-1]
# update tStart for outer loop i
tStart = tEnd
return listRiccati