We will try to pinpoint the location of a moving vehicle with high accuracy from noisy sensor data. We'll do this by modeling the vehicle state as a discrete-time linear dynamical system. Standard Kalman filtering can be used to approach this problem when the sensor noise is assumed to be Gaussian. We'll use robust Kalman filtering to get a more accurate estimate of the vehicle state for a non-Gaussian case with outliers.
A discrete-time linear dynamical system consists of a sequence of state vectors $x_t \in \mathbf{R}^n$, indexed by time $t \in \lbrace 0, \ldots, N-1 \rbrace$ and dynamics equations
\begin{align} x_{t+1} &= Ax_t + Bw_t\\ y_t &=Cx_t + v_t, \end{align}where $w_t \in \mathbf{R}^m$ is an input to the dynamical system (say, a drive force on the vehicle), $y_t \in \mathbf{R}^r$ is a state measurement, $v_t \in \mathbf{R}^r$ is noise, $A$ is the drift matrix, $B$ is the input matrix, and $C$ is the observation matrix.
Given $A$, $B$, $C$, and $y_t$ for $t = 0, \ldots, N-1$, the goal is to estimate $x_t$ for $t = 0, \ldots, N-1$.
A Kalman filter estimates $x_t$ by solving the optimization problem
\begin{array}{ll} \mbox{minimize} & \sum_{t=0}^{N-1} \left( \|w_t\|_2^2 + \tau \|v_t\|_2^2\right)\\ \mbox{subject to} & x_{t+1} = Ax_t + Bw_t,\quad t=0,\ldots, N-1\\ & y_t = Cx_t+v_t,\quad t = 0, \ldots, N-1, \end{array}where $\tau$ is a tuning parameter. This problem is actually a least squares problem, and can be solved via linear algebra, without the need for more general convex optimization. Note that since we have no observation $y_{N}$, $x_N$ is only constrained via $x_{N} = Ax_{N-1} + Bw_{N-1}$, which is trivially resolved when $w_{N-1} = 0$ and $x_{N} = Ax_{N-1}$. We maintain this vestigial constraint only because it offers a concise problem statement.
This model performs well when $w_t$ and $v_t$ are Gaussian. However, the quadratic objective can be influenced by large outliers, which degrades the accuracy of the recovery. To improve estimation in the presence of outliers, we can use robust Kalman filtering.
To handle outliers in $v_t$, robust Kalman filtering replaces the quadratic cost with a Huber cost, which results in the convex optimization problem
\begin{array}{ll} \mbox{minimize} & \sum_{t=0}^{N-1} \left( \|w_t\|^2_2 + \tau \phi_\rho(v_t) \right)\\ \mbox{subject to} & x_{t+1} = Ax_t + Bw_t,\quad t=0,\ldots, N-1\\ & y_t = Cx_t+v_t,\quad t=0,\ldots, N-1, \end{array}where $\phi_\rho$ is the Huber function $$ \phi_\rho(a)= \left\{ \begin{array}{ll} \|a\|_2^2 & \|a\|_2\leq \rho\\ 2\rho \|a\|_2-\rho^2 & \|a\|_2>\rho. \end{array}\right. $$
The Huber penalty function penalizes estimation error linearly outside of a ball of radius $\rho$, whereas in standard Kalman filtering, all errors are penalized quadratically. Thus, large errors are penalized less harshly, making this model more robust to outliers.
We'll apply standard and robust Kalman filtering to a vehicle tracking problem with state $x_t \in \mathbf{R}^4$, where $(x_{t,0}, x_{t,1})$ is the position of the vehicle in two dimensions, and $(x_{t,2}, x_{t,3})$ is the vehicle velocity. The vehicle has unknown drive force $w_t$, and we observe noisy measurements of the vehicle's position, $y_t \in \mathbf{R}^2$.
The matrices for the dynamics are
$$ A = \begin{bmatrix} 1 & 0 & \left(1-\frac{\gamma}{2}\Delta t\right) \Delta t & 0 \\ 0 & 1 & 0 & \left(1-\frac{\gamma}{2} \Delta t\right) \Delta t\\ 0 & 0 & 1-\gamma \Delta t & 0 \\ 0 & 0 & 0 & 1-\gamma \Delta t \end{bmatrix}, $$$$ B = \begin{bmatrix} \frac{1}{2}\Delta t^2 & 0 \\ 0 & \frac{1}{2}\Delta t^2 \\ \Delta t & 0 \\ 0 & \Delta t \\ \end{bmatrix}, $$$$ C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}, $$where $\gamma$ is a velocity damping parameter.
The recurrence is derived from the following relations in a single dimension. For this subsection, let $x_t, v_t, w_t$ be the vehicle position, velocity, and input drive force. The resulting acceleration of the vehicle is $w_t - \gamma v_t$, with $- \gamma v_t$ is a damping term depending on velocity with parameter $\gamma$.
The discretized dynamics are obtained from numerically integrating: $$ \begin{align} x_{t+1} &= x_t + \left(1-\frac{\gamma \Delta t}{2}\right)v_t \Delta t + \frac{1}{2}w_{t} \Delta t^2\\ v_{t+1} &= \left(1-\gamma\right)v_t + w_t \Delta t. \end{align} $$
Extending these relations to two dimensions gives us the dynamics matrices $A$ and $B$.
