Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
4.2 Simulation
single objective contrary to the multi-objective tuning required for non adaptive observers.
The parameters are split into two categories (see Figure 4.6) :
− the ones defining the performance of the system with respect to noise and perturbations:
– matrices Q and R are meant to provide decent noise filtering 4 ,
– θ1 characterizes the performance of the high-gain mode,
− the parameters related to the adaptation procedure:
– computation of innovation: d, the delay, and δ the time discretization used to
calculate Id,
– the rising time of θ: ∆T ( ∆T appears in the function F0(θ) of the observer (4.5)),
– the sigmoid function parameters: β, m = m1 + m2,
– the speed of the decay of θ(t) when innovation is small: λ.
Let us propose a methodology for the tuning of our set of parameters.
Without the
adaptation equation
⎧
⎨
⎩
R
Q
θ1
⎧
⎪⎨
⎪⎩
Parameters of the
adaptation equation
β
λ
∆T
d
m1
� δ
m2
θ1
STEP STEP 2 STEP 3
Figure 4.6: Bold: crucial parameters.
1. Non adaptive parameter tuning (Q, R and θ1).
At this stage simulations/experiments are done using the observer with F(θ, Id) = 0
(i.e. it is an extended Kalman filter, high-gain if θ(0) > 1).
First, we tune the classical EKF by making some simulations with noise, or by
choosing some experimental data sets without large perturbations. Our goal here is to
achieve acceptable smoothing of the noise. The input signal of the DC machine is set
to u(t) = 120 + 12 sin(t). This condition makes the system state oscillate and eases the
burden of the graphical analyses 5 . The observer’s initial state is taken to be equal to
the system’s initial state whenever the initial state is known. For all these simulations,
the load torque is kept constant equal to 0.55. We refer to this situation as the first
scenario.
The matrices Q and R are required to be symmetric definite positive. Usually, in
the continuous case, they are considered diagonal with positive coefficients:
4
In the stochastic setting, Q and R are the covariance matrices of the state (resp. output) noise. In our
deterministic point of view, they constitute additional tuning parameters.
5
This is not a requirement at all. It’s only here to provide a non stationary signal as the output. This step
can also be performed with a stationary system.
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