Page 191 - Innovations in Intelligent Machines

State Estimation for Micro Air Vehicles 183
The Laplace transform representation of a simple unity DC gain low-pass
filter is given by
a

Y (s)= C(s)U(s) = U(s),
s + a
were u(t) is the input of the filter and y(t) is the output. Taking the inverse
Laplace transform we obtain
˙ y = −ay + au. (9)

By introducing an integrating factor, it is straightforward to show that the
solution to this differential equation is given by

T

y(t + T)= e −aT y(t)+ a e −a(T −τ) u(τ) dτ.
0
Assuming that u(t) is constant between sample periods results in the expres-
sion
T
y(t + T)= e −aT y(t)+ a e −a(T −τ) dτu(t)
0
= e −aT y(t)+(1 − e −aT )u(t). (10)
Note that this equation has a nice physical interpretation: the new value of
y (filtered value) is a weighted average of the old value of y and u (unfil-
tered value). We will use the notation C(s){·} to represent the low-pass filter
operator. Therefore ˆx = C(s){x} is the low-pass filtered version of x.

4.2 State Estimation by Inverting the Sensor Model

In this section we will derive the simplest possible state estimation scheme
based on inverting the sensor models. While this method is effective for angu-
lar rates, altitude, and airspeed, it is not effective for estimating the position
and Euler angles.

Position and Heading

The position variables p n , p e and the course heading χ can be estimated by
low-pass filtering the GPS signals:

ˆ p n = C(s){y GPS,n } (11)
ˆ p e = C(s){y GPS,e } (12)
ˆ χ = C(s){y GPS,course }. (13)

Figure 3 shows the actual and estimated states using this scheme. Note
that since the measurements are received at 1 Hz, the estimates have a sampled
data characteristic that includes significant delay.