Source code for filterpy.kalman.square_root

# -*- coding: utf-8 -*-
# pylint: disable=invalid-name, too-many-instance-attributes

"""Copyright 2015 Roger R Labbe Jr.

FilterPy library.

Documentation at:

Supporting book at:

This is licensed under an MIT license. See the readme.MD file
for more information.

from __future__ import (absolute_import, division)

from copy import deepcopy
import numpy as np
from numpy import dot, zeros, eye
from scipy.linalg import cholesky, qr, pinv
from filterpy.common import pretty_str

[docs]class SquareRootKalmanFilter(object): """ Create a Kalman filter which uses a square root implementation. This uses the square root of the state covariance matrix, which doubles the numerical precision of the filter, Therebuy reducing the effect of round off errors. It is likely that you do not need to use this algorithm; we understand divergence issues very well now. However, if you expect the covariance matrix P to vary by 20 or more orders of magnitude then perhaps this will be useful to you, as the square root will vary by 10 orders of magnitude. From my point of view this is merely a 'reference' algorithm; I have not used this code in real world software. Brown[1] has a useful discussion of when you might need to use the square root form of this algorithm. You are responsible for setting the various state variables to reasonable values; the defaults below will not give you a functional filter. Parameters ---------- dim_x : int Number of state variables for the Kalman filter. For example, if you are tracking the position and velocity of an object in two dimensions, dim_x would be 4. This is used to set the default size of P, Q, and u dim_z : int Number of of measurement inputs. For example, if the sensor provides you with position in (x,y), dim_z would be 2. dim_u : int (optional) size of the control input, if it is being used. Default value of 0 indicates it is not used. Attributes ---------- x : numpy.array(dim_x, 1) State estimate P : numpy.array(dim_x, dim_x) State covariance matrix x_prior : numpy.array(dim_x, 1) Prior (predicted) state estimate. The *_prior and *_post attributes are for convienence; they store the prior and posterior of the current epoch. Read Only. P_prior : numpy.array(dim_x, dim_x) Prior (predicted) state covariance matrix. Read Only. x_post : numpy.array(dim_x, 1) Posterior (updated) state estimate. Read Only. P_post : numpy.array(dim_x, dim_x) Posterior (updated) state covariance matrix. Read Only. z : numpy.array Last measurement used in update(). Read only. R : numpy.array(dim_z, dim_z) Measurement noise matrix Q : numpy.array(dim_x, dim_x) Process noise matrix F : numpy.array() State Transition matrix H : numpy.array(dim_z, dim_x) Measurement function y : numpy.array Residual of the update step. Read only. K : numpy.array(dim_x, dim_z) Kalman gain of the update step. Read only. S : numpy.array Systen uncertaintly projected to measurement space. Read only. Examples -------- See my book Kalman and Bayesian Filters in Python References ---------- [1] Robert Grover Brown. Introduction to Random Signals and Applied Kalman Filtering. Wiley and sons, 2012. """
[docs] def __init__(self, dim_x, dim_z, dim_u=0): if dim_z < 1: raise ValueError('dim_x must be 1 or greater') if dim_z < 1: raise ValueError('dim_x must be 1 or greater') if dim_u < 0: raise ValueError('dim_x must be 0 or greater') self.dim_x = dim_x self.dim_z = dim_z self.dim_u = dim_u self.x = zeros((dim_x, 1)) # state self._P = eye(dim_x) # uncertainty covariance self._P1_2 = eye(dim_x) # sqrt uncertainty covariance self._Q = eye(dim_x) # sqrt process uncertainty self._Q1_2 = eye(dim_x) # sqrt process uncertainty self.B = 0. # control transition matrix self.F = np.eye(dim_x) # state transition matrix self.H = np.zeros((dim_z, dim_x)) # Measurement function self._R1_2 = eye(dim_z) # sqrt state uncertainty self._R = eye(dim_z) # state uncertainty self.z = np.array([[None]*self.dim_z]).T self.K = 0. self.S = 0. # Residual is computed during the innovation (update) step. We # save it so that in case you want to inspect it for various # purposes self.y = zeros((dim_z, 1)) # identity matrix. self._I = np.eye(dim_x) self.M = np.zeros((dim_z + dim_x, dim_z + dim_x)) # copy prior and posterior self.x_prior = np.copy(self.x) self._P1_2_prior = np.copy(self._P1_2) self.x_post = np.copy(self.x) self._P1_2_post = np.copy(self._P1_2)
