Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
4.2 Simulation
− the innovation, Id, is defined by the formula:
� t
Id(t) = �y(s) − ˆyt−d(s)� 2 ds (4.6)
where
t−d
– y(s) is the output of the DC machine (the current),
– ˆyt−d(s) =x1 (t − d, z(t − d),s) with x (t − d, z(t − d),s) is the solution of the normal
form equations (4.2), (4.3), (4.4) over the time window [t − d, t], with the
initial condition that x(t − d) =z(t − d), is the estimated state at time t − d.
4.2.2 Implementation Considerations
z(t
dz/dt=
dS/dt=
d!/dt=
u(t
d delay
Innovation
Figure 4.2: Observer Structure.
The simulation of the DC motor is straightforward. Thus, here we only comment on
the implementation of the observer. The computation of a solution for the algorithm above
is decomposed in two main parts (see Figure 4.2):
1. the computation of the innovation, Id(t),
2. the update of the estimated state, the elements of Riccati matrix and the variable θ.
The latter part is common to all continuous time Kalman style observers. It presents no
particular difficulties 3 , and is illustrated in Figure 4.3.
In order to compute the innovation (refer to Figure 4.4) we need to:
3 From theory, we know that the Riccati matrix is symmetric. Therefore we can solve the equation for only
either the upper triangular part or the lower triangular part of the matrix.The introduction of this artifice has
two advantages:
− it saves computational time (the solution of n(n+1)
equations are needed instead of n 2
2 ),
− because of tiny machine inaccuracies, the situation may arise when the coefficients, which are supposed
to be equal, are determined to be unequal. This situation violates the symmetric requirement of the
matrix and algorithm breaks down.
To solve the matrix equations, the implementation of the observer requires the use of a small utility that
transforms (n × n) square matrices into vectors, and vice-versa.
Another device, which can be used when considering discrete-time systems, is to work directly with the square
root of the Riccati matrix. This method is detailed in [43], Chapter 7.2.
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y(t