Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
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Controller
Set
Points
u(t)
System
x(t)
y(t)
Figure 1.1: A control loop.
The study of observers is divided into three subproblems:
Observer
z(t)
Estimated
state
1.1 Context
1. the observability problem is the study of a given mathematical model in order to determine
to which extent it can be useful to estimate state variables,
2. the convergence problem focuses on the observer algorithm, the diminution of the error
between the real and estimated state is studied and the convergence to zero assessed
(see Figure 1.2),
3. the loop closure problem addresses the stability of closed control loops, when the state
estimated by an observer is used by a controller 3 (see Figure 1.1).
The juggler solves the two first problems when he is capable of catching the balls having
one or both eyes closed. He solves the third problem when he goes on juggling.
For systems that are described by linear equations, the situation is theoretically well
known and answers to all those problems have already been provided (see for example [74]).
We therefore concentrate on the nonlinear case.
There is a large quantity of observers available for nonlinear systems, derived from practical
and/or theoretical considerations:
− extended observers are adaptations of classic linear observers to the nonlinear case (e.g.
[58]),
− adaptive observers estimate both the state of the system and some of the model parameters;
a linear model can be turned into one that is nonlinear in this configuration
(e.g [98]),
− moving horizon observers see the estimation procedure as an optimization problem (e.g.
[12]),
3 A controller calculates input values that stabilize the physical process around a user defined set point or
trajectory. The model used in the observer and the possible model used in the controller may not be the same.
3