[docs] def update(self, z, R2=None): """ Add a new measurement (z) to the kalman filter. If z is None, nothing is changed. Parameters ---------- z : np.array measurement for this update. R2 : np.array, scalar, or None Sqrt of meaaurement noize. Optionally provide to override the measurement noise for this one call, otherwise self.R2 will be used. """ if z is None: self.z = np.array([[None]*self.dim_z]).T self.x_post = self.x.copy() self._P1_2_post = np.copy(self._P1_2) return if R2 is None: R2 = self._R1_2 elif np.isscalar(R2): R2 = eye(self.dim_z) * R2 # rename for convienance dim_z = self.dim_z M = self.M M[0:dim_z, 0:dim_z] = R2.T M[dim_z:, 0:dim_z] = dot(self.H, self._P1_2).T M[dim_z:, dim_z:] = self._P1_2.T _, self.S = qr(M) self.K = self.S[0:dim_z, dim_z:].T N = self.S[0:dim_z, 0:dim_z].T # y = z - Hx # error (residual) between measurement and prediction self.y = z - dot(self.H, self.x) # x = x + Ky # predict new x with residual scaled by the kalman gain self.x += dot(self.K, pinv(N)).dot(self.y) self._P1_2 = self.S[dim_z:, dim_z:].T self.z = deepcopy(z) self.x_post = self.x.copy() self._P1_2_post = np.copy(self._P1_2)
[docs] def predict(self, u=0): """ Predict next state (prior) using the Kalman filter state propagation equations. Parameters ---------- u : np.array, optional Optional control vector. If non-zero, it is multiplied by B to create the control input into the system. """ # x = Fx + Bu self.x = dot(self.F, self.x) + dot(self.B, u) # P = FPF' + Q _, P2 = qr(np.hstack([dot(self.F, self._P1_2), self._Q1_2]).T) self._P1_2 = P2[:self.dim_x, :self.dim_x].T # copy prior self.x_prior = np.copy(self.x) self._P1_2_prior = np.copy(self._P1_2)
[docs] def residual_of(self, z): """ returns the residual for the given measurement (z). Does not alter the state of the filter. """ return z - dot(self.H, self.x)
[docs] def measurement_of_state(self, x): """ Helper function that converts a state into a measurement. Parameters ---------- x : np.array kalman state vector Returns ------- z : np.array measurement corresponding to the given state """ return dot(self.H, x)
@property def Q(self): """ Process uncertainty""" return dot(self._Q1_2.T, self._Q1_2) @property def Q1_2(self): """ Sqrt Process uncertainty""" return self._Q1_2 @Q.setter def Q(self, value): """ Process uncertainty""" self._Q = value self._Q1_2 = cholesky(self._Q, lower=True) @property def P(self): """ covariance matrix""" return dot(self._P1_2.T, self._P1_2) @property def P_prior(self): """ covariance matrix of the prior""" return dot(self._P1_2_prior.T, self._P1_2_prior) @property def P_post(self): """ covariance matrix of the posterior""" return dot(self._P1_2_prior.T, self._P1_2_prior) @property def P1_2(self): """ sqrt of covariance matrix""" return self._P1_2 @P.setter def P(self, value): """ covariance matrix""" self._P = value self._P1_2 = cholesky(self._P, lower=True) @property def R(self): """ measurement uncertainty""" return dot(self._R1_2.T, self._R1_2) @property def R1_2(self): """ sqrt of measurement uncertainty""" return self._R1_2 @R.setter def R(self, value): """ measurement uncertainty""" self._R = value self._R1_2 = cholesky(self._R, lower=True) def __repr__(self): return '\n'.join([ 'SquareRootKalmanFilter object', pretty_str('dim_x', self.dim_x), pretty_str('dim_z', self.dim_z), pretty_str('dim_u', self.dim_u), pretty_str('x', self.x), pretty_str('P', self.P), pretty_str('F', self.F), pretty_str('Q', self.Q), pretty_str('R', self.R), pretty_str('H', self.H), pretty_str('K', self.K), pretty_str('y', self.y), pretty_str('S', self.S), pretty_str('M', self.M), pretty_str('B', self.B), ])