Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
1.2 Motivations
− interval observers address models with uncertain parameters. Unlike adaptive observers
they do not estimate the model parameters. They use the fact that any parameter p
lives in an intervals of the form [pmin,pmax] (e.g. [91, 105]).
The present work focuses on high-gain observers, which are a refinement of extended
observers, allowing us to prove that the estimation error converges to zero in a global sense.
The design of such observers relies on a general theory of observability for nonlinear systems
[57]. It is proven that the convergence is exponential. This means that the estimation error
is bounded on the upper limit by a decreasing exponential of time, whose rate of convergence
can be set by the user. The loop closure problem is studied in [57], Chapter 7, and a weak
separation principle is stated in [113] (see also [21]).
Our prototype observer is the Kalman filter. In order to further develop our motivations,
we provide a review of high-gain observer algorithms in the next section. A complete discusion
is the object of Section 2.5.
A non measured state variable
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Initial guess
'()* +,-.)*
/+0,1)0,2. 3,) ). (40(.5(5 6)*1). 7,*0(8
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Time
Figure 1.2: Estimation of a non measured variable:
The initial estimated state is wrong. With time, the observer reduces the estimation error.
1.2 Motivations
The extended Kalman filter is a practical answer to the observation problem for nonlinear
systems. Although global convergence is not theoretically proven, this observer works well in
practice. High-gain observers are based on extended observers, and result from the grouping
of two main ingredients:
1. the use of a special representation of the nonlinear system, provided by the observability
theory (see Chapter 2) and,
4