def stationary_values(self):
"""
Computes the limit of :math:`\Sigma_t` as t goes to infinity by
solving the associated Riccati equation. Computation is via the
doubling algorithm (see the documentation in
`matrix_eqn.solve_discrete_riccati`).
Returns
-------
Sigma_infinity : array_like or scalar(float)
The infinite limit of :math:`\Sigma_t`
K_infinity : array_like or scalar(float)
The stationary Kalman gain.
"""
# === simplify notation === #
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Q, R = np.dot(C, C.T), np.dot(H, H.T)
# === solve Riccati equation, obtain Kalman gain === #
Sigma_infinity = solve_discrete_riccati(A.T, G.T, Q, R)
temp1 = dot(dot(A, Sigma_infinity), G.T)
temp2 = inv(dot(G, dot(Sigma_infinity, G.T)) + R)
K_infinity = dot(temp1, temp2)
# == record as attributes and return == #
self.Sigma_infinity, self.K_infinity = Sigma_infinity, K_infinity
return Sigma_infinity, K_infinity
def stationary_values(self):
"""
Computes the limit of Sigma_t as t goes to infinity by
solving the associated Riccati equation. Computation is via the
doubling algorithm (see the documentation in
`matrix_eqn.solve_discrete_riccati`).
Returns
-------
Sigma_infinity : array_like or scalar(float)
The infinite limit of Sigma_t
K_infinity : array_like or scalar(float)
The stationary Kalman gain.
"""
# === simplify notation === #
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Q, R = np.dot(C, C.T), np.dot(H, H.T)
# === solve Riccati equation, obtain Kalman gain === #
Sigma_infinity = solve_discrete_riccati(A.T, G.T, Q, R)
temp1 = dot(dot(A, Sigma_infinity), G.T)
temp2 = inv(dot(G, dot(Sigma_infinity, G.T)) + R)
K_infinity = dot(temp1, temp2)
# == record as attributes and return == #
self.Sigma_infinity, self.K_infinity = Sigma_infinity, K_infinity
return Sigma_infinity, K_infinity
def dare_test_tjm_1():
A = [[0.0, 0.1, 0.0], [0.0, 0.0, 0.1], [0.0, 0.0, 0.0]]
B = [[1.0, 0.0], [0.0, 0.0], [0.0, 1.0]]
Q = [[10**5, 0.0, 0.0], [0.0, 10**3, 0.0], [0.0, 0.0, -10.0]]
R = [[0.0, 0.0], [0.0, 1.0]]
X = solve_discrete_riccati(A, B, Q, R)
Y = np.diag((1e5, 1e3, 0.0))
assert_allclose(X, Y, atol=1e-07)
def dare_test_tjm_2():
A = [[0, -1], [0, 2]]
B = [[1, 0], [1, 1]]
Q = [[1, 0], [0, 0]]
R = [[4, 2], [2, 1]]
X = solve_discrete_riccati(A, B, Q, R)
Y = np.zeros((2, 2))
Y[0, 0] = 1
assert_allclose(X, Y, atol=1e-07)
def dare_test_tjm_3():
r = 0.5
I = np.identity(2)
A = [[2 + r**2, 0], [0, 0]]
A = np.array(A)
B = I
R = [[1, r], [r, r * r]]
Q = I - np.dot(A.T, A) + np.dot(A.T, np.linalg.solve(R + I, A))
X = solve_discrete_riccati(A, B, Q, R)
Y = np.identity(2)
assert_allclose(X, Y, atol=1e-07)
def dare_test_tjm_3():
r = 0.5
I = np.identity(2)
A = [[2 + r**2, 0],
[0, 0]]
A = np.array(A)
B = I
R = [[1, r],
[r, r*r]]
Q = I - np.dot(A.T, A) + np.dot(A.T, np.linalg.solve(R + I, A))
X = solve_discrete_riccati(A, B, Q, R)
Y = np.identity(2)
assert_allclose(X, Y, atol=1e-07)
def dare_test_tjm_2():
A = [[0, -1],
[0, 2]]
B = [[1, 0],
[1, 1]]
Q = [[1, 0],
[0, 0]]
R = [[4, 2],
[2, 1]]
X = solve_discrete_riccati(A, B, Q, R)
Y = np.zeros((2, 2))
Y[0, 0] = 1
assert_allclose(X, Y, atol=1e-07)
def dare_test_tjm_1():
A = [[0.0, 0.1, 0.0],
[0.0, 0.0, 0.1],
[0.0, 0.0, 0.0]]
B = [[1.0, 0.0],
[0.0, 0.0],
[0.0, 1.0]]
Q = [[10**5, 0.0, 0.0],
[0.0, 10**3, 0.0],
[0.0, 0.0, -10.0]]
R = [[0.0, 0.0],
[0.0, 1.0]]
X = solve_discrete_riccati(A, B, Q, R)
Y = np.diag((1e5, 1e3, 0.0))
assert_allclose(X, Y, atol=1e-07)
def stationary_values(self, method='doubling'):
"""
Computes the limit of :math:`\Sigma_t` as t goes to infinity by
solving the associated Riccati equation. The outputs are stored in the
attributes `K_infinity` and `Sigma_infinity`. Computation is via the
doubling algorithm (default) or a QZ decomposition method (see the
documentation in `matrix_eqn.solve_discrete_riccati`).