In [1]:
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
def plot_state(t,actual, estimated=None):
'''
plot position, speed, and acceleration in the x and y coordinates for
the actual data, and optionally for the estimated data
'''
trajectories = [actual]
if estimated is not None:
trajectories.append(estimated)
fig, ax = plt.subplots(3, 2, sharex='col', sharey='row', figsize=(8,8))
for x, w in trajectories:
ax[0,0].plot(t,x[0,:-1])
ax[0,1].plot(t,x[1,:-1])
ax[1,0].plot(t,x[2,:-1])
ax[1,1].plot(t,x[3,:-1])
ax[2,0].plot(t,w[0,:])
ax[2,1].plot(t,w[1,:])
ax[0,0].set_ylabel('x position')
ax[1,0].set_ylabel('x velocity')
ax[2,0].set_ylabel('x input')
ax[0,1].set_ylabel('y position')
ax[1,1].set_ylabel('y velocity')
ax[2,1].set_ylabel('y input')
ax[0,1].yaxis.tick_right()
ax[1,1].yaxis.tick_right()
ax[2,1].yaxis.tick_right()
ax[0,1].yaxis.set_label_position("right")
ax[1,1].yaxis.set_label_position("right")
ax[2,1].yaxis.set_label_position("right")
ax[2,0].set_xlabel('time')
ax[2,1].set_xlabel('time')
def plot_positions(traj, labels, axis=None,filename=None):
'''
show point clouds for true, observed, and recovered positions
'''
matplotlib.rcParams.update({'font.size': 14})
n = len(traj)
fig, ax = plt.subplots(1, n, sharex=True, sharey=True,figsize=(12, 5))
if n == 1:
ax = [ax]
for i,x in enumerate(traj):
ax[i].plot(x[0,:], x[1,:], 'ro', alpha=.1)
ax[i].set_title(labels[i])
if axis:
ax[i].axis(axis)
if filename:
fig.savefig(filename, bbox_inches='tight')
In [2]:
n = 1000 # number of timesteps
T = 50 # time will vary from 0 to T with step delt
ts, delt = np.linspace(0,T,n,endpoint=True, retstep=True)
gamma = .05 # damping, 0 is no damping
A = np.zeros((4,4))
B = np.zeros((4,2))
C = np.zeros((2,4))
A[0,0] = 1
A[1,1] = 1
A[0,2] = (1-gamma*delt/2)*delt
A[1,3] = (1-gamma*delt/2)*delt
A[2,2] = 1 - gamma*delt
A[3,3] = 1 - gamma*delt
B[0,0] = delt**2/2
B[1,1] = delt**2/2
B[2,0] = delt
B[3,1] = delt
C[0,0] = 1
C[1,1] = 1
We seed $x_0 = 0$ (starting at the origin with zero velocity) and simulate the system forward in time. The results are the true vehicle positions x_true (which we will use to judge our recovery) and the observed positions y.
We plot the position, velocity, and system input $w$ in both dimensions as a function of time. We also plot the sets of true and observed vehicle positions.
In [3]:
sigma = 20
p = .20
np.random.seed(6)
x = np.zeros((4,n+1))
x[:,0] = [0,0,0,0]
y = np.zeros((2,n))
# generate random input and noise vectors
w = np.random.randn(2,n)
v = np.random.randn(2,n)
# add outliers to v
np.random.seed(0)
inds = np.random.rand(n) <= p
v[:,inds] = sigma*np.random.randn(2,n)[:,inds]
# simulate the system forward in time
for t in range(n):
y[:,t] = C.dot(x[:,t]) + v[:,t]
x[:,t+1] = A.dot(x[:,t]) + B.dot(w[:,t])
x_true = x.copy()
w_true = w.copy()
plot_state(ts,(x_true,w_true))
plot_positions([x_true,y], ['True', 'Observed'],[-4,14,-5,20],'rkf1.pdf')
In [4]:
%%time
import cvxpy as cp
x = cp.Variable(shape=(4, n+1))
w = cp.Variable(shape=(2, n))
v = cp.Variable(shape=(2, n))
tau = .08
obj = cp.sum_squares(w) + tau*cp.sum_squares(v)
obj = cp.Minimize(obj)
constr = []
for t in range(n):
constr += [ x[:,t+1] == A*x[:,t] + B*w[:,t] ,
y[:,t] == C*x[:,t] + v[:,t] ]
cp.Problem(obj, constr).solve(verbose=True)
x = np.array(x.value)
w = np.array(w.value)
plot_state(ts,(x_true,w_true),(x,w))
plot_positions([x_true,y], ['True', 'Noisy'], [-4,14,-5,20])
plot_positions([x_true,x], ['True', 'KF recovery'], [-4,14,-5,20], 'rkf2.pdf')
print("optimal objective value: {}".format(obj.value))
In [5]:
%%time
import cvxpy as cp
x = cp.Variable(shape=(4, n+1))
w = cp.Variable(shape=(2, n))
v = cp.Variable(shape=(2, n))
tau = 2
rho = 2
obj = cp.sum_squares(w)
obj += cp.sum([tau*cp.huber(cp.norm(v[:,t]),rho) for t in range(n)])
obj = cp.Minimize(obj)
constr = []
for t in range(n):
constr += [ x[:,t+1] == A*x[:,t] + B*w[:,t] ,
y[:,t] == C*x[:,t] + v[:,t] ]
cp.Problem(obj, constr).solve(verbose=True)
x = np.array(x.value)
w = np.array(w.value)
plot_state(ts,(x_true,w_true),(x,w))
plot_positions([x_true,y], ['True', 'Noisy'], [-4,14,-5,20])
plot_positions([x_true,x], ['True', 'Robust KF recovery'], [-4,14,-5,20],'rkf3.pdf')
print("optimal objective value: {}".format(obj.value))