acm 116: the kalman filter - stanford universitycandes/acm116/handouts/kalman.pdf · acm 116: the...
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ACM 116: The Kalman filter
• Example
• General Setup
• Derivation
• Numerical examples
– Estimating the voltage
– 1D tracking
– 2D tracking
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Example: Navigation Problem
• Truck on “frictionless” straight rails
• Initial position X0 = 0
• Movement is buffeted by random accelerations
• We measure the position every ∆t seconds
• State variables (Xk, Vk) position and velocity at time k∆t
Xk = Xk−1 + Vk−1∆t + ak−1∆t2/2
Vk = Vk−1 + ak−1∆t
where ak−1 is a random acceleration
• Observations
Yk = Xk + Zk
where Zk is a noise term.
The goal is to estimate the position and velocity at all times.
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General Setup
Estimation of a stochastic dynamic system
• Dynamics
Xk = Fk−1Xk−1 + Bk−1uk−1 + Wk−1
– Xk: state of the system at time k
– uk−1: control-input
– Wk−1: noise
• Observations
Yk = HkXk + Zk
– Yk is observed
– Zk is noise
• The noise realizations are all independent
• Goal: predict state Xk from past data Y0, Y1, . . . , Yk−1.
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Derivation
Derivation in the simpler model where the dynamics is of the form
Xk = ak−1Xk−1 + Wk−1
and the observations
Yk = Xk + Zk
The objective is to find, for each time k, the minimum MSE filter based onY0, Y1, . . . , Yk−1
X̂k =k∑
j=1
h(k−1)j Yk−j
To find the filter, we apply the orthogonality principle
E((Xk −k∑
j=1
h(k−1)j Yk−j)Y`) = 0, ` = 0, 1, . . . , k − 1.
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Recursion
The beautiful thing about the Kalman filter is that one can almost deduce theoptimal filter to predict Xk+1 from that predicting Xk.
h(k)j+1 = (ak − h
(k)1 )h(k−1)
j , j = 1, . . . , k.
Given the filter h(k−1), we only need to find h(k)1 to get the filter at the next
time step.
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How to find h(k)1 ?
Observe that the next prediction is equal to
X̂k+1 = h(k)1 Yk +
k∑j=1
(ak − h(k)1 )h(k−1)
j Yk−j
= akX̂k + h(k)1 (Yk − X̂k)
Interpretation
X̂k+1 = akX̂k + h(k)1 Ik
• akX̂k is the prediction based on the estimate at time k
• h(k)1 Ik is a corrective term which is available since we now see Yk
– h(k)1 is called the gain
– Ik = Yk − X̂k is called the innovation
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Error of Prediction
To find h(k)1 , we look at the error of prediction
εk = Xk − X̂k
We have the recursion
εk+1 = (ak − h(k)1 )εk + Wk − h
(k)1 Zk
• ε0 = Z0
• E(εk) = 0
• E(ε2k+1) = [ak − h
(k)1 ]2E(ε2
k) + E(W 2k ) + [h(k)
1 ]2E(Z2k)
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To minimize the MSE εk+1, we adjust h(k)1 so that
∂h
(k)1
E(ε2k+1) = 0 = −2(ak − h
(k)1 )E(ε2
k) + 2h(k)1 E(Z2
k)
which is given by
h(k)1 =
akE(ε2k)
E(ε2k) + E(Z2
k)
Note that this gives the recurrence relation
E(ε2k+1) = ak(ak − h
(k)1 )E(ε2
k) + E(W 2k )
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The Kalman Filter Algorithm
• Initialization X̂0 = 0, E(ε20) = E(Z2
0)
• Loop: for k = 0, 1, . . .
h(k)1 =
akE(ε2k)
E(ε2k) + E(Z2
k)
X̂k+1 = akX̂k + h(k)1 (Yk − X̂k)
E(ε2k+1) = ak(ak − h
(k)1 )E(ε2
k) + E(W 2k )
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Benefits
• Requires no knowledge about the structure of Wk and Zk (only variances)
• Easy implementation
• Many applications
– Inertial guidance system
– Autopilot
– Satellite navigation system
– Many others
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General Formulation
Xk = Fk−1Xk−1 + Wk−1
Yk = HkXk + Zk
The covariance of Wk is Qk and that of Zk is Rk.