Parameters
----------
method : str, optional(default="doubling")
Solution method used in solving the associated Riccati
equation, str in {'doubling', 'qz'}.
Returns
-------
Sigma_infinity : array_like or scalar(float)
The infinite limit of :math:`\Sigma_t`
K_infinity : array_like or scalar(float)
The stationary Kalman gain.
"""
# === simplify notation === #
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Q, R = np.dot(C, C.T), np.dot(H, H.T)
# === solve Riccati equation, obtain Kalman gain === #
Sigma_infinity = solve_discrete_riccati(A.T, G.T, Q, R, method=method)
temp1 = np.dot(np.dot(A, Sigma_infinity), G.T)
temp2 = inv(np.dot(G, np.dot(Sigma_infinity, G.T)) + R)
K_infinity = np.dot(temp1, temp2)
# == record as attributes and return == #
self._Sigma_infinity, self._K_infinity = Sigma_infinity, K_infinity
return Sigma_infinity, K_infinity
def stationary_values(self, method='doubling'):
"""
Computes the limit of :math:`\Sigma_t` as t goes to infinity by
solving the associated Riccati equation. The outputs are stored in the
attributes `K_infinity` and `Sigma_infinity`. Computation is via the
doubling algorithm (default) or a QZ decomposition method (see the
documentation in `matrix_eqn.solve_discrete_riccati`).
Parameters
----------
method : str, optional(default="doubling")
Solution method used in solving the associated Riccati
equation, str in {'doubling', 'qz'}.
Returns
-------
Sigma_infinity : array_like or scalar(float)
The infinite limit of :math:`\Sigma_t`
K_infinity : array_like or scalar(float)
The stationary Kalman gain.
"""
# === simplify notation === #
A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
Q, R = np.dot(C, C.T), np.dot(H, H.T)
# === solve Riccati equation, obtain Kalman gain === #
Sigma_infinity = solve_discrete_riccati(A.T, G.T, Q, R, method=method)
temp1 = dot(dot(A, Sigma_infinity), G.T)
temp2 = inv(dot(G, dot(Sigma_infinity, G.T)) + R)
K_infinity = dot(temp1, temp2)
# == record as attributes and return == #
self._Sigma_infinity, self._K_infinity = Sigma_infinity, K_infinity
return Sigma_infinity, K_infinity
def dare_golden_num_2d(method):
A, B, R, Q = np.eye(2), np.eye(2), np.eye(2), np.eye(2)
gold_diag = np.eye(2) * (1 + np.sqrt(5)) / 2.
val = solve_discrete_riccati(A, B, R, Q, method=method)
assert_allclose(val, gold_diag)
def dare_golden_num_float(method):
val = solve_discrete_riccati(1.0, 1.0, 1.0, 1.0, method=method)
gold_ratio = (1 + np.sqrt(5)) / 2.
assert_allclose(val, gold_ratio)
def dare_test_golden_num_2d():
A, B, R, Q = np.eye(2), np.eye(2), np.eye(2), np.eye(2)
gold_diag = np.eye(2) * (1 + np.sqrt(5)) / 2.
val = solve_discrete_riccati(A, B, R, Q)
assert_allclose(val, gold_diag)
def dare_test_golden_num_float():
val = solve_discrete_riccati(1.0, 1.0, 1.0, 1.0)
gold_ratio = (1 + np.sqrt(5)) / 2.
assert_allclose(val, gold_ratio)