"""Copyright 2015 Roger R Labbe Jr.
FilterPy library.
http://github.com/rlabbe/filterpy
Documentation at:
https://filterpy.readthedocs.org
Supporting book at:
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
This is licensed under an MIT license. See the readme.MD file
for more information.
"""
from __future__ import (absolute_import, division, print_function,
unicode_literals)
from numpy import array, zeros, vstack, eye
from scipy.linalg import expm, inv
def Q_discrete_white_noise(dim, dt=1., var=1.):
""" Returns the Q matrix for the Discrete Constant White Noise
Model. dim may be either 2 or 3, dt is the time step, and sigma is the
variance in the noise.
Q is computed as the G * G^T * variance, where G is the process noise per
time step. In other words, G = [[.5dt^2][dt]]^T for the constant velocity
model.
**Paramaeters**
dim : int (2 or 3)
dimension for Q, where the final dimension is (dim x dim)
dt : float, default=1.0
time step in whatever units your filter is using for time. i.e. the
amount of time between innovations
var : float, default=1.0
variance in the noise
"""
assert dim == 2 or dim == 3
if dim == 2:
Q = array([[.25*dt**4, .5*dt**3],
[ .5*dt**3, dt**2]], dtype=float)
else:
Q = array([[.25*dt**4, .5*dt**3, .5*dt**2],
[ .5*dt**3, dt**2, dt],
[ .5*dt**2, dt, 1]], dtype=float)
return Q * var
def Q_continuous_white_noise(dim, dt=1., spectral_density=1.):
""" Returns the Q matrix for the Discretized Continuous White Noise
Model. dim may be either 2 or 3, dt is the time step, and sigma is the
variance in the noise.
**Paramaeters**
dim : int (2 or 3)
dimension for Q, where the final dimension is (dim x dim)
dt : float, default=1.0
time step in whatever units your filter is using for time. i.e. the
amount of time between innovations
spectral_density : float, default=1.0
spectral density for the continuous process
"""
assert dim == 2 or dim == 3
if dim == 2:
Q = array([[(dt**4)/3, (dt**2)/2],
[(dt**2)/2, dt]])
else:
Q = array([[(dt**5)/20, (dt**4)/8, (dt**3)/6],
[ (dt**4)/8, (dt**3)/3, (dt**2)/2],
[ (dt**3)/6, (dt**2)/2, dt]], dtype=float)
return Q * spectral_density
def van_loan_discretization(F, G, dt):
""" Discretizes a linear differential equation which includes white noise
according to the method of C. F. van Loan [1]. Given the continuous
model
x' = Fx + Gu
where u is the unity white noise, we compute and return the sigma and Q_k
that discretizes that equation.
**Example**::
Given y'' + y = 2u(t), we create the continuous state model of
x' = | 0 1| * x + |0|*u(t)
|-1 0| |2|
and a time step of 0.1:
>>> F = np.array([[0,1],[-1,0]], dtype=float)
>>> G = np.array([[0.],[2.]])
>>> phi, Q = van_loan_discretization(F, G, 0.1)
>>> phi
array([[ 0.99500417, 0.09983342],
[-0.09983342, 0.99500417]])
>>> Q
array([[ 0.00133067, 0.01993342],
[ 0.01993342, 0.39866933]])
(example taken from Brown[2])
**References**
[1] C. F. van Loan. "Computing Integrals Involving the Matrix Exponential."
IEEE Trans. Automomatic Control, AC-23 (3): 395-404 (June 1978)
[2] Robert Grover Brown. "Introduction to Random Signals and Applied
Kalman Filtering." Forth edition. John Wiley & Sons. p. 126-7. (2012)
"""
n = F.shape[0]
A = zeros((2*n, 2*n))
A[0:n, 0:n] = -F.dot(dt)
A[0:n, n:2*n] = G.dot(G.T).dot(dt)
A[n:2*n, n:2*n] = F.T.dot(dt)
B=expm(A)
sigma = B[n:2*n, n:2*n].T
Q = sigma.dot(B[0:n, n:2*n])
return (sigma, Q)
def linear_ode_discretation(F, L=None, Q=None, dt=1):
n = F.shape[0]
if L is None:
L = eye(n)
if Q is None:
Q = zeros((n,n))
A = expm(F*dt)
phi = zeros((2*n, 2*n))
phi[0:n, 0:n] = F
phi[0:n, n:2*n] = L.dot(Q).dot(L.T)
phi[n:2*n, n:2*n] = -F.T
zo = vstack((zeros((n,n)), eye(n)))
CD = expm(phi*dt).dot(zo)
C = CD[0:n,:]
D = CD[n:2*n,:]
q = C.dot(inv(D))
return (A, q)