Two variables:
• X̂k|k estimate of the state at time k based upon Y0, . . . , Yk−1
• Ek|k error covariance matrix, Ek|k = Cov(Xk − X̂k|k)
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Prediction
X̂k+1|k = FkX̂k|k−1
Ek+1|k = FkEk|k−1F Tk + Qk
Update
Ik = Yk − HkX̂k+1|k Innovation
Sk = HkEk+1|kHTk + Rk Innovation covariance
Kk = Ek+1|kHTk S−1
k Kalman Gain
X̂k+1|k+1 = X̂k+1|k + KkIk Updated state estimate
Ek+1|k+1 = (Id − KkHk)Ek+1|k Updated error covariance
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Estimating Constant Voltage
We wish to estimate some voltage which is almost constant except for somesmall random fluctuations. Our measuring device is imperfect (e.g. because ofa poor A/D conversion). The process is governed by:
Xk = X0 + Wk, k = 1, 2, . . .
with X0 = 0.5V , and the measurements are
Zk = Xk + Vk, k = 1, 2, . . .
where Wk, Vk are uncorrelated Gaussian white noise processes, withR := Var(Vk) = 0.01, Var(Wk) = 10−5.
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0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Accurate knowledge of measurement variance, Rest=R=0.01
Iteration
Volta
ge
estimateexact processmeasurements
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0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Optimistic estimate of measurement variance, Rest=0.0001
Iteration
Volta
ge
estimateexact processmeasurements
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0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Pessimistic estimate of measurement variance, Rest=1
Iteration
Volta
ge
estimateexact processmeasurements
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1D Tracking
Estimation of the position of a vehicle.
Let X be a state variable (position and speed), and A is a transition matrix
A =
1 ∆t
0 1
.
The process is governed by:
Xn+1 = AXn + Wn
where Wn is a zero-mean Gaussian white noise process. The observation is
Yn = CXn + Zn
where the matrix C only picks up the position and Zn is another zero-meanGaussian white noise process independent of Wn.
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0 5 10 15 20 25!4
!2
0
2
4
6
8
Time
Posi
tion
Estimation of a moving vehicle in 1!D.
exact positionmeasured positionestimated position
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2D Example
General setup
X(t + 1) = FX(t) + W (t), W ∼ N(0, Q),
Y (t) = HX(t) + V (t), V ∼ N(0, R)
Moving particle at constant velocity subject to random perturbations in itstrajectory. The new position (x1, x2) is the old position plus the velocity(dx1, dx2) plus noise w.
x1(t)
x2(t)
dx1(t)
dx2(t)
=
1 0 1 0
0 1 0 1
0 0 1 0
0 0 0 1
x1(t − 1)
x2(t − 1)
dx1(t − 1)
dx2(t − 1)
+
w1(t − 1)
w2(t − 1)
dw1(t − 1)
dw2(t − 1)
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Observations
We only observe the position of the particle.
y1(t)
y2(t)
=
1 0 0 0
0 1 0 0
x1(t)
x2(t)
dx1(t)
dx2(t)
+
v1(t)
v2(t)
Source: http://www.cs.ubc.ca/∼murphyk/Software/Kalman/kalman.html
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Implementation
% Make a point move in the 2D plane
% State = (x y xdot ydot). We only observe (x y).
% This code was used to generate Figure 17.9 of
% "Artificial Intelligence: a Modern Approach",
% Russell and Norvig, 2nd edition, Prentice Hall, in preparation.
% X(t+1) = F X(t) + noise(Q)
% Y(t) = H X(t) + noise(R)
ss = 4; % state size
os = 2; % observation size
F = [1 0 1 0; 0 1 0 1; 0 0 1 0; 0 0 0 1];
H = [1 0 0 0; 0 1 0 0];
Q = 1*eye(ss);
R = 10*eye(os);
initx = [10 10 1 0]’;
initV = 10*eye(ss);
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seed = 8; rand(’state’, seed);
randn(’state’, seed);
T = 50;
[x,y] = sample_lds(F,H,Q,R,initx,T);
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Apply Kalman Filter
[xfilt,Vfilt] = kalman_filter(y,F,H,Q,R,initx,initV);
dfilt = x([1 2],:) - xfilt([1 2],:);
mse_filt = sqrt(sum(sum(dfilt.ˆ2)))
figure;
plot(x(1,:), x(2,:), ’ks-’);
hold on
plot(y(1,:), y(2,:), ’g*’);
plot(xfilt(1,:), xfilt(2,:), ’rx:’);
hold off
legend(’true’, ’observed’, ’filtered’, 0)
xlabel(’X1’), ylabel(’X2’)
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Apply Kalman Smoother
[xsmooth, Vsmooth] = kalman_smoother(y,F,H,Q,R,initx,initV);
dsmooth = x([1 2],:) - xsmooth([1 2],:);
mse_smooth = sqrt(sum(sum(dsmooth.ˆ2)))
figure;
hold on
plot(x(1,:), x(2,:), ’ks-’);
plot(y(1,:), y(2,:), ’g*’);
plot(xsmooth(1,:), xsmooth(2,:), ’rx:’);
hold off
legend(’true’, ’observed’, ’smoothed’, 0)
xlabel(’X1’), ylabel(’X2’)