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We have a system described at time t by a state vector: x(t);
The state transition formula in matrix form is:
x(t) = F*x(t-1) + B*u(t) + e(t), where
e(t) is a random vector with covariance matrix Q,
Q is a covariance matrix that we re-estimate at each step,
F is a matrix that specifies how the state changes if no controls are applied,
u( t) is a vector of controls that have some effect on the state,
B is a matrix that specifies how controls affect the state;
We may not have a perfect view of the state.
z(t) is a vector of what we measure/see. It is related to the true state by:
z(t) = H*x(t) + w(t), where
H is a matrix that relates how the true state is relate to what we see,
w(t) is a random vector with covariance matrix R;
1) GPS device: We want to predict vehicle location. When it loses GPS
signal, we can update estimated location only by dead reckoning.
When GPS signal is recovered, it gives noisy estimate of true state.
Combine step of Kalman filter combines the extrapolated and measured estimates.
2) Radar target estimation: Radar signal gives estimate of x & y distance.
True state is (x, y), (dx, dy), & (ddx, ddy). We get a fresh radar estimate
of the location every second. How do we update estimated state of object?