Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
The full proof of convergence has been developed in the single output continuous setting,
and afterwards extended to the multiple output and continuous-discrete settings.
A second major concern of this work, was the applicability of the observer. We therefore
extensively described its implementation on a single output system: a series-connected DC
machine. The time constraints where investigated via experiments performed using a real
motor in a hard real-time environment. The testbed was described in Chapter 4 and the
compatibility with real-time constraints assessed.
We conclude this work with some ideas for future investigations.
The Luenberger Case
It is much more direct to prove the convergence of a high-gain Luenberger observer.
This is because of the absence of the Riccati equation. However, it prevents us from
providing a local result of convergence when θ = 1. Therefore the adaptation strategy
has to be different from that used here (Cf. [11]), or performed for a specific class of
nonlinear systems (see for example [19]).
Automatic code generation
The implementation procedure for this algorithm is now well known. It can be roughly
classified into two parts: 1) coding specific to the model, and 2) coding related to the
observer mechanisms. It would be interesting to create a utility that automatically
generates the code of the observer once the model has been provided. We could save
development time, and implementation errors cause by typos.
Dynamic output stabilization
Dynamic output stabilization is considered in the second part of [57]. An extension of
this work to a closed loop containing an adaptive observer is a natural development.
This may be accomplished because the observer presented here is an exponential observer.
Since θ is allowed to increase when convergence is not achieved, we can expect to
deliver a good estimate to the control algorithm. The ability to quickly switch between
modes will be important.
Cascaded systems
Let us consider an observable nonlinear cascaded system of the form:
⎧
⎨
⎩
˙x = f(x, u),
˙ξ = g(x,ξ ),
y = h(x,ξ ).
One could imagine a situation where the state variable x is well known or estimated,
but not the variable ξ. Does the θ parameter of the observer really need to be high
for the part of the estimation that corresponds to x? We could consider a high-gain
observer with two varying high-gain parameters.
Unscented Kalman filter
The unscented Kalman filter is a derivative free, nonlinear observer that has received a
lot of attention recently [71]. This observer is based on the unscented transformation:
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