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State Estimation for Micro Air Vehicles 189
5.1 Dynamic Observer Theory

As a first step in deriving the Kalman filter, we briefly review dynamic observer
theory. Consider the linear time-invariant system modeled by the equations

˙ x = Ax + Bu
y = Cx.

A continuous-time observer for this system is given by the equation
˙
ˆ x = Aˆx + Bu + L (y − Cˆx), (21)
% &' (
% &' (
copy of the model correction due to sensor reading
where ˆx is the estimated value of x. Letting ˜x = x − ˆx we get that
˙
˜ x =(A − LC)˜x
which implies that the observation error decays exponentially to zero if L is
chosen such that the matrix A − LC is Hurwitz [20].
In practice, the sensors are usually sampled and processed in digital hard-
ware at a sample rate T s . How should the observer equation shown in Eq. (21)
be modified to account for sampled sensor readings? The typical approach is
to propagate the system model between samples using the equation

˙
ˆ x = Aˆx + Bu (22)
and then to update the estimate when a measurement is received using the
equation
+
ˆ x =ˆx + L(y(t k ) − Cˆx ), (23)
−
−
where t k is the instant in time that the measurement is received and ˆx − is
the state estimate produced by Eq. (22) at time t k . Equation (22) is then
+
re-instantiated with initial conditions given by ˆx . The continuous-discrete
observer is summarized in Table 3 [16]. The observation process is shown
graphically in Figure 8. Note that a fixed sample rate is not required. The
continuous-discrete observer can be implemented using Algorithm 1 which is
listed below.

5.2 Essentials from Probability Theory
T
Let X =(x 1 ,...,x n ) be a vector whose elements are random variables. The
mean, or expected value of X is denoted by
⎛ ⎞ ⎛ ⎞
µ 1 E{x 1 }
.
⎜ . ⎟ ⎜ . ⎠ = E{X},
⎟
. .
µ = ⎝ . ⎠ = ⎝
µ n E{x n }